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User's Guide - Physikalisch

1. propagation tout 0 01 tfinal 0 50 name hh psi gridpop end run section PBASIS SECTION Label DVR N Parameter X HO 32 0 0 1 0 1 0 K HO 32 0 0 1 0 1 0 end pbasis section SBASIS SECTION X 3 Y 3 end sbasis section OP_DEFINE SECTION title Henon Heiles PES end title nd op_define section PARAMETER SECTION mass_X 1 0 mass_Y 1 0 lambda 0 2 au nd parameter section HAMILTONIAN SECTION modes X Y 1 0 KE 1 0 5 gq 2 1 lambda 3 q 3 il lambda 2 16 q 4 1 1 0 1 KE 0 5 1 q 2 lambda 2 16 iL q 4 lambda q q 2 lambda 2 8 g2 q 2 end hamiltonian section INIT_WF SECTION build X gauss 1 80 0 00 0 75 Y gauss 0 00 1 20 0 50 end build end init_wf section INTEGRATOR SECTION VMF ABM 6 1 0d 7 1 0d 5 end integrator section end input Example A 1 An input file for a wavepacket propagation using the Henon Heiles Hamiltonian Appendix B The Structure of the Programs Figure B 1 displays a flowchart of the MCTDH program package The MCTDH program first reads the input file via the eingabe routines and computes the memory requirements De pending on the input settings it then starts some or all of the calculation types The routines callx allocate the memory the routines runx perform the calculations Communication between these parts of the MCTDH program as well as between the MCTDH and the Potfit and Analysis programs is done employing the files indicated by
2. amp or are allowed The underscore _ how 54 6 Setting up the Hamiltonian ever has a special meaning If one writes a label as label_modelabel where modelabel denotes a modelabel of one of the dof then the program puts the corre sponding operator in the modelabel column of the HAMILTONIAN SECTION assuming unit operators for all other degrees of freedom This feature is frequently used for complex absorbing potentials CAPs See Section 6 10 for an example Remember not to make the underscore part of a label except when using this modelabel feature 6 5 Implementing user defined 1D operators The examples in the previous section have underlined how easily a Hamiltonian operator can be implemented in many cases into the MCTDH program Nevertheless it might happen that a new 1D operator is required for a particular problem When the desired 1D operator is a real function i e a potential then there are three simple ways to implement it The first way is to define the function by a set of data points x f x This data is written in ASCII format to a file called say data When one includes in a LABELS SECTION a statement like V2 externalld data then the data will be read by MCTDH quadratically interpolated and assigned to the label v2 As usual the path given as argument to externalld may be absolute or relative In the latter case it is as usual relative to the location
3. oO Figure 2 7 The vibrational spectrum of CO2 as obtained by Fourier transform of the autocorrelation function and by FD using the same autocorrelation function For better visibility the Fourier spectrum is shifted upwards by 50 units eigenvalues but omits those which lie outside the energy window or which are detected as spurious according to an internal error measure Unfortunately the file filter eig may still contain spurious eigenvalues These are detected by performing several filter diagonalisations with slightly different parameters and keeping only those eigenvalues which are stable To perform the additional filter diagonalisation runs type filter84 file_outputname f1 window_energypoints 150 co2ft filter84 file_outputname f2 vp_principle 1 H filter_function box co2ft The double quoted arguments of filter84 overwrite the values from the input file We used a different number of energy points 150 rather than 125 and a different variational principle 1 H rather than H together with a different damping function box rather than cos as compared to the first filter diagonalisation See the HTML documentation for details The output files are now f1 and f2 x respectively A list of the stable eigenvalues together with an error estimate based on the spread of the eigenvalues is produced by the command fdmatch84 filter eig fl eig f2 eig sort n fdcheck84 3 0 gt results You may inspect th
4. 87 FWHM values of the window functions g times the length of the autocorre lation function Remember that the length of the autocorrelation function is twice the propagation time if the t 2 trick isused 104 Input sections required for different calculation types 129 Description of the calculation types 2 0000 130 Simple one dimensional operators 2 0 044 134 Operator symbols which require no arguments 135 One dimensional operators which require arguments 140 One dimensional potential energy curves 4 4 141 Two dimensional operators surface scattering 142 Two dimensional operators C4 C 2 0 2 2 2 2 000 142 Some general multi dimensional operators 142 One dimensional operators Rf Rfm hKEh hFRh hdgh hdqRh dqR dq2R 143 Matrix operator symbols used for an electronic degree of freedom 145 List of Figures 2 1 2 2 2 3 2 4 2 5 2 6 2 7 11 1 11 2 11 3 12 1 12 2 B 1 D 1 The NOCI S absorption spectrum 0 00004 4 Overlay Plot gs esa ba ek ak a ee BA ewe Mee eat 6 Diabatic state populations of pyrazine 200 7 The pyrazine Sz absorption spectrum 00004 8 H Hp reaction probability 0 0 0000 10 The vibrational spectrum of LICN 22000004 11 The vibration
5. HAMILTONIAN SECTION_H2 usediag modes rd rv theta LO ob KE 1 1 0 li ey ek vh2 1 end hamiltonian section to the operator file In fact the simpler input HAMILTONIAN SECTION_H2 modes rv TO KE LO wvh2 end hamiltonian section works as well because usediag is default for ei genf and for DOF s which are not listed a unit operator is assumed by default The desired functions are then generated by using the eigenf keyword in build block of the INIT_WF SECTION e g INIT_WF SECTION build rd gauss 4 50d0 8 76d0 0 18d0 rv eigenf H2 pop 2 theta leg jbt s10 sym end build end init_wf section generates a three dimensional wavepacket with a Gaussian along mode rd and the second eigenfunction i e the first excited state of the operator H2 for rv NB pop 1 is default and may be dropped For the theta degree of freedom an associated Legendre function is 76 7 Generating the initial wavepacket taken The associated Legendre function is specified by the value of the parameters jb f and s10 If the vei gen keyword has been included in the RUN SECTION then the eigenfunctions and eigenvalues are written to the file veigen In this way this procedure can be used to numerically exactly diagonalise a one dimensional operator If the veigen keyword is not given the eigenvalues are still written to the log file NB This is only possible if the primitive basis f
6. 73 7 5 Generating Wigner functions as initial functions 74 7 6 Generating eigenfunctions of a one dimensional Hamiltonian 75 7 7 Reading the initial wavepacket from file aoaaa 76 7 8 Diagonalising a multi dimensional operator to create multi dimensional SPFs 77 7 9 Generating an initial wavepacket using an operator 77 7 10 Creating a set of initial wavepackets 2 000 78 Contents M 7 11 Setting up a diabatically corrected initial wavepackets 78 8 Choosing an integration scheme 80 8 1 Using the VMF integration scheme in an MCTDH calculation 80 8 2 Using the CMF integration scheme in an MCTDH calculation 81 8 3 Description of the available integrators 2 82 8 4 Fine tuning the integration 2 020000 84 8 4 1 Propagating in natural or interaction picture orbitals 84 8 4 2 Suitable integrator settings for improved relaxation 84 8 4 3 Evaluating potentials using the TDDVR or CDVR method 85 9 Treating non adiabatic systems 86 9 1 Setting up the Hamiltonian for a non adiabatic system 86 9 2 Defining the primitive basis for a non adiabatic system 87 9 3 Defining the single particle basis for a non adiabatic system 87 9 4 Building the initial wavepacket for a non adiabatic system 90 10 Treating bosonic systems 91 10 1 Setting up the H
7. A Ay This should remain close to 1 0 unless a CAP is used in which case the norm will disappear with the wavepacket 98 11 Analysing the results employing the Analyse programs KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKKKKKKKKKKKKKKKKKK MCTDH version 8 Release 1 Revision 6 KKK KK KK KK KKK KKK KKK KKK KKK KKK KKK KKK KK KKK KKK KKK KKK KKK KKK KKK KKK KKKKKKKKKKKKKKKKKKK imc rieieecet Host bose Tue Feb 8 10 58 01 2000 usr people graham runsl nocll NOC1 S1 Propagation sin HO Leg 36 24 60 CAP Time 00 fs CPU lt 16 Sy Norm 1 00000000 E tot 1 188024 eV E corr 1 029664eV Delta E 0000 meV Natural weights 1000 rd 999 2709 1283 0008 0 0000 0 0000 rv 999 3842 6153 0005 0 0000 0 0000 theta 998 9065 1 0845 0088 0002 0 0000 Mode expectation values and variances rd lt q gt 4 3143 lt dq gt 0794 lt n gt 2 1160 lt dn gt 25 2145 rv 2 lt q gt 2 1549 lt dq gt 0670 lt n gt 0349 lt dn gt 2249 theta lt q gt 2 2283 lt dq gt 0767 lt j gt 4 6297 lt dj gt 3 9988 Time 1 00 fs CPU 1 32 s Norm 1 00000000 E tot 1 188024 eV E corr 1 023057eV Delta E 0 0000 meV Natural weights 1000 rd 997 8587 2 1345 0068 0001 0 0000 rv 997 8951 2 0843 0201 0005 0 0000 theta 998 5412 1 4346 0233 0009 0 0000 Mode expectation values and variances rd lt q gt 4 3155 lt dq gt 0798 lt n gt 2 2337
8. J2 momenta Here K k k so this is differ ent from jpm for the two individual angular mo menta Only for two successive KLegs which fur thermore have to be combined in one mode jz Jz 10 Angular momentum operator Only for sphFBR and KLeg For PLeg use dq or p on the DOF jz2 I idg second power of angular momentum operator jz Only for sphFBR and KLeg For PLeg use dq 2 on the DOF jp 2 j Square of angular momentum raising operator Only for KLeg and PLeg jm 2 j_ Square of angular momentum lowering operator Only for KLeg and PLeg jpjm j j Product of angular momentum raising and lower ing operators Only for KLeg and PLeg continued 136 C The built in symbolic expressions Table 2 continued Symbol Operator Notes jmjp j j Product of angular momentum lowering and raising operators Only for KLeg and PLeg sJp sin Jy J is total angular momentum Only for KLeg and PLeg sJm sin x J J is total angular momentum Only for KLeg and PLeg sJpk sin J k k sin 0 J 2 Only for KLeg and PLeg sJmk sin J k kx sin 0 J_ 2 Only for KLeg and PLeg Jp J4 multiplication with C7 and shift k k 1 Jm J_ multiplication with Ce and shiftk gt k 1 Jx Jz Je J4 J 2 Jy iJy iJy J4 J 2 dth1 Og sin 0 first derivative only for Leg KLeg and PLeg DVR no symmetry dth2 z cos0 Og sin
9. To get an overview on the available releases of the mctdh 8 4 branch submit the com mand svnm list SVNM mctdh84 releases This will provide an output similar to 4 ols 4 2 o 6 8 co CO g OoOO BR PP HPP AD A 152 F 1 Useful svn commands 153 8 1 1 9 where of course one may exchange mctdh84 with mctdh83 or mctdh85 to list the contents of those directories 8 4 8 4 If you want to download the latest version type svn export SSVNM mctdh84 releases 8 4 9 mctdh84 9 where the directory mctdh84 9 will be created by svn it should not previously exist Of course one may give any name to the final directory and may give its full path if it is to be created in a directory different from the current one The svn export command will pro vide you with exactly the same data as found on the mctdh84 9 tgz file of http mctdh uni hd de packages A better alternative is to use svnm checkout SVNM mctdh84 releases 8 4 9 mctdh84 9 The difference is that with this command additionally a couple of svn files will be copied to the final directory which almost doubles the size of the latter However the svn files give you access to most of the svn commands E g moving cd the the mctdh directory here mctdh84 9 and submitting the command svnm status will tell you which files are modified or added with respect to the repository Or svnm diff old SVNM mctdh84 releases 8 4 9 new wi
10. 1e 05 5 10 15 20 25 30 time fs 1e 06 Figure 11 1 The natural orbital populations for the single particle function basis for the vibrational degree of freedom as a function of time for the photo dissociation of NOCI 11 4 Checking the efficiency of a calculation The timing file which is obtained by adding the keyword timing to the RUN SECTION contains information about how much time is spent in the various sections and subroutines of the program This information can be used to improve the efficiency of a calculation For instance if in a CMF run the BS integrator used to propagate the single particle functions takes less than one or two percent of the total effort one should combine more single particle functions If on the other hand the BS integrator takes more than 80 of the total effort one should remove some of the combinations If the propagation of one certain mode takes much longer than the propagation of the other modes although the combined grid sizes are comparable then check whether the DVR representation is appropriate The information listed in the timing file can be extremely helpful It is a good practice to always include the timing file 11 5 Watching the system s evolution The program showd1d84 is designed to plot the evolution of the system density along a degree of freedom It reads the gridpop file which can be created by adding the keyword gridpop in the RUN SECTION of the input file T
11. 07 02 V a y z the HAMILTONIAN SECTION reads HAMILTONIAN SECTION modes x y kr 6 9 Incorporating natural potentials 61 1 0 KE 1 ol 14 0 ji KE 1 1 0 1 p il KE 1 0 V end hamiltonian section The label V is defined in the LABELS SECTION LABELS SECTION V newsurf nd labels section in the operator file If the order of the arguments of V differ from the order defined in the mode line then the order has to be explicitly specified E g turning to the above example but cyclic interchang ing the modes the HAMILTONIAN SECTION reads HAMILTONIAN SECTION modes i gZ x y 1 0 KE 1 j ed 1 0 1 KE l 1 1 0 1 ee il l E 1 0 2 amp 3 amp 1 V end hamiltonian section because the first argument of V is the second mode etc See the HTML documentation and Section 6 13 for further details 6 9 Incorporating natural potentials With the aid of the Potfit program potential surfaces can be fitted to the product form 6 4 These fits are known as natural potential fits Natural potentials are the method of choice to employ non separable potential surfaces in an MCTDH calculation How such a fit is generated will be discussed in Sec 12 1 We only mention that the Potfit program requires the non separable potential to be implemented in the operator library in exactly the same way as described in the previous section After having constructe
12. 88 9 Treating non adiabatic systems OP_DE title FINE S ECTION Pyrazine 4 mode model end titl e nd op_define secti PARAME TER SECTION wl0a 0 09357 w6a 0 0740 wl 0 1273 w9a 0 1568 delta 0 46165 k6al 0 0964 k6a2 0 1194 k11 0 0470 k12 0 2012 k9al 0 1594 k9a2 0 0484 lambda 0 1825 on ev ev ev ev ev ev ev ev ev ev ev ev nd parameter section HAMILTONIAN SECTION modes vl0a v6a vl v9a el 1 0 wl0a KE T Al ab 1 0 5 wl0a q 2 l 1 1 1 1 0x w6a 1 KE 1 I 1 0 5xw6a 1 q 2 ab 1 1 1 0 wl 1 KE 1 1 0 5 wl il T q 2 1 1 1 0 w9a 1 1 1 KE 1 0 5 w9a I a 1 q 2 1 delta 1 1 T 1 S1 amp 1 delta 1 1 1 S2 amp 2 k6al q ah 1 S1 amp 1 k6a2 q 1 1 S2 amp 2 k11 T 1 q I S1 amp 1 k12 a 1 q 1 S2 amp 2 k9al T i q S1 amp 1 k9a2 i T i q S2 amp 2 lambda q 1 1 1 S1 amp 2 end hamiltonian section end operator Example 9 1 An operator file for the pyrazine 4 mode 2 state model system In the single set formalism which is the default the wavepackets on each surface are represented by the same single particle function basis As there is thus only one single particle basis the SPF BASIS SECTION has the same form as for adiabatic systems e g SPF BASIS SECTION vl0a voa v1 v9a 5 6 4 ix nd spf basis section 9 3 Defining the single particle basis for a n
13. S A Ndengu F Gatti R Schinke H D Meyer and R Jost Absorption cross section of ozone Isotopologues calculated with the multiconfiguration time dependent Hartree MCTDH method I The Hartley and Huggins bands J Phys Chem A 114 2010 9855 9863 A U J Lode A I Streltsov O E Alon H D Meyer and L S Cederbaum Corrigendum Exact decay and tunneling dynamics of interacting few boson systems J Phys B 43 2010 029802 R Marquardt M Sanrey F Gatti and F L Quere Full dimensional quantum dynamics of vibrationally highly excited NHD2 J Chem Phys 133 2010 174302 162 List of MCTDH references 171 172 173 174 175 176 O Vendrell and H D Meyer Multilayer multiconfiguration time dependent Hartree method Imple mentation and applications to a Henon Heiles Hamiltonian and to pyrazine J Chem Phys 134 2011 044135 D J Haxton K V Lawler and C W McCurdy Multiconfiguration time dependent Hartree Fock treat ment of electronic and nuclear dynamics in diatomic molecules Phys Rev A 83 2011 063416 M Schr der F Gatti and H D Meyer Theoretical studies of the tunneling splitting of malonaldehyde using the multiconfiguration time dependent Hartree approach J Chem Phys 134 2011 234307 T Ernst D W Hallwood J Gulliksen H D Meyer and J Brand Simulating strongly correlated multiparticle systems in a truncated Hilbert space Phys Rev
14. The Heidelberg MCTDH Package A set of programs for multi dimensional quantum dynamics User s Guide Version 8 Release 4 Revision 11 Authors G A Worth M H Beck A Jackle HD Meyer F Otto M Brill and O Vendrell Address Theoretische Chemie Physikalisch Chemisches Institut Im Neuenheimer Feld 229 D 69120 Heidelberg Germany Email Hans Dieter Meyer pci uni heidelberg de September 9 2015 Contents List of Tables List of Figures List of Examples Copyright 1 Introduction 2 An MCTDH tutorial 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 Determining the absorption spectrum for the photodissociation of NOCI Determining state populations for the photo excitation of pyrazine Determining reaction probabilities for the exchange reaction of H H2 Determining the vibrational spectrum of LICN Determining the vibrational spectrum of CO by filter diagonalisation Determining eigenstates by improved relaxation Determining eigenstates by block improved relaxation Using potfit and chnpot to fit a surface to ab initio data points 2 8 1 Transforming the ab initio data to product form 2 8 2 Interpolating the natural potential to a new primitive grid Optimizing an external field with Optimal Control Theory OCT 2 10 Concluding Remarks 3 Defining the type of calculation to be made 3 1 3 2 3 3 3 4 3 5 3 6 3
15. 1 0 eV to 1 0 eV An energy shift of 0 2258 eV has been added due to the zero point energy of the system A phenomenological broadening with a relaxation time of 30 fs has also been added The result is shown in Fig 2 4 Again the same figure is generated more simply by typing plspec e 0 2258 eV 1 0 1 0 eV 30 1 2 3 Determining reaction probabilities for the exchange reaction of H H2 9 2 3 Determining reaction probabilities for the exchange reaction of H H The H Hg2 system is the smallest reactive molecular system but it is the prototype of all three atom reactions As interaction potential we will use the LSTH potential energy surface This is a full 3D surface and as such must be first transformed to MCTDH product form The Potfit program can accomplish this fast and reliably at least as long as the full primitive product grid is not too large After the wavepacket is propagated the reaction probability is determined by flux analysis See the MCTDH review 1 or the original publication 10 for more details Here we will perform a scattering calculation for vanishing total angular momentum J 0 only Thus the result is a initial state selected reaction probability and not a cross section 1 Copy the files MCTDH_DIR pinputs Isth inp and MCTDH_DIR inputs hh2 inp to your tutorial directory and create the directories Isthfit and hh2 2 To perform the potential fit calculation type potfits4 lsth This will take about 5
16. Cederbaum Exact decay and tunneling dynamics of interacting few boson systems J Phys B 42 2009 044018 U Manthe Layered discrete variable representations and their application within the multiconfigurational time dependent hartree approach J Chem Phys 130 2009 054109 O Vendrell F Gatti and H D Meyer Strong isotope effects in the infrared spectrum of the zundel cation Angew Chem Int Ed 48 2009 352 355 O Vendrell M Brill F Gatti D Lauvergnat and H D Meyer Full dimensional 15D quantum dynamical simulation of the protonated water dimer III mixed Jacobi valence parametrization and bench mark results for the zero point energy vibrationally excited states and infrared spectrum J Chem Phys 130 2009 234305 See supplementary material EPAPS document E JCPSA6 130 023924 which can be downloaded from ftp ftp aip org epaps journ chem phys E JCPSA6 130 023924 O Vendrell F Gatti and H D Meyer Full dimensional 15D quantum dynamical simulation of the protonated water dimer IV Isotope effects in the infrared spectra of D D20 i H D20 and D H2 o f isotopologues J Chem Phys 131 2009 034308 M Brill O Vendrell and H D Meyer Shared memory parallelization of the multiconfiguration time dependent Hartree method and application to the dynamics and spectroscopy of the protonated water dimer In Advances in the Theory of Atomic and Molecular Systems P Piecuch J Maruani G Delg
17. For periodic the parameter x is taken as Nth grid point ry xf For s periodic the grid points are additionally shifted by Av 2 i e xo xi Ax 2 and zy zf Ax 2 As an example the lines X fft 32 0 00 3 0434 linear and X fft 32 0 00 3 1416 periodic are equivalent The keywords periodic and s periodic are particularly useful to describe angular degrees of freedom For angular modes ranging from 0 to some integer fraction of 27 one may alternatively use the keywords 2pi or s 2pi The example above is thus equivalent to X fft 32 2pi 2 For a more detailed discussion see the HTML documentation 4 6 Spherical harmonics FBR 41 4 6 Advanced topic Spherical harmonics FBR The spherical harmonics FBR is the appropriate choice when there is rotational motion which must be described by two angles 0 and The spherical harmonic functions 2j 1 j m im Yme yf e cos e 4 7 serve as basis functions where P denotes the Legendre polynomial 4 4 The matrix elements of the angular momentum operators j j j7 and j are then given by simple formulas Examples for a PRIMITIVE BASIS SECTION defining a spherical harmonics FBR for the set of coordinates alpha and beta are PRIMITIVE BASIS SECTION alpha sphFBR 9 nosym beta phiFBR end primitive basis section and PRIMITIVE BASIS SECTION alpha sphFBR 8 sym beta phiFBR 4 2 end primitive basis section The key
18. O Vendrell F Gatti and H D Meyer Dynamics and infrared spectroscopy of the protonated water dimer Angew Chem Int Ed 46 2007 6918 6921 A N Panda F Otto F Gatti and H D Meyer Rovibrational energy transfer in ortho H2 para H2 collisions J Chem Phys 127 2007 114310 F Richter F Gatti C L onard F Le Qu r and H D Meyer Time dependent wave packet study on trans cis isomerisation of HONO driven by an external field J Chem Phys 127 2007 164315 S Woittequand D Duflot M Monnerville B Pouilly C Toubin S Briquez and H D Meyer Classical and quantum studies of the photodissociation of a HX X C1 F molecule adsorbed on ice J Chem Phys 127 2007 164717 O Vendrell F Gatti D Lauvergnat and H D Meyer Full dimensional 15D quantum dynamical simulation of the protonated water dimer I Hamiltonian setup and analysis of the ground vibrational state J Chem Phys 127 2007 184302 O Vendrell F Gatti and H D Meyer Full dimensional 15D quantum dynamical simulation of the pro tonated water dimer II Infrared spectrum and vibrational dynamics J Chem Phys 127 2007 184303 F Otto F Gatti and H D Meyer Rotational excitations in para Hz2 para He collisions Full and reduced dimensional quantum wave packet studies comparing different potential energy surfaces J Chem Phys 128 2008 064305 S Z llner H D Meyer and P Schmelcher Few boson dynami
19. aid of the following two examples The first one is a fairly simple one namely a modified Henon Heiles Hamiltonian i e two coupled anharmonic oscillators The Hamiltonian is 1 a2 82 D ied 3 41 4A 272 Ha pe S T AE ee 6 2m r y 2 3 16 8 The modification with respect to the original Henon Heiles Hamiltonian is the last quartic term It makes the system bound The corresponding operator file is included as Example 6 2 As one can see the Hamiltonian 6 2 is represented by the symbols in the HAMILTONIAN SECTION one product term per line The mode labels have to match with those in the 52 6 Setting up the Hamiltonian Table 6 1 A selection of built in symbolic expressions that can be used to define the Hamiltonian For a complete list please refer to the HTML documentation or to Appendix C The variables x and 0 represent the mode labels associated with the corresponding degrees of freedom Symbol Operator Description 1 1 Unit operator q x Multiply by position coordinate x q r x Multiply by rth power of x sin sin x Sine of coordinate cos cos Cosine of coordinate tan tan x Tangent of coordinate exp e Exponential of coordinate dq Ox First derivative dq n on nth derivative KE z7 amp Kinetic energy term Bee ay Oo 8in 09 Angular momentum squared Derivatives with n gt 2 are only allowed for an FFT primitive basis The variable m stands for the mas s_x label defined in
20. are formally equiv alent this does not hold for their efficiency It is our experience that the exponential DVR performs faster than FFT for small grids NV lt 16 while FFT is faster for large grids N 64 Between these limits both representations are similarly fast FFT and exponential DVR have an identical set of input parameters Supposed there are two degrees of freedom labelled X and Y an exponential DVR and an FFT representation are employed for these coordinates by a PRIMITIVE BASIS SECTION reading PRIMITIVE BASIS SECTION X exp 25 Did Y fft 48 1 5 end primitive basis section linear A oO w As usual the first number defines the number N of grid points Remember that N must be odd for an exponential DVR and factorise into powers of 2 3 and 5 better 2 and 3 only for FFT If the last keyword is missing or linear then the remaining two numbers which will be called x and xy in the following are interpreted as the grid points 79 x and xN 1 xf The grid spacing is Ax af xi N 1 Due to the periodic boundary conditions the first grid point xo and the point xy which is the one following the last point on the grid are to be identified Instead of Linear the keywords periodic or s periodic may be specified which changes the interpretation of x and x p In both cases x and x p are considered identical due to the periodic boundary conditions The grid spacing is now Ax af 2 N
21. in the order of modes and of DOFs within a mode I e a muld potential may operate on mode 2 3 and 4 but must not operate on mode 2 and 4 only In the latter case there is a hole mode 3 Holes are only allowed at the beginning and the end of a muld potential One may fill the holes by incorporating dummy variable as done above However this will make the application of a muld potential even more expensive One rather should try to re order the modes and DOFs to avoid the holes In any case it is advisable to transform a muld potential to a separable natpot by using potfit see Section 12 1 How a natpot is incorporated is discussed in Section 6 9 6 14 Golden rules for writing operator files 69 Finally an important note It is not possible to re order the arguments or to add dummy variables if the special multi dimensional function readsrf is used In this case the re ordering and or addition of dummy variables must be done on the readsrf data file 6 14 Golden rules for writing operator files General remarks File names modelabels labels and parameters are case sensitive Hence most parts of the operator file are case sensitive in contrast to the input file All input is assumed to be in atomic units In contrast to the input file this holds also for times One hence has to explicitly give the unit fs when a parameter value is given in femto seconds Parameter Section See also Section 6 2 Parameters are real numbers Whe
22. lt dn gt 2 3028 rv lt q gt 2 1578 lt dq gt 0668 lt n gt 0588 lt dn gt 2809 theta lt q gt 2 2276 lt dq gt 0767 lt j gt 4 8200 lt dj gt 4 1250 Propagation was successful Total time memis One 70 4 125571 usr people graham runsl nocll NOC1 S1 Propagation sin HO Leg 36 24 60 CAP Example 11 2 A section of an output file from the wavepacket propagation on the S1 surface of NOCI e E tot The total energy i e expectation value of the Hamiltonian This should remain constant unless a CAP is used when the energy will dissappear with the wavepacket e E corr The correlated energy i e the expectation value of the terms in the Hamiltonian which correlate the degrees of freedom The correlated and uncorrelated Hamiltonian terms are listed in the op log file 11 3 Checking the accuracy of a calculation 99 e Delta E The loss in energy during the calculation i e difference between the energy at time and at time 0 Natural weights The natural weights i e eigenvalues of the one dimensional density matrices are given for each mode in the calculation A weight of 1 0000 given here indicates that the least important natural orbital is present in 0 1 of the wavefunction e Mode expectation values q and dq are the expectation values and variances for the position operator where dq q q This gives an idea of the spread of the wavepacket and a check on the grid used If a
23. relaxation or diagonalisation Additionally hereto one may use the MCTDH program to solely set up a primitive basis a Hamiltonian operator or an initial wavepacket This is done with the keywords gendvr genoper or geninwf in the RUN SECTION The RUN SECTION is hence required for all calculation types The generated information can then be read from file in following calculations be using the keywords readdvr readoper or readinwf in the RUN SECTION Each calculation type has a level associated with it which reflects the stages for a cal culation These levels are listed in Tab A 2 Each level keyword automatically contains the lower levels thus the keyword propagation implies gendvr genoper geninwf propagation and a wavepacket propagation will be performed after first setting up a DVR operator and initial wavepacket The listed files are files which contain the informa tion from the lower level calculations In the input file there may appear keywords which have a UNIX filename as argument e g oppath These filenames are interpreted relative to the location of the input file A The concept of the input file 131 HEHE HE HE EEE FE FE HE HE HE HE EE EEE FE FE HE TE EE EE TE ERE EE ERE EE EE E E E RE EE HE EE EEE Propagating a wavepacket using the Henon Heiles Hamiltonian HEHEHE HEHE EEE FE FE HE HE HEH HE HE EEE HE HE EEE EE HE EE E E E E E EE EE E E E E E EE HE EH HE HE EH RUN SECTION
24. single particle basis Secs 4 and 5 as well as the initial wavefunction Sec 7 must also be defined Finally you may select an integration scheme different from the default Sec 8 A wavepacket propagation is then initiated by placing the keyword propagation in the RUN SECTION A typical example is RUN SECTION propagation tfinal 50 0 tout 1 0 name results psi auto gridpop end run section The parameters t final and tout denote the time the propagation will run up to and the time interval after which the data is output in femtoseconds The name directory is results in our example The other parameters have been established in Sec 3 2 A number of ad ditional options may be selected in the RUN SECTION We refer the reader to the HTML documentation for details 3 4 Relaxing a wavepacket to produce the lowest eigenstate In a relaxation calculation a wavepacket is propagated in imaginary time to produce the low est ro vibrational eigenstate Analogous to a real time propagation an operator and an input file are required to define the Hamiltonian the primitive and single particle basis the initial wavefunction and possibly the integration scheme A relaxation calculation is selected in the RUN SECTION by the keyword relaxation instead of propagation The RUN SECTION may read RUN SECTION relaxation tout 10 0 tfinal 100 0 name results end run section The parameters have been introduced in Secs 3 2
25. the number of data points needed to represent the SPFs Otherwise it is likely that one over combines It is always useful to inspect the timing file The time needed to propagate the SPFs should take between 2 and 25 of the total effort CMF scheme assumed Chapter 6 Setting up the Hamiltonian For a quantum dynamical calculation a Hamiltonian operator has to be defined This is typ ically done using the operator file which is a text file that is read and interpreted by the MCTDH program The MCTDH program is capable to parse a variety of mathematical ex pressions This often allows the implementation of a new Hamiltonian without programming any routines 6 1 The operator file The operator file contains all the information the program needs to set up the Hamiltonian It must have the extension op e g hamiltonian op Similarly to the input file the operator file is divided into sections In the OP_LDEFINE SECTION the Hamiltonian is characterised Numerical constants can be specified in the PARAMETER SECTION The definition of the Hamiltonian is contained in the HAMILTONIAN SECTION Finally any labels that are used to describe the Hamilton operator are compiled in the LABELS SECTION The operator file for the propagation of NOCI on the S surface is given as Example 6 1 In order to select a particular operator file an OPERATOR SECTION is required in the input file The keywords opname and oppath then point to this file For insta
26. 10 nd natpot basis section end input Example 11 3 The input file projfit inp for the potfit calculation 4 Create a projection operator file by performing an mctdh genoper run using the just created natpot files For this metdh run an input and an operator file must be written As an example these files for the creation of a projection operator are shown in Example 11 4 and 11 5 In the operator file projector1 op one uses the natpot files which are in the directories projector111 projector112 and projector122 Note that the SPF and PRIMITIVE BASIS SECTIONs must be identical to the ones used in the metdh propagation run where the psi file was created that will be used for the calcualtion of the adiabatic populations 5 Calculate the expectation value of the projection operator with the analyse routine expect This requires the wave function that is contained in the psi file Alternatively the adiabatic populations can be computed on the fly by setting the expectation keyword in the RUN SECTION In this case it is not necessary to store the wavefunc tion Notice that the adproj run provides also the vpot files of the adiabatic surfaces This makes it possible to perform propagation in the adiabatic approximation Typical calculation times for the adpop run are about 20s for 3D up to 9h for 6D Each calculation was performed on a 2 6GHz processor and the psi files contained 121 timesteps 3D and 6D pyrazine models In the case o
27. 121 2004 4585 R van Harrevelt and U Manthe Degeneracy in discrete variable representations General considerations and applications to the multiconfigurational time dependent hartree approach J Chem Phys 121 2004 5623 M D Coutinho Neto A Viel and U Manthe The ground state tunneling splitting of malonaldehyde Accurate full dimensional quantum dynamics calculations J Chem Phys 121 2004 9207 9210 C Cattarius and H D Meyer Multidimensional density operator propagations in open systems Model studies on vibrational relaxations and surface sticking processes J Chem Phys 121 2004 9283 9296 O Vendrell and H D Meyer Proton conduction along a chain of water molecules Development of a linear model and quantum dynamical investigations using the multiconfiguration time dependent hartree method J Chem Phys 122 2005 104505 K Giese H Ushiyama K Takatsuka and O K hn Dynamical hydrogen atom tunneling in dichlorotropolone A combined quantum semiclassical and classical study J Chem Phys 122 2005 124307 S Woittequand C Toubin B Pouilly M Monnerville S Briquez and H D Meyer Photodissociation of a HCI molecule adsorbed on ice Chem Phys Lett 406 2005 202 209 B Pouilly M Monnerville F Gatti and H D Meyer Wave packet study of the UV photodissociation of the Ar2HBr complex J Chem Phys 122 2005 184313 S Z llner H D Meyer and P Schmelcher Multi electro
28. 2 2 2 000 119 12 3 Extra flexibility combining potfitandchnpot 120 12 3 1 Dealing with an arbitrary primitive grid 120 IV Contents 12 3 2 Transforming between two natural potentials withchnpot 120 12 4 Manipulating potentials with the projection program 122 12 4 1 Input and output files 2 2 0 0 0 0 0 0000 122 12 4 2 Generating a Fourier transformed potential 125 12 4 3 Using a Fourier transformed potential in MCTDH 126 12 5 Downsizing previous poffits the cutnpot and rdnpot functions 128 A The concept of the input file 129 B The Structure of the Programs 132 C The built in symbolic expressions 133 D Structure of the WF array 146 E Installing the MCTDH package 147 F The svn repository of the Heidelberg MCTDH package 152 Fl Usefulsvncommands 0 00200 eee eee eee 152 List of MCTDH references 154 Index 163 List of Tables 2 1 4 1 6 1 8 1 8 2 9 1 A l A 2 Cd C 2 C 3 C4 C5 C 6 C 7 C 8 C9 Vibrational energies of CO computed with MCTDH FD 14 Available DVR FBR representations for the primitive basis 37 Selection of built in symbolic expressions 4 4 52 Available integrators in dependence of the calculationtype 81 Optimal orders for the ABM and BS integrators 83 Built in symbolic expressions for non adiabatic systems
29. 9 2 k 1 Yor V22 where f denotes the number of molecular degrees of freedom and no is a one dimensional operator acting exclusively on the xth degree of freedom It is no restriction to assume that the elements of the y matrices are only zero and one i e a 0 1 The program knows a number of built in symbolic expressions that can be used to define the y matrices in the Hamiltonian section of the operator file These symbols are compiled in Tab 9 1 For instance the symbol S1 amp 1 specifies the symmetric matrix that couples states 1 and 1 while Z1 amp 2 stands for the unsymmetric matrix that couples initial state 2 with final state 1 Note that the symbol S1 amp 2 implies that initial state 2 couples with final state 1 and vice versa because the corresponding matrix is symmetric 86 9 2 Defining the primitive basis for a non adiabatic system 87 Table 9 1 The built in symbolic expressions that can be used to define the couplings of a non adiabatic Hamil tonian Two electronic states are assumed Matrix Symbol Matrix Symbol 1 0 0 1 1 6 i S1 amp 2 or S2 amp 1 1 0 0 1 i o S1 amp 1 5 Z1 amp 2 0 0 0 0 a S2 amp 2 A Z2 amp 1 As an example the operator file for the 4 mode 2 state model of the pyrazine molecule is shown in Example 9 1 The Hamiltonian for this system reads a S a H D gt a 06 Q ar a al P k a Qi k 3 Qioa i i 10a 9 3 with 2 10a 6a 1 9a The numerical p
30. AMU end primitive basis section Here we have assumed that the system under consideration has two degrees of freedom labelled X and Y The keyword HO specifies the harmonic oscillator DVR The next entry denotes the number JN of basis functions or equivalently grid points The next three numbers define the equilibrium position z the frequency w and the mass m If the mass entry is missing the program sets m to 1 The above example also demonstrates the use of units The default unit for times is fem toseconds for all other input variables it is atomic units Hence exactly the same DVR is selected for the two modes X and Y A complete list of the available units can be found in the HTML documentation The second way to choose a Hermite DVR is to specify the first and last grid point by employing the keyword xi xf PRIMITIVE BASIS SECTION X HO 36 xi xf 0 528 0 528 Y HO 36 xi xf 0 528 0 528 end primitive basis section 38 4 Selecting a DVR FBR representation for the primitive basis RUN SECTION name nocll propagation tfinal 25 0 tout 1 0 psi double auto steps gridpop title NOC1 propagation CMF 5 6 6 spf 5 5 5 prim 36 24 60 end run section OPERATOR SECTION opname nocll alter labels CAP_rd CAP 5 0d0 0 3d0 3 nd alter labels nd operator section SPF BASIS SECTION rd 5 rv 5 theta 5 nd spf basis section PRIMITIVE BASIS SECTION Label DVR N Parameter
31. Chem Phys 101 1994 5831 5840 J Y Fang and H Guo Four dimensional quantum dynamics of the CH3I MgO photodissociation Chem Phys Lett 235 1995 341 346 156 List of MCTDH references 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 J Y Fang and H Guo Multiconfiguration time dependent Hartree studies of the Cl2Ne vibrational predissociation dynamics J Chem Phys 102 1995 1944 L Liu J Y Fang and H Guo How many configurations are needed in a time dependent Hartree treatment of the photodissociation of ICN J Chem Phys 102 1995 2404 J Y Fang and H Guo Quantum dynamics within the multiconfiguration time dependent Hartree ap proximation J Mol Struct Theochem 341 1995 201 215 A Jackle and H D Meyer Reactive scattering using the multiconfiguration time dependent Hartree approximation General aspects and application to the collinear H H2 gt H2 H reaction J Chem Phys 102 1995 5605 A P J Jansen and H Burghgraef MCTDH study of CH4 dissociation on Ni 111 Surf Sci 344 1995 149 158 A Capellini and A P J Jansen Convergence study of multi configuration time dependent hartree simu lations H scattering from LiF 001 J Chem Phys 104 1996 3366 3372 U Manthe and F Matzkies Iterative diagonalization within the multi configurational
32. DVR is used n and dn are the ex pectation values and variances of the number representation in the corresponding FBR basis a measure of which functions are populated If a Legendre DVR is used i e Leg Leg R or KLeg the symbol is changed to 7 and dj to indicate that the number representation is in this case just the expectation and variance of the angular momen tum If an FFT basis or the exponential DVR is used p and dp the expectation value and variance of the momentum operator are given At the end of the file should be written Propagation was successful and the CPU time used for the calculation 11 3 Checking the accuracy of a calculation The accuracy of an MCTDH calculation depends on both the size of the primitive and the single particle function bases Analyse programs are available for both these tasks Note that one has to be in the name directory when applying the analyse routines and scripts as shown below 11 3 1 Checking the primitive basis size The program rdgpop reads and evaluates the populations of the primitive basis functions e g the grid points This is used to check that enough primitive basis functions have been used for the calculation The program requires the gridpop file which is obtained by specifying the gridpop keyword in the RUN SECTION of the input file The program can be used either to calculate the maximum population or to evaluate the change of population with time of the poin
33. The SPF BASIS SECTION may read 90 9 Treating non adiabatic systems SPF BASIS SECTION multi set vl0a 4 3 v6a 5 4 vil 3 3 v9a 3 3 nd spf basis section The keyword multi set selects the multi set formalism For instance the line vl0a 4 3 requests four functions to be used for the wavepacket in the lower state 1 and three functions for the wavepacket in the upper state 2 The multi set formalism usually requires fewer single particle functions per state than the single set formalism This makes the former more efficient in most cases 9 4 Building the initial wavepacket for a non adiabatic system For a non adiabatic system with o electronic states also o initial wavefunctions have to be built unless they are read from file When generating the initial wavepacket the MCTDH program however assumes that only one electronic state is initially populated and hence sets all wavefunctions on other states to zero The initial wavefunction can therefore be defined in the same way as for adiabatic systems The program has solely be supplied with the information which state is to be populated at the beginning This is achieved using the init_state keyword in the INIT WF SECTION as shown in Example 9 2 If this keyword is missing state 1 is initially populated When a multi set wavefunction is to be read from file it is convenient to use a Read Inwf block because then the wavefunction read and the system wav
34. and 3 3 After a successful run the desired lowest eigenstate is stored in the restart file This eigenstate can then be used as initial wavefunction in following calculations by using the file keyword in the INIT WF SECTION To check the convergence of a relaxation calculation with respect to the propagation time tfinal you may look at the total energy being displayed in the output file If this has not changed significantly during the last outputs the eigenstate is converged Otherwise the calculation should be continued to longer times See Sec 3 9 for continuing calculations 24 3 Defining the type of calculation to be made 3 5 Advanced topic Improved relaxation Generation of excited eigenstates This section may be difficult to understand if one is not familiar with the constant mean field CMF integration scheme For a brief discussion on CMF see Sec 8 2 and 8 4 2 A more comprehensive discussion on CMF can be found in the MCTDH review The Improved Relaxation algorithm is discussed in the MCTDH feature article 4 and more recently and more comprehensively in refs 22 24 Consider a relaxation run where the CMF integration scheme is adopted and the SIL inte grator is used to propagate the A vector In this case it seems to be somewhat cumbersome to generate a relaxed i e propagated in imaginary time A vector by taking a linear combina tion of the eigenvectors of the Lanczos matrix It is more meaningful to replace
35. are declared as complex Hence one is wasting memory and CPU time by not making use of the reality of the wave function To reduce this waste one may use the keywords rDAV or rrDAV rDAV is similar to DAV but it stores the Davidson vectors as real This substantially reduces the memory demand rrDAV performs in addition the matrix vector operation H A in real arithmetic which speeds up this step Note that rrDAV can be used only if each single operator appearing in the Hamiltonian is real This in particular excludes the use of the operator p One may replace it by the operator dq Furthermore rrDAV cannot be used if there are so called muld operators i e multi dimensional operators which act on more than one MCTDH particle simultaneously The use of natural potentials however is allowed Note that the relaxation of the SPFs is always done in complex arithmetic Turning more to the technical side we note that the integrator setting for improved relaxation is quite different from the one for propagation For improved relaxation with Davidson the CMF accuracy is usually set to a rather low value around 1073 whereas high accuracies 10 7 10 8 and 10 8 107 9 are to be used when propagating the SFPs and the A vector respectively See Sec 8 4 2 For propagating the SFPs we recommend the RK8 integrator In contrast to a propagation run one now prefers to have an output after each update step i e after each diagonalisation To achie
36. ascii where data gives the path of the data file and ascii might be replaced by binary if the file is in binary format For POTFIT one would write pes readsrf data ascii 6 8 Implementing non separable potentials potential surfaces 59 in the OPERATOR SECTION of the POTFIT input file The file data must contain the potential energy values at the grid points just the energies and one energy per line The order is implicitly defined by the Primitive Basis Section or in mctdh by a i amp j amp k Gj k 1 2 construct of the Hamiltonian Section Note first index runs fastest There is no check for the consistency of the data Important note Since version 8 3 10 there are four more files which were introduced to separate changes done by a user from changes done by the authors This new proce dure will simplify an update of the package Rather than editing the Makefile please edit now the two files install user_surfdef and install user_surfaces Similarly rather than edit ing opfuncs funcsrf F please edit now the file opfuncs usersrf F For implementing new 1D functions please use the file opfuncs user1d F The implementation of a non separable potential follows the same philosophy as that of separable potentials now with source opfuncs usersrf F being the relevant file to modify Let us again assume that you provide a FORTRAN routine say newsurf stored in a file source surfaces newsurf f to evaluate the non separable potent
37. be plotted with the help of GNUPLOT scripts The installation of the MCTDH package is described in Appendix E This documentation is intended to help the user by explaining with many examples how to set up and run a calculation and analyse the results For a calculation the Hamiltonian operator and the input parameters must be defined This is done in two ASCII files named operator and input file which must have op and inp respectively as extension The required data is put in as keywords In both files the keywords are grouped together into sections each with a specific set of information The sections start with a line containing the keyword XXX SECTION and end with END XXX SECTION where XXX is the name of the section Everything following a is treated as comment How to set up the operator and input file will be detailed in the following chapters Note however that this Guide does not claim to be complete Although the majority of options of the MCTDH package and in particular those being most important for your daily work is described there are probably still options useful for you that are not documented here For the full list of options see therefore the HTML manual The HTML manual also describes the installation process Some parts of the User s guide are labelled as advanced topics indicated by a in the table of contents These parts contain information on features of the MCTDH package that make the pro
38. berd 0 1qd berv70 bew 1 c002 berd 0 1qd berv70 bew 2 c003 berd 0 1lqd berv 0 bew 3 c004 berd 0 1qd berv70 bew 4 c005 berd 0 l1lqd berv70 bew 5 c006 berd 0 1lqd berv 0 bcw 6 c010 berd 1 1qd berv70 bew 0 c011 bcrd 1 iqd berv70 bew 1 end hamiltonian section LABELS SECTION berd expl 1 5 4 315 lqd exp 1 5 4 315 berv gq 2 136 bew expcos 1 1 theta0 end labels section end operator Example 6 4 A surface file containing the MANTHE analytic fit to the NOCI S potential energy surface etm pt eV of the HAMILTONIAN SECTION and defined as LABELS SECTION V mysurf nd labels section in the LABELS SECTION of the operator file Then you may add alter label V natpot name nd label parameter to the OPERATOR SECTION of your input file The program then uses the natural potential fit stored in the name directory rather than the potential surface labeled my surf It is also possible to move some part of the Hamiltonian section e g the potential to a separate file called surface file Use of this is made in Example 6 1 where only the kinetic energy part is defined in the HAMILTONIAN SECTION The separable poten tial is stored in the surface file nocl1 srf part of which is displayed in Example 6 4 See 66 6 Setting up the Hamiltonian MCTDH_DIR operators nocl1um srf for a complete listing and compare it with Eqs 32 33 of Ref 3 The surface is defined in the Hamil
39. directly by a K line and the corresponding degrees of freedom here theta and K must be declared as combined in the SPF BASIS SECTION The first number after KLeg specifies the number N of 6 grid points the following keyword 4 9 Three Dimensional rotational DVR 43 controls which symmetry is to be used For a11 all values 2 Ky K 1 K1 N 1 are employed for odd even only odd even values of are taken ie K K 2 K 2N 2o0rl K 1 K1 3 K 2N 1 Here Ky denotes the minimum of K i e Ay 0 if Kmin Kmax lt 0 and K min Kminl Amax else The numbers following K are the minimal Kmin and maximal Kmax values for K The K grid thus consists of 9 points for the example above The two dimensional Legendre DVR P Leg is similar to KLeg except that a Fourier transformation from K to is performed An example for a PRIMITIVE BASIS SECTION defining a PLeg two dimensional Legendre DVR for the set of coordinates theta and phi is given by PRIMITIVE BASIS SECTION theta PLeg 31 even phi exp 11 2pi end primitive basis section The keyword PLeg selects the two dimensional Legendre DVR and the label phi indicates the second coordinate on which the PLeg representation is based Similarly to KLeg and to the spherical harmonics FBR the PLeg line must be followed directly by a exp line and the corresponding degrees of freedom here theta and phi must be declared as combined i
40. exp i x pl a p2 sinh p1 p2 r sinh pl a p2 cosh p1 p2 r cosh p1 x p2 tanh p1 p2 r tanh p1 x p2 cos1 p1 p2 r cos p1 x cos p1 p2 expl p1 p2 r 1 exp pl x p2 expcos p1 p2 r exp p1 cos x exp p1 cos p2 expcos1 p1 p2 r exp p1 cos x p2 qtanh p1 p2 p3 r tanh p2 arccos x p1 motanh p1 p2 p3 p4 r tanh p3 1 exp p1 x p2 asin p1 p2 p3 r arcsin p1 x p2 p3 acos p1 p2 p3 r arccos pl x x p2 p3 atan p1 p2 p3 r arctan pl x p2 p3 coschirp p1 p2 p3 r cos a p2 p1 p2 exp zx p3 tgauss p1 p2 r exp pl arccos x p2 gauss p1 p2 r exp pl a p2 ngauss o xo r 2n0 exp a z0 20 morse p1 p2 p3 p4 p5 Morse function See Note 1 morsel p1 p2 p3 p4 Morse function See Note 1 CAP p1 p2 p3 p4 iW See Note 2 ACAP p1 p2 p3 p4 p5 iW See Note 3 continued modes theta k sda eeos 7p i cep 22 Jep ecos 1 ep end hamiltonian section is a valid construct Note that the 2D operator j_p may be multiplied from right or left with and operator operating on 0 only However it may be multiplied only from right with a local k dependent function here c_p The operators dth1 dth2 qdq and sdq replace the first derivative operator for the Leg and KLeg PLeg rHO cos and sin when the
41. file by hand one may let the MCTDH program do that This can be accomplished by giving the keywords t cpu and or tstop in the RUN SECTION or by using the options t cpu and or t stop See the HTML documentation for details 3 10 Advanced topic Using parallel shared memory hardware If shared memory hardware is available MCTDH can take advantage of it The parallel fea tures of MCTDH are used if the keyword usepthreads Tissetin the RUN SECTION where I stands for the number of processors that should be used Further arguments can be added which disable the parallelisation of the different MCTDH routines The paralleli sation of the following routines can be disabled phihphi no phihphi calcha funka2 no funka summf no summf mfields no mfields hlochphi no hlochphi hlochphilm no hlochphilm funkphi no funkphi getdavmat no getdavmat and dsyev no dsyev In the MCTDH code the funkphi routine calls several subroutines 30 3 Defining the type of calculation to be made hlochphilm mfsumphilm dichtlphilms hunphilms addhunphilms and projectlms The parallelisation of the first routine is switched off by setting the no hlochphilm keyword Setting the no funkphi keyword switches off the parallelisation of the other routines all together The funka2 routine is only used in the case of a relaxation run with the rrDAV integrator There are more keywords The mem calcha and the mem mfields keywords enable M
42. form can easily be evaluated with the projection program A PROJECTION SECTION of the corresponding input file would look like this Figure 12 2 Jacobi coordinates for a 4 atomic system 126 12 Using the Potfit program PROJECTION SECTION PROJECTOR KO PHI cos 0 nd projector PROJECTOR K1 PHI cos 1 nd projector ROJECTOR K2 PHI cos 2 end projector Ae aas error nd projection section oO 0 U where PHI is the modelabel of the torsional coordinate in question Take note that the defini tion of the cos projector function already includes the factor 1 27 The usage of the error keyword is recommended here as it will calculate the error measure described in the last sec tion so one can check whether one has calculated enough Fourier components to faithfully represent the original potential look at the log file Once this projection run is finished you will find the files vpot_KO vpot_K1 etc which contain the individual Fourier components Vo As these are defined on the full grid of the remaining coordinates Q one must use potfit to bring them into the product form required by MCTDH It will be necessary to write separate input files to fit the separate Fourier com ponents but these input files will look very much alike When writing the input files note the following e in the RUN SECTION use readvpot to read the vpot_Kx files created by projection e in the OPERATOR S
43. fxy will be treated as mode operator Inspect the op log file After the operator terms are summed if possible they are listed under the heading Hamiltonian Operator Terms h htmdof htmmode htmmuld htmtype htmsym string No F m md Typ Sym Term A non zero entry in the column f or m denotes that the operator term acts on the DOF f or the particle m respectively If however x y and z would form one MCTDH particle then fxy would not be recog nised as a mode operator because it does not act on all DOFs of the mode One has to include z as a dummy DOF 68 6 Setting up the Hamiltonian HAMILTONIAN SECTION modes Re ol S S S Z theta const 2 amp 3 amp 4 fxy 5 cos end hamiltonian section or equivalently HAMILTONIAN SECTION modes Rae ape g i Se l Z theta Const 1 amp amp fxy cos end hamiltonian section Assume that there is another symbol fzx which refers to the function f z x An inclusion of this function may look like HAMILTONIAN SECTION modes LCR jh seo ayoul 2 ol sbheta const 4 amp 2 amp 3 fzx 5 cos end hamiltonian section Note that the dummy variable s must be the last one s The symbol fzx is simply inter preted here as f2 z x y with no dependence on y Note that one can freely re order the arguments of a multi dimensional function when using a numbered Hamiltonian line The use of muld potentials is restricted as there must be no holes
44. if np gt 0 then hoprpar 2 hoppar 1 np hoprpar 3 hoppar 2 np else hoprpar 2 1 0d0 hoprpar 3 0 0d0 endif ifunc 2 Finally you have to modify the subroutine ufuncld Find the following comments C newfunc EXAMPLE C Set here the routine you want to call C elseif ifunc eq 2 then C call newfunc x v and replace those comments by the following lines C cot cot ax x b elseif ifunc eq 2 then call cot x v hoprpar 2 hoprpar 3 This completes the modifications for implementing the cotangent function After recompiling the program type compile mctdh the new label cot may be used in the HAMILTONIAN SECTION When the label cot is used with parameters it must as usual be assigned to a simple label in a LABELS SECTION see Sec 6 4 6 7 Advanced topic Implementing separable potentials In favorable cases the potential energy surface may be expressed by the built in symbols known to the program as has been discussed in Sec 6 3 If this is not possible new symbolic 6 7 Implementing separable potentials 57 expressions have to be introduced as detailed in the previous section In some cases however it may be more convenient to set up a set of labels specifically for a complicated separable potential i e sf Vines f S Vata 6 4 k 1i 1 where f and s denote the numbers of degrees of freedom and Hamiltonian terms respec tively Let us further assume
45. in connection with the KE keyword and strange results may occur if KE is used but mas s_modelabel is not set Do not use this construct when the second part is not a valid modelabel E g for the total mass do not use mass_tot but rather use mass tot or mass tot orMassTot instead Labels Section See also Section 6 4 One may use the same set of letters numbers and special characters for labels as are al lowed for parameters Again the colon is not allowed to be part of the name although pre defined labels i e those listed in Appendix C may contain a colon Moreover the un derscore has a special meaning for labels If a label has the structure label_modelabel then the mctdh program will put the corresponding operator in that column of the Hamiltonian Section which refers to modelabel One must not put it explicitly there Unit operators are assumed for all other degrees of freedom This feature which is often used to include CAPS 70 6 Setting up the Hamiltonian excludes the general use of the underscore in a label E g defining alabelas exp_1 may produce an error because 1 may not be a modelabel Note that there must not be a parameter and a label which have the same name E g q cannot be used as parameter because it is pre defined as a label The program checks that parameter and label names are disjoint Hamiltonian Section See also Section 6 3 Only simple labels may appear in a Hamiltonian Section Operators with arg
46. lt 1lori Q gt N For a periodic shift on a periodic grid exp DVR of FFT use shift Q and shift Q N with Q gt 0 Table C 4 One dimensional potential energy curves Symbol Potential Curve v NO NO potential curve v H2 Ho potential link Isth vbmkp H2 H potential link h4bmkp v HO OH potential morse function from h2o0 f v 0H OH potential link hoosrf v CH CH potential link c2h v C2 C potential link c2hasec v OF OF potential curve vrho H3 H Hp potential in hypersphaerical coordinates theta 7 link Isth vthe H3 H Hg potential in hypersphaerical coordinates rho 2 484773 link Isth vdj 000 expansion coeffiecent Vooo for DJ H4 surface link h4dj vdj 022 expansion coeffiecent Vo22 for DJ H4 surface link h4dj vdj 224 expansion coeffiecent V224 for DJ Hy surface link h4dj 142 C The built in symbolic expressions Table C 5 Two dimensional operators used for molecule surface scattering Symbol Operator coshcosth p cosh p cos sinhcosth p sinh p cos cossinthcosphi p cos p sin x cos cossinthsinphi p cos p sin sin sinsinthcosphi p sin p x sin cos sinsinthsinphi p sin p x sin sin reY l m Re Y 9 imY l m Im Y 0 Table C 6 Multi dimensional C C_ symbols defined on truncated k1 ko ka grid See also Hamiltonian Documentation Available Surfaces Symbol Operator cpp jtot J dim d v KIHI EEk ka cmm
47. meigenf and build the initial state as Hartree product from those low dimensional eigenstates Compare with the hono dav inp and H2CS r1r2 inp input files on MCTDH_DIR inputs Furthermore one may use the orthogonalise keyword to purify an initial state from contributions of already converged eigenstates See HTML documentation under INIT WF SECTION When the ClI pace is too small i e when the A vector length is too short the improved relaxation algorithm will not converge due to a variational breakdown Not only the state requested but all states below this one must be representable by the SPF basis sets For highly excited states this may require large Cl spaces One must be careful when using mode combinations and avoid too strongly combined modes because over combination will make the ClI space too small Relaxation to the ground state is always unproblematic If 3 6 Performing a numerically exact calculation 27 a relaxation to an excited state does not converge one has to use more SPFs although the natural populations may already be very low The natural population indicates how important an orbital is for representing the desired state But again here the orbitals have to represent in addition all states below the desired one Rather than single relaxation one may use block relaxation See Sec 2 7 for an example The keyword split rst splits the restart file of the block relaxation into individual restart files for eac
48. of the input file MCTDH reads the data file two columns free format till it finds an end of file Blank lines and lines beginning with a hash are ignored The x values have to be equally spaced Extrapolation is not possible hence the first x point must be smaller than the first grid point and the last x point must be larger than the last grid point For optimal performance there should be two data points below the first and above the last grid point respectively The data points should be dense enough such that the interpolation error is negligible The second way is similar to the first one but here the data must be given precisely at the grid points A file of either ASCII or binary format must be generated such that the first line contains the data for the first grid point the second line for the second point etc Let us call this file again data The LABELS SECTION should now contain the statement V2 readld data ascii where ascii is to be replaced by binary if the file data is in binary format See also HTML documentation Hamiltonian Documentation Labels Section The third way is to use the pre defined label my1d and to edit the subroutine myld which is on source opfuncs funcid F The label my1d is to be used in the same way as e g the label cos i e with or without parameters One may make use of up to five parameters Grid based 1D operators i e matrix representations of 1D operators with respect to
49. one can set environment vari ables in mctdhrc which point to the MCTDH backup and elk directories for users who want to change the code Use the script mklinks to set a link to the PES requested However one should do so only when a PES is needed After a PES is linked to the MCTDH _DIR source surfaces directory one has to compile mctdh or potfit with the i option e g execute the commands mklinks h4bmkp compile i h4bmkp potfit before running potfit84 to bring the BMKP surface of the H4 system to product form Load the URL file home muser mctdh doc index html into your browser to inspect the HTML on line documentation Bookmark this page A simple but quick help is provided through the script mhelp It briefly explains the keywords of the input 150 E Installing the MCTDH package file Try mhelp h All MCTDH scripts and programs know the help option h If you want to inspect the code try the scripts mcb meg mcl and phelp You may go to the AdvancedUser directory e g type cdm Ad and execute make there read the README file first This will give you access to additional scripts and routines meale is quite useful There are four compile configuration files on the install directory compile cnf_le compile cnf_be compile cnflenp and compile cnf_benp The letters le and be stand for little endians and big endians respectively np denotes no parallelization The install_mctdh script copies one of these four config
50. populations of non adiabatic systems 107 100 80 60 20 0 6 Energy eV Figure 11 3 The total full line and projected flux of dissociating NOCI The projection is on the vibrational states of the NO fragment v 0 v 1 and v 2 respectively plistate Note that plstate may also be used for runs evolving on one single potential energy surface In this case the norm squared of the wavefunction is plotted In the presence of CAP s this quantity is non constant In case of a multi state calculation which uses the single set formulation the state populations presently have to be read from the output file Use the command fgrep population output sed s population gt file to write the state populations to the file file 11 8 2 Adiabatic populations computed with adpop Because mctdh usually works in the diabatic representation the diabatic populations can be easily calculated see above But if one is interested in the adiabatic state populations these are more difficult to obtain There are two possible ways The first possibility is to use the analyse program adpop This program reads the psi file of an metdh run and the so called pes file which is a special operator file in which all derivative operators are ignored The pes file is conveniently generated by setting the option pes i e running mctdh84 pes inpfile Moving to the name directory one may then calculate the
51. rd sin 36 3 800 5 600 rv HO 24 2 136 0 272 ev 7 4667 AMU theta Leg 60 0 all end primitive basis section INTEGRATOR SECTION CMF var 0 5 1 0d 5 BS spf 10 1 0d 6 SIL A 12 1 0d 6 nd integrator section INIT_WF SECTION file nocl0d end init_wf section end input Example 4 1 An input file for a wavepacket propagation on the S1 surface of NOCI Here the two parameters define the grid points x and zy The program then computes the corresponding product mw as well as the equilibrium position x xy 2 The above example is thus equivalent to the previous one A radial Hermite DVR can be selected in exactly the same way than a Hermite DVR but with rHO rather than HO as keyword 4 3 Legendre DVR A Legendre or rotator DVR is employed for angular degrees of freedom 0 because the asso ciated Legendre functions P7 cos 0 are eigenfunctions of the angular momentum operator i The basis functions are thus the 2 normalised associated Legendre functions Xi m 1 0 21 1 l m a eu oe 4 3 4 4 Sine DVR 39 with m gt 0 and restricted tom lt l lt m N 1 The parameter m denotes the magnetic quantum number and is treated as a fixed parameter The associated Legendre function P is given by the polynomial 1 j qitm PI x 2 Geer ike im pana 4 4 forO lt m lt l A Legendre DVR is selected for a coordinate named thet a by the line
52. remains unchanged There is a default example from MCTDH in the file opfuncs userid F showing how users can define their own one dimensional potentials There are three subroutines in this file ufdef1d ufuncld and my1d Suppose we use the label my1d in the operator file LABELS SECTION my myld pl p2 p3 p4 p5 pd nd labels section The subroutine my1d will be executed during running MCTDH Although there is no real potential coded in this subroutine one can find instructions on how to implement the user defined potentials To call the subroutine my 1d the program has to identify the label my1d This is done in the subroutine uf def1d where MCTDH reads the label and its parameters e g pl The arrays hoprpar and hopipar serve to pass real or integer arguments to the function definition in subroutine ufuncld Here we only use the real one The program by default stores an exponent r in hoprpar 1 or hopipar 1 depending on its type If any parameters p1 p2 given in square brackets are present this is indicated by the counter np being greater than zero and the parameters are automatically pushed to the array elements hoppar 1 np hoppar 2 np Since hoprpar 1 is reserved for the exponent there is a shift of arguments between hoprpar and hoppar If no parameters have been specified i e np 0 default values are taken for the parameters Besides the issue of parameters each label is given a differ
53. requires that the wavefunction has been stored psi file Alternatively one may compute the adiabatic populations on the fly by setting the keyword expectation in the RUN SECTION of the metdh propagation Here s a list of all steps needed to compute the adiabatic populations 1 Create an pes file by an mctdh genpes run 2 Create vpot files with adproj as explained above Just run adproj in the name direc tory of step 1 adproj will read the pes file 3 Fit the vpot files of the projector with potfit to create natpot files As potfit requires an input file an example is shown in Example 11 3 Here potfit will read the vpot file apr_p1_11 and will create the natpot file in the name directory projector111 In the OPERATOR SECTION the keyword pes none must be given Note that there must not appear an electronic degree of freedom in the PRIMITIVE BASIS SECTION A similar input file must be generated for each matrix element of the pro jector 11 8 Monitoring state populations of non adiabatic systems 109 RUN SECTION name projector111 readvpot apr_pl_ll path of the vpot file end run section OPERATOR SECTION pes none no PES from library needed nd operator section as the PES is read from vpot PRIMITIVE BASIS SECTION vl0a HO 20 0 0 1 0 10 voa HO 30 0 0 1 0 130 v1 HO 20 0 0 1 0 1 0 end primitive basis section NATPOT BASIS SECTION vl0a 10 v6a contr v1
54. script plgpop 11 11 3 Checking the accuracy of a calculation 101 11 3 2 Checking the single particle function basis size The quality of the single particle function basis is reflected in the populations of the nat ural orbitals see Sec 3 3 in Ref 1 If the calculation contains natural orbitals with a low population these are not significant for the representation of the wavefunction and the calculation is of a reasonable quality Unfortunately different properties have different con vergence criteria and it is not possible to give absolute figures for when the natural orbitals are insignificant As a general rule of thumb when the population of the highest least pop ulated natural orbital is below 1 i e a population below 0 01 the calculations will be reasonable although convergence may be a way off Experience has shown that it is important that the single particle function bases for all the modes are balanced i e the lowest natural orbital populations are similar for all There is little point spending effort on converging the single particle function basis for one mode when the dynamics can be seriously affected by the poor representation of another mode The program rdcheck is used to check the natural orbital populations and so show where more functions are required Two basic pieces of information are required by the program a state and a mode for analysis If no arguments are given the program prompts for wh
55. strong increase of computational labour that arises when such potentials are evaluated directly The disadvantage of these procedures is their introduction of an additional unpredictable error in the calculation For a discussion of the TDDVR and CDVR methods see Ref 1 The TDDVR or CDVR method can be used by simply inserting either the keyword TDDVR or the keyword CDVR into the INTEGRATOR SECTION of the input file Presently both TDDVR and CDVR work only in combination with the VMF integration scheme Chapter 9 Treating non adiabatic systems To treat a non adiabatic system i e a system which involves a manifold of coupled potential energy surfaces the Hamiltonian operator has to be set up appropriately Furthermore the primitive and the single particle basis as well as the initial wavepacket have to be defined in a special way when the system is non adiabatic 9 1 Setting up the Hamiltonian for a non adiabatic system The Hamiltonian of a non adiabatic system with electronic states can be written as a x o matrix Ay nee Hio H aie 9 1 Ay Hoo For the sake of simplicity we assume in the following that only two electronic states have to be accounted for i e o 2 The generalisation to larger numbers of states is straightforward To implement the Hamiltonian matrix which is now two dimensional into the MCTDH program it must have the product form required by the MCTDH method s k HES hP ah 4
56. systems J Chem Phys 109 349 1998 J Chem Phys 109 1998 351 H D Meyer Multiconfiguration time dependent Hartree method In The Encyclopedia of Computational Chemistry Chichester 1998 P v R Schleyer N L Allinger T Clark J Gasteiger P A Kollman H F Schaefer II and P R Schreiner Eds vol 5 John Wiley and Sons pp 3011 3018 R Milot and A P J Jansen Ten dimensional wave packet simulations of methane scattering J Chem Phys 109 1998 1966 1975 F Matzkies and U Manthe Accurate quantum calculations of thermal rate constants employing MCTDH H2 OH H H20 and D2 OH D DOH J Chem Phys 108 1998 4828 F Matzkies and U Manthe Accurate reaction rate calculations including internal and rotational motion A statistical MCTDH approach J Chem Phys 110 1999 88 A Jackle M C Heitz and H D Meyer Reaction cross sections for the H D2 v 0 1 system for col lision up to 2 5 eV A multiconfiguration time dependent Hartree wave packet propagation study J Chem Phys 110 1999 241 248 I Burghardt H D Meyer and L S Cederbaum Approaches to the approximate treatment of complex molecular systems by the multiconfiguration time dependent Hartree method J Chem Phys 111 1999 2927 2939 A Raab I Burghardt and H D Meyer The multiconfiguration time dependent Hartree method gener alized to the propagation of density operators J Chem Phys 111 1999 875
57. terms is protocolled in the showpot log file Chapter 12 Using the Potfit program 12 1 Transforming a potential to product form For optimal performance the MCTDH algorithm requires the Hamiltonian to be given as a sum of products of single particle operators MCTDH product form The kinetic energy operator usually is in MCTDH product form but the potential is often given as a multidi mensional function The program potfit is able to transform a given potential energy surface to MCTDH product form For small systems e g 3D it does this job fast and reliable However in contrast to the MCTDH program which avoids using the primitive product grid potfit has to employ the full primitive product grid The computational resources used by potfit thus increase much more strongly with the size of the system than the ones required by MCTDH The numerical effort of potfit can be reduced if the potential surface is partly given in product form For example if a 6 D surface reads V u V Ww Y z Vi u v w Va z y z V3 u v w V x y z 12 1 one should of course apply potfit to V1 V4 individually rather than applying it directly to V Similar to metdh84 potfit84 is started with giving the path of an input file as argument potfit84 lt inputfile gt The input file is structured similar to the MCTDH input file To understand it one must be familiar with the basics of the potfit algorithm The potfit algorithm i
58. that you provide a FORTRAN routine say mypot stored ina file source opfuncs mypot f to evaluate the one dimensional potentials v V a in depen dence of k i and x v and z in a u subroutine mypot k i x v integer k i realx8 x v end The first step then is to add this file to the operator library by inserting the line AR_OPFUNCS PATH_OPFUNCS mypot o at the corresponding position into the install Makefile The second step is to establish labels for the one dimensional potentials Although the choice of these labels is arbitrary we strongly recommend the use of some systematic nomen clature e g mypot1 1 mypot2 1 mypot1 2 etc where the first number denotes k and the second i For instance with s 3 and f 2 the potential would then be defined by the lines 1 0 mypoti 1 mypotl 2 1 0 mypot2 1 mypot2 2 1 0 mypot3 1 mypot3 2 in the HAMILTONIAN SECTION Next a link between these labels and the potential routine is needed This is done by adding the lines 8 call mypot ipar 2 ipar 3 x v return to subroutine callanld in the file source opfuncs callanld f The file number 8 should be the next free number in that subroutine For this to work properly you have to store the in dices k andi in ipar 2 and ipar 3 before Remember that ipar 1 is reserved for an exponent if present This can be accomplished by inserting a new subroutine defmypot into your mypot f file subroutine defmyp
59. the addition of phenomenological broadening The calculation takes the ground state wavefunction here a simple product of gaussians as the ground state surface is harmonic and places it on the Sz excited surface Propagation then takes place and rapid population transfer to the S state is observed Finally the spectrum of the model system is calculated 1 Copy the file MCTDH_DIR inputs pyr4 inp and create the directory pyr4 2 To perform the photo excitation calculation type mctdh84 pyr4 This will take about 20 seconds The calculation can now be analysed Move to the directory pyr4 which contains all the data files from the propagation 8 2 An MCTODH tutorial 70 60 F 50 40 F 30 20 F 1 0 5 0 0 5 1 Energy eV Figure 2 4 The absorption spectrum for the pyrazine molecule on excitation to the S2 state calculated using a 4 mode model with phenomenological broadening 1 To plot the diabatic state populations type rdcheck84 g 1 0 gnuplot persist chk pl or more simply type plstate The result is shown in Fig 2 3 Note that very fast transfer occurs to the S state At around 80 fs the system returns to the conical intersection connecting the two states and a second transference of population occurs 2 To plot the spectrum type autospec84 g 1 0 2258 eV 1 0 1 0 eV 30 1 gnuplot persist spectrum pl The first line produces a GNUPLOT file with data to plot the spectrum from
60. the initial spherical harmonic must be part of the primitive basis set In an MCTDH calculation this again defines not only the initial wavepacket but also the first single particle function Higher single particle functions are other spherical harmonics whose quantum numbers are as close as possible to the quantum numbers of the first one 7 5 Generating Wigner functions as initial functions For rotational motion of polyatomic molecules in three dimensions Wigner D functions can be used for the initial wave function These are defined as j 274 1_ Dm la 8 7 Dy ig Cis By 7 5 LD apa Se a Ble 7 6 d 4 B jm e k 7 7 where Di a 3 7 is the normalized Wigner D function and d p B is the Wigner small d function and j7 lt m k lt j A Wigner D function can be generated as an inital wave function by specifying the wigner keyword followed by two K lines as in the example below INIT_WF SECTION build mode type q n beta wigner 5 nosym excite mkj print j of initial wigner gamma k 3 ae Ts E initial k k range k step siz alpha k 1 a2 2d initial m m range m step siz end build END INIT_WF SECTION The Wigner big D function is generated in the above example as the product of a Wigner small d function for the 8 Euler angle and a Kronecker 6 for the associated momentum quantum number of each of the a and y angles The first number foll
61. the relaxed A vector by the ground state of the Lanczos matrix as one is interested in the ground state but not in a proper propagation in imaginary time This modification leads not only to a faster convergence but more importantly offers the possibility to converge to an excited state One simply takes the n th state of the Lanczos matrix as relaxed A vector However the dimen sion of the Lanczos matrix is in general small compared to the length of the A vector The pseudo spectrum of the Lanczos matrix is thus a poor simulation of the spectrum of the ma trix representation in the set of the single particle functions of the Hamiltonian H Although the n th eigenstate of the Lanczos matrix may be a good approximation to an eigenstate of the H matrix it may not approximate the n th state of H but a higher one Thus the algorithm of improved relaxation will converge to some eigenstate but one is not sure which one it is except when the ground state is sought To deal with this situation the algorithm is set up such that that eigenvector of the Lanc zos matrix is taken which has the largest overlap with the one of the previous CMF step A warning is written to the log file if this overlap squared is smaller than 0 66 There are two ways to select the initial Lanczos vector The first way is to write relaxation n to the RUN SECTION The n th eigenvector of the Lanczos matrix is then taken as starting point of the relaxation The c
62. the script compile cnf has to be modified see above The option lpthread must be added in the line MCTDH_ADD_LIBS and the option pthread must be added in the line MCTDH_CFLAGS in the section of the compiler that is used Then MCTDH must be compiled again For the standard compilers e g GNU or Intel all the necessary extensions of the compile scripts are already done To improve the efficiency of the shared memory parallel MCTDH on NUMA non uni form memory access machines the numa h library can be linked 7 To do so the u option must be given for compilation compile u mctdh Doing so the POSIX threads created during an MCTDH run are distributed and bound cyclicly to the available processors to prevent thread migration Of course this only works if the numa library is available on your computer On NUMA machines e g typical Opteron or Xeon clusters we observed that execution times of identical runs may vary by more than 20 Using the numa h library however these surprising variations vanish and all runs take almost identical execution times These times agree with the shortest times observed without using numa h 3 11 Advanced topic Using parallel distributed memory hard ware Beside parallel shared memory hardware MCTDH can make use of parallel distributed memory hardware For the distributed memory parallelization of MCTDH the Message One should edit compile cnf_be and compile cnf_le as well because compile cnf is overw
63. the spf basis section If the degrees of freedom are not combined in the natpot basis section they still can be combined in a mctdh run There are however restrictions For each combined mctdh mode potfit must use precisely the same combination or may treat all degrees of freedom of this mode uncombined Combining degrees of freedom can reduce the number of natural potentials needed for convergence This will be important for large systems The PRIMITIVE BASIS SECTION must be identical to the PRIMITIVE BASIS SECTION of the following MCTDH calculation except for the ordering of the degrees of freedom This ordering however must be consistent with the ordering of the argu ments of the pes to be fitted If one is insecure about the latter ordering inspect the file source opfuncs funcsrf F and search for the name of the particular pes under discussion See also the HTML documentation NB One may use the order keyword OPERATOR SECTION of the potfit input file to define a new order in which the arguments are passed to the surface routine However this does not work for the readsrf surface See the HTML documentation for details Note that mctdh uses the modelabels to associate the natpot terms with the DOFs If this is not wanted one may give the keyword ignore as a parameter to the natpot keyword in the LABELS SECTION In this case one must use a numbered input e g 1 amp 2 amp 3 V in the HAMILTONIAN tableau to indicate on which DOFs an
64. this is done a ptiming file is created for each MPI prozess containing the timing information for the parallel rou tines Thereby the quality of the shared memory parallelization within the different MPI prozesses can be checked in more detail as by the mpitiming file These files are denoted by ptiming0O ptiming1 and so on This is mainly for testing To combine distributed and shared memory parallelization both the usepthreads and the usempi keywords must be set in the RUN section and MCTDH must be started with the mpirun command For a parallel calculation where each MPI prozess uses 4 POSIX threads the RUN section may look like RUN SECTION usepthreads 4 usempi end run section Here the parallelization for all routines is working The MPI parallel ones are called in each MPI prozess where they run in shared memory parallel mode with 4 POSIX threads Additionally in the master process the other shared memory parallel routines for the SPF propagation are used Example mpirun np 5 mctdh84 mpi g77 lt inputfile gt The g77 built MCTDH is now run with 5 MPI processes and 4 POSIX threads per MPI process i e 20 cores are used The specific command to be given may depend on the queueing system of your computer and hence may differ from the example above 3 11 Using parallel distributed memory hardware 35 To be able to start the MPI parallel MCTDH the program must be compiled with some additional options For the standa
65. time dependent Hartree approach Calculation of vibrationally excited states and reaction rates Chem Phys Lett 252 1996 71 U Manthe A time dependent discrete variable representation for multi configuration Hartree methods J Chem Phys 105 1996 6989 M Ehara H D Meyer and L S Cederbaum Multi configuration time dependent Hartree MCTDH study on rotational and diffractive inelastic molecule surface scattering J Chem Phys 105 1996 8865 8877 K Museth and G D Billing Generalization of the multiconfigurational time dependent Hartree method to nonadiabatic systems J Chem Phys 105 1996 9191 F Matzkies and U Manthe A multi configurational time dependent Hartree approach to the direct calculation of thermal rate constants J Chem Phys 106 1997 2646 T Gerdts and U Manthe The resonance Raman spectrum of CHsI An application of the MCTDH approach J Chem Phys 107 1997 6584 A Jackle Die zeitabhdngige Multikonfigurations Hartree Methode und ihre Anwendung auf reaktive Streuprozesse PhD thesis Universitat Heidelberg 1997 H D Meyer G A Worth and J Y Fang Comment on Generalization of the multiconfigurational time dependent Hartree method to nonadiabatic systems J Chem Phys 105 9191 1996 J Chem Phys 109 1998 349 K Museth and G D Billing Response to Comment on Generalization of the multiconfigurational time dependent Hartree method to nonadiabatic
66. to a bug it otherwise sometimes performs in correctly 4 8 Advanced topic Extended Legendre DVR KLeg and Two Dimensional Legendre DVR PLeg Similarly to the spherical harmonics FBR the extended Legendre DVR employs the spher ical harmonics as basis set It thus defines a two dimensional representation As the name indicates there is a FBR DVR transformation from the angular momentum indices to grid points 0 There is however no such transformation for the m pair The extended Leg endre DVR will typically be used to describe angular motion of a molecule with total angular momentum J gt 0 The projection of the angular momentum of the angular motion under discussion onto the body fixed frame is usually called K or Q rather than m The angle which is the coordinate canonical conjugate to K is an Euler angle and does not appear in the potential There is thus no point to perform a K transformation The 2D single particle functions 0 K are hence given in a mixed DVR FBR representation An example for a PRIMITIVE BASIS SECTION defining an extended Legendre DVR for the set of coordinates theta and K is given by PRIMITIVE BASIS SECTION theta KLeg gi even K K 4 4 end primitive basis section The keyword KLeg selects the extended Legendre DVR and the label K indicates the second coordinate on which the KLeg representation is based Similarly to the spherical harmonics FBR the KLeg line must be followed
67. values of the CAP parameters i e the Cap length CAP strength and CAP order need to be determined It is our experience that the optimal CAP order is 2 or 3 the larger the energy range Eynax Emin the larger the optimal CAP order The CAP length should be as small as possible in order not to waist grid points On the other hand a short CAP requires a large CAP strength which in turn produces unwanted CAP reflexions The program reflex84 computes an estimate of the CAP transmission and CAP reflexion This estimate derived in Ref 25 is very accurate for a free particle To determine the optimal CAP parameters one needs to know the lowest and highest kinetic energy component for the CAP degree of freedom with which the particle enters CAP These energies are sometimes difficult to estimate If one has used FFT or exponential DVR for the CAP degree of free dom the command showd1d84 a pop2 y 0 01 fx x denotes the number of the degree of freedom of the CAP displays the momentum distribution from which the desired energies can be calculated The program reflex84 is most conveniently called through the shell script pleap The parameters necessary for the calculation are given as options and arguments The program prompts for missing input To give an example let us turn to NOCI Type plcap n 3 m 16 1 0 6 e 0 3 0 1 2 0 ev This computes the reflection and transmission probability for a CAP order of 3 a reduced mass of 16 ato
68. wit 105 File number 57 Filter program see Program Finite basis set representation see FBR Flux program see Program Fourier transformed potential see Potential Function Gaussian 71 harmonic oscillator 36 Legendre 38 72 particle in a box 39 spherical harmonic 41 73 Gaussian function see Function Golden rules 69 Gridpop file see File Hamiltonian section see Section Harmonic oscillator DVR see DVR Hermite Harmonic oscillator function see Function Henon Heiles 51 Hermite DVR see DVR Improved block relaxation see Wavepacket Improved relaxation see Wavepacket 84 Init_wf section see Section Initial stepsize for the CMF scheme 82 for the ABM integrator 83 for the BS integrator 83 for the RK5 8 integrator 83 Initial wavefunction see Wavefunction Input file see File see File Installing package 147 Integration order 83 Integration schemes 80 Integrator section see Section Interaction picture orbital see Orbital Iteration file see File KLeg 42 KLeg see DVR Labels section see Section Lanczos algorithm 28 Lanczos integrator see SIL integrator Lanczos Arnoldi integrator see SIL integrator Legendre DVR see DVR extended 73 function see Function Log file see File see File Mode combination 46 Muld potentials 67 Multi packet 78 Multi set 87 Name directory 22 Natpot 61 Natpot file see File Natural orbital see Orbital Natural population see Population N
69. yes yes INITWF no no yes yes yes INTEGRATOR no no no yes no May be in the operator file Only if parameters are used in the DVR specifications Only if parameters are used in the Hamiltonian specification Only if the Hamiltonian is in an operator file Only if label definitions are required in the Hamiltonian specification 129 130 A The concept of the input file Table A 2 Description of the calculation types The table shows the RUN SECTION keyword required for a certain calculation type and the level associated with this type Also given are the files that are created and needed by the different calculation types Level Keyword Description Created files Required files 1 gendvr Sets up primitive bases dvr 2 genpes Sets up a pes file for analysis pes dvr 2 genoper Sets up an operator for use oper dvr 3 geninwf Sets up an initial wavefunction restart dvr oper 4 propagation Propagates a wavepacket User defined dvr oper restart 4 relaxation Relaxes a wavepacket User defined dvr oper restart 4 diagonalisation Diagonalises a Hamiltonian User defined dvr oper restart Which sections are required depends on the type of calculation to be made Table A 1 lists the various sections and indicates which are required for the various types A possible calculation type is for instance a propagation a relaxation or a diagonalisation symbolised in the RUN SECTION by the corresponding keyword propagation
70. 0 0 25 0 0 end build will generate a flat function for the X degree of freedom and a plane wave with momentum 2 5 au for the Y degree of freedom In a numerically exact calculation the initial wavepacket is simply the product of the functions 7 1 or 7 2 for the degrees of freedom involved In an MCTDH calculation however the program interprets each line in the INIT WF SECTION as first single particle function for that degree of freedom Higher single particle functions are then constructed by multiplying the preceding function by x so producing a series of powers of x followed by Schmidt orthogonalisation onto the lower functions The set of all products of the functions of the included modes then defines the initial configurational space The initial wavefunction in an MCTDH calculation is then chosen as one of these config urations The default is to use the product of the first single particle functions of each mode thus arriving at the same initial wavefunction as in a numerically exact calculation the prod uct of the functions 7 1 or 7 2 One may however also populate a different configuration with the aid of the pop keyword e g INIT_WF SECTION build X gauss 4 315 0 0 0 0794 pop 2 Y HO 24151 0 0 0 218 eV 13615 5 pop 3 end build end init_wf section The initial wavepacket is in this example the product of the second single particle function in X and the third in Y 7 2 Setting up Legendre functions as
71. 0 2 An MCTODH tutorial operators the system Hamiltonian including the dipole operator multiplied with the electric field and a second operator containing the dipole operator alone The system Hamiltonian is used to perform the propagations while the dipole operator is used to evaluate the electric field To execute an example create a new empty directory change to it and copy the example input files e g pyrazine inp and pyrazine op see Ref 18 Inspect the input file and run the command optcntrl mnd pyrazine inp The command optcntrl H will provide help about options Recently the a number of new features have been implemented into OCT MCTDH such as optional filtering of the field and the use of different optimization schemes 19 21 Please refer to the HTML documentation for details 2 10 Concluding Remarks This tutorial has shown you some typical applications of MCTDH In order to ensure that within this tutorial all calculations can be done quickly only the optimal control example takes somewhat longer 60 minutes on a 3 GHz PC we have chosen rather small example systems This however should not mislead you MCTDH is for treating large systems The full power of MCTDH is uncovered when turning to problems which require such a large primitive product grid that a standard numerical exact wavepacket propagation becomes impossible on a workstation A good example for such a problem is the calculati
72. 09 0 2785E 09 0 2162E 09 Trace Sum of all preceding Natural Weights eV 2 1 0 3600E 02 0 5371E 03 0 2313E 03 0 6467E 04 7 0 1045E 04 0 6010E 05 0 3147E 05 0 1907E 05 13 0 3154E 06 0 2118E 06 0 1258E 06 0 7847E 07 19 0 2040E 07 0 1315E 07 0 7802E 08 0 4150E 08 25 0 1157E 08 0 7814E 09 0 5028E 09 0 2866E 09 Abs Trace Sum of relevant Natural Weights eVx 2 Sqrt Abs Trace Sum of rel Nat Weights meV KKKK KKK KK KKK Mode rd rv KKKKKKKKKKKKKKKKKKKKKKKEK trace 0 8399 eVxx 2 0 3651E 04 0 1198E 04 0 1052E 05 0 4074E 06 0 3625E 07 0 1433E 07 0 1783E 08 0 7816E 09 0 1371E 09 0 6923E 10 0 2817E 04 0 1618E 04 0 8552E 06 0 4478E 06 0 4222E 07 0 2789E 07 0 2367E 08 0 1585E 08 0 1495E 09 0 8030E 10 2 81665E 05 543072 KKEKKKKKKKKKKKKKKKKKKKKK au au au au au Trace of reduced density matrix 0 8297 au red trace 0 8399 eVxx 2 Number of eigenvalues considered 4 24 Reduced Eigenvalues Natural Weights eVx 2 1 0 8370E 00 0 2502E 02 0 3699E 03 0 2115E 04 0 1505E 05 0 7 O 1717E 06 0 9290E 07 0 4180E 07 0 3013E 07 0 1483E 07 0 13 0 4230E 08 0 3149E 08 0 1693E 08 0 1479E 08 0 9545E 09 0 19 0 1571E 09 0 2029E 10 0 6050E 12 0 1463E 18 0 2231E 36 0 Trace Sum of all preceding Natural Weights eVx x 2 1 0 2896E 02 0 3935E 03 0 2362E 04 0 2474E 05 0 9691E 06 0 7 0 1990E 06 0 1061
73. 2 2863 H Naundorf G A Worth H D Meyer and O K hn Multiconfiguration time dependent hartree dynam ics on an ab initio reaction surface Ultrafast laser driven proton motion in phthalic acid monomethylester J Phys Chem A 106 2002 719 158 List of MCTDH references 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 T N Rescigno W A Isaacs A E Orel H D Meyer and C W McCurdy Theoretical study of resonant excitation of CO2 by electron impact Phys Rev A 65 2002 32716 S Sukiasyan and H D Meyer Reaction cross section for the H D2 vo 1 HD D and D H2 vo 1 DH H systems A multi configuration time dependent Hartree MCTDH wave packet propagation study J Chem Phys 116 2002 10641 10647 H K6ppel M Doscher I Baldea H D Meyer and P G Szalay Multistate vibronic interactions in the benzene radical cation II Quantum dynamical simulations J Chem Phys 117 2002 2657 2671 U Manthe Reaction Rates Accurat quantum dynamical calculations for polyatomic systems J Theor Comp Chem 1 2002 153 F Huarte Larrafiaga and U Manthe Accurate quantum dynamics of a combustion reaction Thermal rate constants of O P CHa X A1 gt OH X I1 CH3 X Aj J Chem Phys 117 2002 4635 M Nest and H D Meyer Benchmark calculations on high dimensiona
74. 2_pottit inp e cO2_pes e co2_r_grid e co2 theta_grid The file co2_pes contains the ab initio energies and the two following files the corre sponding grid points 3 Move to the co2fit directory and execute the command potfit84 mnd co2_potfit A new subdirectory co2_potfit has been created change to it and execute showpot84 vfit In the interactive menu type 3 times 1 return to see the plot of the natural potential fit By definition it is identical to the original data on the grid points since the same number of natural potentials as grid points has been used Inspect the co2_potfit inp file 2 8 2 Interpolating the natural potential to a new primitive grid 1 Copy the file MCTDH_DIR pinputs co2_chnpot inp to the co2fit directory 2 Execute the command chnpot84 mnd co2_chnpot A new subdirectory co2_chnpot has been created which contains new dvr and natpot files As before the newly interpolated potential can be inspected using the utility showpot84 and following the interactive menu The process outlined in the present and previous subsections can be repeated us ing the alternative set of files co2_r_dense_grid co2dense_pes co2_theta_dense_grid 2 9 Optimizing an external field with Optimal Control Theory OCT 19 co2_chnpot dense inp co2_potfit dense inp The initial grid is double as dense as the original one After the potfit stage the data points are interpolated to the same grid as in the previous case T
75. 581 19 1783 998 x 1779 401 1779 466 1779 377 20 1829 101 1829 017 1829 017 1829 013 21 1858 538 1858 223 1858 242 1858 210 22 1901 955 1902 884 1897 807 1897 580 23 1909 695 1897 012 u 1902 886 1902 838 24 1970 004 1966 496 x 1961 701 1961 558 25 2002 997 2002 376 2002 404 2002 323 2 7 Determining eigenstates by block improved relaxation 17 26 2025 521 2025 382 2025 384 2025 381 27 2049 494 2048 983 2049 045 2048 967 28 2120 335 2120 044 2120 019 2120 002 29 2147 592 2136 577 21362967 z 2136 276 30 2173 911 2153 985 2154 080 2153 897 31 2211 437 2210 637 2210 633 2210 622 32 2242353 2240 892 2240 933 2240 825 33 2292 LOT 22 9 1 21 21 2291 196 2291 096 34 2306 560 2306 468 2306 477 2306 460 35 2341 711 2339 301 2322 SOND 28 2321 754 36 2357 212 2340 799 u 2339 333 2339 225 37 2376 415 2376 423 2339 687 2339 416 38 2396 093 2370 751 2370 709 2370 415 39 2402 523 2400 605 u 2376 419 2376 401 SOG 2 jo te WOLD AO ee 2 Oe gt 5 0 gt 10 u unconverged The first run blk 1 used the SPF set 9 4 16 18 and took 46 min CPU time on a 3 2 GHz Pentium 4 and 165 MB RAM the second run with the SPF set 10 5 30 20 took 3 h CPU time and 560 MB RAM the third run with 12 5 42 28 SPFs took 11 h CPU time and 1340 MB RAM The third calculation blk 3 is fully converged and serves as reference Deviations from these results are indicated by and when
76. 7 3 8 3 9 3 10 Using parallel shared memory hardware Specifying the task for MCTDH 0 Specifying the desired output 2 2 2 ee ee Propagating a wavepacket 2 2 2 0 0 2000000200 GG Relaxing a wavepacket to produce the lowest eigenstate Improved relaxation Generation of excited eigenstates Performing a numerically exact calculation Diagonalising the Hamiltonian using the Lanczos algorithm Starting a calculation 2 2 0 0 0 00000000000 Continuing or stopping acalculation 0 VI Vil vill 10 12 14 15 18 18 18 19 20 H Contents 3 11 Using parallel distributed memory hardware o oo aaa 32 4 Selecting a DVR FBR representation for the primitive basis 36 4 1 Available DVR FBR representations 0000 36 4 2 Hermite and radial Hermite DVR 000 36 4 3 Eegendre DVR da ate att a ee ee A tay dota ar a tad 38 44 Sime DVR eiiis hh he Sencha Ae ois a i Bee Gok de a eee hs 39 4 5 Exponential DVR and fast Fourier transform 02 39 4 6 Spherical harmonics FBR 000000000 2 eee 41 4 7 Restricted Legendre DVR 0 0 0 00000000008 41 4 8 Extended Legendre DVR and Two Dimensional Legendre DVR 42 4 9 Three Dimensional rotational DVR 000 43 5 Defining the single particle basis 45 5 1 Specifying the number of single particle f
77. 77 mer This is of course just an example One can give any name to MCTDH_PLATFORM a con venient choice is the output of uname p or uname m The symbol which is given to MCTDH_COMPILER must be listed in the compile cnf file If one wants to use a compiler which is not listed there one has to edit compile cnf to add the new compiler Note that during installation one of the files compile cnf_x is copied to compile cnf where x stand for le be lenp or benp Hence one may whish to edit those files as well Finally let us summarize the commands you now should be familiar with menv compile mhelp cdm and if you use a PES from addsurf mklinks Try the help option h and in spect the HTML on line documentation The Analyse Programs Utility Scripts to learn more about the utility scripts If you want to inspect the code make yourself familiar with mcb mcg mcl and phelp There is also a backup facility and an automatic program test Elk Test see the HTML on line documentation for details For additional information on the install process see the page Installation and Compilation of the HTML on line docu mentation When the installation is completed it is advisable to work through the tutorial Sec 2 A final remark on Apple computers running under Mac OS x Darwin should be made Versions launched 2014 or later install painlessly on a Mac after some additional software is installed One ne
78. 8 2349 090 0 086 3 17E 2 2 26E 4 4 2548 374 2548 349 0 017 1 06E 1 3 65E 5 5 6 7 8 2671 113 2671 152 0 008 9 08E 2 3 00E 5 2797 154 2796 339 0 752 4 23E 3 2 10E 4 3612 845 3612 860 0 013 5 61E 2 7 65E 5 3714 789 3715 040 0 108 1 57E 2 9 32E 5 9 3792 679 3792 531 0 075 6 78E 2 2 16E 4 10 3942 480 3942 562 0 002 8 62E 2 3 35E 4 11 4064 101 4064 190 0 040 1 34E 2 2 31E 4 12 4225 043 1 05E 4 13 4673 332 6 17E 5 14 4853 622 4853 747 0 046 4 27E 2 1 62E 4 15 4977 828 4977 548 0 572 2 73E 2 3 41E 4 16 5022 273 5022 408 0 492 3 41E 2 1 61E 4 17 5099 668 9 38E 4 18 5197 251 5197 442 0 055 4 47E 2 8 24E 6 19 5329 746 5329 986 0 029 1 31E 2 6 07E 5 20 5475 283 5480 947 0 839 2 19E 4 1 84E 5 21 5667 488 1 80E 7 22 5915 216 1 41E 4 23 6016 687 4 45E 4 24 6075 984 6076 471 0 554 2 12E 2 1 50E 3 25 6227 915 6233 344 0 074 3 08E 2 8 89E 5 26 6239 852 1 31E 2 27 6347 956 6333 856 2 964 2 15E 3 3 33E 4 28 6435 398 6434 692 0 261 1 59E 2 6 00E 4 29 6503 081 3 09E 6 30 6588 730 6588 345 0 745 5 48E 3 2 46E 5 2 6 Determining eigenstates by improved relaxation Improved relaxation is a MCSCF variant where the SPFs are optimised by relaxation prop agation in negative imaginary time but the A vector is determined by diagonalisation of the Hamiltonian matrix evaluated in the set of the present SPFs In contrast to filter diagonalisa tion improved relaxation yields not only the eigenenergies but also the eigenstates Impr
79. 9 8772 List of MCTDH references 157 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 R Milot and A P J Jansen Energy distribution analysis of the wave packet simulations of CH4 and CD4 scattering Surf Sci 452 2000 179 190 R Milot and A P J Jansen Bond breaking in vibrationally excited methane on transition metal catalysts Phys Rev B 61 2000 15657 15660 G A Worth Accurate wave packet propagation for large molecular systems The multi configuration time dependent Hartree MCTDH method with selected configurations J Chem Phys 112 2000 8322 8329 A Raab Untersuchung der Dynamik quantenmechanischer Systeme in Wechselwirkung mit Umgebungen mit Hilfe der zeitabhdngigen Multikonfigurations Hartree Methode PhD thesis Universitat Heidelberg 2000 M H Beck Berechnung von Schwingungsrotationsspektren mit Hilfe der zeitabhdngigen Multikonfigu rations Hartree und der Filter Diagonalisierungs Methode PhD thesis Universitat Heidelberg 2000 A Raab On the Dirac Frenkel McLachlan variational principle Chem Phys Lett 319 2000 674 678 A Raab and H D Meyer Multi configurational expansions of density operators Equations of motion and their properties Theor Chem Acc 104 2000 358 369 A Raab and H D Meyer A numerical study on the performance of the multiconfi
80. A 84 2011 023623 K Giri E Chapman C S Sanz and G Worth A full dimensional coupled surface study of the photodis sociation dynamics of ammonia using the multiconfiguration time dependent Hartree method J Chem Phys 135 2011 044311 L Blancafort F Gatti and H D Meyer Quantum dynamics study of fulvene double bond photoisomer ization The role of intramolecular vibrational energy redistribution and excitation energy J Chem Phys 135 2011 134303 Index ABM integrator 80 82 Adiabatic correction see Correction Adiabatic population 107 Analysis flux analysis 105 of accuracy 99 of efficiency 102 of electronic populations 106 of PES 112 of primitive basis 99 of PSI 111 of results 96 of single particle basis 101 of system evolution 102 of system spectrum 104 reaction probabilities 105 analysis interface 96 Auto file see File Autospec program see Program Auxiliary Operators 66 Basis electronic 87 primitive 36 single particle 45 87 Bosons 91 BS integrator 80 82 Calculations continuing 29 distributed memory 32 parallel 29 32 shared memory 29 starting 28 stopping 29 CAP 62 order 62 starting point 62 strength 62 CDVR see Correlation DVR Check file see File CMF scheme see Constant mean field scheme Colbert Miller DVR see DVR sine Complex absorbing potential see CAP Constant mean field scheme 81 Continuing a calculation see Calculations Corr
81. AMILTONIAN SECTION The labels cap1 and cap2 or any other labels you have chosen are then defined in the LABELS SECTION LABELS SECTION capl CAP 5 0 0 375 3 1 cap2 CAP 1 0 0 240 2 1 nd labels section The parameters in square brackets are from left to right the starting point ze strength 7 both in a u and order b of the CAP The last number specifies whether the CAP lies to the right 1 which is also the default or left 1 of ze Note the input may consist of numbers parameters or algebraic expressions containing numbers and parameters Hence 6 10 Using complex absorbing potentials CAPs 63 capl CAP 3 0 x0 1 0d 3x strength 3 1 is a valid statement provided the parameters x0 and strength are defined The described way of including CAPs has the disadvantage that the CAPs are hard wired in the operator file The MCTDH program therefore offers the opportunity to switch on CAPs from the input file without any change of the HAMILTONIAN or LABELS SECTION in the operator file To this end include the lines alter labels cap_x CAP 5 0 0 375 3 I cap_y CAP 1 0 0 240 2 1 nd alter labels in the OPERATOR SECTION of the input file The special keyword cap1_x where x stands for one of the mode labels tells the program to add a CAP to the corresponding degree of freedom See Section 6 4 for more details on the special meaning of the underscore _ within a label The optimal
82. ARAMETER SECTION Thus the section PARAMETER SECTION a 1 0 b 2 0 c 3 0 d 2 0xa bxc 1 0 e EXP d 9 0 a nd parameter section sets d to 9 0 and eto 1 0 The algebraic expression must not contain spaces or brackets Only the operators are allowed An exponent acts only on the label to which it is attached and are evaluated before Otherwise the order of operation is strictly from left to right Exponents must be numbers not parameters they may be signed and real e g alpha 0 5 is a valid construct The exponential form e g 2 3d 5 is allowed for numbers only The decimal exponent must be indicated by a d not by a D e or E See the HTML documentation Hamiltonian Documentation Parameter Section for further details The last line of the example above demonstrates the use of functions in parameter arith metic Available functions are EXP LOG LOG10 SIN COS TAN ASIN ACOS ATAN SINH COSH TANH ABS INT ATAN2 MIN MAX The definitions of these functions are the usual FORTRAN definitions Note that the last three functions depend on two arguments The two arguments which may be numbers parameters or numerical expressions must be separated by a blank or a comma The keyword for the function must follow directly the equal sign not even a minus is allowed in between No operations must follow the closing bracket of the function except possibly a unit Some labels have a
83. CES newsurf f S FC FFLAGS c o into install user_surfaces Note that the second line starts with a tab and not with a series of blanks The two files install User_surfdef and install user_surfaces will be sourced i e read by the Makefile They hence contain the personal additions to the Makefile Similarly the 60 6 Setting up the Hamiltonian files opfuncs user1d F and opfuncs usersrf F contain the personal additions to the FORTRAN code The subroutine newsurf will be called from the MCTDH or Potfit program via the subroutine uvpoint i e user V point which is located on the file opfuncs usersrt F At the end of the subroutine uvpoint a call to the new surface routine must be added For this in comment elseif ifunc eq 2 then near line 107 of file opfuncs usersrf F and replace the line following this if statement with call newsurf gpoint v The array gpoint contains the coordinates in the order specified in the HAMILTONIAN SECTION via the mode line and the i amp j amp k construct See Section 6 13 The surface number hopilab 2 in our example then has to be defined Note hopilab is called ifunc in some routines This is done in subroutine udefsrf which also is stored on source opfuncs usersrf F Add lines such as else if label 1 11 eq newsurf then write ilog a newsurf lt remarks gt hopilab 2 to this subroutine If the surface depends of parameters or option
84. CTDH package We list them in chronological order M Ehara M C Heitz A Raab S Wefing S Sukiasyan C Cattarius F Gatti F Otto M Nest A Markmann M R Brill O Vendrell M Schr der D Pelaez Ruiz and Phillip S Thomas We are very very grateful to all of them Citations When citing the MCTDH program package in the literature the following citation should be used G A Worth M H Beck A J ckle and H D Meyer The MCTDH Package Version 8 2 2000 University of Heidelberg Heidelberg Germany H D Meyer Version 8 3 2002 Version 8 4 2007 See http mctdh uni hd de A comprehensive description of the methods incorporated in the programs is in 1 M H Beck A J ckle G A Worth and H D Meyer The multiconfiguration time dependent Hartree MCTDH method A highly efficient algorithm for propagating wave packets Phys Rep 324 1 2000 1 The original paper is 2 H D Meyer U Manthe and L S Cederbaum The multi configurational time dependent Hartree approach Chem Phys Lett 165 1990 73 These two papers should be cited as well You may further wish to include the references VIII List of Examples Ix 3 U Manthe H D Meyer and L S Cederbaum Wave packet dynamics within the multiconfiguration Hartree framework General aspects and application to NOCI J Chem Phys 97 1992 3199 4 H D Meyer and G A Worth Quantum molecular dynamics Propagating wavepa
85. CTDH to use more memory for a more efficient parallelisation of the calcha and mfields routines In default mode the parallelisation is optimized for low memory requirements Furthermore there is the summf 2 keyword If this keyword is set a differently parallelised summf routine is used This may increase the efficiency of the parallelisation if a very large combined mode is present dominating the calculation of summf A parallel version of the LAPACK dsyev routine is used in MCTDH The dsyev I keyword can be used to set the minimum size of the matrix that is diagonalized in parallel The results of a parallel calculation may slightly differ from those of a non parallel one due to numerical reasons Furthermore a parallel calculation needs more memory This is one of the reasons why the parallel use of the routines can be disabled Depending on the type and the parameters of the calculation some routines may only marginally improve the perfor mance of the parallelisation They can be turned off to save memory Moreover depending on the calculations made some routines can produce overhead which overcompensates the gain of their parallelisation These routines should also be turned off Example RUN SECTION usepthreads 4 no funkphi end run section In this example a parallel calculation with 4 processors will be performed but the paralleli sation of the funkphi routine is disabled Sometimes disabling the parallelisation of routines ev
86. DH method A highly efficient algorithm for propagating wave packets Phys Rep 324 2000 1 105 H D Meyer U Manthe and L S Cederbaum The multi configurational time dependent Hartree ap proach Chem Phys Lett 165 1990 73 78 U Manthe H D Meyer and L S Cederbaum Wave packet dynamics within the multiconfiguration Hartree framework General aspects and application to NOCI J Chem Phys 97 1992 3199 3213 H D Meyer and G A Worth Quantum molecular dynamics Propagating wavepackets and density operators using the multiconfiguration time dependent Hartree MCTDH method Theor Chem Acc 109 2003 251 267 H D Meyer F Gatti and G A Worth Eds Multidimensional Quantum Dynamics MCTDH Theory and Applications Wiley VCH Weinheim 2009 G A Worth H D Meyer and L S Cederbaum The effect of a model environment on the S2 absorption spectrum of pyrazine A wavepacket study treating all 24 vibrational modes J Chem Phys 105 1996 4412 G A Worth H D Meyer and L S Cederbaum Relaxation of a system with a conical intersection coupled to a bath A benchmark 24 dimensional wavepacket study treating the environment explicitly J Chem Phys 109 1998 3518 3529 A Raab G Worth H D Meyer and L S Cederbaum Molecular dynamics of pyrazine after excitation to the S2 electronic state using a realistic 24 mode model Hamiltonian J Chem Phys 110 1999 936 946 G A Worth H D Meye
87. E 06 0 6434E 07 0 3421E 07 0 1938E 07 0 13 0 7838E 08 0 4688E 08 0 2995E 08 0 1516E 08 0 5612E 09 0 Abs Trace Sum of relevant Natural Weights eV 2 2 47378E 06 Sqrt Abs Trace Sum of rel Nat Weights meV 1 5728 KKK EK K KKK KK EK Mode 3 theta KOK OK OK RK KK KKK RK RK OK OK OK Contracted mod Dimension 60 60 Global weighted L 2 error estimated from neglected natural weights 5 5354 meV 44 646cm 1 2 0342E 04 a u Weighted rms error on rel grid points meV 3 9734 1 4602E 04 Weighted rms error on all grid points meV 5 4941 2 0190E 04 Unweighted rms error on rel grid points meV 31 2338 1 1478E 03 Unweighted rms error on all grid points meV 133 0489 4 8895E 03 Max absolute error on rel grid points meV 313 2904 1 1513E 02 Max absolute error on all grid points eV 2 2438 8 2457E 02 Example 12 2 An excerpt of a potfit output file for the NOCI S1 surface au 5984E 06 7315E 08 3832E 09 2449E 18 3708E 06 1207E 07 1780E 09 118 12 Using the Potfit program compares well with the numerically evaluated error Weighted rms error on all which is 5 4941 meV By the way the choice 5 4 for the number of natural potentials is a bit unbalanced as the sums of neglected weights of the two DOFs are quite different A more balanced choice would be 5 3 or 9 4 which would lead to estimated errors of 7 2 meV or 2 4 meV respectively The following 10 iterations reduce the fit error to 2 202 meV
88. ECTION use pes none e in the PRIMITIVE BASIS SECTION use exactly the basis definitions from the projection input file but only for the remaining coordinates Q omit all the coordi nates that were projected out After the potfit runs are complete you can optionally change the primitive basis by using chnpot 12 4 3 Using a Fourier transformed potential in MCTDH The last section described how to generate the Fourier components Vo in a form usable by MCTDH however this is only one ingredient of 12 11 The other is the shifting of the k coordinates Q k k2 gt U Q k 2 k2 Q In MCTDH this can be accomplished with the simple shift operator which we will denote here formally as oe where q is the DOF on which it operates and A is the amount by which the DOF is shifted SW Gis as Weg Se 12 13 In this notation 12 11 becomes recall that Va V_q VW Q k k2 Q Q k k2 12 14 ValQ 2 5 HQ ky k2 Oe Si TQ k k2 II o Me ke ll 1 12 4 Manipulating potentials with the projection program 127 HAMILTONIAN SECTION modes R xl r2 thi k1 th2 k2 a eas kinetic energy omitted 1 0 1 amp 2 amp 3 amp 4 amp 6 VO Se Hes SU 1 0 162636486 V1 5 kpl 7 km1 1 0 162836486 V1 5 km1 7 kpl 1 0 1 amp 2 amp 3 amp 4 amp 6 V2 5 kp2 7 km2 1 0 1 amp 283648 amp 6 V2 5 km2 7 kp2 end hamiltonian section LAB ELS SECTION vo V1 V2
89. EGRATOR SECTION is INTEGRATOR SECTION BS 8 1 0d 6 end integrator section Here we have omitted the VMF keyword since it is the default Note that one may also not define at all the INTEGRATOR SECTION This is equivalent to specifying the VMF and ABM keywords together with some default parameters for the ABM integrator that can be found in the HTML documentation 80 8 2 Using the CMF integration scheme in an MCTDH calculation 81 Table 8 1 Available integrators in dependence of the calculation type The table displays which of the integrators ABM BS RKz and SIL can be chosen depending on whether a VMF CMF or numerically exact calculation is being made An underlined checkmark v indicates the default Integrator Calculation type ABM BS SIL RKz VMF vO 4 Vv CMF A vector v V wv v CMF vector v v V Numerically exact Vv V vy vV 8 2 Using the CMF integration scheme in an MCTDH calculation In many cases an MCTDH calculation is more efficient if the VMF scheme is replaced by the constant mean field or CMF scheme In the CMF scheme the numerical effort is reduced by holding the mean fields density matrices and Hamiltonian matrix elements constant for some time rather than evaluating them in each integration step Note that the CMF scheme does presently not work in combination with the CDVR approximation The CMF scheme is detailed in Sec 5 2 of Ref 1 and in Ref 26 Table 8 1 displa
90. F BASIS SECTION For instance if five modes V W X Y and Z are specified in the PRIMITIVE BASIS SECTION PRIMITIVE BASIS SECTION V HO 16 0 0d0 1 0d0 1 0d0 W HO 22 0 0d0 1 0d0 1 0d0 X HO 32 0 0d0 1 0d0 1 0d0 Y HO 21 0 0d0 1 0d0 1 0d0 Z HO 12 0 0d0 1 0d0 1 0d0 end primitive basis section and you wish to include only W X and Z in your calculations this can be accomplished by the SPF BASIS SECTION SPF BASIS SECTION W 7 x 8 Drees nd spf basis section or by commenting out the corresponding lines for V and Y The approximation being made here is that the coupling between the included and ex cluded modes is negligible As a result the Hamiltonian is built simply ignoring all terms that include contributions from the excluded degrees of freedom 5 3 Combining modes to produce multi dimensional single par ticle functions For large systems or when certain degrees of freedom are strongly coupled it may be advan tageous to combine degrees of freedom together and use multi mode single particle functions see Sec 4 5 in Ref 1 for further details A combination scheme can be easily specified by grouping together the degrees of free dom to be combined in the SPF BASIS SECTION As an example we consider a system consisting of eight degrees of freedom named r1 r8 The SPF BASIS SECTION SPF BASTIS SECTION el 4 15 r2 r3 ET 10 r5 r6 r8 12 nd spf basis section then combines for
91. I file eigval 1 To see the results type e g less eigval The first line describes the entries of the eigval file 12 2 An MCTODH tutorial 2 To plot the spectrum type pleigval a 6 6 x 6 1 This displays the spectrum in the converged energy range The result is shown in Fig 2 6 Note that energies with very small intensities are not visible To display all lines add the option 1 in order to use a logarithmic scale for the intensities 2 5 Determining the vibrational spectrum of CQO by filter diagonalisation To Fourier transform the autocorrelation function is the straightforward procedure to extract eigen energies from a time evolved wavepacket This however requires a very long prop agation time T as the resolution improves only like A T This limit set by the uncertainty relation can be overcome when employing the filter diagonalisation FD method introduced by Neuhauser Our particular version of the FD method is discussed in Refs 12 13 The following example shall show how filter diagonalisation and MCTDH propagation can be combined The example is similar to the problem studied in Ref 13 however here we sacrifice some accuracy in order to gain speed 1 Copy the file MCTDH_DIR inputs co2t inp and create the directory co2 2 Copy the file MCTDH_DIR finputs co2ft inp to the directory co2 3 To perform the MCTDH propagation type mctdh84 co2t This will take less than 2 minutes As done in t
92. KKKKKKKKK KKK KKK KKK KKK KOK KK KR RR RK CHECK SYSTEM eee ee a eo KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK KKK KKK KK KKK Mon Dec 9 14 42 09 CET 2013 The path of the MCTDH directory is home dieter MCTDH mctdh84 9 System Linux Platform 1686 Operating System GNU Linux Machine cauchy Processor unknown Kernel Linux This is a 32 bit system These are the variables determined by platform cnf MCTDH_PLATFORM i686 MCTDH_COMPILER gfortran MCTDH_GFORTRAN_VERSION 4 7 2 These are the default compilers Fortran compiler gfortran C compiler geg make command make If you want to use other compilers please edit platform cnf and possibly also compile cnf or use the c option when compiling The Fortran compiler gfortran is apps gcc 4 7 bin gfortran The C compiler gcc is apps gcc 4 7 bin gec The make command make is usr bin make Congratulation you have a GNU make GNUPLOT exist on your system gnuplot 4 4 patchlevel 0 Python exist on your system Python 2 6 6 Your bash is GNU bash version 4 1 5 1 release Distributed memory parallelization with MPI seems to be possible For this use Fortran compiler consistent with gcc version 4 7 2 GCC Shared memory parallelization with POSIX Threads seems to be possible The use of the NUMA library for the shared memory parallelization with POSIX Threads seems to be possibl
93. MILTONIAN SECTION_OPER section and setting nodiag see Sec 6 12 This sets up an operator labelled OPER any other string except SYSTEM can be chosen for this name Once this has been done adding the keyword operat e OPER to the INIT WF SECTION generates a wavepacket by ap pling this operator to the initial packet For a numerically exact wavefunction this procedure is simple For an MCTDH wavefunc tion however the optimal single particle functions for the final wavepacket may be different from those of the initial wavepacket To optimise the basis functions for the new wavefunc tion an iterative procedure is used Details of the iterations are output in the log file 78 7 Generating the initial wavepacket 7 10 Advanced topic Creating a set of initial wavepackets Instead of propagating only a single wavepacket one may also define a set of P initial wavepackets W Wp which are then propagated simultaneously This is called a multi packet calculation For example to propagate P 2 wavefunctions with coordinates X and Y one first has to add the line packets 2 to the SPF BASIS SECTION in order to specify the number P of packets The definition for the initial wavefunction has to be given for each wavepacket e g INIT_WF SECTION build X gauss 4 315 0 0 0 0794 pack 1 gauss 322 0 0 0 053 pack 1 x HO Zils 5 A 0 0 0 218 eV 13 645 5 pack 2 Y gauss 3 840 0 0 0 1378 pack 2 end build end init_wf s
94. OPERATOR SECTION specifies the potential energy surface to be used and defines cuts which remove large positive and or negative potential values which if kept would slow down the integrator 116 12 Using the Potfit program The NATPOT BASIS SECTION defines how many natural potentials are used for the fit The more natural potentials one includes the more accurate is the fit but the slower is the MCTODH calculation One of the degrees of freedom should have the argument contr This is the mode over which a contraction is performed see Refs 1 28 29 Contract over that degrees of freedom which converges most slowly Inspect the output to see which one it is One should avoid to contract over a mode if it is defined on a much larger grid than the other modes To decide how may natural single particle potentials should be included in the potfit one should inspect the output file which displays the natural populations as well as the sums of neglected natural populations This will be discussed below MODE COMBINATION is also possible In the NATPOT BASIS SECTION the degrees of freedom can be combined into a single mode in the same way as it is done in the spf basis section for an mctdh run If the resulting natpot file is used in a mctdh run the degrees of freedom combined in potfit must also be combined in the spf basis section note that the order of the degrees of freedom in the mode must be identical in the natpot basis section and in
95. T BASIS SECTION rd 5 rd 15 These are the values for nat pot I rv 4 rv 15 review Section 9 1 theta contr nd natpot basis section PRIMITIVE BASIS SECTION rd sin 36 3 80 5 60 rv HO 24 2 3 6 0 272 ev 7 4667 AMU theta Leg 60 0 all end primitive basis section SEPARABLE WEIGHT SECTION rd D 3 904 5 83d 03 rv 2 v NO Lads ledo theta 3 222 6 d0 nd separable weight section CORRELATED WEIGHT SECTION v lt 2 0 eV rd lt 5 d0 nd correlated weight section end input Example 12 1 A potfit input file for the NOCI S1 surface The output files of potfit are structured similarly to the MCTDH ones There are the files output input log and timing The prodwei file lists the separable weights and the file iteration compiles various error measures for each iteration step The script plpweight reads prodwei and plots the separable weights and the script plpit reads iteration and plots the error measures versus the number of iterations The file natpot finally contains the natural potential fit which may be read by the MCTDH program A potfit input file noclpot inp is shown in Example 12 1 The RUN SECTION defines the number of iterations to be performed the name directory and the files to be opened The potential is first evaluated on the full product grid and written to the file vpot Note vpot is needed by showpot when plotting the exact potential The
96. TION of the input file If this file has not been generated one dimensional densities may be still be plotted from the gridpop file using the showd1d program see Sec 11 5 The options are accessed by typing in the number given and responding to the questions The words in brackets indicate what the present option is Some options lead to a change in options being displayed Option 10 can be used to change between various plot tasks What is possible depends on the system for a non adiabatic system the choices include not only plotting the reduced density but plotting a cut through the adiabatic or diabatic wavefunction These plots should be used with caution as the values are often extremely low in a cut If a pes file is present see 112 11 Analysing the results employing the Analyse programs Sec 11 10 cuts through the PES may also be plotted If there is more than one electronic state one may chose between a adiabatic or diabatic representation of the potential Option 20 allows a different cut to be chosen Enter either x y or a number for each DOF Information about the mode boundaries is given by option 40 The program then chooses the grid point nearest to the selected coordinate for the plotted cut Note that when densities are plotted the values of the numbers given are irrelevant They merely serve as space holders The density is the integral of over all those DOFs which are labelled with a number Option 400 allows
97. This information is useful for checking that the absorption process has finished The shell script plwtt visualises the function Wy while plflux and plflux r visualise the flux and the transition probability respectively The program flux84 cannot only determine the total energy resolved flux going into a particular arrangement channel i e going into a particular CAP but can also determine the flux which is projected onto final quantum states or which is weighted by an operator This is probably best demonstrated by an example Copy the file MCTDH_DIR operators nocl1 op to your tutorial directory and add the following lines to this operator file HAMILTONIAN SECTION_vib usediag modes rd rv theta 1 0 1 KE 1 JO 1 v NO 1 end hamiltonian section Then edit the input file nocl1 inp and set the propagation time to t final 60 and re run Since flux84 analyses the wavepacket as it is absorbed by the CAP a longer propagation time is required as for converging the spectrum Then execute the commands flux84 w s 19 lo 12 0 61 2 0 ev rd mv flux flux 0 Flux84 w s 19 lo 12 O vib u 200 0 61 2 0 ev rd mv flux flux op Flux84 ed flux 0 s 19 lo 12 O vib u ev 0 61 2 0 ev rd mv flux flux op_r flux84 w s 19 lo 12 P 2 eigenf vib 1 0 61 2 0 ev rd mv flux flux 1 flux84 w s 19 lo 12 P 2 eigenf vib 2 0 61 2 0 ev rd mv flux flux 2 fFlux84 w s 19 lo 12 P 2 eig
98. ackage RUN SECTION name lt S1 gt dvrdir lt S2 gt natpotdir lt S3 gt end run section PRIMITIVE BASTIS SECTION x sin 45 2 4 2 4 y sim A5 T72 5 35 end primitive basis section Fit Section x spline y spline end fit section end input The RUN SECTION contains keywords that control the program execution e g where the old initial files are found where to store the new results etc See the HTML documenta tion for a detailed description of every keyword The PRIMITIVE BASIS SECTION spec ifies the new DVR and hence the new primitive grid The Fit Section describes how each degree of freedom has to be interpolated A detailed description of the chnpot command line and input file options is maintained in the HTML documentation Here some hints are given in how to use this utility after chnpot has been executed the user should check that the natpot obtained has been correctly interpolated This can be done conveniently with the showpot utility Original potentials with discontinuities or regions where the function slope changes abruptly can lead to oscillatory behavior of the interpolated function Depending on the starting data points the user may have to find the most convenient interpolation scheme for each degree of freedom The program can perform spline and essentially non oscillatory interpolation ENO for degrees of freedom without special boundary conditions and fourier sine and cosine inte
99. ado Barrio and S Wilson Eds vol 20 Springer Verlag 2009 p 69 F Otto F Gatti and H D Meyer Erratum Rotational excitations in para H2 para Hg2 collisions Full and reduced dimensional quantum wave packet studies comparing different potential energy sur faces J Chem Phys 131 2009 049901 S Woittequand C Toubin M Monerville S Briquez B Pouilly and H D Meyer Multiconfiguration time dependent Hartree and classical dynamics studies of the photodissociation of HF and HCL molecules adsorbed on ice Extension to three dimensions J Chem Phys 131 2009 194303 J Seibt T Winkler K Renziehausen V Dehm F W rthner H D Meyer and V Engel Vibronic transitions and quantum dynamics in molecular oligomers A theoretical analysis with an application to aggregates of perylene bisimides J Phys Chem 113 2009 13475 M Brill O Vendrell and H D Meyer Distributed memory parallelisation of the multi configuration time dependent hartree method In High Performance Computing in Science and Engineering 09 Heidel berg 2010 W E Nagel D B Kr ner and M Resch Eds Springer pp 147 163 S Bhattacharya A N Panda and H D Meyer Multiconfiguration time dependent Hartree approach to study the OH H2 reaction J Chem Phys 132 2010 214304 M Eroms M Jungen and H D Meyer Nonadiabatic Nuclear Dynamics after Valence Ionization of H20 J Phys Chem A 114 2010 9893 9901
100. al spectrum of CO9 2 a 13 The natural orbital populations as a function of time for NOCI 102 The density as a function of time for NOCI 103 Total and projected flux of dissociating NOC 107 Main concepts involved in the usage of ab initio data with the MCTDH package 121 Jacobi coordinates for a 4 atomic system 20 0 125 The structure of the MCTDH programs 0 132 Structure of wave function 2 2 2 0 000 000 00004 146 VI List of Examples 4 1 6 1 6 2 6 3 6 4 9 1 9 2 10 1 10 2 11 1 11 2 11 3 11 4 11 5 11 6 12 1 12 2 12 3 12 4 A l An input file for a wavepacket propagation of NOC 38 An operator file for the NOCI S state 2 20 0 0028 49 An operator file for a propagation using the modified Henon Heiles Hamil COMED cs onset el oe teas Pees Be cat Seas Wiehe verte Ge aa eae A ete Gane Grd 52 A parameter file for the Henon Heiles Hamiltonian 64 A surface file for the NOCI S potential 0 65 An operator file for the pyrazine 4 mode 2 state model system 88 An input file for the pyrazine 4 mode 2 state model system 89 An operator file for N 3 one dimensional bosons ina harmonic trap 94 An input file for N 3 one dimensional bosons in a harmonic trap 95 The analysis startup menu 000000008 0s 97 An output file from a w
101. am SIL integrator 82 Sine DVR see DVR Single particle basis see Basis Single particle function 45 multi mode 46 Single particle operator 51 Single particle basis section see Section spf basis Single set 87 Spectrum see Analysis Spf basis section see Section Spherical harmonic function see Function Spherical harmonics FBR 41 Starting a calculation see Calculations Stop file see File Stopping a calculation see Calculations Structure of the WF array 146 Surface file see File svn repository Subversion 152 Symbolic expression built in 51 86 133 user defined 54 System bosonic see Bosons System non adiabatic 86 TDDVR see Time dependent DVR TDH see Time dependent Hartree 166 Index Temperton FFT see FFT Three Dimensional rotational DVR see DVR Time dependent DVR 85 Time dependent Hartree 45 Time dependent operators 70 Timing file see File see File Two dimensional Legendre DVR see DVR Variable mean field scheme 80 VMF scheme see Variable mean field scheme Vpot file see File Wavefunction initial 71 90 structure of 146 Wavepacket improved block relaxation 15 24 improved relaxation 14 24 propagation 23 relaxation 23 Wigner see DVR
102. amiltonian 0 000 91 10 2 Modifying the input 2 2 0 00 00000000000 92 11 Analysing the results employing the Analyse programs 96 11 1 The Analysis Interface 00 00 0022 ee eee 96 11 2 Interpreting the MCTDH output 2 00 97 11 3 Checking the accuracy ofacalculation 0 4 99 11 3 1 Checking the primitive basis size 2 00 99 11 3 2 Checking the single particle function basis size 101 11 4 Checking the efficiency of a calculation 0 102 11 5 Watching the system s evolution 0 2000004 102 11 6 Determining photo dissociation and photo absorption spectra 104 11 7 Computing excitation and reaction probabilities 0 105 11 8 Monitoring state populations of non adiabatic systems 106 11 8 1 Diabatic populations 0 20 00 02 0000 106 11 8 2 Adiabatic populations computed with adpop 107 11 8 3 Adiabatic populations computed with adproj 108 11 9 Plotting 2D cuts through the system density 111 11 10Plotting cuts through the potential energy surfaces 112 12 Using the Potfit program 114 12 1 Transforming a potential to productform 0 2 114 12 2 Using abinitiodata 2 ee 118 12 2 1 Using ab initio data directly with the metdh program 118 12 2 2 Using the potfit program
103. arameters are defined in the PARAMETER SEC TION e g w10a corresponds to w10a and k6a1 to H This Hamiltonian is represented in the operator file by the HAMILTONIAN SECTION see Example 9 1 The modelabel e1 labels the electronic states For more details of the pyrazine calculations see Ref 7 9 2 Defining the primitive basis for a non adiabatic system For the treatment of a non adiabatic system not only the operator but also the input file has to be set up appropriately One modification concerns the PRIMITIVE BASIS SECTION where an electronic basis has to be specified for the mode labelled e1 in the operator file PRIMITIVE BASIS SECTION vl0a HO 22 0 0 1 0 1 0 vo6a HO 32 0 0 1 0 1 0 v1 HO 21 0 0 1 0 1 0 v9a HO T2 0 0 1 0 1 0 el el 2 end primitive basis section The primitive basis type for the electronic basis is e1 and the number denotes the number of states in the system in this case two This sets up a discrete vector representation for the o States The complete input file for the 4 mode 2 state pyrazine model from which the above lines were taken is displayed in Example 9 2 9 3 Defining the single particle basis for a non adiabatic system In an MCTDH propagation or relaxation a single particle basis is needed for the representa tion of the wavefunction For non adiabatic systems there are two possible representations termed single or multi set see Sec 3 5 of Ref 1 for details
104. articularly interested in block improved relaxation 1 Copy the file MCTDH_DIR inputs bIKHONO inp to your tutorial directory 2 To perform the block relaxation execute the command mctdh84 mnd b1kHONO amp Edit the input file such that the numbers and keywords which appear after or HP become valid Then run the input again The convergence of the eigenenergies is most con veniently visited by running the script rdrlx The convergence can be inspected graphically by running plbrlx which is very similar to plrlx but requires as argument the number of the state to be plotted The converged eigenenergies in cm7 obtained from these runs read as follows blk 1 sing 1 b1lk 2 b1lk 3 0 0 006 0 000 0 000 0 000 1 93 992 93 3973 93 974 93 972 2 600 920 600 872 600 873 600 871 3 710 781 710 623 fLOR62 5 710 621 4 796 056 795 999 796 000 795 997 5 944 422 944 111 944 116 944 108 6 1055 391 1055 384 1055 385 1055 384 7 1188 605 1188 073 1188 079 1188 070 8 1267 671 1267 600 1267 609 1267 598 9 1306 671 1306 601 1306 604 1306 595 10 1313 852 1312 748 1312 761 1312 736 11 1386 094 1385 262 1385 263 1385 247 12 1405 637 1405 519 1405 545 1405 510 13 1549 283 1547 471 1547 458 1547 431 14 1575 849 1574 831 1574 851 1574 821 15 1640 938 1640 887 1640 887 1640 884 16 1690 148 1690 009 1690 034 1690 006 17 1726 574 1726 015 1726 050 1726 009 18 1770 187 1761 572 u 1761 638 1761
105. at it requires The basic information is provided by typing rdcheck84 0 0 when in the directory containing the data files from a calculation The arguments 0 0 select no particular state or mode The program then prints some information about the system and most importantly the maximum population of the highest natural orbital lowest natural weight is displayed Maximum over time of lowest nat weight final time 30 00 fs mode s 1 1 1 810E 03 2 2 472E 04 3 9 870E 05 This information says that the calculation should be of reasonable quality as all mode contain natural orbitals that remain fairly insignificant If a mode is selected the populations as a function of time can also be graphically dis played The modes are numbered in the order in which they are listed in the SPF BASIS SECTION of the input file For NOCI the order corresponds to the degrees of freedom rd rv theta and so 2 selects the vibrational mode rv The NOCI system has only one state and so rdcheck84 1 2 produces a file nat pl which contains the natural populations as a function of time The GNUPLOT program can be conveniently used by including GNUPLOT data in this file i e rdcheck84 g 1 2 gnuplot persist nat pl would produce the plot shown in Fig 11 1 Again a more convenient way to produce this plot is provided by a pl script Just type plnat 1 2 102 11 Analysing the results employing the Analyse programs 0 01 0 001 0 0001
106. atural potential see Potential Non adiabatic system see System Numerically exact calculation 27 Op define section see Section Operator file see File Operator section see Section Operator 1D user defined 54 Optcntrl program see Program Orben file see File Orbital energies 25 interaction picture 84 natural 84 Output file see File see File Parallel calculation see Calculations see Calculations Parameter section see Section Particle in a box function see Function Plall program see Program Plane wave 40 Plane wave DVR see DVR exponential Plauto program see Program Plbrlx program see Program see Program Plcap program see Program PLeg 43 PLeg see DVR Plfdspec program see Program Plpit program see Program Plpweight program see Program Plqdq program see Program Plrlx program see Program see Program Plspec program see Program Plspeed program see Program Plstate program see Program Plupdate program see Program Population Index 165 natural 101 of grid points 99 Potential ab initio 118 Fourier transform of 125 multi dimensional 58 67 natural 61 114 non separable 58 one dimensional 57 separable 56 Potfit program see Program Primitive basis see Basis Primitive basis section see Section Product form 51 86 Prodwei file see File Program adpop 107 adproj 108 analysis 96 autospec 4 8 104 105 efield 19 fdcheck 13 fdmatch 13 filte
107. avepacket propagation of NOC 98 An input file for a potfit calculation 0000 109 The input file for the metdh genoper run 2 204 110 An operator file fora projection operator 2040 110 The showsys Menu sare Tare AA a RB a oe ew es a ed a aes 111 A potfit input file for the NOCI S1 surface 2 2 20 0 002 115 A potfit output file for the NOCI S1 surface 02 117 A projection input file for the BMKP surface for Ha 2 123 An operator file showing the use a Fourier transformed potential 127 An input file for a propagation using the Henon Heiles Hamiltonian 131 vil Copyright The software and documentation in the MCTDH package is copyright 1996 2000 Graham A Worth Michael H Beck Andreas Jackle and Hans Dieter Meyer Permission is granted to use and copy this software and its documentation Further distri bution requires the agreement of the authors Permission to modify the software is granted The authors would welcome if additions and bug fixes are made available to them for inclu sion in future releases of the package This software is provided as is and without warranty of any kind Acknowledgements The very first MCTDH program later called version 1 was written by Uwe Manthe as part of his PhD work in Heidelberg Over the years several graduate students post docs and visitors have made contributions to the M
108. become necessary to implement the same operator twice once with usediag once with nodiag 6 13 DOF mode and muld potentials The potential functions from which the Hamiltonian is build are mostly one dimensional They are accessed through symbols like q sin or vh2 However MCTDH can also use multi dimensional potential functions These are defined in funcsrf F of usersrf F If the multi dimensional potential operates on one particle i e on one combined mode it is a particle or mode operator Otherwise it is a so called muld potential The use of muld potentials makes the propagation slow and in general one will use potfit to generate a nat ural potential which is a sum of products of DOF or mode potentials However there is no disadvantage in using multi dimensional mode potentials The MCTDH program will recognise a multi dimensional potential as mode potential if it operates on all DOFs of one mode but on no other DOF Assume that there is a symbol fxy which refers to a potential function fj y A Hamiltonian Section may then read HAMILTONIAN SECTION modes R x y Z theta const 2 amp 3 fxy 5 cos end hamiltonian section or equivalently HAMILTONIAN SECTION modes Roo ek i y n Z theta const 1 amp fxy Lra Heos end hamiltonian section If x and y are uncombined then fxy will be treated as a muld potential but when x and y are combined to form a MCTDH particle then
109. ble Moreover natpots cannot be multiplied with other operators See Section 6 9 133 134 C The built in symbolic expressions Table C 1 Simple one dimensional operators The expression x is the coordinate r can be replaced by any real number positive or negative If r 1 it is not required e g q and q 1 are synonemous The expression n can be replaced by any non negative integer Symbol Operator Notes 1 1 Unit operator I i 1 Imaginary unit times unit operator qt x Multiply by rth power of x qs r 1 z Multiply by rth power of V1 x sin r sin x rth power of sine of coordinate cos r cos x rth power of cosine of coordinate tan r tan x rth power of tangent of coordinate cosh r cosh x rth power of hyperbolic cosine of x sinh r sinh x rth power of hyperbolic sine of x acos r arccos x rth power of acosine of coordinate asin r arcsin x rth power of asine of coordinate atan r arctan x rth power of atangent of coordinate expr exp z exponential of coordinate gauss r exp z gaussian of coordinate ngauss r exp z 2 V2T normalized gaussian legth n P cos x nth order Legendre polynomial of cosine of x asleg 1_m P x associated Legendre polynomial of x see function plgndr in sorce lib utilities legendre f aslegth lm P cos x associated Legendre polynomial of co sine of x cp VI J 1 K k 1 Ctx symbol appearing with
110. bosonic atoms should of course be symmetric in all particles For bosonic atoms with no more than binary interactions it usually has the form H h pi zi gt V x zj i lt j so the one particle operator h including kinetic energy has to be listed in the operator file for any boson 7 while the interaction potential V requires a manual entry for any combination lFor simplicity we assume a system of spin polarized one dimensional bosons The extension to higher dimensions is slightly more complicated insofar as one has to distinguish between particles i 1 N and degrees of freedom x 1 f To match these two different modes x belonging to one and the same physical particle i have to be combined cf Sec 5 3 91 92 10 Treating bosonic systems i lt j Altogether these are N N 1 2 terms which naturally limits the application to few particles As an example the operator file for N 3 one dimensional bosons in a harmonic trap is shown in Example 10 1 for more details the reader is referred to 27 In that case h p 4p 527 the two body potential V x gd x shaped as a normalized Gaus sian of width o has to be fitted to the direct product form imposed by MCTDH This is carried out as usual via potfit set there e g pes gaussid width 0 05 the re sulting natpot is included in the LABELS SECTION The numerical parameters are defined in the PARAMETER SECTION even though som
111. case of the ABM BS and SIL integrators the first one is the integration order and the second one the error tolerance while the last one depends on the integrator Typical error tolerances range from 1073 or 1074 low accuracy over 1075 or 1076 normal accuracy to 1077 or 1078 high accuracy For the RK5 and RK8 integrators where the order is fixed the first parameter specifies the error tolerance and the second one the initial stepsize For all integrators it is in general not useful to work with an error tolerance less accurate than 1075 When performing a calculation one should first select the desired error tolerance The second step is to define the integration order The meaning of this parameter is slightly dif ferent for the three integrators which provide it For the ABM integrator which runs with a fixed order the order parameter in the INTEGRATOR SECTION is the true integration or 8 3 Description of the available integrators 83 Table 8 2 Optimal orders for the ABM and BS integrators in dependence of the error tolerance The optimal ABM order was found empirically and might differ slightly in other cases The values for the BS integrator on the other hand can be proved to be the optimal orders What is called optimal BS order in this guide is actually the maximum number of extrapolations Error tolerance Optimal ABM order Optimal BS order 107 3 4 1074 4 5 1075 5 7 1076 5 8 1077 6 9 1078 6 10 d
112. ck ets and density operators using the multiconfiguration time dependent Hartree MCTDH method Theor Chem Acc 109 2003 251 5 H D Meyer F Gatti and G A Worth Eds Multidimensional Quantum Dynamics MCTDH Theory andApplications Wiley VCH Weinheim 2009 A list of publications on the MCTDH method itself and on applications of MCTDH is given at the end of this Guide The latest version of this list can be found on the MCTDH homepage http mctdh uni hd de From this URL a review on the MCTDH scheme Ref 1 and the MCTDH feature article Ref 4 can be downloaded There you will also find a small bibtex file mctdh bib which contains references to most MCTDH articles This is for your convenience Chapter 1 Introduction The MCTDH method is an efficient algorithm for the solution of the time dependent Schr dinger equation For a full description of the theory see the review 1 You may also wish to read the MCTDH book 5 The MCTDH program has been developed to per form quantum mechanical wavepacket propagations employing this method All the options and variants of the MCTDH method presented in the review are implemented Furthermore the MCTDH program can be used to propagate wavefunctions numerically exactly and to diagonalise a Hamiltonian by the Lanczos algorithm A variety of programs included in the MCTDH package serve to analyse the results of a calculation and compute observable quantities which can directly
113. cs in double wells From single atom to correlated pair tunneling Phys Rev Lett 100 2008 040401 M Brill O Vendrell F Gatti and H D Meyer Shared memory parallelisation of the multi configuration time dependent hartree method and application to the dynamics and spectroscopy of the protonated water dimer In High Performance Computing in Science and Engineering 07 Heidelberg 2008 W E Nagel D B Kroner and M Resch Eds Springer pp 141 156 M Basler E Gindensperger H D Meyer and L S Cederbaum Quantum dynamics through conical intersections in macrosystems Combining effective modes and time dependent Hartree Chem Phys 347 2008 78 B Br ggemann P Person H D Meyer and V May Frequency dispersed transient absorption spectra of dissolved perylene A case study using the density matrix version of the MCTDH method Chem Phys 347 2008 152 165 S Z llner H D Meyer and P Schmelcher Tunneling dynamics of a few bosons in a double well Phys Rev A 78 2008 013621 S Z llner H D Meyer and P Schmelcher Composite fermionization of one dimensional bose bose mixtures Phys Rev A 78 2008 013629 G A Worth H D Meyer H Koppel and L S Cederbaum Using the MCTDH wavepacket propagation method to describe multimode non adiabatic dynamics Int Rev Phys Chem 27 2008 569 606 O Vendrell and H D Meyer A proton between two waters insight from full dimensional quantum dy
114. d first derivative only for Leg 09 sin cos 6 KLeg and PLeg DVR symmetry qdq 5 x Op Oy x for rHO DVR this replaces the first derivative sdq z sin x O r sin x for cos DVR this replaces the first derivative sdq2 sin x 0 x sin x cdq z cos x r Ay cos x cdq2 cos x 0 Oz cos x csdq nee x cos x Or Oy sin x cos z udq z vV 1 z r r V1 2 uqdq z V1 x r r V1 x x udq2 5 1 2 s s 1 x continued C The built in symbolic expressions 137 Table 2 continued Symbol Operator Notes ite 0 cot 8 s sin 6 02 Wigner DVR angular momentum o 2 cos 8 a 0 j p i BF ei za g i cot 8 jm j BF i ey a5 a g Og icot B a j ps J4 sF i cot 8 a Og tay j ms j sF ei i cot 3 0q Og ain jpm Ce j Br Cg 7 BF jpms Chur G4 se Copy J sr jp 2 J Br jp 2s 94 5 jm 2 j BF jm 2s j Sr squared operator with matrix elements J J K M J J 1 J K M Wigner DVR body fixed angular mo mentum lowering operator which operates as j pr J K M J J J 1 K K 1 J K 1 M Wigner DVR body fixed angular mo mentum raising operator which operates as j sr J K M V I J 1 K K 1 J K 1 M Wigner DVR space fixed angular momentum raising operator which operate
115. d a natural potential fit it may be used in the MCTDH program by correspondingly defining the label V in the above example LABELS SECTION V natpot directory nd labels section Here directory denotes the path to the directory containing the natpot file which is created by the Potfit program Replacing the path by the keyword name i e natpot name indicates that the natpot file is in the name directory i e the directory the output is directed to Note that MCTDH uses the modelables to let the potential operate on the degrees of freedom or on combined modes i e MCTDH particles in the correct order It is hence 62 6 Setting up the Hamiltonian recommended to use an unnumbered Hamiltonian line e g V rather than a numbered one e g 1 amp 2 amp 3 V_ when V refers to a natural potential One may put the symbol V in any column and it may be convenient to place it in the first column If a numbered Hamil tonian line is used however the numbers must be consistent with the modelabels Otherwise the program will stop If one gives the keyword i gnore as argument to natpot in the Labels Section then the modelabels are ignored and the assignment of the modes or DOFs is done exclusively by using the numbers As a final remark we note that the modelabels of a natpot may be altered by running chnpot84 There are some restrictions when using natural potentials natpots natpots cannot be multiplied with an
116. d above for the DOFs on which one projects one is free to use any basis definition For the remaining DOFs one must use the same basis definitions as for the subsequent potfit run which in turn must be the same as for the following mcetdh run unless one uses chnpot in between The PROJECTION SECTION contains the definitions of the projector functions y q During a single projection run it is possible to use several different sets of projection functions One such set of functions is defined in one PROJECTOR section After the 124 12 Using the Potfit program PROJECTOR keyword one must give an arbitrary string which is used as a label for the projected potential in example 12 3 this would be the labels O00 200 and 222 The definition of the individual projector functions is done by first stating the modelabel of the DOF on which to project here e g AL and then giving the type and parameters of the projection function here e g leg 2 0 which stands for the Legendre polynomial P49 Please see the HTML documentation for which projector function types are available and what parameters they take For each defined projector projection produces a vpot file which contains the projected potential on the product grid of the remaining DOFs The name of this file depends on the given projector label e g in case of the 200 projector the corresponding file will be called vpot_200 To use this file in th
117. d in which order the natpot shall operate See the HTML documentation for details The SEPARABLE WEIGHT SECTION and the CORRELATED WEIGHT SECTION finally define the separable and correlated weights respectively See the HTML documenta tion for details The OUTPUT FILE contains important information on the natural populations and on error measures Shown in Example 12 2 is an excerpt of an potfit output file which is generated by running the input Example 12 1 The block named Trace Sum of all preceding Natural Weights eV 2 displays the sum of neglected reduced natural weights e g the second entry 0 5371E 03 is the sum of eigenvalues 3 to 36 This sum is directly related to the fit er ror As in this example we took 5 and 4 single particle potentials into account the estimated error is given by 0 2817E 04 0 2474E 05 0 5535E 02 in eV i e 5 5 meV This estimate is printed in the output below the line Global weighted L2 error estimated from neglected and it 12 1 Transforming a potential to product form 1 17 KKKKKKK KK KKK Mode Trace of reduced density matrix 0 8297 au red Number of eigenvalues considered 5r of 36 Reduced Eigenvalues Natural Weights eVx 2 1 0 8363E 00 0 3063E 02 0 3058E 03 0 1667E 03 7 0O 5736E 05 0 4436E 05 0 2863E 05 0 1240E 05 13 0 1324E 06 0 1037E 06 0 8599E 07 0 4732E 07 19 0 7491E 08 0 7248E 08 0 5348E 08 0 3652E 08 25 0 4284E 09 0 3753E
118. d iterate One should note the following the described procedure implies that both the x and y coordinates belong to the same combined mode and there are no other coordinates are present in this mctdh particle Otherwise the program would treat the new potential as a muld potential i e a multi dimensional potential This slows down the performance of MCTDH Therefore the procedure outlined above is most useful when the potential operates on one combined mode MCTDH particle exclusively 12 2 2 Using the potfit program A second alternative is to use the potfit program to convert the ab initio data to product form and then use mctdh as it is shown in Fig 12 1 This possibility circumvents the inconveniences described at the end of the previous section since the different degrees of freedom can then be used in different combined modes or be simply uncombined The usage of potfit has been covered in chapter 12 1 of this guide so here it will only be covered how to make it read an ab initio surface defined on a primitive grid The readsrf keyword see previous section for details comes now into play in the OPERATOR SECTION of the input file of potfit 120 12 Using the Potfit program OPERATOR SECTION pes readsrf path to file S nd operator section As in the previous case the file containing the data values has to be written with the index of the first specified degree of freedom running fastest the most internal loo
119. d rl xunit cm 1 2534 528194 to display directly the excitation energies Inspect the input files and try to understand every line The tiny 3D problem COz is of course too simple to show the strengths of improved relaxation If you wish to solve some 6D problems run the input files hono dav inp and H2CS x inp 2 7 Determining eigenstates by block improved relaxation The block variant of improved relaxation is very useful if several low lying states are to be computed It makes use of the single set multi packet feature of MCTDH i e the different 16 2 An MCTODH tutorial packages are formally put on different single set electronic states True electronic states either in multi set or single set formalism can be added as well Because the packets are treated in single set the SPFs are propagated or relaxed on a state averaged mean field As there is only one set of SPFs the SPFs cannot be optimal for one eigenstate they are optimized for the full block of eigenstates to be computed Hence the block form will in general require more SPFs to achieve the same accuracy as a single improved relaxation But because the block form generates several eigenstates at once it is often more conve nient and sometimes even numerically more efficient However a block relaxation requires considerably more memory than a single relaxation The following example takes more computation time than the previous ones It may be skipped if one is not p
120. diabatic and adiabatic state populations by running adpop84 If needed one and two dimensional adiabatic densities can be calculated by adding the modelabels to the program call adpop84 v9a vl0a v6a In this case the one dimensional adiabatic density for the v9a degree of freedom and the two dimensional density for the v10a v6a degrees of freedom will be created as well as 108 11 Analysing the results employing the Analyse programs the diabatic and adiabatic state populations The created files are adp for the populations adp_v9a for the one dimensional density adp_v10a_v6a for the two dimensional density and the log file adp log To create two dimensional densities the two modelabes must be separated by acomma The modelabels for one dimensional densities stand alone To visu alise the density files created by adpop use the shell script pladpop that can plot one and two dimensional densities For plotting the state populations simply use plgen As the calculation runs over the full primitive grid the calculation is slow Analyzing one wavefunction takes about grid dimension A vector length 1 5 x 1078 s ona 3 GHz P4 The adpop calculations can be accelerated by setting the options q or mc The first option enables the quick modus where all points are ignored for which the product of the one particle grip populations i e the 1D densities are smaller than some threshold The loss in accuracy is usually negligible and can b
121. dimensionale Wellenpaketdynamik nach elektronischen Anregungen PhD thesis Uni versitat Heidelberg 1991 U Manthe H D Meyer and L S Cederbaum Multiconfigurational time dependent Hartree study of complex dynamics Photodissociation of NO2 J Chem Phys 97 1992 9062 9071 U Manthe and A D Hammerich Wavepacket dynamics in five dimensions Photodissociation of methyl iodide Chem Phys Lett 211 1993 7 H D Meyer U Manthe and L S Cederbaum The multi configuration Hartree approach In Numerical Grid Methods and their Application to Schr dinger s Equation Dordrecht 1993 C Cerjan Ed Kluwer Academic Publishers pp 141 152 A P J Jansen A multiconfiguration time dependent Hartree approximation based on natural single particle states J Chem Phys 99 1993 4055 4063 U Manthe Comment on A multiconfiguration time dependent Hartree approximation based on natural single particle states J Chem Phys 101 1994 2652 A P J Jansen Response to Comment on A multiconfiguration time dependent Hartree approximation based on natural single particle states J Chem Phys 101 1994 2654 A D Hammerich U Manthe R Kosloff H D Meyer and L S Cederbaum Time dependent photodis sociation of methyl iodide with five active modes J Chem Phys 101 1994 5623 J Y Fang and H Guo Multiconfiguration time dependent hartree studies of the CH3I MgO photodisso ciation dynamics J
122. duct of body fixed angular momentum operators which operates as j sF jz Br J K M KyJ J 1 K K 1 J K 1 M Wigner DVR product of space fixed angular momentum operators which operates as j sr jz se J K M MV J J 1 M M 1 J K M 1 Wigner DVR product of body fixed an gular momentum operators which oper ates as jz pr J pr J K M K 1 J J 1 K K 1 J K 1 M Wigner DVR product of space fixed an gular momentum operators which oper ates as j sr J sF J K M M 1 J J 1 M M 4 1 J K M 1 Wigner DVR_ product of body fixed angular momentum operators which operates as j_ pr jz BF J K M Ky J J 1 K K 4 1 J K 1 M Wigner DVR_ product of space fixed angular momentum operators which operates as j sr jz se J K M MV J J 1 M M 1 J K M 1 continued C The built in symbolic expressions 139 Table 2 continued Symbol Operator Notes jzjm jz BF J BF Wigner DVR product of body fixed an gular momentum operators which oper ates as j pr j_ pr J K M K 1 J J 1 K K 1 J K 1 M jzjms jz sF J sF Wigner DVR product of space fixed an gular momentum operators which oper ates as jz sr J sr J K M M 1 J J 1 M M 1 J K M 1 Notes to Table C 2 The volume element assumed for Leg KLeg PLeg Wigner is sin d0 whereas all other DVRs i
123. e Compile with u option to enable the use of NUMA See compile h The standard installation should work without problems If not already done you should now run install_mctdh E Installing the MCTDH package 149 KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK KR RR KK RK RRR RK Finished System Check ORK kk I ak KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK This is just for fun but if you read the message The standard installation should work without problems then there should be no problems By the way please note that the GNU compilers GCC 3 4 0 3 4 3 and GCC 4 0 0 4 1 0 are buggy DO NOT USE THEM We recommend to use GNU GCC 4 4 2 or higher For the 8 5 branch where we start to use some fancy features of FORTRAN95 2003 we recommend to use even higher versions e g 4 7 2 however currently 2013 4 4 2 is still sufficient Next to GNU the intel pgi and some other compilers are also fine See install compile cnf for possible compilers You may edit the compile scripts and add new compilers to it Please read the README file of the install directory and start with the installation i e type install_mctdh while being in the install directory The script will ask you several questions and in general you should answer yes i e type y In fact one may run yes installmctdh but this is not recommended for beginners Please try to und
124. e controlled by setting the parameter qt 01 The Monte Carlo integration is of course less accurate but allows to attack problems which are not feasible for adpop using direct integration 11 8 3 Adiabatic populations computed with adproj If one is interested in the adiabatic state populations alone one should use the second variant which is more efficient especially for larger systems gt 5 degrees of freedom However this way is more cumbersome than the adpop calculation The procedure is explained in the following First a pes file must be created often this has already been done by the previous mctdh run After this running adproj generates several vpot files which contain the adiabatic sur faces and the projection operator matrix elements for each electronic state As an example the file apr_p1_12 contains the matrix element 1 2 of the projector p projector for the first adiabatic electronic state In contrast the file apr_v2 contains the second adiabatic poten tial energy surface v potential Then the vpot files must be fitted by potfit to bring them into the MCTDH product representation To calculate the expectation value of the projector one needs the projector in form of an MCTDH oper file This oper file can be created by an mctdh run with the keyword genoper in the RUN SECTION After the oper file has been created the expectation value of the projector can easily be calculated with the expect anal yse routine This
125. e directories noclO and nocl1 2 To perform the ground state relaxation calculation type mctdh84 nocld You will have to wait about 2 seconds The timings given in this manual are for a 3 GHz PC running under Linux 4 2 An MCTODH tutorial 100 90 80 70 60 50 40 30 20 10 10 0 6 0 8 1 1 2 1 4 1 6 1 8 2 Energy eV Figure 2 1 The absorption spectrum for the NOCI molecule on excitation to the S state 3 To perform the photo dissociation calculation type mctdh84 nocll This will again take about 2 seconds NB There is now the option mnd make name directory which allows you to skip the creation of the name directory E g mctdh84 mnd nocll will make the name directory before starting the calculation The calculation can now be analysed Move to the directory nocl1 which contains all the data files from the propagation 1 To watch the system dissociating type showd1d84 a M y 5 sm fl In order to understand the options and parameters type showd1d84 h and see the HTML documentation Try the other format options S T and inspect the motion of the other degrees of freedom f2 f3 The program showd1d also supports interactive plotting Start the program with showd1d84 inter and follow the menu options to select and alter the plot 2 To plot the spectrum type 2 1 Determining the absorption spectrum for the photodissociation of NOC1 5 autospec84 g 1 0 6 2 0 ev 0 0 1 gnu
126. e input file The program evaluates this section and stores the information in the dvr file The PRIMITIVE BASIS SECTION also serves to define the system coordi nates by allocating a label to each degree of freedom in the system These labels are then used in other sections such as the INIT WF SECTION the SPF BASIS SECTION and the HAMILTONIAN SECTION to map the different sets of information onto the system coor dinates A complete input file for the photo dissociation process of NOCI is shown in Ex ample 4 1 There three Jacobian coordinates are defined labelled rd dissociative degree of freedom rv vibrational degree of freedom and theta angular degree of freedom Their DVR representations are sine Hermite and Legendre respectively The parameters associated with these representations are explained in the following sections 4 2 Hermite and radial Hermite DVR In the Hermite or harmonic oscillator DVR the harmonic oscillator functions ao aij mw m 4 Hi Vines x i ema w 2 4 1 are taken as basis functions The Hermite DVR is thus typically used for vibrational modes In the above equation H denotes the jth Hermite polynomial and j starts from zero 36 4 2 Hermite and radial Hermite DVR 37 Table 4 1 The DVR FBR representations for the primitive basis which are available in the MCTDH package Also displayed are the corresponding keywords that are used in the input file and a typical applicati
127. e number of grid points to be summed over is changed to 2 i e rdgpop84 2 0 then the output is Maximal values all times final time 30 00 fs dof grid begin grid end basis begin basis end T ra 001164114 000002005 593797892 000013545 2 TY 0 000000000 010178351 993075095 000088905 3 theta 0 000000000 0 000000000 316521645 000008258 The start of the rv grid is still unpopulated and so 2 grid points could be removed at the start of this degree of freedom By increasing the number of points over which the sum is made it is possible to evaluate how many grid points can be removed Whether it is possible to remove the grid points depends on the primitive basis being used Thus the Leg DVR used for the theta degree of freedom by definition runs from 7 to 0 if symmetry is used theta runs from 7 2 to 0 and these points cannot be removed However the use of the restricted Legendre DVR Leg R allows to remove unused angular grid points To chart the population of the end grid points select a degree of freedom Thus rdgpop84 1 1 produces the file gpop pl which contains the populations of the end of grid points as a func tion of time The option g writes information for GNUPLOT to this file and so rdgpop84 g 1 1 gnuplot persist gpop pl produces a plot of the time dependence of the end of grid points A more convenient way to plot the time evolution of the population of the end grid points is provided by the plgpop
128. e of the values are conveniently reset in the input file see below 10 2 Modifying the input The symmetrization mentioned above brings about some minor modifications of the way the wave function Y is handled The MCTDH ansatz 10 1 is now simplified insofar as the single particle functions are now identical i e we have Y Yj with a single set of functions y j lt n This reflects in the input file as illustrated in Example 10 2 The SPF BASIS SECTION only gives the first orbital p1 while all others are mapped via the entry x2 id 1 etc It goes without saying that by extension the primitive basis also has to be identical for ev ery boson Again this is reflected in the input file where the PRIMITIVE BASIS SECTION contains repetitions of the very same line x1 HO 125 xi xf 4 0 4 0 for any zi A little less trivial is the choice of the initial wave function Vo As stated above it must be permutation symmetric a demand which can be met by the following standard choices e A Hartree state Yo y is implemented trivially by including the following lines in the INIT WF SECTION build xl eigenf spo x2 map x1 x3 map x1 end build Here eigenf spo specially selects the lowest eigenfunction y of the single particle operator spo previously defined in the operator file This particular feature is not essential in this case one may as well use harmonic oscillator functions HO instead but we have incl
129. e operator file At present only one target operator can be specified in the operator file it is however possible to use it with multiple initial states Multi target optimizations are possible by using the multi packet algorithm for target states Multi packet wave functions are treated within a multi target optimization procedure For target states the control functional J can be chosen either as Ntar fj T E t E Nar 9 Y T Mean a0 TO 2 1 i 0 or as F ple N SE oT Ta an 2 tar ek i JE tari ao dt S t 2 2 2 0 Here ao is the so called penalty factor that penalizes for strong fields and S t serves as a pulse envelope that can be defined in the operator file The functional 2 1 leads to optimiza tions of the target state populations only while within 2 2 the phases are also aligned 17 If Niar 1 both functionals are identical Example inputs are provided under MCTDH_DIR inputs optcnirl The Python script optcntrl parses the input file and invokes OCT related programs from the MCTDH pack age Note the script reqires Python 2 4 or 2 5 and relies on the Python executable be ing found in usr bin env python This path can be changed in the first line of the script MCTDH_DIR bin python optcntrl py From the input file a number of temporary input files for the actual MCTDH calculations are created The same applies for the operator file The operator file must contain at least two 2
130. e subsequent potfit run one must read it with the readvpot keyword in the RUN SECTION and also specify pes none in the OPERATOR SECTION Example input files for potfit runs which use projection output can be found in MCTDH_DIR pinputs omkpe_fit_ inp In the case where the projected potentials ve where 7 numbers the different sets of projector functions constitute a series expansion of V it is worthwhile to re assemble the projected potentials into another full dimensional potential V which can then be compared to the original potential V Thus one can check whether enough terms of the series expansion i have been taken into account to faithfully represent the original potential To this end let x Ny be the projector function for the 2 th DOF in the i th set and let z0 be the corresponding complementary projector function such that fon Pla Oa 1 12 4 If we assume that the projector sets 7 and j are orthonormal i e p II uw x ae XD ae 6 12 5 x 1 then we can define the re assembled potential ViGies a7 F Pla 2 ap VS liin 3 f 12 6 An error measure that depends on the remaining coordinates and which has the unit of energy is eee V i oe AV ieee dp qr fdu dgy If you give the error keyword in the PROJECTION SECTION this error measure will be calculated and stored in another vpot file named projerr vpot This can then be inspected with the showpot program Finally the l
131. eas the accuracy of the integrators is rather high Note also that the parameter for regularizing the density matrices eps_inv is also set to a low value its default value is 1078 This is because the lowest natural SPF populations are in the range 107 10710 for improved relaxation runs whereas they are typically in the range 107 1076 for propagation runs The RK8 integrator was found to perform best for SPF relaxation If a hight ac curacy of the results is not required one may set the RK8 accuracy to 1 0d 8 and eps_inv 1 0d 9 or even remove the eps_inv line As orbital type natorb was chosen in this example The default for improved relaxation is energyorb may be abbreviated to enorb Energy orbitals make the Hamiltonian matrix more diagonal dominant than other orbital choices which accelerates the convergence of the Davidson digonalizer However the computation of the energy orbitals is a bit costly and we noticed that in particular when a preconditioner is used it is often more efficient to use natural orbitals Standard orbitals st dorb which are default for propagation can also be used Note that the use of energy orbitals requires that the keyword orben is set in the Run Section 8 4 3 Advanced topic Evaluating potentials using the TDDVR or CDVR method The time dependent DVR TDDVR and Correlation DVR CDVR methods offer the possi bility of employing non separable potentials within the MCTDH scheme without the
132. ectation value of a positive definite operator J Chem Phys 109 1998 385 J P Palao and R Kosloff Optimal control theory for unitary transformations Phys Rev A 68 2003 062308 154 List of MCTDH references 155 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 L Wang H D Meyer and V May Femtosecond laser pulse control of multidimensional vibrational dynamics Computational studies on the pyrazine molecule J Chem Phys 125 2006 014102 M Schr der J L Carreon Macedo and A Brown Implementation of an iterative algorithm for optimal control of molecular dynamics into MCTDH Phys Chem Chem Phys 10 2008 850 M Schr der and A Brown Realization of the cnot quantum gate operation in 6d ammonia using the oct mctdh approach J Chem Phys 131 2009 034101 M Schr der and A Brown Generalized filtering of laser fields in optimal control theory application to symmetry filtering of quantum gate operations New J Phys 11 2009 105031 H D Meyer F Le Qu r C L onard and F Gatti Calculation and selective population of vibra tional levels with the Multiconfiguration Time Dependent Hartree MCTDH algorithm Chem Phys 329 2006 179 192 L J Doriol F Gatti C lung and H D Meyer Computation of vibrational energy levels and eigenstates of f
133. ection The pack keyword defines the packet to which an input line belongs It is possible to specify more initial packets in this section than given by the packets argument All data with pack gt packets will then be ignored Note that the auto file now contains the cross correlation matrix Cop 2t Wa H Ust a8 1 5P 7 8 rather than the auto correlation function See the HTML documentation for the exact format of the auto file 7 11 Advanced topic Setting up a diabatically corrected initial wavepackets The flux analysis and similarly twprob require the knowledge of the energy distribution of the initial wavepacket This energy distribution is written to the enerd file if the keyword correction is given in the INITWF SECTION NB The file enerd is called adwkb in older versions The keyword correction requires an argument If correction edstr is given it is assumed that the wavepacket is located far out side such that the interaction potential can savely be neglected The energy distribution is then given by Eq 140 of the MCTDH review 1 i e essentially by a fourier trans form of the single particle function of the translational degree of freedom It is assumed that the translational degree of freedom is the dof number 1 i e is the first item in the PRIMITIVE BASIS SECTION Otherweise the keyword trans I1 12 must be given where I1 denotes the number of the translational dof and optional I2 its elec t
134. ection adiabatic 78 diabatic 78 Correlation DVR 85 CSIL integrator see SIL integrator Diagonalisation 28 Discrete variable representation see DVR Distributed memory see Calculations DOF mode and muld potentials 67 DVR 36 exponential 39 extended Legendre KLeg 42 Hermite 36 Legendre 38 radial Hermite 36 restricted Legendre 41 sine 39 three dimensional rotational Wigner 43 two dimensional Legendre PLeg 42 DVR file see File Efield program see Program Electronic basis see Basis Energy cut off 66 Energy distribution 78 Energy weights 25 Error estimate of the SIL integrator 83 Error message 28 Error tolerance of the ABM integrator 82 of the BS integrator 82 of the CMF scheme 82 of the RK5 8 integrator 82 of the SIL integrator 82 Exponential DVR see DVR Extended Legendre see Legendre Extended Legendre DVR see DVR Fast Fourier transform see FFT FBR 36 Fdcheck program see Program Fdmatch program see Program FFT 39 Temperton 40 File auto 22 104 check 101 106 chk pl 7 dvr 36 eigval 11 enerd 78 105 flux 105 163 164 Index flux log 105 gridpop 99 102 gtau 105 input 115 129 iteration 115 log 22 115 natpot 115 operator 48 orben 25 output 22 97 115 pes 5 112 prodwei 115 psi 22 111 ptiming 29 32 restart 76 rlx_info 15 27 spectrum pl 4 8 104 stop 29 surface 65 timing 102 115 vpot 115
135. ed flux 0 was used to input the file flux 0 as energy distribution the default is the file enerd the generation of which however is only useful for a scattering not half scattering problems To visualise the quotient weighted flux total flux type plflux G r f flux op_r The structures below 0 8 eV are numerical noise because one divides a very small number by another very small number Obviously the higher the energy of the absorbed photon the larger is the vibrational energy of the NO fragment Compare this plot with the eigenenergies displayed in the flux log file Finally we demonstrate the use of projectors Type plflux G f flux 0 d flux 1 e flux 2 plflux G a 1 0 y 25 f flux 0 d flux 2 e flux 3 The first plot shows the flux projected onto the vibrational ground state and the first excited state respectively in comparison with the total flux The second plot is similar but shows the flux projected onto the first and second excited state respectively The Fig 11 3 displays the total and projected projected flux similarly to the plots generated above 11 8 Monitoring state populations of non adiabatic systems 11 8 1 Diabatic populations The electronic state populations of amulti set run are written to the check file and are read by rdcheck84 which writes them to the file chk pl One then may use GNUPLOT to plot the state populations but it is easier to use plstate 11 8 Monitoring state
136. ed kinetic energy hFRh step KE Rf step Truncated kinetic energy times reflection hdgh step dq step Truncated first derivative hdqRh step dq Rf step Truncated first derivative times reflection dqR dq R First derivative times reflection dq2R dq 2 R Second derivative times reflection Notes to Table C 3 All input variables are numbers parameters or arithmetic expressions containing num bers and parameters See Hamiltonian Documentation Parameter Section for details The use of units is not allowed here Note that these symbolic expressions with parameters must not appear in a HAMILTONIAN SECTION They rather have to be linked to simple symbols without parameters in a LABELS SECTION Compare with Example 6 4 1 A morse curve can be given by D exp a ax xo 1 Ep where D is the dissociation energy depth parameter defines the curvature xo the equilibrium position and Ep is an energy shift parameter If one uses the symbol morsel these are precisely the input parameters i e D a o Eo For the symbol morse the in put parameters are D w o exo m where m is the mass w is the frequency of the related harmonic oscillator and exo is the position at which the potential is zero Note that w and a are related by a mw 2D while exo and Eo are related by Eo D exp a exp x9 1 2 A CAP Complex Absorbing Potential is an imaginary negative potential used to absorb a wavepacket as i
137. eds Apple s Xcode to obtain make the GNU compilers gec and gfortran with version GCC 4 2 x or higher and gnuplot Currently the parallelization with pthreads does not work although the pthread libraries are installed However after installing the openMPI software one can use MCTDH with MPI parallelization Appendix F The svn repository of the Heidelberg MCTDH package We use Subversion or short svn for version control of the MCTDH package As svn is likely to be available on your computer installation we open the possibility to download the MCTDH package directly from our svn repository rather than from the MCTDH web site http mctdh uni hd de packages If you are new to svn you may wish to consult the svn book which can be downloaded from the URL http svnbook red bean com en 1 6 svn book pdf To access the MCTDH svn repository a username and password are needed These are given in the Letter to the new MCTDH user and are the same as the ones requested to access the MCTDH web site http mctdh uni hd de packages F 1 Useful svn commands In order to abbreviate the commands we suggest to add the following lines to your bashrc or alias file alias svnm svn username lt user gt password lt psswd gt non interactive export SVNM svn www pci uni heidelberg de mctdh where lt user gt and lt psswd gt are to be replaced with the username and password given in the Letter to the new MCTDH user
138. efunction do not need to have the same number of electronic states See Section 7 7 Chapter 10 Treating bosonic systems While MCTDH is designed for distinguishable particles it also allows for the treatment of indistinguishable particles The only conceptual complication that needs to be taken care of is the permutation symmetry encoded in the general rules of quantum mechanics More precisely Y is a valid wave function only if for any permutation P of two particles we have P j W The sign holds for bosonic particles which are the subject of this chapter for fermions there already exist specially modified versions of MCTDH The consequence is that bosons live only in the symmetry restricted Hilbert space H Pj v V Vi j CH Obviously the most elegant way to extend the MCTDH ansatz TQ t X AJBI t 10 1 J would be to include only basis function H a demand clearly not met by the Hartree products employed in MCTDH However it is possible to circumvent this by simply pro jecting any wave function onto H This amounts to keeping the expansion coefficients Az symmetric rather than the basis vectors themselves In fact any wave function will usually stay permutation symmetric under real or imaginary time evolution if both the initial state and the Hamiltonian are chosen as outlined below and if numerical errors are kept at bay 10 1 Setting up the Hamiltonian The Hamiltonian for identical
139. els section end operator Example 6 1 An operator file for the NOCI S state OP_DEFINE SECTION title NOC1 S1 surface end title nd op_define section There is no restriction on the format or number of lines used The next step for creating a new operator file is described in the following section 6 2 Defining numerical constants A Hamiltonian typically contains some numerical constants such as reduced masses fre quencies or coupling strengths In the PARAMETER SECTION of the operator file labels can be associated with these constants which can then be used in the definition of the Hamil tonian This makes the operator file not only easier to read but also very simple to change as is depicted in Sec 6 11 For example a PARAMETER SECTION reading PARAMETER SECTION mh 1837 15 hydrogen mass mc 21874 7 carbon mass nd parameter section 50 6 Setting up the Hamiltonian defines the labels mh and mc as the masses of the hydrogen and carbon nuclei in a u Note that it is possible to specify the unit of a parameter The just mentioned example can hence also be written PARAMETER SECTION mh 1 0 H mass mc 12 0 AMU nd parameter section The keywords H mass and AMU stand for hydrogen mass and atomic mass unit See the HTML documentation for available units Furthermore elementary algebraic operations can be performed in the P
140. ement of the G matrix of the kinetic energy operator A kinetic energy operator can always be written in the form f f 1 T 3 X pi Gipj Pipi Vertra 8 1 ia i where G F and Vertra are potential like terms This equation defines the G matrix Note that the kinetic energy defined in the Hamiltonian Section does not need to be of this par ticular form A further useful task is performed if the keyword test is used In this case all input files are checked and all output files are created even a psi file for t 0 like in a propagation run but no propagation step is done A test run may hence be used to convert a restart file into a psi file It is possible to start a continuation run see Sec 3 9 after a test run 3 2 Specifying the desired output The output produced by the MCTDH program is sent to a directory called the name directory The absolute or relative path of this directory is specified by the name keyword in the RUN SECTION of the input file The name directory should already exist when the MCTDH program is started Otherwise one may use the option mnd make name directory to create a new directory During run time a number of files are generated in the name directory Some of these are always created while others have to be selected by the user The most interesting file of the former category is the log file which records what happens at the various stages of a run Frequently used files of the second categor
141. en increases the computational speed as it is the case for propagation of C2H4 If the cal culation is made with all routines running in parallel mode the parallel part of the program according to Amdahl s law is 61 whereas the parallel part is 75 if the parallelisation of the funkphi routine is disabled In general the parallelisation works better for larger systems i e systems with many Hamiltonian terms and many single particle functions This is due to the fact that either loops over the Hamiltonian terms or loops over the single particle functions are parallelised We have observed that the performance of the parallelisation does not only depend on the system studied but also on the computer platform and compiler The C2H4 system has been propagated on an Itanium cluster using the Intel compiler This resulted in a speedup of 2 30 4 processors for which Amdahl s law states a parallel parallel part of 75 The same system propagated on a quad opteron also with the Intel compiler showed a speedup of 2 58 4 processors and hence a parallel part of 81 7 One of our better examples is Hy H inelastic scattering Here the Hamiltonian consists of many terms because there is a large potfit A speedup of 6 19 is observed when running this system on 8 processors in parallel This implies that 95 5 of the work is done in parallel The memory used increased form 56 MB one processor to 60 MB 8 processors The performance of the paralleli
142. enf vib 3 0 61 2 0 ev rd mv flux flux 3 106 11 Analysing the results employing the Analyse programs autospec84 FT 0 61 2 0 ev 0 1 The first flux run evaluated the total flux To display it type plflux f flux 0 As you will notice the plot looks very similar to the absorption spectrum shown in figure 2 1 It is not identical though as the definition of an absorption spectrum contains a factor w the energy of the absorbed photon This multiplication is omitted when autospec84 or plspec is run with the FT option as we have done above Now the spectra are identical as one observes when typing plflux G f flux 0 d spectrum pl This shows the flux and the Fourier transform of the autocorrelation function on top of each other Next we modify the flux by letting the operator of the vibrational energy act on it Type plflux G f flux 0 d flux op and you will see the total flux in comparison with the vibrational energy weighted flux It is now clear that the structures in the spectrum are due to vibrational excitation of the NO fragment By the way via the option u 200 the modified flux was multiplied by the factor 200 This was done to make it comparable with the total flux Usually the option u is followed by an energy keyword e g u ev to transform the weighted flux from a u to a desired energy unit One may divide the weighted flux by the total flux to observe the vibrational energy con tent The option
143. ent function number ifunc and the program will call different subroutine according to the i func number For example we have defined ifunc equals to 1 if the label is my1d in subroutine ufdef1d and in subroutine ufuncld subroutine my 1d will be called if i func equals 1 As an example if you wish to add the cotangent function cot a x x b to the pro gram using the label cot You would then have to edit both the subroutines ufdef1d and ufuncld in the file opfuncs user1d F and additionally write a subroutine of the cotangent function First the new function must be coded subroutine cot x v a b C cot cot ax x b C this subroutine is called in subroutine ufuncld C the parameters are defined in subroutine ufdefld TEAL S Xy Vo ay oDye r ax x b v 1 d0 tan r end subroutine 56 6 Setting up the Hamiltonian Any valid FORTRAN expression can be employed to define a new function Then you have to define the label cot in the subroutine ufdef1d and give it a function number ifunc Go to the end of the subroutine ufdefid where the code reads like near line 60 of file opfuncs user1d F ifunc 1 C newfunc This is of course just an example elseif label l ilbl eq newfunc then C ifunc 2 Cc C end of if loop C endif The next free function number is 2 so you should replace the three out commented lines with C cot cot ax x b else if label 1 ilbl eq cot then
144. ents in MCTDH_DIR pinputs bmkpe_proj inp 12 4 Manipulating potentials with the projection program 123 RUN SECTION name bmkpproj directory where output is written timing write timing information to file timing output write output to file output end run section OPERATOR SECTION pes h4bmkp jacobian surface to use BMKP H4 surface veut lt 0 0 ev cut off at 0 0eV PES zero point is at 9 5eV nd operator section PRIMITIVE BASIS SECTION R FFT 128 1 80 VE oO RAB HO 8 1 4483 0 019216 0 5 h mass RCD HO 8 1 4483 0 019216 0 5 h mass AL leg 10 0 even BE leg 10 0 even PHI exp 27 2p end primitive basis section PROJECTION SECTION PROJECTOR 000 AL leg 0 0 BE leg 0 0 PHI cos 0 end projector PROJECTOR 200 AL leg 2 0 BE leg 0 0 PHI cos 0 end projector further projectors omitted PROJECTOR 222 AL leg 2 2 BE leg 2 2 PHI cos 2 end projector error end projection section end input Example 12 3 A projection input file for the BMKP surface for H2 2 excerpt The RUN SECTION only specifies the name directory and which output files to be opened The OPERATOR SECTION is the same as in potfit it specifies the PES to be used and defines cuts to remove large potential values The PRIMITIVE BASIS SECTION has the same structure as in metdh and potfit It defines the grid points of the product grid on which 12 3 is evaluated As mentione
145. ep is accomplished by means of the cutnpot function This is invoked by including the cutnpot keyword in the RUN SECTION of the potfit input file The NATPOT BASIS SECTION should contain the desired smaller values of the expansion coefficients while preserving the definitions of the contracted mode and the particles The natpot file to be reduced cut should either be placed in the working directory as natpot1 or under a different name or location in which case the name with absolute path should be provided as cutnpot path Then potfit will produce a new shrinked natpot file within the same directory Finally the subsequent measurement of the fitting error for sufficiently small primitive grids is obtained by using the rdnpot function which is invoked by adding the RUN SECTION keyword rdnpot and running again potfit As previously indicated for the sake of consistency the DVR definitions the contracted mode and the mode combination scheme if any must remain unaltered during the whole process Hence if a new particle scheme is needed then a new potfit or a new mgpf calcu lation has to be performed The cutnpot and rdnpot keywords appear in the RUN SECTION otherwise the input is the standard one Note that cutnpot yields the same results as a normal potfit with the reduced number of single particle potentials provided no correlated weights are used Appendix A The concept of the input file With the exception of a continuat
146. er For the BS and SIL integrators which continously adapt their integration order during a run the order parameter denotes the maximum number of extrapolations and the maximum integration order respectively What the order parameter hence defines is actually the mem ory being allocated as all three integrators have in common that with each increase of the order parameter by one one additional wavefunction vector must be stored The order parameter of the ABM and BS integrators should be chosen according to Tab 8 2 Larger values do not increase the efficiency but only enlarge the memory requirements In the case of the ABM integrator a larger value in fact decreases the efficiency Smaller values for the order parameter lead to longer CPU times They might however be used if memory must be saved The optimal order parameter of the SIL integrator unfortunately cannot be predicted but has to be found out empirically for each system Typical values range from 6 to 16 After a calculation the largest order the SIL integrator has used is given in the log file If this value is smaller than the order parameter you should decrease the order parameter accordingly to avoid the waste of memory in future calculations If the largest order equals the order parameter this indicates that the efficiency might become higher if a larger order parameter is chosen so increase the order parameter for optimal performance When memory intensive systems are investi
147. erstand all the questions before giving an answer If you think you have made a wrong choice you can always stop the installation process with the ctrl c command and then start anew The install_mctdh script LaTeX compiles the guide compiles the code sets some environment variables and writes the path of the MCTDH directory to your bashrc file or some other configuration file if you are not running under bash Note that you have to source your bashrc or other configuration file to activate the changes The path to the MCTDH directory is stored in the environment variable MCTDH_DIR to inspect it type echo S MCTDH_DIR But it is more convenient to run the script menv which writes a list of all MCTDH environment variables to screen If you are running under bash the install script will also create the mctdhre file This en ables the powerful cdm command try cdam h and sets a link mctdh which points to the currently active MCTDH directory in the present case to home muser MCTDH mctdh84 x One may install several MCTDH packages but only one will be active Use the script minstall to switch between the different versions If you want to make use of the surfaces library move the file addsurf tgz to your MCTDH directory i e in our example to nome muser MCTDH and untar it Then you should edit the mctdhrc In this case simply remove the in front of export MCTDH_ADDSURF and source mctdhrc to activate this change Similarly
148. es that the ground state is taken as the first SPF In contrast to eigenf meigenf counts the eigenfunctions from zero The Lanczos iteration is stopped when the selected state here the ground state is converged Adding the argument full forces meigenf to perform as many iterations as there are grid points leading to a numerically exact full diagonalisation of the operator This is not recommended if the particular mode under discussion is repre sented by many grid points more than 300 say With an additional integer argument one may limit the number of iterations Finally if the integer argument for the selected eigenstate is replaced by the argument follow then that eigenfunction which has the largest overlap with the initial function will be taken as first SPF Example meigenf 3 oper follow full select write 125 In this example the maximum number of arguments is given See the HTML documenta tion for explanation and more information 7 9 Advanced topic Generating an initial wavepacket using an operator It is also possible to first generate an initial wavepacket and then to apply an operator to this wavepacket before starting the propagation This is required e g when the initial wave packet to be generated is the dipole operator acting on a ground state wavefunction To do this a wavepacket must be build or read in as described in this section above The operator must also be defined in the operator file using a HA
149. f using adproj it takes about 10s 3D up to Smin 6D for the same Hamiltonians and wavefunctions Note that in the potfit step of the second variant there are as usual less natpot terms than grid points See Example 11 3 The numbers in the NATPOT BASIS SECTION were chosen such that there was virtually no difference to the numerically exact adpop run 110 11 Analysing the results employing the Analyse programs RUN SECTION name projl genoper title pyrazine 3D projector end run section OPERATOR SECTION opname projectorl nd operator section SPF BASIS SECTION multi set vl0a tend v6a tO TO v1 6 5 nd spf basis section PRIMITIVE BASIS SECTION vl0a HO 20 0 0 1 0 1 0 v6a HO 30 0 0 1 0 1 0 v1 HO 20 0 0 1 0 1 0 el el 2 end primitive basis section end input Example 11 4 The input file proj1 inp for the metdh genoper run OP_DEFINE SECTION title projection operator for 3D pyrazine end title nd op_define section LABELS SECTION P11 natpot projector111 P12 natpot projector112 P22 natpot projector122 end labels section HAMILTONIAN SECTION_projectorl usediag modes el vlOa v6a v1 1 0 S1 amp 1 Pll 1 0 S2 amp 2 P22 1 0 S1 amp 2 P12 end hamiltonian section end operator Example 11 5 The operator file projection1 op for the metdh genoper run 11 9 Plotting 2D cuts through the sys
150. formance of the integration plall prints a list of all pl scripts but for more information see the HTML documentation Note that all pl scripts support the h option We do recomment the use of the pl scripts The program showsys84 is a powerful tool for plotting 1D and in particular 2D views on wavepackets and potentials To plot the potential one first has to generate a so called pes file To do so move up to the tutorial directory where nocl1 inp is located and type mctdh84 pes nocll This will generate the files pes log pes and op log pes in the nocll name directory The pes file is an operator file in which all terms containing derivative operators or CAPs are deleted The WARNING message which appears can be ignored It just tells you that the mcetdh program will not perform a propagation although there is a keyword propagation in the input file Now move back to the name directory and type showsys84 A menu appears see Example 11 6 which allows various options to be set Go to menu point 10 type 10 and change the plot task to 2 plot pes type 2 Next input a 1 three times and a 2D cut through the surface with theta fixed to 1 545 radians will pop up Now use menu point 20 change coordinate section i e chose another cut If one gives x and two numbers a 1D plot will appear After you have played around enough go back to menu point 20 and input x y 2 1 Then use menu point 5 You will be asked for a file name Cho
151. gated it again might become necessary to use a smaller than optimal order parameter at the price of a longer computation time If the ABM BS or RKz integrator is employed the last parameter to be specified is the initial stepsize in fs In the case of the BS integrator the output interval tout defined in the RUN SECTION is normally a good choice For the ABM integrator the initial stepsize should in general be by a few orders of magnitude smaller than the output interval This is because the ABM integrator although being a multi step method has to be started as a one step method i e with an order of two since initially the wavefunction is given for a single point of time only In case of the RKz integrators the initial stepsize can also be omitted or set to zero The integrator then tries to guess a suitable value for the initial stepsize by employing a single explicit Euler step and estimating the second derivative of the solution However in our experience this guess is often too conservative Whether the initial ABM BS or RKz stepsize was chosen reasonably can be decided with the aid of the steps file which is generated when the steps keyword in the RUN SECTION is set From this ASCII file it can be seen how large the first successful step actually was This value may then be used as initial stepsize in future calculations For the complex SIL method two different error estimates called standard and improved estimate are i
152. grams more convenient to use but do not extend their functionality The advanced topics also deal with options of the MCTDH package which are needed in special cases only You may skip these parts until you got more experienced with the MCTDH package Please keep in mind the following typographical conventions which are designed to help you reading the User s Guide 2 1 Introduction Typewriter The typewriter font is used for literal characters such as keywords and labels given in the input files the names of routines and variables and extracts of the source code Italics The italics font indicates arguments which are supposed to be substituted by the user Bold face Bold face emphasises the names of programs and scripts in the MCTDH Package and their options Sans serif The sans serif font is employed for files directories and paths UPPERCASE The different sections that arrange the input and operator files are given in uppercase SMALL CAPS Small capital letters are used for the names of persons as well as programs that are not part of the MCTDH package Chapter 2 An MCTDH tutorial When you have successfully installed the MCTDH package see Appendix E you have var ious programs in the field of multi dimensional quantum dynamics at your disposal Before we go into the details of how to use these programs we would like to invite you to a short tour of the MCTDH package by performing some exemplary calculations On th
153. grd f for further information Non adiabatic operators If the system contains more than one electronic state the Hamiltonian can be written in matrix form i e Ay Hi2 Ha Hz C 1 To input such a form the symbols in Table C 9 can be used Thus the operator n o r e i o t oo C 2 can be represented symbolically as modes X el T0 Bat 1 T0 h_2 S1 amp 2 1 0 h_3 21 amp 2 See also Sec 9 for more examples Table C 9 Matrix operator symbols used for an electronic degree of freedom Symbol Operator Sf amp i Symmetric matrix element Zf amp i Unsymmetric matrix element 1 Unit matrix Appendix D Structure of the WF array psi dgldim U psi adim psi totphidim asa ah A p zpsi 1 zetf 1 1 psi block s psi phidim m i _ OS ONT A ee so if pli zpsi s zetf m 1 psi subdim m dim m s pe zetf m s Figure D 1 The structure of the wave function 146 Appendix E Installing the MCTDH package The installation of the MCTDH package is is very easy if you install it on a PC with a not too old Linux system On more fancier machines one may need to edit the compile scripts to set compiler options appropriately and or install some open source software like gnuplot or a GNU make It is essential to have a bash shell version 3 x is recommended but lower versions may also work and it is highly recommended that yo
154. guration time dependent Hartree method for density operators J Chem Phys 112 2000 10718 10729 F Huarte Larrafiaga and U Manthe Full dimensional quantum calculations of the CHa H CH3 He2 reaction rate J Chem Phys 113 2000 5115 U Manthe and F Matzkies Rotational effects in the H2 OH H H20O reaction rate Full dimensional close coupling results J Chem Phys 113 2000 5725 F Matzkies and U Manthe Combined iterative diagonalization and statistical sampling in accurate reaction rate calculations Rotational effects in O HC1 OH Cl J Chem Phys 112 2000 130 H Wang Basis set approach to the quantum dissipative dynamics Application of the multiconfiguration time dependent Hartree method to the spin boson problem J Chem Phys 113 2000 9948 M C Heitz and H D Meyer Rotational and diffractive inelastic scattering of a diatom on a corrugated surface A multiconfiguration time dependent Hartree MCTDH study on N2 LiF 001 J Chem Phys 114 2001 1382 1392 G A Worth Quantum dynamics using pseudo particle trajectories A new approach based on the multi configuration time dependent Hartree method J Chem Phys 114 2001 1524 1532 S Sukiasyan and H D Meyer On the effect of initial rotation on reactivity A multi configuration time dependent Hartree MCTDH wave packet propagation study on the H D2 and D Hz reactive scattering systems J Phys Chem A 105 2001 2604 2611 F Gat
155. h eigenstate Block relaxation does not allow the use of the keywords arguments follow or lock only relaxation 0 is allowed Hence one computes the b lowest eigenstates where b denote the block size However one may set the keyword rlxemin see the HTML docu for a full description This will force the code to compute the b lowest states above the argument of rlxemin An initial wavefunction for a block improved relaxation run can be read by using the keywords block spf block A or if there is already a wavefunction in correct block form simply by using the file keyword The keyword autoblock is particularly useful See the HTML docu for a comprehensive description of these keywords Finally we note that the rlx_info file is most conveniently read with the aid of the script rdrlx Type rdrlx h for obtaining more information The script plrlx or plbrlx for block relaxation plots the energy versus relaxation time 3 6 Performing a numerically exact calculation The MCTDH program although originally developed for wavefunction dynamics within the MCTDH scheme also allows one to perform numerically exact wavefunction propa gations Employing the MCTDH program for such calculations has the advantage that one can benefit from the easy way a Hamiltonian can be set up A numerically exact calculation may also be useful for comparison with an MCTDH calculation Of course a numerically exact propagation is only feasible for rather small syste
156. he input file it is often more convenient to use options E g mctdh84 pes nocll will generate a pes file of the NOCI S1 surface and stores it in the name directory Now change to the directory where the pes file has been stored and start the showsys program The program is also able to generate two dimensional plots of the system density from the psi file If only pes plotting is required then start the program using the pes option showsys84 pes If both pes cuts and density plots are wanted this option should not be used A menu appears which allows the interactive generation of plots This is shown in Example 11 6 When the potential used is given by a natpot file generated by the potfit84 program see next Section then it is more convenient to use the showpot84 program to visualise the potential energy surface showpot84 is menu driven similar to showsys84 and it allows to plot 1D and 2D cuts of the original surface of the natural potential fit and of differences between them When the parameter natpot cut menu point 500 is larger than zero then all natural potential terms the supremums norm of which is smaller than natpot cut are removed A proper use of natpot cut may reduce the number of potential terms by a factor 1 2 or so with only marginally reducing the accuracy of the fit The parameter natpot cut is 11 10 Plotting cuts through the potential energy surfaces 113 also available in mctdh84 The number of omitted
157. he m third DOF are denoted j_ps j_ms jpms jp 2s jm 2s jpjms jmjps jpjzs jzjps jmjzs and jzjms Note that j and j are defined as j Jz ijy and j Jx ijy for both the SF and BF system Due to the anomalous commutation relation for the BF operators j p j Br decreases k by one whereas j_ps j sp increases m by one The operator cjpm is a 4D mode operator two successive KLegs It only works if the two KLegs are combined into one mode and for this case it replaces the use of natural potentials of the cpp cmm surfaces see Table C 5 Note that the operators j p j m jpm cjpm jp 2 jm 2 sJp sJm sJpk and sJmk as well as j 2 if the latter operates on a KLeg PLeg combined mode are 3D tensors in MCTDH and not matrices Hence care must be taken when multiplying these operators with other operators To give an example HAMILTONIAN SECTION 140 C The built in symbolic expressions Table C 3 One dimensional operators which require arguments The expression x stands for the coordinate p can be replaced by any parameter from the PARAMETER SECTION or any real number The exponent r can be any real number If r 1 it is not required e g q p and q p 1 are synonemous Symbol Operator q p r p qs p r Vp x2 sin p1 p2 r sin p1 x p2 cos p1 p2 r cos pl a p2 tan p1 p2 r tan pl a p2 exp p1 p2 r exp p1 a p2 Exp p1 p2 r
158. he previous examples you should study the log output timing etc files and in vestigate natural and grid populations In particular it is useful to investigate the spectrum Thus type plspec e 2534 52981 cm 1 200 7000 cm 1 The option e 2534 52981 cm 1 shifts the zero point of the energy scale by 2534 52981 cm which is the ground state energy Thus the ground state is now ex pected at zero Try the options g 0 g 1 and g 2 and you will understand why g 1 is the default The plot depicts the spectral lines having a width of almost 100 cm This demonstrates that a precise determination of eigen energies by Fourier transform of the au tocorrelation function is difficult See Fig 2 7 To continue with the tutorial move to the co2 directory and type filter84 co2ft This runs the filter diagonalisation and creates the files filter eig filter inp and filter log The file filter inp repeats the input file but additionally shows all default and computed param eter values The file filter log displays what filter84 has been doing It also contains a list of all computed eigenvalues and intensities The file filter eig again contains the computed 2 5 Determining the vibrational spectrum of COz2 by filter diagonalisation 13 350 i i 1 L L i i 300 250 M 200 M 150 4 M 100 M o Aen e a T pl ts LH Lh T 1000 2000 3000 4000 5000 6000 700C Energy cm 1
159. he rms of the difference of the two interpolations to the same grid is 9 55 x 1076 au i e 0 26 meV This is a very good value considering that the potential spans an energy interval of 8 eV Indeed using showpot84 to compare the outcome of both chnpot84 runs shows that the obtained potential energy surfaces are virtually equal which constitutes a nice example of the usefulness of the chnpot84 utility when preparing an MCTDH calculation from ab initio data points We have used this method quite succesfully to create multi dimensional 3D fits to ab initio data The only restriction is that the data is to be supplied on a product grid Equidistance of the grid points however is not required 2 9 Optimizing an external field with Optimal Control Theory OCT MCTDH can be used to perform coherent control calculations within the OCT scheme The OCT algorithm was developed by Tannor and coworkers 15 and by Rabitz and coworkers 16 For this purpose metdh and the routine efield are called from the script optentrl OCT maximizes the expectation value Y T O Y T at the final time T where O denotes some positive semidefinite hermitian operator The control target O can be defined in two different ways If O is the projector onto a target quantum state i e O Vtar Wtar then it is sufficient to specify the target state Wiar If O is a general operator e g a projector onto an electronic state O S S it has to be specified in th
160. her files necessary for a propagation will be created but no propagation step will be performed see below propagation 4 Propagation in real time see Sec 3 3 relaxation 4 A relaxation or an improved relaxation will be performed see Sec 3 4 or Sec 3 5 respectively continuation 4 A continuation of the run will be performed see Sec 3 9 diagonalisation 4 The Hamiltonian will be diagonalised see Sec 3 7 When a high level task is to be performed the necessary tasks of lower order are auto matically included E g by setting the keyword propagation one implicitly also sets gendvr genoper and geninwf 21 22 3 Defining the type of calculation to be made The pes files created with the keywords genpes or gengmat or the options pes or gmat are special operator files which include only potential like operators Note that all non diagonal i e kinetic energy terms are automatically removed from the Hamiltonian when setting the keyword genpes The pes file is usually used in the context of the analyse routines vminmax or showsys see Sec 11 10 The showsys program can plot 2D cuts through potential energy surfaces or other multidimensional functions provided in the form of a pes file The pes file created with the genpes keyword contains the potential energy surface of the system If the keyword gengmat I1 I2 or the option gmat I1 I2 is used then the pes file contains the 11 12 matrix el
161. here is a master user who installs the package and clients who only need to add the line source SMCTDH_DIR install MCTDH_ client to their bashrc The file MCTDH_ client is generated during installation Of course SMCTDH_DIR must be replaced by the full path of the MCTDH directory which for the present example reads home muser MCTDH mctdh84 x Alternatively the clients may sim ply copy the file MCTDH_client to their bashrc If the automatic detection of platform and compiler does not work one has to edit the platform cnf script Note that during installation platform cnf def is copied to platform cnf Hence one may whish to edit platform cnf def as well Around line 100 platform cnf reads SET MACHINE DEPENDENT OPTIONS system uname s system MYSYSTEM Incomment if automatic dection doesnt work case system in MYSYSTEM Here you may set the variables by hand MCTDH_PLATFORM Please set try uname p or uname m MCTDH_COMPILER Please set if not listed in compile cnf aH have to edit compile cnf as well Change this to e g you E Installing the MCTDH package 151 ET MACHINE DEPENDENT OPTIONS He ER wn system uname s system MYSYSTEM Incomment if automatic dection doesnt work case system in MYSYSTEM Here you may set the variables by hand MCTDH_PLATFORM cruncher MCTDH_COMPILER pg
162. his file contains the one dimensional 11 5 Watching the system s evolution 103 0 3 0 25 AN KIN H be WN eg Nee 0 15 I ee Mini Figure 11 2 The density along the dissociative degree of freedom as a function of time for the photo dissociation of NOCI densities output at intervals specified by the tout keyword This density can be plotted using the showd1d84 program in conjunction with GNUPLOT For example the NOCI photo dissociation calculation has been run using the example input file 4 1 The command showd1d84 a T fl requests that the density for the first degree of freedom as listed in the PRIMITIVE BASIS SECTION of the input file i e the dissociative mode rd is written to a file denid_f1 The options make this file into a GNUPLOT grid file complete with commands The option a automatic lets showd1d84 call GNUPLOT The 3D plot of density is shown in Fig 11 2 Try also the other format options A complete list of options is obtained trough the command showd1d84 h The similar program is showspf84 which displays the single particle functions uncom bined modes only of course Note that showd1d84 reads the gridpop file while showspf84 reads the psi file Finally there is showrst84 which plots the single particle functions of the restart file 104 11 Analysing the results employing the Analyse programs 11 6 Determining photo dissociation and photo absorption spec tra The spectrum of a
163. his information for every external entry Optionally a unit may be given to convert e g Angstroem to au or degree to radian As before the information concerning where the ab initio data points are found is given in the OPERATOR SECTION of the same input file After execution the program yields a natpot file and a dvr file One should note that these natpot and dvr files cannot be used directly in a simulation since only the grid points of the DVR are found on the dvr file and not the matrices representing the derivative operators However any natpot whose primitive grid is known through the corresponding dvr file can be interpolated to a desired DVR by the chnpot utility yielding new natpot and dvr files to be used in the simulation phase This is covered in the next subsection As a remark the external keyword is more powerful and the non local part of an arbitrary DVR can also be supplied see the HTML documentation for this advanced feature 12 3 2 Transforming between two natural potentials with chnpot The chnpot utility interpolates a given natpot file into a new primitive grid corresponding to the desired DVR It needs to know where to find the initial natpot and dvr files An example input file for chnpot reads 12 3 Extra flexibility combining potfit and chnpot 121 Grid Values ab initio data y Potfit MCTDH Coma Figure 12 1 Main concepts involved in the usage of ab initio data with the MCTDH p
164. i END PRIMITIVE BASIS SECTION The keyword wigner is used to select the L normalized Wigner small d functions as the basis functions for the G angle The number following the wigner keyword denotes the number of 7 values to be used and the al1 keyword denotes that both odd and even j values are to be included For the y and a angles the coordinate representation is declared by an exp line alternatively a K line can be used to declare the momentum representation In the example above the alpha DOF is used in coordinate representation while the gamma DOF is used in momentum representation The input line for the 8 angle must appear first and be immediately followed by a line for the y angle which must then be followed by a line for the a angle that is the order of the degrees of freedom must be given as J K M All three DOF must be declared as combined in the SPF BASIS SECTION Chapter 5 Defining the single particle basis In an MCTDH calculation propagation or relaxation the single particle basis has to be defined This is done in the SPF BASIS SECTION of the input file The SPF BASIS SECTION also enables one to treat degrees of freedom as a combined mode or to select only a subset of the system degrees of freedom 5 1 Specifying the number of single particle functions In the SPF BASIS SECTION the number n of single particle functions to be used for each degree of freedom x of the system are listed Supposed tha
165. ia tive adsorption of Hz on Cu 100 J Chem Phys 121 2004 3829 3835 List of MCTDH references 159 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 D J Haxton Z Zhang H D Meyer T N Rescigno and C W McCurdy Dynamics of dissociative attachment of electrons to water through the B metastable state of the anion Phys Rev A 69 2004 062714 T Wu H J Werner and U Manthe First principles theory for the H CH4 H2 CH reaction Science 306 2004 2227 2229 B Lasorne F Gatti E Baloitcha H D Meyer and M Desouter Lecomte Cumulative isomerization probability studied by various transition state wave packet methods including the mctdh algorithm bench mark HCN CNH J Chem Phys 121 2004 644 654 F Gatti and H D Meyer Intramolecular vibrational energy redistribution in Toluene A nine dimensional quantum mechanical study using the MCTDH algorithm Chem Phys 304 2004 3 15 M Petkovi and O K hn Ultrafast wave packet dynamics of an intramolecular hydrogen transfer system from vibrational motion to reaction control Chem Phys 304 2004 91 E V Gromov A B Trofimov N M Vitkovskaya H Koppel J Schirmer H D Meyer and L S Cederbaum Theoretical study of excitations in furan Spectra and molecular dynamics J Chem Phys
166. ial v V 2 in dependence of the coordinate vector x i 7 v and Z in a u subroutine newsurf x v real 8 x v end As a first step one has to provide a default routine in case the new surface is not linked I e one creates a file DEF_newsurf f on the directory source surfaces and writes to it e g see source surfaces for examples subroutine newsurf x v real 8 x v write 6 a a ttttt HEHEH HEHEH HEHEH Py newsurf is not linked Run compile with i option write 2 a a EHEF Het HH Het HH HEE HH newsurf is not linked Run compile with i option stop end Note that the file DEF_newsurf f may also need to include dummy routines for other subrou tines which are on the source surfaces newsurf f file In particular if the routine newsurf contains an entry point the dummy routine must have a corresponding entry point as well If the program compiles but does not link after having added a new surface it is likely that there is a mistake in the dummy routine The next step then is to ensure that these two file will be compiled by inserting the line SURFDF1 AR_SURFDEF PATH_SURFACES DEF_newsurf o into install user_surfdef Finally insert newsurf o S PATH SURFA
167. id A unit can only be appended to the end of an expression and it will modify the value of the full expression e A function name must be in upper case letters and the argument s of the function must be in square brackets No arithmetic is possible with the function Do the arithmetic with the parameter which was defined by the function e The parameters and their values are protocoled in the op log file One may check if the program has performed the parameter arithmetic in the way expected 6 3 Using symbolic expressions to define the Hamiltonian The MCTDH and Potfit programs are able to parse a variety of mathematical symbols such as powers exponential and trigonometric functions and first and second derivatives A com plete list of these built in symbols can be found in the HTML documentation A selection of these symbols is compiled in Tab 6 1 For a complete list please refer to App C The symbolic expressions are used in the HAMILTONIAN SECTION to define the Hamiltonian Within the MCTDH framework the Hamiltonian must in general be given in product form BSS GP ec he 6 1 r 1 where f and s denote the numbers of degrees of freedom and Hamiltonian terms respectively This structure is reflected in the HAMILTONIAN SECTION which is mainly a table with s lines and f 1 columns containing the coefficients c and single particle operators h To make things concrete let us explain the usage of the symbolic expressions with the
168. iming is created This file contains timing information for each MPI prozess An mpitiming file for a run with 8 prozesses is shown here Process cpu cpu 0 14501 84 12 40 al 15288 35 13 08 2 14399 39 12 32 34 3 Defining the type of calculation to be made 3 13243 32 1133 4 17475 32 14 95 5 13699 10 11 72 6 13507 77 T1235 7 14790 03 12 65 The first column denotes the MPI process 0 being the master process the second column shows the cpu time spent in each process and the third column gives the percentage of the cpu time The Example shows a well paralellized case where the cpu time is equally distributed over the processes If the MPI parallelization is combined with the usage of POSIX threads the fourth column appears that shows the sum of real time spent in the threads of each prozess This is similar to the summation done for the pt iming files but here the summation was additionally done over the different routines The following example is an MCTDH run with 2 MPI processes with 4 POSIX threads each denoted by 4 Process cpu cpu sum real time 4 0 51811 34 50 31 51574 81 1 51168 49 49 69 51095 79 If the sum of real time is comparable to the cpu time the shared memory parallelization works well for each prozess see the comments to pt iming file in sec 3 10 For combined calculations distributed and shared memory parallelization a further op tion for the pt iming keyword can be set ptiming all If
169. in one operator However when using multiple instances of the same externalld function do this by defining a label and referring to it in the operator rather than declaring externalld file with the same file file repeatedly as this wastes buffer memory due to duplication C The built in symbolic expressions 145 9 An arbitrary real 1D function may be defined through a set of points The points must coincide with the grid points The potential values on the grip points are read from file file one value per line The file may be in ascii or binary format binary is de fault Give ascii or binary as second argument after file For multi dimensional surfaces use readsrf 10 A real 1D function may be defined through a user written subroutine Edit the sub routine myld on SMCTDH_DIR source opfuncs funcld F 11 The flux operator 0 7 is set up in a sine or exponential basis and then trans formed to DVR representation This operator might be used with eigenf to pro duce flux eigenstates as initial wavefunctions To regularize the flux operator and to make its eigenfunctions more localized it is multiplied from right and left with cos np 2Pmax cosh tp 2pPmax The exponent power may be zero or any pos itive real number power 1 is recommended Special operators There is a number of operators especially defined for the methyl iodine CH3I system Their labels all start with MI See opfuncs ch3i f and opfuncs ch3i
170. ingle particle functions or only the A vector respectively Finally the extension fix enforces the use of a fixed stepsize To discriminate these CMF stepsizes from the integrator step sizes the former are often called update times 8 3 Description of the available integrators The integrators that are available are an Adams Bashforth Moulton ABM predictor corrector method with fixed order and variable stepsize a Bulirsch Stoer BS extrapola tion scheme with polynomial extrapolation and variable order and stepsize two Runge Kutta RK5 8 integrators with adaptive stepsize and fixed order 5 or 8 respectively and a short iterative Lanczos SIL algorithm with variable order and stepsize Note that the hermi tian SIL integrator is automatically replaced by a complex SIL integrator also known as Lanczos Arnoldi integrator if the Hamiltonian is complex More precisely the Lanczos Arnoldi routine is automatically chosen if there is a complex potential e g a CAP in one of the separable parts of the Hamiltonian However the Hamiltonian may be non hermitian for various other reasons In these case one has to replace the SIL keyword by CSIL which enforces the use of the Lanczos Arnoldi integrator Note that the integration will be incorrect if the hermitian SIL integrator is used for a non hermitian Hamiltonian Which integrator has been used is protocoled in the log file Each integrator is associated with up to three parameters In
171. initial functions An initial function frequently employed for angular modes is an associated Legendre function 21 1 l m d1 m 41 9 9 Fm Pi cos 0 7 3 with 0 lt m lt l The parameter m denotes the magnetic quantum number and is treated as a fixed parameter P is the unnormalised associated Legendre function 4 4 The following example defines the initial wavepacket of a system with two degrees of freedom rd and theta as product of a Gaussian function in rd and a Legendre polynomial in theta INIT_WF SECTION build rd gauss 4 50d0 8 76d0 0 18d0 theta Leg 0 0 sym end build end init_wf section 7 3 Setting up extended Legendre functions as initial functions 73 The numbers after the keyword Leg denote m and l respectively If the corresponding type of the primitive basis is not Leg then m must be zero if it is Leg then m must coincide with the value in the PRIMITIVE BASIS SECTION In an MCTDH calculation the program uses the Legendre polynomial specified in the INIT_WF SECTION to define not only the initial wavepacket but also the first single particle function Which higher single particle functions are used depends on the last parameter which can be sym or nosym In the latter case all values of l both even and odd are employed in the former case only those Legendre functions having the same symmetry as either even or odd are taken 7 3 Setting up extended Legendre functions a
172. instance the modes r1 and r4 together and treats them as a single coordinate in the calculation The single particle functions for this coordinate are then two dimensional functions in the system coordinates The number of two dimensional single particle functions is 15 for the combined r1 r 4 mode In the example given eight modes have been combined together to produce three modes for the MCTDH calculation As the length of the expansion coefficient vector grows ex ponentially with the number of modes included in a calculation this drastically reduces the computational resources required 5 3 Combining modes to produce multi dimensional single particle functions 47 It is important to choose a good combination scheme One should combine those DOF which are most strongly coupled with each other Then these correlations are taken care of at the SPF level and the number of SPFs necessary for convergence is decreased Some times physical intuition and knowledge of the system helps to identify the strongly coupled DOFs but most of the time this information is missing Then one takes a more practical approach Firstly one combines those DOFs which have similar frequencies because modes with very different frequencies are less likely to couple strongly Secondly one should set up a combination scheme such that the combined grids have similar size One must neither over nor under combine The A vector length should be larger than
173. ion run where a previous wavepacket propagation is carried on to longer times all calculations require an input file name inp This file is a text file with the required options input as keywords As all lines beginning with a are treated as comments title and other text can be usefully added to make the file easier to understand Example input A 1 shows the input file required for a simple wavepacket propagation calculation using a modified Henon Heiles Hamiltonian see Ref 2 for more details of this calculation As the example shows it is possible to include the information of the operator file in the input file This is particularly useful for systems having a simple Hamiltonian The keywords in the input file are grouped together into sections each with a specific set of information The sections start with a line containing the keyword XXX SECTION and end with END XXX SECTION where XXX is the name of the section The possible sections are compiled in Tab A 1 Table A 1 The possible sections in the input file Also displayed is whether a section is required for a certain calculation type Section Calculation Type gendvr genoper geninwf propagation diagonalisation genpes relaxation RUN yes yes yes yes yes PRIMITIVE BASIS yes yes yes yes yes PARAMETER yes yes yes yes yes SPF BASIS no yes yes yes yes OPERATOR no yes yes yes yes OP_DEFINE no yes yes yes yes HAMILTONIAN no yes yes yes yes LABELS no yes yes
174. is data cat results and plot a stick spectrum plfdspec c a 200 results The content of the file results is compiled in Tab 2 1 columns 3 6 and compared with experimental data column 2 To obtain all eigenvalues in this energy range one has to run the propagation with different initial states either sequentially or more efficient in parallel by performing a multi packet propagation Also increasing the accuracy more SPFs e g 16 16 14 and increasing the propagation time e g 250 fs will help to detect states of low intensity See Ref 13 for details 14 2 An MCTODH tutorial Table 2 1 Vibrational energies J 0 of CO2 The MCTDH FD energies EF p are compared with experi mental ones Fexzp AE and AJ denote internal error estimates of the eigen energies and intensities respectively Missing entries specify states that have not been detected In this case the intensity is taken from a larger cal culation and is shown in brackets The missed states are all of very low intensity except for state 26 Here the computed state represents the two neighbouring states 25 and 26 A calculation with a longer propagation time or with several wavepackets will detect more states See Ref 13 All energies are given in cm with respect to the ground state energy No Eezp Erp AE Intensity AI 0 0 000 0 002 0 054 4 49d 2 3 32d 5 1 1285 414 1285 393 0 096 1 07E 1 3 05E 5 2 1388 188 1388 276 0 264 3 48E 2 7 77E 5 3 2349 14
175. is section INTEGRATOR SECTION CMF varphi 0 2 1 0d 2 RK8 spf 1 0d 8 0 1 RRDAV A 200 1 0d 9 end integrator section INIT_WF SECTION file p3d1_19 symcoeff end init_wf section end input Example 10 2 An input file for N 3 one dimensional bosons in a harmonic trap Chapter 11 Analysing the results employing the Analyse programs The set of Analyse programs can be used to analyse the information from a calculation which is stored in the various data files For a complete list of programs see the HTML documentation If a program is started using the h option i e analyse84 h where analyse is the name of the program e g rdgpop a brief description of how to use the program and a list of options will appear The programs are designed to be used together with the GNUPLOT program In many cases the option g will produce a file complete with GNUPLOT commands ready for immediate plotting Some programs also support interactive plotting in conjunction with GNUPLOT In these cases starting the program with analyse84 inter brings up a menu with options to choose what is to be plot to change plotting boundaries etc Here a brief overview of only the most important programs will be given As an example we take the results from the wavepacket propagation of the NOCI system First the system is relaxed on the Sg surface using the input file inputs nocl0 inp Propagation is then made using the file i
176. is trip you will get an overview of the opportunities the MCTDH package offers The tour shall also demonstrate the ease of employing the program and give you an impression of the efficiency of the code First set up and move to a suitable directory in which to run the tutorial calculations e g MCTDH_DIR tutorial or SHOME tutorial then follow the instructions below The tutorial uses standard problems Once a calculation has been made try to understand the input files they can be used as templates for other calculations The expression MCTDH_DIR occurring in the following examples stands for the path of the MCTDH directory 2 1 Determining the absorption spectrum for the photodissocia tion of NOCI The photodissociation of NOCI is a simple photo chemical reaction After excitation from the ground to the first excited state S S4 the chlorine atom dissociates on a femto second time scale This results in a broad band for the absorption spectrum This system was used for the first application of MCTDH to a realistic system 3 The calculation consists of two stages First the ground state wavefunction is generated by energy relaxation of an initial guess wavefunction on the ground state surface So The second stage then places this wavepacket on the excited state surface S leading to pho todissociation 1 Copy the files MCTDH_DIR inputs nocl0 inp and MCTDH_DIR inputs nocl1 inp to your tutorial directory and create there th
177. jtot J dim d v NIAE a aa Table C 7 Some general multi dimensional operators Here parameters are to be given in curly brackets E g coulomb1d a 1 3 b 2 0 c 0 0 d 1 5 See also Hamiltonian Documentation Available Surfaces Symbol Operator readsrf file F Potential values on grid points are read from file file of format F F ascii or binary See HTML Docu For 1D potential use read1d gaussld width w S exp 0 5 1 x2 w V27 w If the optional string S is set to periodic then a 2 m periodic grid is assumed In this case the DVR lines should read e g xl FFT 128 2pi x2 FFT 128 2pi gauss2d width w exp 0 5 a1 2 w y1 ya w 27 w coulomb1d 1 az bz c d C The built in symbolic expressions 143 Table C 8 One dimensional operators for treating symmetric double well potentials by mapping each side on an artificial single set electronic state Note the grid must be a sin DVR which when doubled lies symmetrically to zero but does not contain zero The differential operators which are truncated are firstly defined on this doubled grid but then projected to the working grid Symbol Operator Notes Rf Rf y 2 y n 41 1 Reflection operator Must not be multiplied with other operators Rfm Rfm zi y a n41 i Reflection operator Must not be multiplied with other operators hKEh step KE step Truncat
178. k for instance at the last line of the log file by typing tail 1 results log where results has to be substituted by the name directory Any error messages that might be raised during run time are sent directly to the screen and additionally written to the log file See there in case of problems If the calculation was successful the results can be analysed by the Analyse programs and scripts This is detailed in Chap 11 3 9 Continuing or stopping a calculation 29 3 9 Advanced topic Continuing or stopping a calculation A calculation that has been finished may be continued again so propagating to longer times or performing a larger number of Lanczos iterations This is done by adding the keyword continuation to the RUN SECTION Alternatively one may start a contin uation run using the c option on the command line see the HTML documentation for details In a propagation run the final time must also be increased For instance if the previous calculation finished after 50 fs then you may set tfinal 75 0 to propagate over the next 25fs Similarly if 10000 iterations were made in a diagonalisation run set diagonalisation 15000 for the next 5000 Again it is often more convenient to use options E g the command mctdh84 c tfinal 75 0 lt inputfile gt will start a continuation run where the final time is set to 75 fs The use of other options like I tcpu or t stop may be useful Type metdh84 h to see the list of o
179. keyword sdq is given DVR repectively In these cases the operator dq cannot be used C The built in symbolic expressions 141 Table 3 continued charfun p1 p2 regcoul p1 p2 switch1 p1 p2 switch2 p1 p2 low m w s rai m w s num m w s lwmorse m w readld file F myld p1 p2 p3 flux xe power pgauss o xo Symbol Operator step p O a p Step function See Note 4 rstep p O p x Reverse step function See Note 4 characteristic function if x p1 p2 then charfun 1 else it is zero regularized coulomb function 1 4 x pl p2 0 5 x 1 tanh pl x p2 0 5 x 1 tanh pl a p2 lowering operator See Note 5 raising operator See Note 5 number operator See Note 5 ramorse m w A a zo raising operator for Morse potential See Note 6 A a zo lowering operator for Morse potential See Note 6 cspot J K csmax m centrifugal potential See Note 7 external ld file external 1D function read from file file See Note 8 external 1D function read from file file of format F See Note 9 p4 p5 user supplied routine See Note 10 Flux operator for Cartesian kinetic energy e location of dividing surface power exponent of smoothing function See Note 11 Projector G gt lt G where G denotes a L normalized Gauss G 210 Y4 exp a zo 40 shift Q simple shift on the grid by Q aj Y zi o Yle O0ifi Q
180. kpl km1 kp2 km2 natpot fitK0 natpot fitK1 natpot fitK2 shift 1 shift 1 shift 2 shift 2 end labels section Example 12 4 An operator file showing the use a Fourier transformed potential excerpt Only Fourier compo nents up to Q 2 are included or even more formally on the operator level Vi V gt Vases iL SE 12 15 Q 1 This form can be easily translated into an MCTDH operator file Of course one will truncate the series at a certain Qmax Parts of the corresponding HAMILTONIAN and LABELS SECTIONS are shown in example 12 4 A few explanations are in order e During the following MCTDH propagation we will be using the two dimensional KLeg DVR for the combined coordinates 01 k1 and 62 k2 The use of this DVR is part of the technical reason why we did the Fourier transform in the first place This makes it necessary to order the coordinates differently than in the above theoretical considerations Namely ko can not directly follow k but the k must be paired with the 6 This different coordinate ordering makes it necessary to use the slightly cumbersome 1 amp 2 amp 3 amp 4 amp 6 syntax for the potfitted Fourier components VO V1 V2 This tells the program to apply them to the set of DOFs 1 2 3 4 6 which corresponds to R 71 r2 01 92 Also see section 6 13 Here we assume that the natpot files of the potfitted Fourier components are located in the directorie
181. l Henon Heiles potentials with the Multi Configuration Time Dependent Hartree MCTDH Method J Chem Phys 117 2002 10499 10505 J Trin M Monnerville B Pouilly and H D Meyer Photodissociation of the ArHBr complex inves tigated with the Multi Configuration Time Dependent Hartree MCTDH approach J Chem Phys 118 2003 600 609 G Worth and I Burghardt Full quantum mechanical molecular dynamics using Gaussian wavepackets Chem Phys Lett 368 2003 502 508 C McCurdy W A Isaacs H D Meyer and T Rescigno Resonant vibrational excitation of CO2 by electron impact Nuclear dynamics on the coupled components of the 7II resonance Phys Rev A 67 2003 042708 1 19 F Huarte Larrafiaga and U Manthe Quantum mechanical calculation of the OH HCl gt H20 Cl reaction rate Full dimensional accurate centrifugal sudden and J shifting results J Chem Phys 118 2003 8261 M Nest and H D Meyer Dissipative quantum dynamics of anharmonic oscillators with the Multi Configuration Time Dependent Hartree MCTDH Method J Chem Phys 119 2003 24 H Wang and M Thoss Multilayer formulation of the multiconfiguration time dependent Hartree theory J Chem Phys 119 2003 1289 1299 H Wang and M Thoss Theoretical study of ultrafast photoinduced electron transfer processes in mixed valence systems J Phys Chem A 107 2003 2126 2136 D Egorova M Thoss W Domcke and H Wang Modeling of u
182. le relaxations we took excellent starting vectors namely the eigenstates obtained by the first block relaxation and that the second block relaxation yield eigenenergies of better accuracy than the sing 1 calculations one may conclude that single and block relaxation take similar amounts of CPU time for obtaining similar accuracy But the memory consumption of the block relaxation is consid erably larger 20 MB compared to 560 MB However it requires much less human effort to run a block relaxation as compared to run 40 single relaxations 18 2 An MCTODH tutorial 2 8 Using potfit and chnpot to fit a surface to ab initio data points In this example it is shown how to use ab initio data points to generate a natural potential and how this natural potential can then be interpolated into a more suitable grid for a MCTDH simulation To perform such tasks the programs potfit84 and chnpot84 will be used respec tively A detailed description of how such tasks are accomplished is found in chapter 12 2 of this guide The data used in this example taken from Ref 14 is a 2D cut corresponding to the PES of the CO anion in C2 symmetry The two coordinates are the length of the two CO bonds and the angle between them i e symmetric stretch and bending 2 8 1 Transforming the ab initio data to product form 1 Create a new directory for example co2fit 2 From the directory MCTDH_DIR pinputs copy to the directory co2fit the files e co
183. ler steps The interaction picture may allow the equations of motion to be integrated more efficiently than the standard VMF scheme This is especially true if an operator has a large separable part It is however usually less efficient than the CMF scheme The type of orbitals to be used is selected in the INTEGRATOR SECTION of the input file since the orbital type affects the form of the equations of motion to be integrated Place the keyword natorb or interpic into the INTEGRATOR SECTION i e INTEGRATOR SECTION natorb end integrator section or INTEGRATOR SECTION interpic end integrator section in order to move from standard to natural or interaction picture orbitals respectively 8 4 2 Suitable integrator settings for improved relaxation The integrator settings for improved relaxation are somewhat different from those for prop agation Improved relaxation requires a CMF fix or CMF varphi integration scheme The 8 4 Fine tuning the integration 85 best is simply to use CMF this defaults to CMF var for propagation runs and to CMF varphi for improved relaxation Improved relaxation furthermore requires a Davidson integrator actually a diagonalizer i e the keyword DAV rDAV rrDAV or cDAV A typical setting might read INTEGRATOR SECTION CMF 1 0 3 0d 3 RK8 spf 1 0d 9 rrDAV A 200 1 0d 8 natorb eps_inv 1 0d 10 end integrator section Note that the CMF accuracy is rather low wher
184. ll display the differences between your code and the one on the repository You may pipe this output to less If you have already checked out a previous version and want to merge with a newer one type e g svnm merge SSVNM mctdh84 releases 8 4 8 SSVNM mctdh84 releases 8 4 9 This command merges the differences between release 8 4 8 and 8 4 9 to your mctdh directory which must be the current directory Here we are assuming that you are working with release 8 4 8 and are updating to 8 4 9 Moreover rater than downloading a release one may download the current developers code svnm checkout SSVNM mctdh84 trunk mctdh84 dev This makes life easier as one can simply run svnm update to merge with the most recent changes For this the current directory must be mctdh84 dev However this way is recommended only for experienced users as the current developers code may not be bug free To be on the safe side one may run the command svnm cat SSVNM mctdh84 trunk changelog less and then update to an appropriate revision by setting the option r lt number gt Finally if one is interested in the branches 8 3 or 8 5 rather than 8 4 one simply replaces the version numbers accordingly List of MCTDH references 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 M H Beck A J ckle G A Worth and H D Meyer The multi configuration time dependent Hartree MCT
185. ltrafast electron transfer processes Validity of multilevel Redfield theory J Chem Phys 119 2003 2761 M Petkovi and O K hn Multidimensional hydrogen bond dynamics in Salicylaldimine Coherent nuclear wave packet motion versus intramolecular vibrational energy redistribution J Phys Chem A 107 2003 8458 8466 M Thoss W Domcke and H Wang Theoretical study of vibrational wave packet dynamics in electron transfer systems Chem Phys 296 2004 217 229 G Worth H D Meyer and L Cederbaum Multidimensional dynamics involving a conical intersec tion Wavepacket calculations using the MCTDH method In Conical intersections Electronic structure dynamics and spectroscopy W Domcke D Yarkony and H Koppel Eds World Scientific Singapore 2004 pp 583 617 F Richter M Hochlaf P Rosmus F Gatti and H D Meyer A study of mode selective trans cis isomerisation in HONO using ab initio methodology J Chem Phys 120 2004 1306 1317 F Richter P Rosmus F Gatti and H D Meyer Time dependent wavepacket study on trans cis iso merisation of HONO J Chem Phys 120 2004 6072 6084 C Iung F Gatti and H D Meyer Intramolecular vibrational energy redistribution in the highly excited Fluoroform molecule A quantum mechanical study using the MCTDH algorithm J Chem Phys 120 2004 6992 6998 R van Harrevelt and U Manthe Multiconfigurational time dependent Hartree calculations for dissoc
186. lue The initial wavefunction is chosen arbitrarily the intensities thus have no physical meaning To keep the CPU time short only a small number of Lanczos iterations will be made The number of iterations is sufficient to converge the lowest 0 5 eV of the spectrum 2 4 Determining the vibrational spectrum of LiCN 11 0 05 1 i i i 0 045 H 0 04 4 L 0 035 0 025 L Intensity 0 02 4 L 0 015 4 L 0 im i 6 2 6 6 6 5 6 4 6 3 Energy 6 1 Figure 2 6 The vibrational spectrum of LiCN 1 The LiCN surface is not linked by default It must be first linked to the program by re compiling MCTDH compile i licn mctdh It might be that you need to copy licnsrf f from the addsurf directory to source surfaces Alternatively you may set a link run the script mklinks Type mctdh84 ver to inspect which surfaces are included See the HTML documen tation Installation and Compilation Compiling the Programs and Hamiltonian Docu mentation Available Surfaces for more details 2 Copy the file MCTDH_DIR inputs licn inp and create the directory licn 3 To diagonalise the Hamiltonian type mctdh84 licn This will take less than 20 seconds The calculation can now be analysed Move to the directory licn which contains all the data files from the diagonalisation The eigenvalues intensities and error estimates for the energies are stored in the ASCI
187. luoroform using the multiconfiguration time dependent Hartree method J Chem Phys 129 2008 224109 H D Meyer Studying molecular quantum dynamics with the multiconfiguration time dependent Hartree method Wiley Interdisciplinary Reviews Computational Molecular Science 2 2012 351 U V Riss and H D Meyer Investigation on the reflection and transmission properties of complex absorbing potentials J Chem Phys 105 1996 1409 M H Beck and H D Meyer An efficient and robust integration scheme for the equations of motion of the multiconfiguration time dependent Hartree MCTDH method Z Phys D 42 1997 113 129 S Z llner H D Meyer and P Schmelcher Ultracold few boson systems in a double well trap Phys Rev A 74 2006 053612 A J ckle and H D Meyer Product representation of potential energy surfaces J Chem Phys 104 1996 7974 A Jackle and H D Meyer Product representation of potential energy surfaces II J Chem Phys 109 1998 3772 S Sukiasyan Investigation of three and four atomic reactive scattering problems with the help of the multiconfiguration time dependent Hartree method PhD thesis Universitat Heidelberg 2005 F Gatti F Otto S Sukiasyan and H D Meyer Rotational excitation cross sections of para Hz para He collisions A full dimensional wave packet propagation study using an exact form of the kinetic energy J Chem Phys 123 2005 174311 U Manthe Mehr
188. maximum FWHM of the energy filters which are the Fourier transforms of the time filters The entries in Table 11 1 are to be divided by the length of the autocorrelation function to yield the FWHM Note that the filters gj and g which are Go al 2 vy GH GH unit 2 49 3 38 4 14 4 78 3 66 4 91 eV fs 20 1 27 3 33 2 38 6 29 5 39 6 cm ps Table 11 1 FWHM values of the window functions g times the length of the autocorrelation function Remember that the length of the autocorrelation function is twice the propagation time if the t 2 trick is used called g4 and g5 in plspec are non negative The energy filters may be inspected by running plspec gn r t 100 0 1 0 1 ev withn 0 1 5 To introduce an additional damping i e a Lorentzian or Gaussian broadening of the spectrum the autocorrelation function may be further multiplied with exp t 7 where T and 1 2 are the two last arguments of autospec84 The exponential is ignored if 7 0 For plspec this is the default Finally if the option EP is set the Fourier transform is multiplied with the photon energy w to arrive at an absorption spectrum This multiplication in general requires to shift the spectrum option e by the ground state or initial state energy The multiplication with w is omitted if the option FT is given this is the default Note that only in this case one may use an energy interval con
189. meters are used to concretise the CMF calcula tion The first one is the initial stepsize in fs A good guess for the initial stepsize is to use the output interval tout specified in the RUN SECTION Whether the initial stepsize was chosen reasonably can be checked by looking at the update file which will be generated in a calculation if the update keyword in the RUN SECTION is set The update file indicates whether repetition steps were necessary in the beginning of the propagation If so one should use a smaller initial stepsize in the following calculations The second parameter defines the CMF error tolerance which controls the stepsizes dur ing the propagation Typical values lie between 1074 very low accuracy and 1078 very high accuracy For many applications an error tolerance of 10 or 1076 will be sufficient The convergence of a calculation with respect to the CMF error can be checked by comparing the results of two calculations performed with different error tolerances We finally note that it is possible to perform a CMF calculation with fixed or variable step sizes To choose among the possible options use the keywords CMF var CMF varphi CMF vara or CMF fix respectively With the extension var the stepsize becomes vari able and is controlled by both the single particle functions and the A vector As var is default CMF var is identical to CMF Using the extension varphi or vara the stepsizes are controlled only by the s
190. mic mass units a CAP length of 0 6 a u a CAP strength of 0 3 a u and for the energy interval 0 1 2 0 eV A GNUPLOT window pops up which displays the reflection and the absorption probabilities and their sum plcap then prompts you for new options Type h to see the list of options or type z 1 e 7 to arrive at a more convenient scale If one inputs e the program will compute the optimal CAP strength for the given energy interval The distribution of kinetic energies of dissociating NOCI lies between 0 2 eV and 1 6 eV thus the reflection and absorption probabilities are below 1074 which is a fairly good value The precise values of the CAP parameters are usually not very critical except when very low kinetic energies are present Very low kinetic energies may appear when an internal excitation takes almost all of the available energy The low kinetic energy contributions are more strongly reflected from the CAP and these reflections lead to artificial oscillatory structures in e g the reaction probability Fig 3 of Ref 10 shows such an small artificial structure near 1 25 eV i e close to the v 2 threshold As discussed in Ref 10 this small artificial structure disappears when using a longer and weaker CAP 64 6 Setting up the Hamiltonian Parameter file for the Henon Heiles system lambda 0 4 nd parameter fil Example 6 3 A parameter file for the Henon Heiles Hamiltonian 6 11 Advanced topic Alte
191. mplemented which can be specified by the third parameter The standard error 84 8 Choosing an integration scheme criterion is based on the product of the sub diagonal elements of the Lanczos matrix The improved one uses the norm of the difference between the wave functions propagated with two consecutive orders The improved estimate requires slightly more computation time but is more reliable when the stepsize is large For details we refer the reader to Refs 1 26 The estimates can be activated by the keywords st andard or novel respectively The former is the default 8 4 Fine tuning the Equations of Motion and the Integration Scheme There are a number of keywords that can be added to the INTEGRATOR SECTION that change the form of the equations of motion or change the way the integration is performed Examples of these are given in this section 8 4 1 Advanced topic Propagating in natural or interaction picture orbitals Instead of the standard single particle functions one may employ natural or interaction picture orbitals Natural orbitals are those single particle functions that diagonalise the MCTDH density matrices Interaction picture orbitals are obtained by moving from the Schr dinger to the interaction picture For details see Secs 3 3 and 3 4 of the review 1 In normal use natural orbitals have no advantages over normal single particle functions they span the same space and may even force the integrator to take smal
192. ms For a numerically exact wavepacket propagation an operator file and the same input sec tions as in an MCTDH calculation are required A numerically exact calculation can be made by including the keyword exact in the RUN SECTION in addition to the calculation type keyword e g propagation or relaxation Rather than using the low dimensional MCTDH single particle functions this sets up a wavepacket or for a non adiabatic system one for each electronic state on the full product primitive basis The operator is also set up on this full grid To propagate this wavepacket any of the integrators listed in Tab 8 1 can be used Al though the ABM integrator is the default the best performance is typically obtained by the SIL integrator because it exploits the fact that the equations of motion in a numerically exact calculation are linear To select the SIL integrator insert for instance INTEGRATOR SECTION SIL 20 1 0d 6 end integrator section into your input file The parameters are discussed in Sec 8 3 The ABM BS or RK5 8 integrator can be chosen as described in the examples in Sec 8 28 3 Defining the type of calculation to be made 3 7 Diagonalising the Hamiltonian using the Lanczos algorithm Although the MCTDH method is a time dependent one the MCTDH program is also capable of diagonalising a Hamiltonian using the Lanczos scheme In such a diagonalisation run the wavefunction is automatically represented on a primi
193. n MCTDH assume the simple volume element dg Because of the non trivial volume element Og is not an anti hermitian operator only Og sinf i e dthl is Note that sinf Og Og sin cos dth1 cos The operators j p j m jpm jp 2 and jm 2 are KLeg or PLeg 2D mode operators Simi larly j 2 becomes a 2D mode operator when operating on a KLeg or PLeg mode Similarly the operators sJp sJm sJpk and sJmk are also 2D KLeg PLeg operators where J denotes the total angular momentum The 1D operators jz jz 2 operate on the k dof of the KLeg mode only The 2D mode operator j p performs a similar multiplicative and shift operation on k but additionally per forms a derivative and k dependent multiplicative operation on the 6 dof of the KLeg mode Similar operations are done by the j_m sJp sJm sJpk and sJmk operators When applied to Wigner functions the operators j 2 j p jm jpm jp 2 jm 2 jpjm jmjp jpjz jzjp jmjz and jzjm are 3D mode operators and are represented as 4D tensors in MCTODH so care must be taken when multiplying these operators with other operators The Wigner operators j p j m jpm jp 2 jm 2 jpjm jmjp jpjz jzjp jmjz and jzjm operate in the BODY fixed axis system that is these operators perform multiplicative operations and shifts depending on the k second degree of freedom in the combined 3D mode The corresponding SPACE fixed operators which perform multiplications and shifts depending on t
194. n Run Section and Output Documentation Data files More recently Aug 2003 a Davidson diagonaliser has been implemented to replace the SIL for improved relaxation The Davidson algorithm is much superior to Lanczos when the generation of a single excited eigenstate is asked for For the Davidson there is a new keyword relaxation lock This is similar to relaxat ion fo1llow but more stable because lock searches for the eigenvector with the largest overlap with the initial wavefunc tion whereas follow takes the eigenvector with the largest overlap with previous vector The keyword ortho is not allowed when using the Davidson and fu11 is useful in special cases only See the HTML documentation for more details A typical input for improved relaxation with Davidson is provided by inputs hono dav inp Note that in this example one is computing the about 120th eigenstate in A symmetry There are several versions of the Davidson diagonaliser implemented With the keyword DAV one calls a routine which works for hermitian Hamiltonians With the aid of the keyword 26 3 Defining the type of calculation to be made cDAV one may diagonalise non hermitian e g CAP augmented Hamiltonians and compute complex resonance energies However in most applications the Hamiltonian will not only be hermitian it will also be real A Hamiltonian is called real if HW is real for every real W As MCTDH is written to propagate wave packets almost all variables
195. n giant dipole resonances of atoms in crossed electric and magnetic fields Eur Phys Lett 71 2005 373 379 K Giese and O Kiihn The all Cartesian reaction plane Hamiltonian Formulation and application to the H atom transfer in tropolone J Chem Phys 123 2005 054315 R van Harrevelt and U Manthe Multidimensional time dependent discrete variable representations in multiconfiguration hartree calculations J Chem Phys 123 2005 064106 S Z llner H D Meyer and P Schmelcher N electron giant dipole states in crossed electric and magnetic fields Phys Rev A 72 2005 033416 A Markmann G Worth S Mahapatra H D Meyer H Koppel and L Cederbaum Simulation of a complex spectrum Interplay of five electronic states and 21 vibrational degrees of freedom in C5H7 J Chem Phys 123 2005 204310 C Crespos H D Meyer R C Mowrey and G J Kroes Multiconfiguration time dependent Hartree method applied to molecular dissociation on surfaces H2 Pt 111 J Chem Phys 124 2006 074706 G Pasin F Gatti C Jung and H D Meyer Theoretical investigation of Intramolecular Vibrational Energy Redistribution in highly excited HFCO J Chem Phys 124 2006 194304 D V Tsivlin H D Meyer and V May Vibrational excitations in a helical polypeptides Multiexiton self trapping and related infrared transient absorption J Chem Phys 124 2006 134907 S Z llner H D Meyer and P Schmelcher Correla
196. n real a number appears in the Parameter Section it should include a dot E g one should use 2 0 ratherthan 2 Numbers in exponential format e g 1 0d 2 should be avoided if the exponent is not large Use 0 01 in this case The letter indicating the exponent must be a lower case d a D e or E will not work One may perform simple arithmetic with the parameters see Section 6 2 In particular one may exponentiate a parameter The exponent however must be a number and cannot be another parameter Use the function EXP for exponentiation If the exponent is integer write it as integer e g write par 3 ratherthanpar 3 0 Note that the exponent can be real and also negative e g par 0 5 is possible The use of brackets is not allowed Rather than writing cent j j 1 2 mass one has to write cent 0 5 j 2 mass 0 5 j mass The string which is used to specify a parameter may consist of upper or lower case letters numbers and the special characters Note in particular that the colon is not allowed to be part of a parameter name It is recommended to choose names which start with a letter Note that there is a pre defined parameter PI with obvious meaning There is a special parameter called mass_modelabel where modelabel is a label which was assigned to one of the degrees of freedom in the Primitive Basis Section This special parameter should be set to the reduced mass of the indicated degree of freedom This parameter is used
197. n the SPF BASIS SECTION The number of grid points of the exponential DVR must be odd The 11 points chosen here allow to represent K in the interval 5 lt K lt 5 4 9 Advanced topic Three Dimensional rotational DVR Wigner Wigner DVR uses the L normalized Wigner D functions Di a B 7 as a basis set These are defined as OF EL 2g Dm kla 8 7 8r2 Dh klb 4 8 D lap e da e 4 9 where a and y are the Euler angles representing rotation around the space fixed SF and body fixed BF z axes respectively and is the Euler angle between the SF and BF z axes where the z y z right handed axis convention is used The Wigner small d function ark 8 is defined as 3 j m ea k 4 10 Wigner DVR defines a three dimensional representation which can be used to model rota tion of polyatomic molecules An FBR DVR transform is used to convert between the angular momentum index j and the grid points 8 The m a and k yi momentum coordinate conjugate pairs can be used in either momentum or coordinate representation in MCTDH in the latter case a discrete Fourier transform switches between grid points and momentum indices A PRIMITIVE BASIS SECTION input block defining Wigner DVR uses three input lines one for each Euler angle For example 44 4 Selecting a DVR FBR representation for the primitive basis PRIMITIVE BASIS SECTION Mode DVR N beta wigner 20 all gamma k 7 7 alpha exp 15 2p
198. namics simulations of the H20 H OH2 cluster Phys Chem Chem Phys 10 2008 4692 4703 S Faraji H D Meyer and H Koppel Multistate vibronic interactions in difluorobenzene radical cations II Quantum dynamical simulations J Chem Phys 129 2008 074311 List of MCTDH references 161 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 J M Bowman T Carrington Jr and H D Meyer Variational quantum approaches for computing vibrational energies of polyatomic molecules Mol Phys 106 2008 2145 2182 G Pasin C Iung F Gatti F Richter C L onard and H D Meyer Theoretical investigation of in tramolecular vibrational energy redistribution in HFCO and DFCO induced by an external field J Chem Phys 129 2008 144304 U Manthe The state averaged multi configurational time dependent Hartree approach vibrational state and reaction rate calculations J Chem Phys 128 2008 064108 U Manthe A multilayer multiconfigurational time dependent Hartree approach for quantum dynamics on general potential energy surfaces J Chem Phys 128 2008 164116 M Eroms O Vendrell M Jungen H D Meyer and L S Cederbaum Nuclear dynamics during the resonant Auger decay of water molecules J Chem Phys 130 2009 154307 A U J Lode A I Streltsov O E Alon H D Meyer and L S
199. nce with the OPERATOR SECTION OPERATOR SECTION oppath usr people mctdh operators opname nocll nd operator section the MCTDH or Potfit program would look in the directory usr people mctdh operators for the operator file nocl1 op If the oppath item is not given the program first looks in the startup directory i e the directory where the input file is located then in the directory specified by the default operator path The default operator path is displayed when typing mctdh84 max The operator files that are available in the MCTDH package are tabulated in the HTML documentation If you desire to create a new operator file the first step is to write the OP_DEFINE SECTION In this section the Hamiltonian is given a title between the keywords title and end title The title will be printed e g in the log file In the NOCI example 6 1 the operator is labeled NOCI S1 surface by employing the OP_DEFINE SECTION 48 6 2 Defining numerical constants 49 OP_DEFINE SECTION title NOC1 S1 surface end title nd op_define section PARAMETER SECTION mass_rd 16 1538 AMU mass_rv 7 4667 AMU nd parameter section HAMILTONIAN SECTION modes rd rv theta 0 5 mass_rd dq 2 1 1 0 5 mass_rv 1 dq 2 1 0 5 mass_rd o Ma a ek p 0 5 mass_rv iE e a 22 1 0 V end hamiltonian section LABELS SECTION V srffile nocllum default end lab
200. ng the system Hamiltonian The RUN SECTION keyword genoper Scan then be used to set up the operators in the file S op to the read write file oper_S To check that these operators are used correctly it is necessary to understand the working of the program internal flag diag assigned to each operator of which the total operator is composed If this flag is set to true then the operator is a unit operator and it will not be explicitly evaluated This has an obvious advantage for the efficiency of the program The program also uses these flags to determine which operator terms are separable i e product terms in which all operators except for one are unit operators In some cases however unit operators must be explicitly evaluated An example is when the matrix elements a h yp are required where and yp are different basis sets which happens when e g the operate keyword is used For this reason there is a flag nodiag assigned to each operator which when set to true turns off the use of the diag flag This flag is set by the program but can also be set by hand using nodiag or usediag as the very first keyword in the HAMILTONIAN SECTION_XXX How this flag is set is listed in the log and op log files Note that the system Hamiltonian and the operators used for eigenf meigenf expect and pexpect must be of usediag type For operate 6 13 DOF mode and muld potentials 67 fmat and flux the nodiag variant is required It could
201. nitial wave function The symcoeff statement is added just to be one the safe side and make sure that the initial state is absolutely symmetric The input for potfit is very simple One usually uses as many natpot terms as grid points The fit is therefore exact R e OF pes P UN SECTION name ngaussHOfit_N125 nd run section ERATOR SECTION gaussld width 0 05 nd operator section RIMITIVE BASIS SECTION 2 HO 125 xi xf 4 0 nd primitive basis section P x1 HO 125 xi xf 4 0 x e NATPOT BASTS SECTION xl contr x2 125 end natpot basis section end input A A OO 94 10 Treating bosonic systems Hat tH Hat HEE HEE HE EE EE HE aE HE HE EE HE aE HE EE aE EE HE aE HE a EE aE EE HE aE EE EE HE aE EEE HE aE aE EEE aE EE EE 3 bosonic particles 1D in a harmonic trap HHH Hat tH a HE HE aE HE aE HEE HE aE HE HE EE HE aE HE EE EE aE HE EE aE EE EE HE EEE EE HE aE EE aE EE aE EEE aE EE EE OP_DEFINE SECTION title p3dl1 3 one dimensional bosons in a harmonic trap end title nd op_define section PARAMETER SECTION mass_xl 1 0 mass_x2 1 0 mass_x3 1 0 g 1 0 int strength only dummy value reset in INP file nd parameter section HAMILTONIAN SECTION modes x1 x2 x3 Kinetic energy T0 E il 1 0 1 KE 1 15 0 1 E Harmonic trap 0 5 a i 1 0 5 1 q2 1 0 5 1 q 2 Two par
202. nputs nocl1 inp The data files containing all the information about the calculation and the system evolution are then contained in the directory nocl1 This is the system used in the first tutorial 11 1 The Analysis Interface The analysis program provides a menu driven interface for running many of the ANALYSE programs On typing analysis84 a menu appears as shown in Example 11 1 An option is selected by entering the appropriate number This may lead to further menus which allow the examination or plotting of various quantities of interest 96 11 2 Interpreting the MCTDH output 97 KKKKKKKKKKKKKKK KKK KKK KKK KKK KKK KKK KKK KK KKK KKK KKK KKK KKK KKK KKK KKKKKKKKKKKKK THE HEIDELBERG MCTDH PROGRAM ANALYSIS PACKAGE Program Version 8 Release 2 kkkxkxkxkxkxkxk xkxkxkxkxk xkxkxkxkxkxkxkxkxkxk xkxkxkxkxkxkxkxkxkxkxkxkxkxk xkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxkxxkxx Present directory is workb graham mctdh82 0 stop list change directory analyse convergence analyse integrator analyse results analyse system evolution analyse potential surface compare calculations YAO BWwNEF OO ll Example 11 1 The start up menu in the analysis program which provides an interface for running the various ANALYSE programs A browse function is included to move between directories containing data option 1 Keep typing the name of the new directory either absol
203. og file lists some brief information about the projected potentials and in case that error was requested the mean minimum and maximum values of AV 12 4 Manipulating potentials with the projection program 125 12 4 2 Generating a Fourier transformed potential In the case of a four atomic system which is described in Jacobi coordinates see Fig 12 2 it is of technical advantage for the MCTDH package to replace the relative torsional angle y by two torsional angles p and yo such that y y p2 and then describe the system in terms of their conjugate momenta k and k2 For more details see 30 or 31 In this case the transition from 12 to ky is simply done by a Fourier transform in the following we will abbreviate the set of coordinates R r1 r2 01 02 simply by Q 2m Qn E 1 Q ki k2 z fae ee dyze 22 P Q p1 p2 12 8 0 27 0 Likewise we Fourier transform the potential V Q p Pe 1 P l ValQ z ave V Q 9 12 9 0 or vice versa 2 XO A Va Q 12 10 Q 20 Then it is straightforward to see that the action of the potential operator V on the wavefunc tion in terms of kj is given by VW Q ki k2 3 Vo Q Q k 2 k2 9 12 11 Q 0o Since the potential V is real and often symmetric V Q p V Q the Fourier transformed potential is also symmetric Vo Q V_a Q and 12 9 simplifies to 2T PQ de cos Qp V Q 9 12 12 0 The latter
204. on adiabatic system 89 HEHEHE HEHE HHH HE HEE HH FH HE HE EH EE EH EE HH EE HE EH HE HE HE EE HE EEE HE t pyrazine 4 mode multi set EH HEHEHE HEHE HHH HE HH HE FEE HE FH EH EE HEHE EE HE EE HE EE HE EH HEE EH HE HE HEE RUN SECTION name pyr4mode propagate tfinal 120 0 tout 0 50 tpsi 1 00 psi auto twice steps gridpop end run section OPERATOR SECTION opname pyrmod nd operator section SPF BASIS SECTION multi set vl0a 4 3 voa 5 4 v1 3 3 v9a 3 3 nd spf basis section PRIMITIVE BASIS SECTION vl0a HO 22 0 0 1 0 Ls 0 voa HO 32 0 0 1 0 1 0 v1 HO 22 0 0 1 0 10 v9a HO 12 0 0 1 0 1 0 el el 2 end primitive basis section INTEGRATOR SECTION CMF var 0 5 1 0d 5 BS spf 7 1 0d 5 2 5d 4 SIL A 5 1 0d 5 end integrator section INIT_WF SECTION build init_state 2 vl0a HO 0 0 0 0 1 0 v6a HO 0 0 0 0 1 0 v1 HO 0 0 0 0 1 0 v9a HO 0 0 0 0 1 0 end build end init_wf section end input Example 9 2 An input file for the pyrazine 4 mode 2 state model system Note that no single particle basis needs to be specified for the electronic degree of freedom as this is a complete basis set In the multi set formalism which is often more efficient than the single set formalism the wavepackets on each surface are represented in a different single particle function basis The number of functions desired for each state must therefore be given
205. on follow full ortho 3 5 Improved relaxation Generation of excited eigenstates 25 As the A vector now changes discontinuously the standard variant of the CMF step size control CMF var does not work One must either work with fixed CMF steps CMF fix or let the step size control depend on the single particle functions only CMF varphi The latter choice is usually to be preferred For convenience CMF is set equivalent to CMF varphi when the run type is improved relaxation In all other cases CMF is interpreted as CMF var Note that the integrator parameters initial CMF step size CMF accuracy only for varphi maximal Lanczos space and SIL accuracy may have decisive effects on the convergence The SIL accuracy and space should be chosen higher than for propagation Use an SIL accuracy between 1078 and 107 and a Lanczos space between 20 and 200 the maximal size is 500 The higher the sought eigenstate lies the larger must be the Lanczos or Krylov space The convergence to higher lying states also requires more SPFs than needed for propagation and as indicated by the natural weights The single particle functions are propagated in imaginary time i e they converge toward the lowest states of the mean field operator irrespectively whether they are needed for repre senting the wavefunction or not If for example there is no node less single particle function approximating the ground state of the mean field operator then the pr
206. on for each basis type DVR FBR representation Keyword Typical applications Hermite harmonic osci DVR HO Vibrational modes Radial Hermite DVR rHO Vibrational modes Legendre rotator DVR Leg Angular modes 0 Restricted Legendre DVR Leg R Angular modes 0 Sine DVR sin Vibrational angular and dissociative modes Laguerre DVR Lagu Boundary condition y r rt Exponential plane wave DVR exp perodic boundaries angular and diffractive modes Fast Fourier transform FFT FFT Dissociative modes with large grids Spherical harmonics FBR sphFBR Combined 6 angular modes Extended Legendre DVR KLeg Combined 0 angular modes Two Dimensional Legendre DVR PLeg Combined 0 angular modes Three Dimensional rotational DVR wigner Combined a 6 7 angular modes j 0 1 N 1 Note that the Hermite DVR depends only on the product mw which determines the width and on x which defines the centre of the grid A variant of the Hermite DVR is the radial Hermite DVR which is an appropriate DVR when the wavefunction is defined on a half axis oo only and satisfies the boundary con dition 0 The odd harmonic oscillator functions x 2 V2 x90 a 4 2 are chosen as basis with j 1 N and ad given by Eq 4 1 A Hermite DVR may be selected in two ways One is given by the following PRIMITIVE BASIS SECTION PRIMITIVE BASIS SECTION X HO 36 0 00 0 10 1822 89 Y HO 36 0 00 2 721 eV 1 0
207. on of the absorption spectrum of pyrazine as discussed in Ref 7 There the primitive product grid amounted to 6 6 x 102 points whereas the MCTDH calculation required only 3 76 x 10 configurations 687 MByte RAM and 52h CPU time This was done in 1998 on an IBM RS 6000 power2 workstation which is much slower than any modern PC Chapter 3 Defining the type of calculation to be made In this chapter we present how to define and start the calculation to be made Possible types are propagation relaxation or diagonalisation Propagation and relaxation calculations can be performed either using the MCTDH method or numerically exactly i e using the full primitive product grid We also give a brief overview of the output to be produced 3 1 Specifying the task for MCTDH The MCTDH program package can perform different tasks specified in the RUN SECTION The following tasks are possible Keyword Level Description gendvr 1 A DVR file will be generated see Sec 4 genoper 2 An operator file will be generated see Sec 6 genpes 2 A special operator file called pes will be generated that contains the potential energy surface see below gengmat 2 A special operator file called pes will be generated that contains an element of the G matrix of the kinetic energy see below geninwf 3 An initial wavefunction restart file will be generated see Sec 7 test 4 All input files will be checked and all ot
208. on the relevant region whereas the global error on all grid points increases to 9 067 meV Note that weighted al ways refers to separable weights and relevant refers to correlated weights For large systems the numerical evaluation of the rms error may become costly In such a situation it may be useful to switch this evaluation off or perform it only every n th iteration step See the HTML documentation for details 12 2 Using ab initio data An interesting feature of the MCTDH package is the possibility to define the potential energy operator or a part of it or in general any local operator in configurational space directly from ab initio data in a way that can later be used by a MCTDH calculation The multidimensional grid on which the ab initio data is collected however must be a product grid In general there are no further restrictions e g an equidistant distribution of the grid points is not required The potfit program can be used in order to transform the supplied data into a product form practically a natpot file that can later be used in the MCTDH simulations This is similar to the transformation of general multidimensional functions into product form that has been covered in chapter 12 1 Also the primitive grid where the ab initio data is collected happens to be usually rather sparse due to the cost of evaluating the desired property on each point The MCTDH package provides the chnpot utility which allows to interpolate bet
209. on with the input file hh inp but with the coupling parameter twice as strong as defined in the operator file The D option means that the results are written this time to the name directory new There may be more than one parameter definition and the parameters may carry a unit Thus mctdh84 p lambda 10 9 eV p mass_Y 1 5 hh is also a valid command The order of precedence of parameters defined from different sources is command line input file parameter file operator file Thus parameters in the operator file are the default values which can be altered in a run in a variety of ways The labels in the LABELS SECTION of the operator file may also be modified without altering the operator file This is done using the alter labels keyword One example for this was presented before in Sec 6 10 where complex absorbing potentials were added to the Hamiltonian Another typical example is to switch between different implementations of a potential Supposed the potential to be used is represented by the label V in the line 6 11 Altering a Hamiltonian from input file or command line 65 Uwe Manthe fit NOC1 S1 surface PARAME TER SECTION theta0 127 4 deg c000 0 0384816 c001 0 0247875 c002 0 0270933 c003 0 00126791 c004 0 00541285 c005 0 0313629 c006 0 0172449 nd parameter section HAMILTONIAN SECTION modes rd rv theta LeU 1 v NO I c000 berd 0x 1lqd berv70 bew 0 c001
210. opagation in imaginary time will rapidly change the single particle functions in order to generate such a function Due to this rapid change the method then fails to converge One thus may have to select the initial single particle functions somewhat more carefully However there is no implemented algorithm to do so and one may therefore be forced to use more single particle functions for improved relaxation than finally needed to represent the converged wavefunction When the keyword orben is set in the RUN SECTION then the so called orbital ener gies are computed and output to the file orben The orbital energies are obtained by diagonal ising an appropriate mode Hamiltonian in the set of the single particle functions The mode Hamiltonian is conveniently defined as the trace over the mean fields of the particular mode under discussion I e Hay D2 E See the MCTDH review for a definition of the mean field operators H 09 The eigen functions of Hav called energy orbitals allow to define energy weights as the diagonal values of the density matrix expressed in the basis of the energy orbitals The energy weights are very useful for assigning quantum numbers to a relaxed wavefunction If for each mode there is one weight that is close to one larger than 2 3 say then the full wavefunction is characterised by the quantum numbers of the dominant energy orbitals For further infor mation see the HTML documentation under Input Documentatio
211. or the degree of freedom is a DVR basis i e not an FFT as the program generates the eigenfunctions by diagonalising the operator represented as a real matrix 7 7 Reading the initial wavepacket from file Instead of building a new initial wavepacket one may also read a wavefunction that has been created in a previous calculation from the restart file This is done by the file keyword in the INIT_WF SECTION INIT _WF SECTION file oldrun orthopsi end init_wf section Here oldrun is the path of the directory where the restart file is stored If no path is spec ified the restart file is searched for in the name directory The second optional parameter which can be orthopsi the default or noorthopsi specifies whether or not the single particle functions are Schmidt orthogonalised after being read The primitive basis must be defined in the PRIMITIVE BASIS SECTION identically to the one of the previous calculation from which the initial wavepacket is being read This can be ensured by reading the definition of the primitive basis of the previous run from file using the readdvr keyword in the RUN SECTION rather than defining the primitive basis in a PRIMITIVE BASIS SECTION The number of single particle functions however may differ Since version 8 3 10 there is also a Read Inwf end read inwf block In contrast to the simple file keyword this allows to distribute the SPFs and the blocks of the A vector freely among the elect
212. ot label file k 1i integer file k i a b c characterx x label if label 1 5 eq mypot then a 5 b index label c index label read label at tl b 1 x k read label1 b 1 c 1 i 58 6 Setting up the Hamiltonian file 308 endif return end The number 5 must match the number of characters in the name mypot and file must be 300 plus the file number introduced above Finally add the lines call defmypot buffer ifile hopipar 2 hopipar 3 if ifile ne 0 go to 99 to the subroutine defan1d in the file source opfuncs callanid f and recompile We close in noting that it is possible analogous to what was said in Sec 6 6 to pass additional parameters to the potential routine mypot by employing the arrays hopipar hoprpar and hoppar 6 8 Advanced topic Implementing non separable potentials po tential surfaces It is also possible to include non separable potentials into the MCTDH program i e po tentials that cannot be written in the product form 6 4 Because the direct evaluation of a non separable potential makes an MCTDH calculation extremely inefficient they are typi cally used in numerically exact calculations propagation relaxation or diagonalisation or to generate a separable potential fit using the Potfit program The MCTDH program can however use them as they are which may be useful for checking purposes Note that the use of a mul
213. other operator except a constant Hence a construct like cos V_ is not allowed However if the natpot does not operate on all DOFs or particles it may be combined with operators acting on the remaining DOFs or particles E g if the natpot V does not operate on the first DOF the following Hamiltonian line is allowed const dq 2 V This however does not hold if the other operator is also a natpot I e const Vl V2 is not a valid Hamiltonian line if V1 and V2 are both natpots One cannot multiply natpots with each other 6 10 Using complex absorbing potentials CAPs Complex absorbing potentials CAPs can be employed to reduce the lengths of the primitive grids CAPs are also useful for computing S matrix elements in scattering processes Please refer to Chaps 4 7 and 8 6 of the review 1 for details The CAPs 2W that can be employed in the MCTDH program are one dimensional and monomial i e of the form iW x in x zel O x ze 6 5 The parameters e 7 and b denote the starting point strength and order of the CAP respec tively O specifies Heaviside s step function When the positive sign is used the CAP lies to the right of e otherwise it is located to the left of xe Let us assume that your system under investigation has three degrees of freedom labeled x y and z To turn on CAPs for say the first two modes add the lines 1 0 capl 1 ol 1 0 1 cap2 1 to the H
214. ounting starts from zero i e n 0 1 2 98 Note that the Lanczos matrix depends on the starting vector which in this case is the initial wavefunction defined in the INIT_WF SECTION The second way is to specify relaxation follow In this case the starting vector of the improved relaxation is that eigenvector of the Lanczos matrix that has the largest overlap with the initial wavefunction defined in the INIT WF SECTION The Krylov space and thus the dimension of the Lanczos matrix grows with each step of the Lanczos iteration This process is stopped when the accuracy criterion is satisfied the SIL accuracy parameter is now interpreted as the tolerated error in milli Hartree of the energy of the desired Lanczos eigenvalue or when the specified maximal dimension of the Lanczos matrix is reached If the keyword ful1 is given as a second argument of the relaxation keyword then during the very first build up of the Krylov space the iteration will be continued till the maximal dimension is reached This feature is useful to ensure that the improved relaxation starts from the correct Lanczos eigenvector There is also the keyword ortho which may appear as an argument to the relaxation keyword ortho forces the SIL integrator to perform a full re orthogonalisation of the Krylov space This often significantly improves the convergence but for long A vectors it may take some CPU time In short the improved relaxation command may read relaxati
215. ovals in the diagram MCTDH include files 4 4 i CALLDVR CALLOPER CALLINWF CALLPROP EINGABE auto 7 Y t Y Y psi check RUNDVR RUNOPER RUNINWF RUNPROP oe ANALYSE G pnn if POTFIT oper _ restart Figure B 1 The structure of the MCTDH programs See text for details 132 Appendix C The built in symbolic expressions The following tables describe the symbols and related operators that can be used to set up a Hamiltonian operator General Remarks With the aid of the caret one may apply a power to operators The power may be integer or real and may carry a sign This however works only for potential like operators Inspect the Tables below to learn which operators can be exponentiated Note that symbols like dq 2 or j 2 are operator labels of their own right they do not denote that the second power of the operators dx or j is taken literally Compare with Appendix B Discrete Variable Representation of the MCTDH review Phys Rep 324 2000 1 105 to learn how dq and dq 2 are defined One may multiply operators e g a construct like dq cos dq_ is allowed However multiplication is allowed only among potential like operators and operators with a simple matrix representation This excludes all KLeg and PLeg operators from multiplication See Table C 2 and notes to this ta
216. oved relaxation is more accurate than filter diagonalisation but also more elaborate because one has to perform a separate calculation for each state 1 Copy the file MCTDH_DIR inputs co2_gs inp to your tutorial directory and similarly the files co2_sym inp co2_asym inp and co2_excite inp 2 7 Determining eigenstates by block improved relaxation 15 2 To perform the relaxation execute the command mctdh84 mnd co2_gs and after the job has finished run the inputs co2_sym co2_asym and co2_excite Move to the directory co2_gs and type rdrlx e to read the rlx_info file This produces the following output time order q betax1000 Energy cm 1 ovl 1000 Delta E 2 3 0 0 0 0 000E 00 2922 190 202 862 0 0000 0 000E 00 0 000 15 0 804 71296 2534 937 314 025 0 0000 3 873E 02 0 500 a 0 999 98713 2534 663 002 346 0 0000 2 743E 01 1 250 10 0 999 99715 2534 558 864 426 0 0000 1 041E 01 2 000 9 0 999 99971 2534 535 729 282 0 0000 2 314E 02 4 000 8 0 999 99990 2534 528 474 396 0 0000 7 255E 03 6 000 6 0 0 293E 08 2534 528 204 472 0 0000 2 699E 04 8 000 4 0 0 874E 10 2534 528 194 788 0 0000 9 684E 06 WARNING Davidson did not converge for 1 diagonalisations The star x at 15 indicates that that this Davidson diagonalisation did not converge 16 David son iterations are needed for convergence but in the input file the maximum number of iterations was limited to 15 The non convergence is of no
217. owing wigner denotes the j value of the initial wavefunction initial values of the k and m quantum numbers along with their ranges and step sizes are given in the first and second K lines respectively The corresponding DVR for wigner initial wavefunctions must be wigner for the first degree of freedom in the combined mode either exp or K are allowed DVR FBR types for the second and third degrees of freedom The degrees of freedom are assumed to be given in 7 6 Generating eigenfunctions of a one dimensional Hamiltonian 75 the order J K M The excite keyword can be used to choose between two different schemes for generating unoccupied single particle functions excite mk j preferentially excites m states then k then j while excite kmj preferentially excites k then m then j states 7 6 Generating eigenfunctions of a one dimensional Hamiltonian It may be useful to start a calculation with the wavepacket in a particular eigenstate of a zero th order Hamiltonian This occurs for example in an atom diatom scattering calculation when the diatom starts in a particular vibrational eigenstate To do this one must first define the zero th order operator This is done by including a HAMILTONIAN SECTION OPER section in the operator file see Sec 6 12 to define an operator labelled OPER any other string except SYSTEM can be chosen for this name As an example a one dimensional H Hamiltonian operator can be defined by adding the section
218. p the second degree of freedom in the second most internal loop and so on The order of the degrees of freedom is given by the PRIMITIVE BASIS SECTION in the potfit input file The program potfit will generate a natpot file to be used directly by mctdh Remember that mctdh uses the modelabels to associate the natpot terms with the DOFs 12 3 Extra flexibility combining potfit and chnpot The maximum flexibility in the usage of ab initio data is accomplished by combining potfit with the chnpot utility There are two main reasons why the initial primitive grid in which the ab initio data points are given should be transformed into a more suitable one First the given points may be too sparse and a more dense grid is desired for the dynamical simulation phase Second the primitive grid where the points are supplied does not correspond to a DVR defined in the MCTDH code 12 3 1 Dealing with an arbitrary primitive grid In case that the ab initio data values are given in a grid that does not correspond to a DVR known to MCTDH the external keyword can be used in the PRIMITIVE BASIS SECTION of the potfit input file PRIMITIVE BASIS SECTION x external 16 path_to_x_grid lt unit gt y external 16 path_to_y_grid lt unit gt end primitive basis section path to x grid is the absolute or relative path to a file containing in one column the values x of the x coordinate in this case a total of 16 entries One needs to create a file with t
219. plot persist spectrum pl The first line produces a GNUPLOT file with data to plot the spectrum This is done from 0 6 eV to 2 0 eV and using a simple cosine cutoff function to allow for the finite propagation time The result is shown in Fig 2 1 In order to understand the options and parameters type autospec84 h and see the HTML documentation Note that the spectrum shown is the Fourier transform of the autocorrelation function times the energy Hence it is assumed that the ground state energy is at zero such that energy equals excitation energy If this is not the case use option e to shift the energy scale The FT option suppresses the multiplication with the energy showing directly the Fourier transform NB The option FT is now default Use option EP to switch on the energy prefactor or use Mb lt dipole moment gt to plot the properly normalized absorption spectrum in mega barns To make life easier there exist a number of bash scripts so called pl scripts which auto matically call an analyse routine and plot the results The above commands are equivalent to plspec 0 6 2 0 ev One may alternatively call plspec without arguments The script will then prompt you for the missing input Finally the command plauto plots the autocorrelation function the command plnat plots the natural populations plqdq plots the expectation values of the coordinates and the commands plupdate plupdate e and plspeed show information on the per
220. previous one The grid spacing is Ax xy z1 N 1 in the case of short and Ax y4 1 z0 N 1 in the case of Long 4 5 Exponential DVR and fast Fourier transform The exponential DVR uses plane waves as basis functions It is therefore often used for dissociative degrees of freedom Moreover exponential DVR and FFT are the only primitive basis representations within the MCTDH program that satisfy periodic boundary conditions These occur for instance for angular motion or motion parallel to a corrugated surface 40 4 Selecting a DVR FBR representation for the primitive basis As our implementation of the exponential DVR requires an odd number of basis functions we set N 2n 1 The basis functions are then written as xj L exp 2imj a xo L 4 6 with n lt j lt nand L xy Xo The wavefunctions to be represented satisfy periodic boundary conditions xo Y zy The fast Fourier transform FFT method may be considered as an exponential DVR where however the derivative matrices are not built but the action of them on the wave function is evaluated by two FFTs The MCTDH program uses a Temperton FFT which allows one to use grids the length of which can be factorised into powers of 2 3 and 5 i e N 2 3 5 where j k and l are non negative integers For optimal performance one should work with a grid length of N 2 3 Although the exponential DVR and the fast Fourier transform FFT
221. program What to do when this is not the case will be covered in the following sections As usual the primitive grid is defined in the PBASIS SECTION of the input file For example one could have something like 12 2 Using ab initio data 119 PRIMITIVE BASIS SECTION x sin 17 2 4 2 4 y sin 21 250 BS end primitive basis section for coordinates x and y The information concerning where the actual data values are found is given in the LABELS SECTION of the operator file One has to define a new label making use of the readsrf keyword LABELS SECTION vdat readsrf pathtofile S nd labels section s can be either ascii or binary depending on the file to be read pathtofile is the absolute or relative path to the file containing the data The newly defined vdat label can then be used in the HAMILTONIAN SECTION of the operator for example HAMILTONIAN SECTION modes lx y 1 0 1 amp 2 vdat other operator lines end hamiltonian section The file containing the data values consists of a single column of numeric entries written so that the first index runs fastest The order of the indexes is defined by the i amp j amp ks construct in the HAMILTONIAN SECTION as shown above In our example the file has to be created so that it could be read by the following pseudo code where the i index runs on the y coordinate iterate i in range 1 to 21 iterate j in range 1 to 17 read v j 1 end iterate en
222. ptiming keyword is used but the Irt library is not linked the following error mes sage appears 32 3 Defining the type of calculation to be made aR REE AE E FE HH FE AE AE E FE EH FE AE AE E FE FE EEE HEHEHE EEE HHH no clock_gettime command lrt not linked 4 If lrt is available modify the script compile cnf dea HE a AE FE AE aE aE EE EE EE EH HHH HH HH RR RR HH a a a a E FE H If your compiler supports this library the script compile cnf has to be modified The option lrt must be added in the line MCTDH_ADD_LIBS in the section of the compiler that is used Then MCTDH must be compiled again A similar error message appears if the pthread library was not linked for compilation of MCTDH HEHEHE HEHE EE EE TE AE E TE TE E EEE HEHE EERE EE HE HEH no XXX_yyyy command pthread library not linked If your compiler supports pthread modify the script compile cnf HEHEHE HE HE HEREREREREE EEE HERR EEE HE HEH Se E OE OE OE HE xxx_yyyy is replaced by the name of the routine the program tried to execute but could not be found If your compiler supports pthreads
223. ptions Note that a continuation run reads only the RUN SECTION and ignores all the other sections of the input file as well as the operator file All the necessary information is read from the read write files of the name directory Sometimes it is desirable that a calculation is continued with a different integrator setting This can be accomplished by giving the keyword continuation integrator or by the option ci In this case the INTEGRATOR SECTION will be read Finally the continuation keyword can be used in order to try to complete a crashed calculation The program however does not check the output files for consistency The continuation thus might fail if some relevant data was lost due to the crash Another option of the MCTDH program is to stop a calculation during run time in a controlled manner such that it can be resumed later To this end include the stop file by adding the keyword stop to the RUN SECTION To halt a calculation after the next output edit the stop file and write st op to its first line Alternatively one may just remove the stop file The stop is automatically deleted when the run finishes It thus may serve as a lock file As long as the stop file exists the run is not finished Rather then simply writing stop to the stop file one may supply more specific commands which will halt the program after a certain real time or CPU time has passed This allows e g to outwit CPU time limits Rather than editing the stop
224. r 12 flux 105 optcntrl 19 plall 5 plauto 5 plbrlx 16 27 plcap 63 plfdspec 13 plflux 9 105 106 plpit 115 plpweight 115 plqdq 5 plrlx 15 16 27 plspec 5 8 104 105 plspeed 5 plstate 7 106 plupdate 5 plwtt 9 105 potfit 9 114 projection 122 rdcheck 7 101 106 rdgpop 99 rdrlx 15 16 27 reflex 63 showd1d 4 9 102 showpot 112 115 showrst 103 showspf 103 showsys 5 111 112 Program structure 132 Projection program see Program Propagation see Wavepacket Psi file see File Radial harmonic oscillator DVR see DVR radial Her mite Radial Hermite DVR see DVR Rdcheck program see Program Rdgpop program see Program Rdrlx program see Program see Program readsrf 118 Reflex program see Program Relaxation see Wavepacket relevant region 114 Restart file see File Restricted Legendre DVR see DVR RKS5 8 integrator 80 82 Rotator DVR see DVR Legendre Run section see Section Section correlated weight 116 Hamiltonian 51 87 129 init_wf 71 129 integrator 80 129 labels 61 129 natpot basis 115 op define 48 129 operator 48 115 129 parameter 49 129 primitive basis 36 87 116 129 run 21 115 129 separable weight 116 spf basis 45 88 129 Shared memory see Calculations Showd1d program see Program Showpot program see Program see Program Showrst program see Program Showspf program see Program Showsys program see Progr
225. r and L S Cederbaum State filtering by a bath Up to 24 mode numerically exact wavepacket propagations Chem Phys Lett 299 1999 451 A Jackle and H D Meyer Time dependent calculation of reactive flux employing complex absorbing potentials General aspects and application within MCTDH J Chem Phys 105 1996 6778 A J ckle and H D Meyer Calculation of H H2 and H Dz reaction probabilities within the multicon figuration time dependent Hartree approach employing an adiabatic correction scheme J Chem Phys 109 1998 2614 M H Beck and H D Meyer Extracting accurate bound state spectra from approximate wave packet propagation using the filter diagonalization method J Chem Phys 109 1998 3730 3741 M H Beck and H D Meyer Efficiently computing bound state spectra A hybrid approach of the multi configuration time dependent Hartree and filter diagonalization methods J Chem Phys 114 2001 2036 2046 T Sommerfeld H D Meyer and L S Cederbaum Potential energy surface of the CO anion Phys Chem Chem Phys 6 2004 42 45 D J Tannor V Kazakov and V Orlov Control of photochemical branching Novel procedures for finding optimal pulses and global upper bounds In Time Dependent Quantum Molecular Dynamics J Broeckhove and L Lathouwers Eds Plenum New York 1992 pp 347 360 W Zhu and H Rabitz A rapid monotonically convergent iteration algorithm for quantum optimal control over the exp
226. rd compilers g77 gfortran pgf77 and ifort additional sections are included in the script compile cnf Depending on the compiler the command is compile m mctdh or compile c mpi xxx mctdh where xxx must be replaced by g77 gfortran pgf77 or intel respectively If an other compiler should be used an appropriate section in the compile cnf script must be cre ated The m option is a shorthand form It requires that your default compiler is one out of the four above mentioned compilers Note that it is essentially that your MPI package was created with a compiler compatible to the compiler you would like to use gt Try to check mpif77 v and see the manpages for mpif77 Chapter 4 Selecting a DVR FBR representation for the primitive basis DVR FBR representations are used to set up the single particle functions of an MCTDH or the product grid of a numerically exact calculation The DVR FBR basis is also called primitive basis in the following A comprehensive discussion of the DVR FBR technique can be found in Appendix B of the review 1 4 1 Available DVR FBR representations The DVR FBR representations that are implemented in the MCTDH package are compiled in Tab 4 1 along with examples for typical applications of them Also given are the keywords by which the corresponding primitive basis types are selected in the input file The representations of the primitive basis are specified in the PRIMITIVE BASIS SECTION of th
227. re The FBR DVR transformation matrix is now rectangular rather than square The propagation however is performed exclusively in the smaller set of DVR grid points and some speed up is obtained For example one may replace the theta line in the PRIMITIVE BASIS SECTION of the nocl1 inp input file see Example 4 1 and the Tutorial to 42 4 Selecting a DVR FBR representation for the primitive basis theta Leg r 60 0 all 1 4 2 The two last numbers define the range in radians to be covered by the grid points In the log file one finds xxxxxx Primitive Basis x x mode kappa DVR N xi SEE dx p max rd 1 sin 36 3 800 5 600 0 051429 59 436 rv 2 HO 24 1 620 2 652 0 044849 70 171 theta 3 Leg R 27 60 2 739 1 389 0 051924 m 0 sym 0 The last line tells us that there are 60 FBR functions but only 27 DVR grid points For tech nical reasons the locations of the first and last grid points differs slightly from the inputted numbers Comparing the timing files one finds that the restricted Legendre DVR reduces the total CPU time by a factor of 1 4 and reduces the time spend for propagating the single particle functions of the theta degree of freedom by more than a factor of 4 The analyse programs plgpop and showd1d84 are useful for determining an appropriate theta range Tryplgpop z 1 e 10 2 3andshowd1d84 a y 0 000001 3 When using the restricted Legendre DVR it should be the last entry in the PRIMITIVE BASIS SECTION Due
228. relaxation calculation an ordinary differential equation has to be solved This can be accomplished by different integration methods The first two sections 8 1 and 8 2 are dealing with integration techniques developed especially for the MCTDH method Section 8 3 describes general integrators which can be used in both MCTDH and numerically exact calculations 8 1 Using the VMF integration scheme in an MCTDH calculation In an MCTDH calculation one possible integration method is the variable mean field or VMF scheme which is described in Sec 5 1 of Ref 1 and in Ref 26 As the name implies in the VMF scheme the mean fields are determined in each integration step The VMF scheme is the default in an MCTDH calculation As can be seen from Tab 8 1 the equations of motion in the VMF scheme can be solved with an ABM the default BS or RKz integrator ABM stands for Adams Bashforth Moulton predictor corrector method BS for Bulirsch Stoer extrapolation scheme and RKz for a Runge Kutta integrator or fixed order x where x 5 and x 8 are available We recommend the use of the ABM method because it is generally more efficient To choose a VMF calculation employing the ABM integrator the INTEGRATOR SECTION in the input file should read e g INTEGRATOR SECTION VME ABM 6 1 0d 7 0 01d0 end integrator section The parameters after the ABM keyword are explained in Sec 8 3 When the BS integrator is desired a possible INT
229. relevance here because all later iterations converged However if non convergence of the Davidson happens frequently one cannot trust the results but has to repeat the calculation with a different integrator setting The first data line with the negative time gives the energy expectation value of the initial wavefunction The second line t 0 0 gives the energy obtained by diagonalising the Hamiltonian represented in the orbitals of the initial wavefunction Then the orbitals SPFs are relaxed and the Hamiltonian matrix built from the new orbitals is diagonalised again This procedure is repeated till convergence is reached For the present example the convergence of the improved relaxation scheme is fast beta denotes the squared overlap of the current A vector with the previous one If beta is very close to one the difference from 1 is printed ovl denotes the squared overlap of the current wavefunction with the initial wavefunction This data is evaluated only for relaxation lock runs More information on the performance of the improved relaxation run is obtained when dropping the option e from rdrlx Try rdrlx h A graphical visualisation of the conver gence is provided by plrlx Try plrlx plrlx a 3 and plrlx E l Inspect the outputs of the other relaxation runs in a similar way Note that considerably more Davidson iterations are needed for converging higher excited states Note also that the energy scale is shifted via the keywor
230. ring a Hamiltonian from input file or command line The concrete form of the Hamiltonian i e the values of parameters defined in the PARAM ETER SECTION and the meaning of labels specified in the LABELS SECTION of the oper ator file can be overridden in different ways Let us first consider the change of parameters One possibility is to write the parameters to be changed into a file Example 6 3 shows an example for such a parameter file again for the Henon Heiles system To read this file and use its settings insert the keyword parfile mypara hh par into the OPERATOR SECTION of your input file Here it was assumed that the parameter file is named hh par and resides in the directory mypara relative to the path of the input file A second way is to include the alter parameter and end alter parameter keywords in the OPERATOR SECTION of the input file All parameter definitions in between replace the corresponding parameters in the PARAMETER SECTION of the op erator file For instance the lines alter parameter lambda 0 4 nd alter parameter in the OPERATOR SECTION of the input file set the coupling parameter A of the Henon Heiles potential Example 6 2 to 0 4a u The format of the parameters is the same as in the PARAMETER SECTION of the operator file The third method makes use of command line parameters Starting a calculation employ ing the p option e g mctdh84 D new p lambda 0 4 hh would run a new calculati
231. ritten by one of the latter files when install_ mcetdh is executed You may visit http oss sgi com projects numa to find more information about NUMA API 3 11 Using parallel distributed memory hardware 33 Passing Interface MPI was used To use the MPI parallel MCTDH the keyword usempi must be set in the RUN SECTION and the MCTDH program must be started with the mpirun command Further the MPI compilation of MCTDH has to be used see below To invoke an MPI parallel run with 32 MPI prozesses the command could look like this mpirun np 32 mctdh84 mpi g77 lt inputfile gt Further arguments can be added to the usempi keyword which disable the par allelisation of the MPI parallel routines These routines are calcha no calcha funka2 no funka2 mfields no mfields getdavmat no getdavmat dsyev no dsyev phihphi no phihphi hlochphi no hlochphi and mpir davstep mpibdavstep no dav These keywords are similar to those used in shared mem ory parallelisation with the usepthreads keyword but the last one no dav is different This keyword disables the usage of the MPI parallel routines mpirdavstep mpibdavstep This means that the davidson vectors that are built during an improved relaxation step are stored on one node and are not distributed Additionally there is the keyword dav 1 where I stands for the maximum number of davidson vectors that are stored on one node If this keyword is not set the maximum allowed number of
232. rmonics are employed which are two dimensional The MCTDH program offers a number of function types to be used for each factor in the product i e each degree of freedom 7 1 Building Gaussian functions as initial functions A possible choice for the one dimensional initial functions are Gaussian functions These can be defined in two manners which differ only by the way the width is specified namely either as a N e V 4 e 0 Az eipole zo 7 1 or aS y x N e7 h 2 mw a ao eipo x xo 7 2 with corresponding keywords gauss and HO Here N is a normalisation constant x9 and po are the centre and initial momentum Az is the width and m and w denote mass and frequency Suppose there are two degrees of freedom X and Y then the initial wavepacket may be defined by an INIT WF SECTION reading INIT_WF SECTION build X gauss 4 315 0 0 0 0794 Y HO 221 571 0 0 0 218 eV 13 6155 end build end init_wf section The keywords build and end build enclose the lines that specify how to build the initial wavefunction The first and second number in each line denote xo and po respectively The next numbers are Ax in the first case and w and m in the second As the example shows one may add a unit to the parameters Note that Ax and w may be complex 71 72 7 Generating the initial wavepacket Plane waves may be generated by setting the frequency within the HO line to zero E g build X HO 0 0 0 0 0 0 Y HO
233. ronic state The influence of the interaction potential is partly corrected for when giving correction dia The energy distribution is now evaluated as the overlap of the translational single particle function with a distorted wave rather than a plane wave The 7 11 Setting up a diabatically corrected initial wavepackets 79 distorted wave is the solution of a 1D Schr dinger equation employing the translational mean field as interaction See the MCTDH review 1 Chapter 7 2 for details The distorted wave is no longer calculated through the WKB approximation but evaluated numerically using the Numerov method The quality of the energy distribution may be further improved by replacing the argument dia by ad An adiabatic correction is now performed which modifies the single particle functions of the internal degrees of freedom This improves the mean field and in turn the distorted wave Adiabatic correction however is presently only implemented for the H H2 system and its isotopic variants Remarks e The translational degree of freedom must not be combined with other dof s e There are special routines for the H H system and its isotopic variants These are used when the argument hh2 is additionally given with the keyword correction e g correction hh2 ad e Presently the adiabatic correction works only in combination with the hh2 argument Chapter 8 Choosing an integration scheme In a propagation or
234. ronic states In particular the current system and the wave function read in do no longer need to have the same number of electronic states Example INIT_WF SECTION Read Inwf file gs SPF 1 gt 1 2 3 A 1 gt 2 end read inwf end init_wf section Here the file which is read in gs restart has only one electronic state Its SPFs are copied to all the three states of the current system and its A vector is copied to state 2 The A vector blocks for state 1 or 3 are hence zero See the HTML documentation for more information 7 8 Diagonalising a multi dimensional operator to create multi dimensional SPFs 77 7 8 Advanced topic Diagonalising a multi dimensional operator to create multi dimensional SPFs With the meigenf feature it is possible to diagonalise one or multidimensional hermitian Hamiltonians to create one or multi dimensional SPFs As meigenf uses the Lanczos algorithm with full re orthogonalisation for diagonalisation it needs some initial guess for the SPF Hence the keyword meigenf must not be given in a build block it may come after a build block However meigenf can also alter the SPFs of a wavefunction which is read from file Example meigenf 3 oper 0 Here meigenf will diagonalise the operator oper which must be defined in a Hamiltonian Section similar to eigenf The eigenfunctions of oper will then replace the SPFs of the third mode first argument The third argument O finally indicat
235. rpolation for angular or other degrees of freedom The program can han dle combined modes combined in potfit when the combination is up to 2D type as it is capable of 1D and 2D interpolations If the original natpot has been obtained from a sparse primitive grid it may be a good idea to potfit it using as many natural potentials as grid points in each degree of freedom in case it is computationally feasible In this way the resulting 122 12 Using the Potfit program natpot is exact on the original grid points What follows is the already described interpolation to a suitable grid using chnpot 12 4 Advanced topic Manipulating potentials with the projection program While it is fairly straightforward to include program code for new potential energy surfaces into the MCTDH program package it is sometimes desirable to use an existing PES routine to generate a related PES Examples for this would be e Models with reduced dimensionality where you need an effective potential which is derived from the full dimensional potential by averaging over some degree s of free dom e Series expansion of a PES e g multipole expansions or Fourier transforms of angular degrees of freedom These tasks can be accomplished with the projection program Basically this program takes the potential V q1 qs from an existing PES routine and projects out some degrees of freedom say q1 qp by integrating over them with projection function
236. rv CAP dashed line and the energy distribution of the initial wavepacket dotted line showd1d84 a y 10 f3 and repeatedly press RETURN to step through the pictures Initially the molecule is in the j 0 rotational state and the density is evenly distributed over all angles After about 20 fs the wavepacket reaches the saddle point region and the system is in the transition state The transition state is collinear and consequently the angular distribution is now strongly peaked at zero degrees At later times a more evenly angular distribution is again assumed You may also inspect the motion of the other two degrees of freedom 2 4 Determining the vibrational spectrum of LiCN The MCTDH program is not only capable of propagating wavepackets but also of diago nalising a Hermitian Hamiltonian operator by employing the Lanczos algorithm The time independent Schr dinger equation is then solved rather than the time dependent one This feature and similarly the possibility of performing a numerically exact propagation has been implemented into the mctdh package because then the very convenient operator generation is available for these tasks Lanczos diagonalisation and exact propagation are of course possible only for comparatively small problems As a small example of this feature let us determine the vibrational spectrum of a two dimensional model of the LICN electronic ground state with the CN bond length frozen at its equilibrium va
237. s and the operator file Thus it tells you exactly what you have been doing Since an NOCI run is so fast NOCI is ideally suited for testing Just play around with it You may e g change the numbers of single particle functions or alter the integrator accuracies You also may try the options e g to start a continuation run type mctdh84 c tfinal 50 nocll Type mctdh84 h to obtain the list of options 2 2 Determining state populations for the photo excitation of pyrazine 7 State 1 State 2 0 8 p ee J O 6 F O 4 F O 2 Oo 1 i f oO 20 40 60 80 100 450 time fs Figure 2 3 The diabatic state populations of the pyrazine molecule after excitation to the S2 state calculated using a 4 mode model 2 2 Determining state populations for the photo excitation of pyrazine The pyrazine molecule contains a classic example of vibronic coupling Two states which are close in energy are coupled by motion along one vibrational mode resulting in a broad spec trum for the upper state This system can be described using the simple vibronic coupling model Hamiltonian The vibronic coupling model Hamiltonian is well suited to the MCTDH method being already in the product form required for maximum efficiency For further details of this system see Refs 6 9 and the references therein In this tutorial we use a simple 4 mode 2 state model This qualitatively reproduces the experimental spectrum after
238. s these options have to be read here See subroutine defsrf on opfuncs funcsrf F for examples Finally we want the program to write some information to the log file when the multi dimensional potential energy surface is used To this end one has to edit the subroutine usersurfinfo on the file opfuncs usersrf F One should briefly describe the surface and then name the coordinates The string mlabel j contains the modelabel of the j th coor dinate of mysurf as assigned in the Hamiltonian Section For example the code added to usersurfinfo may read elseif hopilab eq 2 then write ilog a newsurf V x y Z beta function write ilog 2a x mlabel 1 write ilog 2a y mlabel 2 write ilog 2a z mlabel 2 34 4 This is dimension l Finally recompile Include the new surface by running compile with the option i newsurf See HTML documentation Installation and Compilation Compiling the Programs Please be reminded again that only real multi dimensional potential functions should be implemented on usersrf F For one dimensional real functions please use user1d F or funcanld F for complex functions use funcanlz F and for grid based operators use funcgrd f The new potential surface is selected in the HAMILTONIAN SECTION using the V or any other not pre defined label and the LABELS SECTION Assuming that the Hamiltonian is given by H 1 2m 02
239. s as j sr J K M JVJ J 1 M M 1 J K M 1 Wigner DVR space fixed angular mo mentum lowering operator which operates as j_ sr J K M JJ J 1 M M 1 J K M 1 Wigner DVR body fixed combined angu lar momentum operator CFg are defined in Table C 1 Wigner DVR space fixed combined angu lar momentum operator CF are defined in Table C 1 but here M replaces K Wigner DVR squared body fixed angular momentum raising operator Wigner DVR squared space fixed angular momentum raising operator Wigner DVR squared body fixed angular momentum lowering operator Wigner DVR squared space fixed angular momentum lowering operator continued 138 C The built in symbolic expressions Table 2 continued Symbol Operator Notes jpjm jpjms jmjp jmjps jpjz jpjzs jzjp jzjps jmjz jmjzs j BF J BF j sF j sF j er J BF j sr j sF j BF Jz BF J sF jz sF jz BF J BF Jjz sF J SF j BF Jz BF j sF Jz SF Wigner DVR product of body fixed an gular momentum raising and lowering operators Wigner DVR product of space fixed an gular momentum raising and lowering operators Wigner DVR product of body fixed an gular momentum lowering and raising operators Wigner DVR product of space fixed an gular momentum lowering and raising operators Wigner DVR_ pro
240. s described in Ref 1 and in the original papers 28 29 In short the fit to product form is performed in two steps First the potential density matrices are diagonalised to obtain the natural potentials The thus generated product representation of the potential minimises to a very good approximation the overall L error As there are regions of greater and lesser physical importance a better representation can be achieved by introducing weights which emphasise the regions of phys ical importance Separable weights i e weights that act on one degree of freedom only can be incorporated into the first step More powerful however are correlated weights i e weights which cannot be written in product form In a second step one thus may iteratively improve the representation by employing correlated weights The correlated weights are im plemented as relevant regions A relevant region may be defined e g as those areas where the potential energy is below some threshold V R lt Vmax The program then iteratively improves the fit in the relevant region while making it worse in the non relevant region 114 12 1 Transforming a potential to product form 115 RUN SECTION niteration 10 name noclfit iteration prodwei end run section OPERATOR SECTION pes noclisch veut lt 5 0d0 ev veut gt 1 0d0 ev Schinkes surface Potential is cutted above 5 eV and below 1 ev nd operator section NATPO
241. s fitKO etc That is fitKO should contain the potfit of vpot_KO fitK1 that of vpot_K1 etc The shift operators must be put into the LABELS SECTION and then referenced by their labels kp1 km1 etc because of a syntactical restriction of MCTDH it is not allowed to use operators which take an argument directly in the HAMILTONIAN SECTION 128 12 Using the Potfit program This completes the setup of the operator file for a Fourier transformed potential 12 5 Downsizing previous potfits the cutnpot and rdnpot func tions Under certain circumstances it may be necessary to make use of previous potfitted potentials i e natpot files Possibly it may be convenient to reduce the number of single particle potentials and the corresponding contracted coefficients For instance this may be useful to investigate the quality of the potential expansion e g the fitting error upon reduction of the expansion coefficients or simply in order to decrease the cost of the energy evaluation For such a purpose the cutnpot and rdnpot functions have been developed It should be highlighted that they play a major role in the mgpf program under development The process can be summarized in two steps i elimination of unwanted single particle potentials and the corresponding contracted coefficients from a previously existing natpot file and ii reading of the newly generated reduced natpot file for the evaluation of error measurement The first st
242. s initial functions When the extended Legendre DVR is used an initial associated Legendre function and the corresponding K function which is a Kronecker should be generated via the KLeg and K keywords INIT_WF SECTION build theta KLeg 2 sym K K 1 end build end init_wf section The number after the keyword KLeg denotes the initial 4 and the keyword s ym accomplishes that for the generation of higher single particle functions only every second one will be taken I e there will be only even or only odd s depending on whether the initial is even or odd The keyword sym may be replaced by nosym with obvious meaning The number following K denotes the initial value of K Note that the KLeg and K keywords of the INIT_WF SECTION may also be used when the PLeg DVR is employed 7 4 Generating spherical harmonics as initial functions If spherical harmonics have been employed as primitive basis functions for a combined mode alpha and beta a normalised spherical harmonic Ys ae de m im Yim a8 nee cosa e 7 4 is the appropriate initial function for that mode Here P denotes the associated Legendre function 4 4 Spherical harmonics can be selected similarly to the primitive basis by the INIT_WF SECTION INIT _WF SECTION build alpha sphfbr 0 beta phifbr 0 74 7 Generating the initial wavepacket end build end init_wf section The two numbers specify 7 and m Of course
243. s x q to yield an f p dimensional projected potential Visi Gti ep an ddp X1 41 Xp 4p V q1 qf 12 2 Numerically this is done by evaluating the PES and the projection functions on the DVR grid points 0 and employing the DVR weights wh Volgi oa So wx a wl xp 2 Va a a amp i Qp 12 3 projection will store the resulting Vproj as a vpot file which must then be processed in a subsequent potfit run During the projection run the first p degrees of freedom are removed and so one is absolutely free in choosing the DVR basis i e basis type as well as basis parameters and basis size for these DOFs For the f p remaining DOFs however the DVR basis must match the one for the subsequent potfit run As one is free to choose the DVR basis for the DOFs on which one projects one can en hance the accuracy of the integration scheme 12 3 by choosing it according to the projection functions E g if the projection function is a Legendre polynomial one should use a Legen dre DVR as this will turn the integration into a Gauss quadrature For some of the available projection functions the HTML documentation gives hints on which DVR to choose 12 4 1 Input and output files As both projection and potfit take a PES on the full grid as primary input their input files are quite similar An example for a projection input file is given in example 12 3 You can find a full version with more extensive comm
244. s_r1 and mass_r2 have been correspondingly defined in the PARAMETER SECTION See also Example 6 1 Note that all blank lines and all lines which start with 5 minus are ignored Another example the operator file for a 4 mode model of pyrazine is discussed is Sec 9 This example also demonstrates how to treat non adiabatic systems 6 4 Defining labels In the tableau of the HAMILTONIAN SECTION there must appear only simple labels with or without exponents and products of those I e cos q 2 is a valid entry but q q 2 is not Simply use two lines to perform the sum Symbolic expressions with parameters must not appear in the tableau one rather must link them to a simple label This is done in a LABELS SECTION For example LABELS SECTION bew expcos 1 1 theta0 capl CAP 5 0 0 3 3 J dcos cos1 2 0 5 0 t1 2 V natpot name nd labels section See the Appendix C for a list of symbolic expressions In the parameter list there may appear numbers parameters or simple algebraic expressions of those The different entries may be separated by a comma or a blank The use of units is not allowed here File names etc are given in curly brackets If a path is relative it is interpreted as relative to the location of the input file The entry name simply is a shortcut for the path of the name directory A label may consist of upper or lower case letters case sensitive and numbers Even special characters like
245. sation can be monitored using the keyword pt iming If this keyword is set in the RUN SECTION an additional timing file called ptiming is created containing information about the time spent in each thread and routine This keyword also can be used if usepthreads is not set In this case no ptiming file is created but the timing 3 10 Using parallel shared memory hardware 31 file contains information about the timing of the parallelised routines This helps to decide what routine should be used in parallel mode if the parallelisation is turned on The ptiming file is structured in the following way H 0F propagation Subroutine phihphi calcha mfields summf hlochphilm funkphi Calls oy 168 51 ST 5617 5237 cpu 822 72 11297 70 8054 95 4991 97 11071 72 TEGL sum 822 83 11363 55 8055 37 9529 60 11204 26 726 63 thread 1 thread 2 411 44 411 39 5680 55 5683 00 4026 99 4028 38 4764 80 4764 80 5667 22 5537 04 329493 396 70 The first column shows the name of the parallel subroutine the next column gives the num ber of calls to this routine The column cpu shows how much cpu time is spend for the computation The columns thread p give the real time spend in each thread here 2 for the computations made these values are summed up in column sum In this example the parallelisation of the summf routine works badly This can be seen because the cpu time for the summf evaluation is much lo
246. se any convenient name e g xyz The plot data is then written to the file xyz for later use 6 2 An MCTODH tutorial rv au Figure 2 2 Overlay plot wavepacket on potential The Wavepacket density is shown for the times t 0 10 20 30 fs The density is obtained by integrating over all angles whereas the potential contour lines are obtained by fixing the angle to 2 1 rad Next we want to inspect the wavefunction Go to menu point 10 and chose 5 plot reduced density The density i e integrated over all coordinates except those specified by x and y that is integrated over all angles in the present case will be shown Input a 1 three times and you will see the initial density Pressing RETURN will display the density propagated by one time step and so on After you have returned to the menu chose point 400 Overlay plots and then 410 File for overlay and enter the file name xyz After inputting 1 s you will now see an overlay plot i e the wavepacket on top of the contour lines of the potential With menu points 240 and 245 one may switch off the legend or keys and the title Menu point 285 allows to take larger time steps and with point 280 one may switch to different plot forms e g to plot all time slices at once Such a plot is shown in Fig 2 2 Inspect the ASCII files of the name directories in particular output log and timing The file input contains a copy of the input file the option
247. seconds 3 To perform the scattering calculation type mctdh84 hh2 This will take less than 5 minutes 4 To perform the flux analysis move to the directory hh2 and type flux84 e Isth 0 4 2 0 ev rv This will take less than 10 seconds The option e lsth sets the zero point of the energy to the minimum of the H potential curve The other arguments set the energy interval to 0 4 2 0 eV and select the rv CAP for analysis The results of the calculation can now be inspected Type plflux and you will see the reactive flux i e the quantum flux going into the rv CAP and the energy distribution of the initial wavepacket The reaction probability is just the quotient of these two data sets It can be seen by typing plflux r The results are shown in Fig 2 5 One may compare them with those of reference 11 Inspect the ASCII files of both name directories Isthfit and hh2 The Potfit program will be described in more detail later in this guide The motion of the wavepacket can again be visualised with the aid of showd1d84 In particular the 0 degree of freedom is interesting Type 10 2 An MCTODH tutorial 0 8 m 3 S 0 6 5 xe 9 Q D Q ko 2 3 ic o 0 4 D 5 o y Miss x i oO to 1 Energy eV Figure 2 5 This picture shows the reaction probability of the system H H2 v 0 7 0 for total angular momen tum J 0 solid line which is the quotient of the quantum flux going into the
248. special pre defined meaning A complete list of these is given in the HTML documentation Here we only mention the mass_x label cf Example 6 1 The value of this parameter is taken as the mass of the degree of freedom specified by x as defined in the PRIMITIVE BASIS SECTION of the input file By default the mass is otherwise set to 1 0au The mass_x label can then be used to define the kinetic energy by employing the KE operator The symbol KE belongs to the list of expressions the MCTDH program can interpret These will be explained in the following section Other parameters of special meaning are PI jtot and jbf Remarks e Parameters cannot be re defined A second definition is simply ignored but protocoled in the log file E g if a parameter is defined on the command line then a following re definition in the operator file will be ignored This however makes operations like 6 3 Using symbolic expressions to define the Hamiltonian 51 par part l invalid e Since version 8 2 2 it is allowed to put spaces around and However such spaces are allowed only in a PARAMETER SECTION but not in parameter arithmetic any where else e g not within a parameter bracket of an symbolic expression or function One better does not mix parameter arithmetic and units The statement par parlxpar2 par3 ev is valid The whole expression not only par3 is divided by 27 211 However the statement par parl ev par2 AMU is inval
249. system can be generated using autospec84 This reads the autocorrelation function from the file auto which is created if the keyword auto is included in the RUN SECTION of the input file The spectrum is then created over a chosen interval in energy space by Fourier transform The shell script plspec calls autospec84 and then GNUPLOT It is often more convenient to use this script For example the NOCI spectrum can be displayed by typing plspec 0 6 2 0 ev This spectrum was produced in the tutorial and is shown in Fig 2 1 Before the autocorrelation function is Fourier transformed it gets modified To reduce artifacts of the Gibbs phenomenon the autocorrelation function is multiplied with a filter function cos mt 2T where n 0 1 2 and where T denotes the final time plus one time step of the autocorrelation function Due to the t 2 trick see Eq 167 of the MCTDH review the propagation time is only T 2 The columns 2 4 of the spectrum pl file list the results for the different n s When the option lin is set for autospec84 then a second set of filters is used In plspec the choice of the filter is made trough the option g0 g5 the filter g1 is default See the HTML docu For a comprehensive discussion of the filters see the lecture notes INTRODUCTION TO MCTDH chapter 1 3 The lecture notes can be downloaded from the literature downloads site which is part of the MCTDH web site Here we only list the full widths at half
250. t approaches the end of the grid It is defined as i W where W O k a 20 k x zxo and where denotes the Heaviside s step function The input parameters are x o n n k where k is used to choose to which end of the grid the CAP is placed k 1 puts the CAP at the left and k 1 at the right of the grid k 1 is default and may be left out 3 ACAP symbolises an automatic CAP The ACAP is useful when one wants to place the initial wavepacket at a position where it overlaps with the CAP The ACAP remains disabled as long as the wavepacket overlaps with the CAP The ACAP is enabled only when the wavepacket starts to re enter the region where the CAP is defined There is a fifth parameter timecap If this optional parameter is set the ACAP will remain dis abled at least as long as time lt timecap where time is the propagation time in fs The parameter timecap is useful because the automatic enabling of the CAP may some times happen too early The time at which the ACAP is switched on is protocoled in 144 C The built in symbolic expressions the log file Use this information to set the option 1o in flux84 appropriately When flux84 is run it must not evaluate matrix elements of the CAP for times at which the CAP is switched off The symbols step and rstep symbolise a Heaviside s step function and the reverse of it I e step p O a p and rstep p 1 O a p O p z The lo
251. t there are two degrees of freedom the SPF BASIS SECTION SPF BASIS SECTION X 3 Y 3 end sbasis section assigns three single particle functions to each mode Here X and Y are the mode labels which must coincide with those in the PRIMITIVE BASIS SECTION If only one single particle function is used for each degree of freedom a calculation ac tually employs the time dependent Hartree TDH method rather than the MCTDH scheme Another special case is obtained for equal numbers of single particle and primitive basis functions in each degree of freedom x i e ny N with N being the number of prim itive basis functions An MCTDH calculation is then equivalent to a numerically exact one but note that numerically exact calculations can be performed much more efficiently by using the exact keyword see Sec 3 6 Typical numbers of single particle functions range from n N 20 ton N 3 Note that the numbers of single particle functions should obey f ny lt x 5 1 wt 1 since otherwise there will be redundant configurations in the MCTDH wavefunction Hence one must choose n n if there are two degrees of freedom 45 46 5 Defining the single particle basis 5 2 Advanced topic Selecting degrees of freedom from a large system For some systems it may be useful to treat only a subset of the specified coordinates The program allows one to select coordinates simply by listing only those required in the SP
252. taining negative energies compare with Fig 2 4 11 7 Computing excitation and reaction probabilities 105 11 7 Computing excitation and reaction probabilities The analysis of scattering processes i e the computation of excitation and reaction prob abilities is performed by evaluating the quantum flux going into a particular channel The quantum flux is determined by the interaction of the time dependent wavepacket with a CAP The program flux84 performs the necessary analysis The psi file is read and the energy re solved flux is computed and written to the flux file Additionally this flux is divided by the energy distribution of the initial wavepacket to obtain the transition or reaction probabilities The latter step requires in general that the mctdh84 program has computed the energy distri bution of the initial wavepacket and has written it to the enerd file see Section 7 11 NB The file enerd is called adwkb in older versions Besides the flux file flux84 creates the files flux log gtau and wtt When the file gtau is present the flux program skips the time con suming evaluation of the integrals Y t W W t 7 but reads the function g r from the gtau file See the MCTDH review section 8 6 3 Eq 199 for more details This allows to re do the Fourier integrals for another energy interval very quickly The option w enforces the re calculation of g r The file wtt contains the expectation values Wy U t W W t
253. tem density 111 0 stop 1 plot to screen 2 print plot 3 save plot to a postscript file 4 save plot to a gnuplot file use after 1 or 2 5 save data to an xyz file 9 toggle re plot get new set of contours 10 change plot task plot reduced density 20 change coordinate section rd x rv y theta 1 545 Time 0 000 30 change coordinate bounds 40 show coordinate info 50 change a single coordinate 80 change coordinate units 90 change Z axis units au 110 toggle contour mode linear 120 change number of contours 21 150 toggle grid off 160 toggle surface off 170 toggle contour lines on 240 toggle key on 245 toggle title on 250 toggle show points off 260 toggle stick spectrum off 270 toggle smooth curve off 280 change time slice format e g movie step through 285 change time step 1 290 toggle no weigths off 400 Overlay plots off 900 toggle gnuplot output format on 910 change printer lpr 920 change GNUplot command Example 11 6 The start up menu in the showsys program to enable interactive plotting of the system density and potential energy surfaces 11 9 Plotting 2D cuts through the system density The program showsys can be used to display one and two dimensional cuts through the system density using the psi file This must have been generated during a propagation run by including the psi keyword in the RUN SEC
254. the DVR functions may also be read in See the HTML documentation Hamiltonian Documen tation Labels Section for details 6 6 Advanced topic Defining new symbolic expressions When a new operator with a new label is to be implemented this operator must be added to the operators in the source opfuncs directory Operators are divided into four classes which 6 6 Defining new symbolic expressions 55 are handled differently by the program The first of these are grid based operators such as the kinetic energy operator in a DVR basis or a natural potential expansion operator which are only defined with respect to a grid The remaining three classes are all different types of analytic functions complex functions e g CAPs multi dimensional functions e g non separable potential energy surfaces and real one dimensional functions In this and the following section we will give some examples as to how to implement real one dimensional functions The implementation of multi dimensional functions is the topic of Sec 6 8 Important note Since version 8 3 10 there are four new files install user_surfaces install user_surfdef opfuncs user1d F and opfuncs usersrf F These were introduced to sep arate changes done by a user from changes done by the authors This new procedure will simplify an update of the package When implementing a new one dimensional potential function please use now opfuncs user1d F Otherwise the procedure
255. the PARAMETER SECTION The given formula is only valid if a Legendre DVR with magnetic quantum number m 0 is used OP_DEFINE SECTION title Henon Heiles PES end title nd op_define section PARAMETER SECTION mass_X 1 0 mass_Y 1 0 lambda O22 nd parameter section HAMILTONIAN SECTION modes X Y L O KE 1 1 0 1 KE 0 5 q 2 1 0 5 1 q 2 lambda q q 2 lambda 3 q 3 1 lambda 2 16 q 4 i lambda 2 16 1 q 4 q 2 q2 lambda 2 8 end hamiltonian section end operator Example 6 2 An operator file for a wavepacket propagation using the modified Henon Heiles Hamiltonian 6 4 Defining labels 53 PRIMITIVE BASIS SECTION of the input file The coupling parameter A and the masses used for the KE keyword are defined in the PARAMETER SECTION The second example is the kinetic energy of a three atomic molecule with total angular momentum J 0 described by Jacobian coordinates r1 r2 and 0 The kinetic energy reads sin 0 0 g 06 o gt 1 8 1 8 amp 2 1 1 _ 1 1 Qi r Quo Or2 2 mr mr where u and u2 specify the reduced masses associated with r and r2 The representation of T in the HAMILTONIAN SECTION is given by HAMILTONIAN SECTION modes i 2d x2 theta J0 KE il T T0 it Si KE ol 0 5 mass_rl q 2 1 472 0 5 mass_r2 1 q 2 4 72 end hamiltonian section Here we have assumed that the keywords mas
256. the j oper ator see Table C 2 J is fixed cm yI J 1 K K 1 Cy symbol appearing with the j oper ator see Table C 2 J is fixed myld user supplied routine See Note 8 of Table C3 C The built in symbolic expressions 135 Table C 2 Operator symbols which require no arguments The expression n can be replaced by any positive integer Finally m is the mass of the relevant degree of freedom In the MCTDH input this mass is given by the reserved parameter mas s_modelabel If mass_modelabel is not explicitely set it is 1 by default Symbol Operator Notes dq Or first derivative Cannot be used for rHO Leg KLeg PLeg Wigner or sphFBR dq 2 82 second derivative Cannot be used for Leg KLeg PLeg Wigner or sphFBR p 10z momentum deprecated use dq KE 5z 0 Kinetic energy term Cannot be used for modes with a Legendre DVR or sphFBR j2 sin t 6 3 sin Op Angular momentum squared In this form sin 0 o used for sphFBR and PLeg For Leg and KLeg a is replaced by m or K jp et Oy icot A 3p Angular momentum raising operator j7 Only for KLeg and PLeg For Wigner see below jm e 09 icot 0 p Angular momentum lowering operator j_ Only for KLeg and PLeg For Wigner see below jpm CyK I CK j Combined angular momentum operator C Fg are defined in Table C 1 Only for KLeg and PLeg cjpm Cy G14 52 4 Combined operator for two angular CyK f1
257. theta Leg 60 0 all in the PRIMITIVE BASIS SECTION The first number specifies the number N of basis functions or grid points the second one denotes the magnetic quantum number m The last keyword controls which symmetry is to be used For all all values 1 m m 1 m N 1 are employed for odd even only odd even values of l are taken i e l m m 2 m 2N 2o0rl m 1 m 3 m 2N 1 4 4 Sine DVR The sine or Colbert Miller DVR uses the particle in a box eigenfunctions as a basis The box boundaries are xo and xy 41 and L xy 4 1 xo denotes the length of the box The basis functions are thus wtp ae sin jr x xo L vee STILEN 4 5 Note that the grid boundaries xo and xy 1 do not belong to the grid since the wavefunc tion vanishes there by construction The sine DVR has been successfully used in MCTDH calculations for vibrational angular and dissociative degrees of freedom A sine DVR is turned on for a coordinate named say r by the line xr sin 24 1 0 24 0 short in the PRIMITIVE BASIS SECTION Again the first number denotes the number N of grid points The meaning of the next two numbers depends on the last optional keyword which can be short the default or long When short is selected the two numbers are the first and last grid point x and xy If the keyword long is present the two numbers specify the box boundaries zo and xy 1 The line r sin 24 0 0 2530 long is hence equivalent to the
258. they are larger than 0 2 0 5 1 0 2 0 5 0 and 10 cm respectively As one notices the eigenenergies obtained with the first run are not too accurate In particular the last four states are quite bad showing deviations of more than 10 cm The second calculation blk 2 however shows rather good results All deviations are below 1 cm and in most cases the deviations are even below 0 1 cmt To compare block with single relaxation separate single relaxations were performed us ing the eigenstates of the block relaxation as start vectors The small set 9 4 16 18 of SPFs was used and the results are shown in the column sing 1 The results of the single relax ations are much improved as compared to the corresponding block relaxation except for the last four states and for the excited states no 18 and 23 These states as well as state 36 did not converge i e the energy kept on oscillating The energies displayed in this case are some mean value In the block relaxation the SPFs are optimized to represent all 40 states under consideration whereas in the single relaxation they are optimized for a partic ular eigenstate Obviously more SPFs are needed in a block relaxation to obtain results of similar quality However the single relaxations took between 30 s and 3 min each depending on the state to be relaxed In total they took 1 h CPU time which is similar to the 3 h used by the second block relaxation Remembering that for the sing
259. ti M H Beck G A Worth and H D Meyer A hybrid approach of the multi configuration time dependent Hartree and filter diagonalisation methods for computing bound state spectra Application to HO2 Phys Chem Chem Phys 3 2001 1576 1582 S Mahapatra G A Worth H D Meyer L S Cederbaum and H Koppel The APE BBs photoelectron bands of allene beyond the linear coupling scheme An ab initio dynamical study including all fifteen vibrational modes J Phys Chem A 105 2001 5567 5576 C Cattarius G A Worth H D Meyer and L S Cederbaum All mode dynamics at the conical inter section of an octa atomic molecule Multi configuration time dependent Hartree MCTDH investigation on the butatriene cation J Chem Phys 115 2001 2088 2100 H Wang M Thoss and W Miller Systematic convergence in the dynamical hybrid approach for complex systems A numerical exact methodology J Chem Phys 115 2001 2979 M Thoss H Wang and W H Miller Self consistent hybrid approach for complex systems Application to the spin boson model with debye spectral density J Chem Phys 115 2001 2991 C Meier and U Manthe Full dimensional quantum study of the vibrational predissociation of the Iz Neg cluster J Chem Phys 115 2001 5477 F Huarte Larrafaga and U Manthe Vibrational excitation in the transition state The CH4 H gt CH3 Hz2 reaction rate constant in an extended temperature interval J Chem Phys 116 200
260. ti dimensional potential is not of disadvantage if it operates on the coordinates of one MCTDH particle combined mode exclusively See Section 6 13 Another application of non separable potentials is the MCTDH method in combination with the CDVR scheme see Sec 8 4 3 If the CDVR method is to be used during the prop agation the keyword analytic_pes should be included in the OPERATOR SECTION This means that the generated potential operator will not be explicitly calculated on the prim itive grid points but will be stored in the oper file in an analytic form which can be evalu ated on the fly at any point in coordinate space For the convenience of the user there is already a dummy routine source surfaces mysrf f and one merely places the code of the new potential energy surface there When editing mysrf f one finds a brief description of how to make the necessary modifications on the file source opfuncs usersrf F This simplified procedure is convenient for a quick implementa tion of a new surface However it does not allow to pass surface options to the program If options are needed or if more than one potential energy surface is to be implemented one has to go the proper slightly more elaborate way described next However the most easiest way to include a multi dimensional potential into MCTDH or POTFIT is via the readsrf keyword In the LABELS SECTION of an MCTDH operator file there would appear the statement Vmd readsrf data
261. ticle interaction N N 1 2 entries 1 amp 2 vv g 1 amp 3 vv g 2 amp 3 vv end hamiltonian section HAMILTONIAN SECTION_spo modes x1 Kinetic energy 1 0 KE Harmonic trap 0 5 q72 end hamiltonian section LABELS SECTION vv natpot ngaussHOfit_N125 ignore pot fitted interaction potential end labels section ignore is set to ignore the modelabels of the fit end operator Example 10 1 An operator file for N 3 one dimensional bosons in a harmonic trap 10 2 Modifying the input 95 Hat ot a a a a HE aE HE aE HE aE EE HE aE HE HE EE HE aE HE AE HE EE aE HEE HE a aE EE aE HE aE EE aE HE aE aE aE EEE EEE aE EE 3 bosonic particles 1D in a harmonic trap HHH Heat Ht a HE a aE EE HE aE HE aE EE HE aE HE A HE EE HE aE HE AE HE EE aE HEE HE a aE aE EE HE aE EE aE HE aE EEE aE EE EE EE EE RUN SECTION name p3d1_20 nergy not ev time not fs A dimensionless model is treated relaxation 0 rlxunit au tfinal 20 0 tout all tpsi 1 0 gridpop steps cross orben title 3 particles 1D in harmonic trap p3d1_DW end run section OPERATOR SECTION opname p3dl alter parameters g 10 nd alter parameters nd operator section SPF BASIS SECTION xl 15 x2 id 1l x3 id 1 nd spf basis section PRIMITIVE BASIS SECTION x1 HO 125 xi xft 40 4 0 x2 HO 125 xi xf 4 0 4 0 x3 HO 125 xi xf 4 0 4 0 end primitive bas
262. tions in ultracold trapped few boson systems Tran sition from condensation to fermionization Phys Rev A 74 2006 063611 160 List of MCTDH references 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 G Pasin C Iung F Gatti and H D Meyer Theoretical investigation of highly excited vibrational states in DFCO Calculation of the out of plane bending states and simulation of the intramolecular vibrational energy redistribution J Chem Phys 126 2007 024302 T S Venkatesan S Mahapatra H D Meyer H K ppel and L S Cederbaum Multimode Jahn Teller and Pseudo Jahn Teller interactions in the cyclopropane radical cation Complex vibronic spectra and nonradiative decay dynamics J Phys Chem A 111 2007 1746 S Z llner H D Meyer and P Schmelcher Excitations of few body systems in one dimensional har monic and double wells Phys Rev A 75 2007 043608 C Matthies S Z llner H D Meyer and P Schmelcher Quantum dynamics of two bosons in an anharmonic trap Collective versus internal excitations Phys Rev A 76 2007 023602 M R Brill F Gatti D Lauvergnat and H D Meyer Photoinduced nonadiabatic dynamics of ethene Six dimensional wave packet propagations using two different approximations of the kinetic energy operator Chem Phys 338 2007 186 199
263. tive product grid i e in the same way as in a numerically exact calculation A diagonalisation run therefore requires the same input sections as a numerically exact calculation and of course an operator file A possible RUN SECTION reads RUN SECTION diagonalisation 10000 name results end run section The number associated with the diagonalisation keyword indicates the number of Lanczos iterations to be made The name keyword has been explained in Sec 3 2 Other keywords that may be specified in a diagonalisation run can be found in the HTML docu mentation The computed eigenenergies and intensities together with an error estimate of the eigenenergies are compiled in the eigval file If the error of the desired eigenvalues is to large the calculation can be continued to increase the number of iterations See Sec 3 9 for continuing calculations Note that one dimensional Hamiltonians can be numerically exactly diagonalised using a DVR basis See Sec 7 6 for more details of this 3 8 Starting a calculation The MCTDH program is started by typing mctdh84 myinput on your console where we assumed that your input file is named myinput inp If the input file is not stored in the current directory add the correct path to the input file s name A variety of options may be used on starting the MCTDH program Type mctdh84 h to get an overview To find out during run time how far a propagation is proceeded you may loo
264. to generate overlay plots i e plotting a density on top of the contour lines of the potential Before using this menu point the potential plot data has to be stored to some file by using menu point 5 See the tutorial Sec 2 1 and Fig 2 2 for an example 11 10 Plotting cuts through the potential energy surfaces The program showsys can be used to display one and two dimensional cuts through the potential energy surfaces of the system Before this can be done the MCTDH program must be used to generate a pes file from the Hamiltonian information in the operator file see Chapter 6 The pes file is an operator file from which all terms are removed which contain derivative operators or CAPs To generate a pes file set up an input file specifying the system and operator with a PRIMITIVE BASIS SECTION A SPF BASIS SECTION and an OPERATOR SECTION While no information is required about the single particle function basis the SPF BASIS SECTION is required as it also defines which degrees of freedom defined in the PRIMITIVE BASIS SECTION are included In the RUN SECTION the keyword genpes is then re quired along with the name of a directory in which to store the new file Now the MCTDH program is run and the file name pes generated The op log pes file contains information on the function that has been set up and the log is now called log pes This makes it possible to use the same name directory as the propagation run Rather than editing t
265. tonian by LABELS SECTION V srffile nocll directory nd labels section Here directory denotes the path to the directory containing the nocl1 srf file Replacing the directory by the keyword oppath indicates that the surface file is in the same directory as the op file The keyword default will make the program look for the srf file in the default operator directory Using again the alter label keyword in the OPERATOR SECTION of the input file one may select a different potential with a minimum of effort Finally we note that is is possible to impose an energy cut off on a non separable potential energy surface by using the v keyword in the operator section of the input file This is detailed in the HTML documentation 6 12 Setting up Auxiliary Operators In addition to the system Hamiltonian other operators may be required e g to generate eigenfunctions of a zero order Hamiltonian for the initial wavepacket see Sec 7 6 or to calculate the time evolution of an expectation value either using the expect keyword in the RUN SECTION or the ANALYSE program EXPECT Operators needed during a run must be included in the op file They are defined exactly as the system Hamiltonian but are delimited by HAMILTONIAN SECTION_XXX end hamiltonian section where XXX is a label to distinguish the operator Operators to be used in any post propagation analysis can also be set up in a separate op file i e one not containi
266. ts at the ends of the grid In the directory containing the data files typing rdgpop84 results in some information about the calculation and the primitive basis used for each degree of freedom in the calculation The question Number of grid points to be summed over nz then appears If 1 is input then the populations output are those on the end grid points If 2 is input the output population for the beginning of the grid is the sum of the populations of the first and second points while the output population for the end of the grid is the sum of the populations of the last and last but one points And so on The next question asked is 100 11 Analysing the results employing the Analyse programs Runing number of degree of freedom dof dof 0 gt Print only maximum over time Selecting 0 here results in the maximum population at the end grid points being displayed Maximal values all times final time 30 00 fs dof grid begin grid end basis begin basis end 1 rd 000245781 000000064 260884702 000002414 2 ew 0 000000000 003499307 972981513 000017179 3 theta 0 000000000 0 000000000 150721535 000005773 We see that the beginning of the rv grid is unpopulated and this grid point could be re moved without affecting the propagation quality Likewise the ends of the theta grid are unpopulated To avoid answering the questions this result could be obtained by calling the program as rdgpop84 1 0 If th
267. u work under bash although it is possible to work under C shell or kshell as well A bash however must exist as there are several bash scripts Moreover for some tasks like OCT or Cluster Expansion python scripts are used Hence python 2 4 or higher but NOT python 3 should be installed One should begin with creating a directory MCTDH which eventually will contain all the MCTODH stuff but not at least I prefer to do so the output of production runs Move the MCTODH tar ball to the MCTDH directory and unzip and untar it I e mkdir home muser MCTDH cd home muser MCTDH mv lt path gt mctdh84 x tgz tar xzvf mctdh84 x tgz Here it is assumed that your login name is muser and that you have a GNU tar If you do not have a GNU tar you first have to gunzip the tar ball and then untar it without the option z The symbol x stands for the revision number of the particular package which was downloaded When the tar command is finished there should exist a directory mctdh84 x under home muser MCTDH If you are familiar with Subversion svn it will be more convenient to download the code form the svn repository of the Heidelberg MCTDH package For details see Appendix F After the code is downloaded move to the directory mctdh84 x install and run check system cd mctdh84 x install check_system This will create an output like 147 148 E Installing the MCTDH package KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKK
268. uded it simply to give a realistic example e More generally a number or Fock state n1 n2 can be selected which denotes how many particles na N occupy a given single particle mode Ya where Xa Ng N Cast in standard MCTDH form this a sum over all permutations of the single configuration J Chesney teers ees r vr n X Nn X This is a little more cumbersome since we have to keep track of all permutations of J A typical statement from the INIT _WF SECTION now reads 10 2 Modifying the input 93 build xl eigenf spo x2 map x1 x3 map x1 end build A coeff 2n 2 3 CE 0700 end A coeff symcoeff Note that the last block is the same as for the Hartree product above telling MCTDH to select eigenstates Ya of the single particle Hamiltonian spo On top of that the block A coeff defines just which orbitals a should be selected In our example this produces a single configuration J 2 2 3 forget about normalization To make this permutation symmetric i e a number state n 0 n2 2 n3 1 the statement symcoef f has been added If one performs a series of relaxations e g with increasing interaction parameter g then the most convenient choice for the initial state is one already obtained in some previous MCTDH calculation This is done by simply casting the INIT WF SECTION as follows file p3d1_19 symcoeff This ensures that the restart file from the directory p2d1_19 is read as i
269. uments must be assigned to a simple label in the Labels Section With the aid of the caret one may apply a power to operators The power may be integer or real and may carry a sign This however works only for potential like operators Inspect Appendix C to learn which operators can be exponentiated Note that symbols like dq 2 or j 2 are operator labels of their own right they do not denote that the second power of the operators dx or j is taken literally Time dependent operators can easily be implemented The time is simply treated as an additional DOF of the Hamiltonian Section The modelabel of this additional DOF must be Time See the HTML documentation Hamiltonian Liouvillian Documentation and then Time dependent Operators for further details See also the operator file nocl1T op on MCTDH_DIR operators Chapter 7 Generating the initial wavepacket For a quantum dynamical calculation an initial wavepacket 0 is required This is done in the INIT_WF SECTION of the input file The initial wavepacket must have a particular form depending on the method to be used If the MCTDH scheme is employed 0 has to be represented as a multi configurational Hartree product i e a linear combination of products of single particle functions while in a numerically exact calculation it must be mapped onto a product of DVR grids Usually 0 is a simple Hartree product i e a product of one dimensional functions unless spherical ha
270. unctions 45 5 2 Selecting degrees of freedom from a large system 46 5 3 Combining modes to produce multi dimensional single particle functions 46 6 Setting up the Hamiltonian 48 6 1 TPhe operator files niorse eb ee A ea eae av Ed 48 6 2 Defining numerical constants 2 000000004 49 6 3 Using symbolic expressions to define the Hamiltonian 51 6 4 D fininglabels ar sonden he a OR E OE BOE Aa 53 6 5 Implementing user defined 1D operators 54 6 6 Defining new symbolic expressions 004 54 6 7 Implementing separable potentials 0 56 6 8 Implementing non separable potentials potential surfaces 58 6 9 Incorporating natural potentials 2 000 61 6 10 Using complex absorbing potentials CAPs 62 6 11 Altering a Hamiltonian from input file or command line 2 64 6 12 Setting up auxiliary operators 2 2 ee ee ee 66 6 13 DOF mode and muld potentials 0000 4 67 6 14 Golden rules for writing operator files 0 69 7 Generating the initial wavepacket 71 7 1 Building Gaussian functions as initial functions 71 7 2 Setting up Legendre functions as initial functions 72 7 3 Setting up extended Legendre functions as initial functions 73 7 4 Generating spherical harmonics as initial functions
271. uration files to compile cnf the default choice is compile cnf_le The file compile cnf is then read by the compile script which is used to compile individual programs e g compile mctdh or the full package compile all If your compiler does not support pthreads you have to choose compile cnf_lenp or compile cnf_benp We have tried to find reasonable options for the compilers see in particular MCTDH_FFLAGS_OPT but we cannot account for any hardware and software installation on which MCTDH may run Hence depending on your particular hardware software installation the choice of compiler options may not be optimal To inspect the compiler option run compile config Feel free to adjust the compiler options to your particular installation If you add a new compiler please send us the updated compile cnf file If you are working on a system where computers of different kind 32 bit 64 bit Linux other Unix including Mac OS X are interconnected by a common file system you may store and install the MCTDH package only once but run compile on each kind MCTDH is smart and will load automatically the correct executables Run menv on interconnected computers of different kind and you will see that the paths are set differently In general each MCTDH user works with his own package This allows him to change the code according to his demand However sometimes it may be wanted that several users have access to the same package In this case t
272. ute or relative names are allowed until the directory of choice is found Then type no to return to the main menu This option may also be used to list the contents of the present directory If one knows the MCTDH package well it is more convenient and faster to use directly the routines which are called by the analysis interface But for the beginner analysis can be a big help as one is guided through the large selection of analyse tools Note how ever that there are more tools available than accessible through analysis84 See the HTML documentation 11 2 Interpreting the MCTDH output During the propagation of a wavepacket information about the system evolution is output to allow an easy visual check of how the calculation is progressing If the keyword output was included in the RUN SECTION of the input file this information is written to the file output in the directory specified by the name keyword If this keyword was omitted this information is written to the screen during the calculation A section of the output from our example NOCI propagation is shown in Example 11 2 After information about the program version used and where and when the calculation was run starts the information about the system This is output every t fs where tout t is the time specified in the RUN SECTION of the input file At each output step the following information is give e Norm The norm of the wavefunction expansion coefficients
273. ve this one sets tout all In this case one should also give tpsi even if the psi file is not written The tpsi time is taken as the upper limit for the update interval When computing the ground state the Davidson diagonaliser is usually quite fast i e requires only few iterations Turning to excited states however the number of Davidson iterations may become quite large In order to speed up the calculation in this case a better pre conditioner was implemented If one sets precon N then an N x N block of the Hamiltonian matrix is diagonalised and used to improve the pre conditioner One should be careful not to use a too large value for N otherwise the build up of the pre conditioner takes more time as saved by performing a smaller number of Davidson iterations It is usually not useful to use precon when the ground state is computed When computing higher excited states on the other hand the pre conditioner can be very helpful When excited states are to be computed one usually uses relaxation lock This however requires that an initial state is provided which has a decent overlap with the state to be computed There are two convenient ways to generate such an initial state The first one is to operate with some excitation operator on the ground state or on some converged excited state Compare with the co2_x inp input files on MCTDH_DIR inputs The other way is to diagonalise appropriate 1D operators with eigenf or mode operators with
274. vectors integration order is equally distributed over the available nodes intorder of MPI prozesses If I is set to a higher value this can lead to smaller communication costs because the vecors are held on a smaller number of prozessors As in the case of the shared memory parallelisation the results of a parallel calculation may slightly differ from those of a non parallel one due to numerical reasons An example for a RUN SECTION in an MPI parallel run is shown in the following Example RUN SECTION usempi no dsyev end run section Here the parallelisation of the diagonalisation routine DSYEV is turned off The number of prozesses is not specified in the RUN section as is must be done in the shared memory parallel case This is done via the mpi run command see above In the case of the distributed memory parallelization not all parts of MCTDH are paral lelized only the mean field computation and the A vector propagation are parallel but not the SPF propagation This is due to the fact that large calculations are A vector dominated ones in general Hence calculations with a considerable contribution from the SPF propaga tion to the cpu time are not suited for the MPI parallel MCTDH The best tested case was the H5O propagation where up to 1024 prozessors were used This calculation showed a maximum speedup of 118 this corresponds to a parallel part of over 99 If the keyword ptiming is set for an MPI parallel run the file mpit
275. ween natural potentials defined in different primitive grids Therefore the user may collect the ab initio data in a rather sparse primitive grid that can be later be interpolated into a more suitable one for the dynamical simulation phase These operations will be discussed in detail in this chapter The practical implementation of the MCTDH algorithm uses DVR s for the primitive basis whose points define the primitive grid see Chapter 4 It is assumed in our discussion that the value of some property e g the potential energy has been collected on the points defined by the primitive grid using some external program One has to differentiate between the actual points of the primitive grid in each coordinate and the associated value of a certain property As will be immediately seen both pieces of information are given separately to the programs that have to use them There are mainly three ways to use ab initio data in a MCTDH calculation 12 2 1 Using ab initio data directly with the mctdh program This is the least flexible of the possibilities being discussed but the concepts that will be introduced apply equally to the other procedures It corresponds to the direct path to the usage of MCTDH as depicted in Fig 12 1 First let s assume that we have the ab initio data values at the points defined by some primitive grid The primitive grid being used should correspond for each coordinate with some of the DVR s defined in the MCTDH
276. wer than the sum of the realtime spend for this task Using the summf2 keyword this problem can be fixed The corresponding line in a summf2 run indicates that the parallelisatioin now works better Subroutine Calls cpu sum thread 1 thread 2 summf 188394 3809 27 3868 90 1917 64 1951 25 Ignore the number of calls It is increased because now not summf is timed but an internal routine called by summf More things can be seen with help of the ptiming files Some times the parallelisation creates cpu time overhead This cannot be discoverd by checking only one ptiming file In the case of the C2H4 propagation the routines controlled with the no funkphi keyword produce overhead But the corresponding ptiming file is Subroutine Calls cpu sum thread 1 thread 2 funkphi 121389 606 69 627 99 308 03 319 96 Here the columns cpu and sum compare very well But the ptiming file for one thread i e usepthreads 1 reads Subroutine Calls cpu sum thread 1 funkphi 120780 174 36 LI S23 175 23 Comparing the cpu column of both files a strong increase of cpu time is discovered 174 36s 1 proc to 606 69s 2 proc Hence the no funkphi keyword should be used to avoid this overhead Finally we give some overall timings 1 processor 4 processors default no summf summf2 H503 9h 52m 20s 3h 50 m 52s 3h 52m 06s 3h 09m 49s default no funkphi C2H4 22m 21s 12m 20s 8m 49s If the
277. wering operator corresponding to a harmonic oscillator is given by i i mu FP gt d 40 V2mw 2 where p denotes the momentum operator q denotes the position operator m is the mass w is the frequency and qo is the equilibrium position The input parameters are m w s which means the mass m the frequency w and the shift Mw Qq g 40 Note the minus sign The corresponding raising operator is given by bt and the number operator by btb The parameters have the same meaning as for lowering operators NB The lowering raising and number operator require the use of a simple DVR with an ordinary first derivative e g sin HO or exp but not FFT rHO Leg KLeg PLeg or sphFBR b The approximate raising lowering operators R L for a Morse Hamiltonian H p 2m D e200 gale are defined as and R L with w a 2D m and A V2Dm ha The centrifugal potential given by J J 1 2K 2MT2 Veent min esma An arbitrary real 1D function may be defined through a set of points The points are read from file file and are then interpolated to define a general 1D function The data is in free format with one x y data pair per line Blank lines and lines which start with a are ignored Currently the x data called time in the code must increase linearly i e must be equally spaced An arbitrary number of these with different data files may be used
278. word sphFBR selects the spherical harmonics FBR and the label phiFBR indi cates the second coordinate on which the FBR is based The number jmax after the keyword sphF BR or defines the maximum value for 7 The j values are 7 0 1 Jmax for nosym and j 0 2 jmax Or j 1 3 Jmax for sym depending on the parity of jmax With out the optional data max which follows the keyword phiFBR m takes on the values m j j 1 1 0 1 7 1 7 With mmax and possibly also Am given m takes the values m min Mmax J Min Mmax j Am min mmax j Am min Mmax j In the second example we thus have j values of j 0 2 4 6 8 The corresponding values for m are m 0 for j 0 m 2 0 2 for 7 2 and m 4 2 0 2 4 for 7 4 6 8 giving an overall number of 19 basis functions The order in which the primitive basis sets are declared is in general arbitrary However a sphFBR line must be followed directly by a phiFBR line Moreover the corresponding degrees of freedom alpha and beta in the example above must be declared as combined in the SPF BASIS SECTION 4 7 Advanced topic Restricted Legendre DVR The restricted Legendre DVR Leg R is very similar to the ordinary Legendre DVR but can make use of the fact that the angular motion may be restricted to an interval smaller than 0 7 In such a case one may drop the unused grid points assuming that the wave function vanishes the
279. y are the output auto and psi files The output file contains some basic physical quantities of the wavefunction such as norm energy state populations and the position and momentum expectation value of each coordinate The auto file contains the auto correlation function as a function of time Both files are in ASCII for mat In the psi file the wavefunction as a function of time is stored in binary format The three files are selected by placing the keywords output auto or psi in the RUN SECTION Examples for the RUN SECTION are given in the following sections A complete list of the available files and options can be found in the HTML documentation The files gridpop check steps update timing and speed only serve to check the effi ciency or accuracy of an MCTDH calculation They are very useful during the test phase of your calculations Since some of them might become rather large however you may turn them off for your production calculations NB the files output timing speed and for CMF runs update are opened by default To turn them off use the keywords no out put or screen no timing etc 3 3 Propagating a wavepacket 23 3 3 Propagating a wavepacket For performing a wavepacket propagation using the MCTDH method you first have to set up the Hamiltonian in an operator file see Sec 6 This operator file must then be specified in the OPERATOR SECTION of the input file see Sec 6 1 In the input file the primitive and
280. ys the integrators being compatible with the CMF method Since the MCTDH coefficients i e the A vector and the single particle functions i e the y vector are propagated separately in the CMF scheme different integrators can be chosen for each of them This is indicated by appending A or spf to the ABM BS RK5 RK8 or SIL keyword The default is STL A and BS spf which is in general the most efficient combi nation An example for the INTEGRATOR SECTION in the input file is INTEGRATOR SECTION CMF 0 5d0 1 0d 6 SIL A 15 1 0d 7 BS spf 9 1 0d 7 end integrator section This starts a CMF calculation with an initial stepsize of 0 5 fs and an error tolerance of 10 The parameters for the SIL and BS integrator are described in Sec 8 3 If the same integrator e g ABM is to be used for the MCTDH coefficients and the single particle functions the shortcut al1 can be appended to the integrator keyword INTEGRATOR SECTION CMF 1 0d0 1 0d 5 ABM all 5 1 0d 4 0 05d0 end integrator section Note however that the ABM integrator typically will not give you the optimal performance of the CMF scheme As a final example the CMF scheme may also be selected by INTEGRATOR SECTION CMF end integrator section The program then uses default integrators and parameters which are compiled in the HTML documentation 82 8 Choosing an integration scheme In the above examples two optional para

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