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nMOLDYN: User's Guide - Theoretical Biophysics, Molecular

1. ox Figure 2 21 Incoherent Scattering Function Gaussian Approx of the required q vectors through the text field named Q values using the form Qmin AQ Gmax In this way the intermediate scattering function will be calcultad for qm values defined as qm Qmin m Aq with m running from 0 to N 1 and qmaz Ng 1 Ag NMoldyn can calculate for each atom the mean square displacement with respect to a given axis by entering in the Projection Vector field the com ponents of the corresponding unit vector in the form z y z Otherwise writing None the MSD will be calculated by averaging over all the directions Attention the values Vectors per shell and Q shell width are not used to carry out the computation of the intermediate incoherent scattering function in the Gaussian aproximation since no averaging over different g vectors needs to be performed The relative fields must be ignored 40 In the panel Weights the user can select how to weight the different contri butions to the Intermediate coherent scattering function coming from all atoms of the system see Section 2 2 The panel window width for FFT percento of trajectory lenght al lows one to select the width of the Gaussian function to be used in the smoothing procedure for the calculation of the Fourier spectrum of the intermediate scat tering function see Section I The width is defined with respect to the l
2. coherent Q units 1mm v V Output file text EISF_spce300K_1 bar_1C Browse OK Cancel Figure 2 24 EISF 50 Chapter 3 Theoretical background 3 1 Dynamic structure factor The quantity of interest in neutron scattering experiments with thermal neu trons is the dynamic structure factor S q w which is closely related to the double differential cross section I d 0 dQdE The double differential cross section is defined as the number of neutrons which are scattered per unit time into the solid angle interval Q Q dQ and into the energy interval E E dE It is normalized to dQ dE and the flux of the incoming neutrons do k N S 5 A 3 1 Here n is the number of atoms and k k and ko ko are the wave num bers of scattered and incident neutrons respectively They are related to the corresponding neutron energies by E h k 2m and Eo h k2 2m where m is the neutron mass The arguments of the dynamic structure factor q and w are the momentum and energy transfer in units of h respectively ko k gt 3 2 q 3 3 2 Eo E a 3 3 w 3 3 The modulus of the momentum transfer can be expressed in the scattering angle 0 the energy transfer and the energy of the incident neutrons hw hw 4 2 2 04 2 A a 4 Es cos 04 Eo 3 4 The dynamic structure factor contains information about the structure and dynamics of the scattering system 21 It ca
3. ical differentiation of the input data The program implements the following differentiation schemes e Fast e order 2 order 3 e order 4 12 e order 5 Using option fast the first time derivative of each point r t is calculated as lt r tig1 r t At where At is the time step Choosing option order N with N 2 5 nMoldyn calculates the first time derivative of each point r t r x y z using the N order polynomial interpolating the N 1 points across r t where r t belongs to this set I5 The user can select the suited differentiation scheme 2 3 2 2 1 Input File Pressing the button File brings up a menu from which it is possible to choose the following options e Open trajectory e Open trajectory set Open a DLPOLY trajectory Open scattering function e Write PDB file e Show running calculations e Quit The Open trajectory option allows one to select the netCDF file of the trajectory which will be used to perform the calculations Clicking it a file directory selection window like that of Fig 2 2 pops up listing all the sub directories and the files in the working directory To select a directory double click it By doing this the complete pathname of the selected directory appears on the directory selection button at the bottom of the window To select the parent directory left click the directory selection button When a file in the list is highlighted clicked its name is sho
4. 4 20 k 0 64 This allows one to construct the following efficient scheme for the computation of mean square displacements 1 Compute DSQ k r k k 0 N 1 DSQ 1 DSQ N 0 2 Compute SUMSQ 2 55 DSQ k 3 Compute Sag m using the FFT method 4 Compute mean square displacement MS D m in the following loop SUMSQ SUMSQ DSQ m 1 DSQ N m MSD m SUMSQ 2 Sag m N m m running from 0 to Ny 1 It should be noted that the efficiency of this algorithm is the same as for the FCA computation of time correlation functions since the number of operations in step 1 2 and 4 grows linearly with M 4 3 Rigid body fits We discuss now briefly how rigid body motions i e global translations and rotations of molecules or subunits of complex molecules can be extracted from a molecular dynamics trajectory A more detailed presentation is given in 43 We define an optimal rigid body trajectory in the following way For each time frame of the trajectory the atomic positions of a rigid reference structure defined by the three cartesian components of its centroid e g the center of mass and three angles are as close as possible to the atomic positions of the corresonding structure in the MD configuration Here as close as possible means as close as possible in a least squares sense Optimal superposition We consider a given time frame in which the atomic positions of a sub molecule
5. Differentiation fast Length units 4 nm Time steps in output 2000 Output file text VACF_spce300K_1bar_1 Browse OK Cancel Figure 2 10 VACF from coordinates The output file is a file ascii plot containing the DOS of the selected system as a function of the frequency in the selected units Since the VACF is symmetric in time its Fourier spectrum is stored only for values on the positive frequency axis The default name of the output file contains the string DOS followed by the name of the trajectory The user can then change the name and or select the destination directory through the Browse button Pressing the OK button switches to the Almost Done window containing all settings in python language see Fig 2 12 These settings can be saved in a python script file via the Save button and then run later using the command line interface of nMoldyn or run immediately through the button Run Pressing the button cancel closes the window without saving settings 24 y f tat aN Density of states from velocities Projection vector None Weights none 4 incoherent mass coherent Window width for FFT fi 0 of trajectory length Length units 4 nm wv A Frequency units 1ips y 1icm Points in spectrum 2000 Output file text DOS_spce3D0K_1bar_1C Browse ox Figure
6. Output file netCDF CSF_spce300K_1har 10 Browse Output file for Dynamic Structure Factor output file netCDF CSF_SPECT_spce300K_ Browse ox Figure 2 16 Coherent Scattering Function Isotropic the moduli of the required g vectors through the text field named Q values using the form Gin AQ dmax In this way the intermediate scattering function will be calcultad for qm values defined as qm Qmin m Aq with m running from 0 to Ng 1 and qmar Ny 1 Ag Through the text field Vectors per shell the user can enter the number 32 of g vectors N that have to be used in the average of Eq 2 10 The expected value is un integer The required tolerance on the q moduli can be entered via the text field Q shell width The prgram looks for N q vectors whose moduli deviate from the grid of the selcted values within the given tolerance in order to carry out the average of Eq 2 10 The Q shell width fix the g resolution In the panel Weights the user can select how to weight the different contri butions to the Intermediate coherent scattering function coming from all atoms of the system see Section 2 2 The panel window width for FFT percento of trajectory lenght al lows one to select the width of the Gaussian function to be used in the smoothing procedure for the calculation of the Fourier spectrum see Section The width is defined with respect t
7. This option allows the output functions to be displayed in a 2D plot For the functions calculated over a q t gride the plots are done kiping fixed one of the two variables Display 3D The output functions calculated aver a q t gride can be displayed in a 3D plot through option Display3D Direct basis Pressing this button a window containing the basis vectors b1 b2 b3 which span the MD cell pops up Any position vector in the MD call can be written as R 2 b1 y b2 2 b3 with 2 y z having values between 0 and 1 Reciprocal basis Pressing this button a window containing the dual basis vectors bt b b pops up They are defined by the relation b b 97 where by b2 bg are the direct basis vectors 2 2 7 Help Pressing the button Help brings up a drop down menu from which it is possible to choose the option About nMoldyn Pressing this button a window con taining the e mail addresses of all of the people which have contributed to the developpement of the code Gerald Kneller author of the original fortran ver sion of nMoldyn release 1 0 Tomasz R g Krzysztof Murzyn Konrad Hinsen authors of the python version of nMoldyn release 2 1 Paolo Calligari author of a revised version of nMoldyn based on the release 2 1 2 3 Command Line Interface pMoldyn In some situations a graphical interface cannot be used for technical reasons e g text mode connection to remote machines or it is not the most conve nient
8. AR Coef_spce300K_1ba Browse Figure 2 14 Autoregressive model window from velocities 27 The user can enter the order poles number of the Autoregressive model in the field Model Order see Section A A priori the autocorrelation function and its power spectrum can be approximated to almost arbitrary precision by increasing the order of the autoregressive model In practice it has been proven that reliably computation can be carried out up to P of the order of 1000 poles As for the standard calculation of VACF DOS and MSD it is possible to calculate these functions isotropically or along a specific axis This can be made through the Projection Vector field typing None or the components of the corresponding unit vector in the form z y z respectively In the panel Weights the user can select how to weight the different contri butions coming from all atoms of the system see Section 2 2 The weights are normalized to one The length units the frequency units the time steps in the output files of VACF MSD and VACF memory function and the number of points in the output file of DOS can be selected in the panels Length units Frequency units Time steps and Points in spectrum respectively Five output files in ascii format containing the memory function of the VACF the VACF and the MSD as a function of time in ps the DOS as a func tion of frequency in the selected units and the
9. MMTK Proteins import Protein proteins trajectory universe objectList Protein pick the lysozyme lysozyme proteins 0 pick a part of the first chain subchain lysozyme 0 4 25 select the sidechains return subchain sidechains We recall that it is possible to create an input file via the graphical user interface as well Such input file provides a convenient starting point for customization To run a calculation via the command line interface type pMoldyn without input arguments in order to get usage instructions 1 the two examples of specification file shown above are taken from Ref 48 incoherent Saattering AR analysis Oy Model order 50 Extended precision memory function None Q values 0 2 100 Q shell width fr Vectors per shell 50 Weights none incoherent y Mass coherent Q units 1mm v V Time steps in output 2000 Points in spectrum 2000 Scattering function output file output file netCDF AR ISF_spcea00K_1bar_ Browse Structure factor output file output file netCDF AR ISF_SPECT_spcea0t Browse Memory function output file output file netCDF AR ISF_Memory_spce30 Browse OK Cancel Figure 2 23 Coherent Scattering AR analysis 49 Q values 0 2 100 Q shell width lr Vectors per shell 50 Weights none incoherent mass
10. above expression reads n 00 2 MSDar n 2DnAt D At v DI 2 4 70 0 which allows one to compute the MSD within the AR model from the poles and the 8 coefficients of the non normalized VACF Friction coefficient within the AR model The friction coefficient is de fined as the integral over the memory function In the discrete case we write o Y At n At 1 4 71 As shown in 33 the AR model allows us to express W gt z as P Yen z mi yon 2 4 72 5v a 4 72 where the coefficients 3 are given by eq 1351 and the roots z must fulfill the stability criterion 52 Inserting 172 into yields EA 2 the z transform of the discrete memory function within the AR model AR 1 2 to ad 4 73 Using 473 we obtain thus within the AR model 1 g AR _ EM 1 0 P This shows that o can be obtained from the zeros zj of the characteristic polynomial p z defined in 49 4 74 Fine and Feon within the AR model The computation of Fince and Foon within the AR model is carried out using same algorithm emploied for the calculation of the VACF within the AR model Here for each q values the discrete time signal is N S ba ine exp iq Ra t 4 75 a 1 For each q vector one obtains a set of P coefficients 6 Discrete memory function of the Finc and Feon within the AR model For the computation of the discrete memory functions of Fine and Feon nMoldyn uses the same algorit
11. looks for N q vectors whose moduli deviate from the grid of the selcted values within the given tolerance in order to carry out the average The Q shell width fix the g resolution In the panel Weights the user can select how to weight the different contri butions coming from all atoms of the system see Section2 2 The weights are normalized to one The units of the q vectors can be selcted in the panel q units The number N of points in the output file containing the intermediate coherent scattering functions and the number of frequency points in the output file containing their fourier spectra can be entered in the panels time steps in output and Points in spectrum respectively The output files are three netCDF files containing the Coherent intermediate scattering functions as a function of the time in ps for all q values their Fourier spectra as a function of the frequency in THz and their memory functions in ps Since Feon dm k At is symmetric in time it is stored only for the positive time axis and its Fourier spectrum is stored only for values on the positive frequency axis The default names of the output files contain the strings AR CSF AR CSF_SPECT and AR CSF_Memory followed by the name of the trajectory The user can then change the names and or the destination directory through the Browse buttons Pressing the OK button switches to the Almost Done window containing all settings in python language see Fig 2 20 Th
12. of a set of reference structures to the corresponding set of given structures in a Molecular Dynamics trajectory The fit algorithm is described in Section 3 The reference struc tures stored in a pdb file can be selected via the group selection option 28 In general the reference structure is defined in an orthonormal body fixed co ordinate system Usually this is the principal axis system in which the tensor of inertia is diagonal For each time frame and for each group the program calculates e three cartesian components describing the optimal translation of the ref erence structure e four normalized quaternion parameters describing the optimal rotation of the reference structure e the fit error per site Then the angular trajectories are calculated numerically as time integrals over the cartesian components of the angular velocity in the body fixed frame t Wa Wy We according to ex dr wi rT 2 9 where the angular velocity in the body fixed frame is related to the time deriva tive of the quaternion parameters via Eq 5 42 The output file is a file netCDF containing the angular trajectory of the selected system with respect to the rigid body frame The default name of the output file contains the string AT followed by the name of the trajectory The user can then change the name and or select the destination directory from the window angular trajectory through the Browse button in the Angular tra jec
13. of states DOS G n Av which is the Fourier spectrum of the VACF see Paragraph ZZA G n Av Y WaCoaaln Av n 0 N 1 2 8 N is the total number of time steps and Av 1 2N At is the frequency step G n Av can be computed either for the isotropic case or with respect to a 22 Almost done Your settings will stored in an input file whose contents are shown below You can Save them and run the calculations later or Run the calculations immediately from MMIK import title Velocity Autocorrelation Function from velocities trajectory home vania trajectories spce300K_lbar_100ps log_file vacf vel_spce300K_lbar_100ps log time_info 0 9990 1 atoms Water Hydrogen projection_vector None weights incoherent units length Units nm time_steps 2000 Save Run Figure 2 9 VACF from velocities Python script user defined axis see Section BJ The quantities wa are user defined weights which are normalized to one gt wa 1 The spectrum G n Av is computed from the unnormalized VACF such that G 0 gives an approximate value for the diffusion constant D Y Da see Eqs 539 and BAD G n Av is smoothed by applying a Gaussian window in the time domain 38 see Section 11 Its width in the time domain is sigma alpha T where T is the length of the simulation We remark that the diffusion constant obtained from DOS is biased due to the spec
14. qmin M Aq with m running from 0 to N 1 and qmaz N 1 Aq Through the text field Vectors per shell the user can enter the number of g vectors N4 oriented isotropically that have to be used in the average The expected value is un integer The required tolerance on the q moduli can be entered via the text field Q shell width The program looks for N q vectors isotropically directed whose moduli deviate from the grid of the selcted values within the given tolerance in order to carry out the average The Q shell width fix the q resolution In the panel Weights the user can select how to weight the different contri butions coming from all atoms of the system see Section 2 2 The weights are normalized to one The units of the q vectors can be selcted in the panel q units The number N of points in the output file containing the intermediate coherent scattering functions and the number of frequency points in the output file containing their fourier spectra can be entered in the panels time steps in output and Points in spectrum respectively The output files are three netCDF files containing the Incoherent interme 44 diate scattering functions as a function of the time in ps for all q values their Fourier spectra as a function of the frequency in THz and their memory func tions in ps Since Finclqm k At is symmetric in time it is stored only for the positive time axis and its Fourier spectrum is stored only for val
15. required values and lying along explicitly listed directions Pressing this button the window Coherent Scattering Function Explicit List pops up see Fig EIS The input parameters to be specified in order to carry out the computation the coherent intermediate scattering function and its Fourier spectrum are the same as for the Isotropic option In addition here one has to specify the directions along which the q vectors have to be taken This can be done via the text field Q List entering a list of unit vectors in the form L Y 230 Y zi 34 CP mm Coherent Scattering Function Explicit List 2 Q List f1 0 0 0 1 A Q values Q shell width Vectors per shell Weights none incoherent mass coherent Window width for FFT 10 of trajectory length Q units 1 nm v M Time steps in output 2000 Points in spectrum 2000 Output file netCDF CSF_spce300K_1bar_10 Browse Output file for Dynamic Structure Factor output file netCDF CSF_SPECT_spce300K_ Browse OK Cancel Figure 2 18 Coherent Scattering Function Explicit List Coherent Scattering AR analysis In the framework of the Autoregressive model nMoldyn allows the intermedi ate coherent scattering function its Fourier spectrum the coherent dynamical structure factor and its memory function to be computed on a rectangular grid of equidistantly sp
16. use RPM install from the tar file The tar file of nMoldyn can be downloaded from the site http dirac cnrs orleans fr nMOLDYN download html The instructions are tar xzf nMOLDYN 3 0 1 tar gz cd nMOLDYN 3 0 1 python setup py build python setup py install The last command may require administrator privileges Before installing nMoldyn please check the list of prerequisites below 1 the Python interpreter version 2 2 or higer 35 2 the Tk library the netCDF library 36 3 4 Numeric Python recommended version Numeric 23 8 2 13 5 Scientific Python 37 6 MMTK version 2 2 or higher 16 Python itself is part of all modern Linux distributions as is Tk Numerical Python is provided as an optional package with most Linux versions usu ally called python numeric Furthermore it is advisable to install the FFTW library and its Python interface to speed up the FFT operations inside nMOLDYN nMOLDYN uses FFTW automatically if it is installed and the FFTPACK routines from Numerical Python otherwise Please compile and in stall the packages one by one in the order given above followed by nMOLDYN To dowload the tar files of all of the prerequisites see the sites cited in the corresponding references 1 2 Macintosh users There are ready made installers for everything you need Please install these packages in the indicated order 1 MacPython 2 4 1 2 netCDF 3 6 0 TclTkAqua 8 4 9 Numeric 23 7 from
17. 2 11 DOS from velocities From coordinate Pressing the button coordinates will open the window Density of states from coordinates From this window it is possible to enter the same quantities as for the calculation of the DOS from velocities In addition the paneldifferentiation allows one to select how to perform the differentiation of the coordinates trajectories see Fig 2 T3 to obtain the velocities trajectory Refer to Section 2 2 for more details about the differentiation schemes Autoregressive model AR In the framework of the autoregressive model nMoldyn allows one to calculate the VACF the VACF memory function the DOS the MSD and the Autore gressive coefficients al of the velocity trajectory averaged over all selected atoms and three Cartesian coordinates The user is referred to Section 3 9 and Section LA for more details Pressing the button Autoregressive model in the menu Dynamics of nMoldyn a drop down menu allows one to select how to calculate the above quantities from coordinates or from velocities From velocities choosing the option from velocities the window Autore gressive Model from velocities in FigB IA pops up 25 SO Your settings will stored in an input file whose contents are shown below You can Save them and run the calculations later or Run the calculations immediately from MMTK import title Density of states from velocities trajectory home vania trajectorie
18. ASCII representa tion of a specified netCDF file on standard output The ASCII representation is in a form called CDL network Common Data form Language that can be viewed edited or serve as input to ncgen A CDL description consists of three optional parts dimensions variables and data The dimentions part contains the shapes of one or more of the multidimensional variables The variable part may contain variable declarations name data type and shape of a variable and attribute assignments units special values maximum and minimum valid values and packing parameters See Ref TA for a description of the CDL form The output from ncdump is intended to be acceptable as input to ncgen which can generate a binary netCDF file from a CDL file Moreover the ncgen tool can generate a C or FORTRAN program that creates a netCDF file For a description of the ncdump and ncgen utilities see Ref IA Here we recall some useful Unix comands for invoking ncdump as a simple browser for netCDF trajectory file to display the dimension names and sizes variable names types and shapes attribute names and values and optionally the values of data for all variables or selected variables ncdump c h v vari input file where c Show the values of coordinate variables variables that are also dimensions as well as the declarations of all dimensions variables and attribute values Data values of non coordinate variables are not incl
19. Autoregressive coefficients of the velocity trajectory 03 op aP are produced Since the VACF is symmetric in time its Fourier spectrum is stored only for values on the positive frequency axis The default name of the output files contain the string AR Memory AR VACF AR DOS AR MSD AR Coeff followed by the name of the trajectory The user can then change their names and or select the destination directory through the Browse button Pressing the OK button switches to the Almost Done window containing all settings in python language These settings can be saved in a python script file via the Save button and then run later using the command line interface of nMoldyn or run immediately through the button Run Pressing the button cancel closes the window without saving settings From coordinates Pressing the button coordinates will open the window Density of states from coordinates From this window it is possible to enter the same quantities as for the Autoregressive model from coordinates In addition the paneldifferentiation allows one to select how to perform the differentiation of the coordinate trajectories to obtain the velocities Refer to Section BQ for more details about the differentiation schemes Angular Trajectory Option Angular Trajectory allows the angular trajectory of the atomic molecular group s selected via the group selection option to be calculated Choosing this option nMoldyn performs a positional least square fit
20. To analyze the dynamics of complex molecular systems it is often desirable to consider the overall motion of molecules or molecular subunits We will call this motion rigid body motion in the following Rigid body motions are fully deter mined by the dynamics of the centroid which may be the center of mass and the dynamics of the angular coordinates describing the orientation of the rigid body The angular coordinates are the appropriate variables to compute angular corre lation functions of molecular systems in space and time In most cases however these variables are not directly available from Molecular Dynamics simulations since MD algorithms typically work in cartesian coordinates Molecules are ei ther treated as flexible or if they are treated as rigid constraints are taken into 56 account in the framework of cartesian coordinates 25 In nMOLDYN rigid body trajectories can be extracted from a Molecular Dynamics trajectory by fitting rigid reference structures defining a sub molecule to the corresponding structure in each time frame of the trajectory Here fit means the optimal superposition of the structures in a least squares sense The corresponding al gorithm is described in Section 3 It uses quaternion parameters 26 for the rotational fit Quaternions are not only useful for the fitting procedure but also for the computation of angular correlation functions 27 23 3 7 Angular VACF angular trajectories Sim
21. We define now the extended 62 periodic time series As Ay ean aa o ee 45 which have the period 2N A k A k m 2N B k B k m 2N m 0 1 2 4 6 The discrete cyclic correlation of A k and B k is defined as 2N4 1 Sap m Y A k B k m 4 7 k 0 It is easy to see that 1 _ N 1 lt m lt N 1 Cav m N p 942 M 1 lt m lt N 1 4 8 Using the correlation theorem of discrete periodic functions 40 Sag m can be written as 2N 1 1 mn f n sn n 0 where A gy and B 4 are the discrete Fourier transforms of A k and B k respectively gt 27i A k 4 10 k 0 a x DE 27i B k 4 11 If the Fourier transforms of the signals A k and B k as well as the inverse transform in are computed by FFT Sag m can be computed by Nilog Ns instead of x N multiplications It is sometimes said that the FFT method induces spurious correlations We emphasize that this is only the case if the time series a k and b k are not properly extended as indicated in Eqs 4 and 5 The FFT method and the direct scheme 42 give apart from round off errors identical results In many cases not only the computation of a correlation function is required but also the computation of its Fourier spectrum In principle one could use the product A st B 4 which is already available as an intermediate step 63 in the computation of Sag m
22. a thermal average relation can be written as Phan t D Din Del 3 48 The p coefficients can also be expressed as time correlation functions of irre ducible tensor components This is convenient for numerical purposes since time correlation functions of discrete and finite time series can be very efficiently computed by Fast Fourier Transform techniques see Section LT Consider the general form of the time correlation function TI t1 T 7 to f fam do p Q4 t1 Qo to TI Q1 T 1 Qo 3 49 where TY are the components of an irreducible tensor B0 BI From the trans formation properties of irreducible tensors it follows that T Q1 I Pinal NT Qo 3 50 58 For an isotropic system in thermal equilibrium we may now write 1 8m2 p Q t1 Mo to p Q t 0 0 3 51 Inserting this in 349 performing a change in the integration variables from 21 00 to Q Qo and using the orthogonality of the Wigner functions one can show that TAOTI O phn DER 3 52 l where the components f el are referred to a convenient reference frame In prac tice only tensors with integer j are relevant In this case the well known spher ical harmonics B0 BI may be used to define irreducible tensors They are related to the Wigner functions by 23 1 AT Y a B D o a B Y 3 53 where 5 y are Euler angles Following ROSE 32 the Wigner functions can be expressed as complex polynomials i
23. according to 9 This would however not be a good estimate for the spectrum of cay m MI In nMOLDYN all spectra are smoothed by applying a window in the time domain 11 N 1 n nm 1 Pap At 2 exp 27i W m N h u Im Sap m 4 12 The time step At in front of the sum yields the proper normalization of the spectrum In nMOLDYN a Gaussian window is used W m exp 5 0 Iml ut mem ES Its widths in the time and frequency domain are o a T and o 1 2ro respectively We recall that T N 1 At is the length of the simulation co corresponds to the width of the resolution function of the Fourier spectrum 4 2 Mean square displacements The FCA method described above may also be used to compute mean square displacements 42 In the discrete case the mean square displacement of a particle is given by N m 1 Y r k m r k m 0 N 1 414 k 0 u 1 Nim A m where r k is the particle trajectory We define now the auxiliary function Ne m 1 S m Y r k m r k m 0 N 1 4 15 k 0 which is split as follows S m Saarpe m 2S4p m 4 16 Ne m 1 Saarpalm Y Pim 4 17 k 0 N m 1 Saplm X r k r k m 4 18 k 0 The function S4p m can be computed using the FCA method described in Section LI For S14 B m the following recursion relation holds Saar BB m Saarpa m 1 r 2 m 1 r 2 M m 4 19 N 1 Saa BB 0 gt gt r k
24. aced points along the time and the q axis repectively The user is referred to Section BJ for more theoretical details The dynamical variable of the correlation function under consideration SiG ba coh expl iq Ra nAt is considered as a discrete signal which is modeled by an autore gressive stochastic process of order P For each q values the program calculates the set of the relevant P complex coefficients an of the stochastc process av eraging aver all atoms of the system and over all cartesian components The correlation functions and their Fourier spectra are then computed according to the algorithm described in Section 44 Starting from the discretized memory 35 function equation which relates the time evolution of the correlation function to its memory function and using the correlation function calculated by the AR model the program computes for each q value the discretized memory func tion see Section LA The program performs the above calculations isotropi cally Pressing the button Coherent Scattering AR analysis the relevant window pops up see FigB T9 36 Coherent Scattering AR analysis O Model order 50 Extended precision memory function None Q values 0 2 1 00 Q shell width fi Vectors per shell 50 Weights none w incoherent mass coherent Q units 4 1 nm v M Time steps in output 2000 Points in spectrum 2000 Scattering fu
25. aining only very limited functionality to analyze MD trajectories In this NOTE we present the new release 3 0 of the modular program package nMOLDYN which replaces the original version nMOLDYN 2 0 for the analysis of Molecular Dynamics trajectories The program nMOLDYN was developed mainly for use in connection with neutron scattering experiment although many of the quantities are also used in other contexts The combination of neutron scattering experiments and molec ular dynamics MD simulations is a powerful tool to study the structure and dynamics of complex molecular systems Neutron scattering is sensitive to time and space correlations of atomic positions on the ns time scale and the length scale I 2 These are exactly the time and space domains covered by classi cal molecular dynamics simulations On the length scale under consideration the neutron target interaction can be modelled by pseudopotentials with zero range which are centered on the atomic nuclei of the targets The coupling be tween neutron and target is described by so called scattering lengths describing the strength of the neutron nucleus interaction I The differential scattering cross section can be expressed in terms of quantum time correlation functions of the spatially Fourier transformed particle density The corresponding clas sical time correlation function can be easily obtained from molecular dynamics simulations This enables a direct comparison between
26. al over Poo t 3 9 Memory functions Memory functions have been used for a long time in theoretical statistical physics to describe the time dependence of autocorrelation functions Nevertheless the use of memory functions in the context of Molecular Dynamics simulations has been hindered by the lack of a suitable numerical algorithm for their calculation Such an algorithm has been published recently 33 and is now implemented in nMOLDYN The key point is that a reliable estimates for memory functions can be obtained by assuming an Autoregressive AR model for the underlying stochastic process and not for the memory function itself see Section LA Memory function equation of VACF In the context of VACF memory function one usually considers the normalized VACF which is defined as v amp u 0 v 0 Here v t is the x y or z component of the velocity of a tagged atom The memory function t of w t is defined by the relation Y t 3 59 Sut a dr Elt r r 3 60 Eq is called the memory function equation ME For its numerical calculation the dynamical variable under consideration the velocity of a fluid particle is considered as a discrete signal v n v nAt which is modeled by an autoregressive stochastic process of order P An overview of this algohritm is done in section LA The reader is referred to Ref 33 for more details Memory function equation of Coherent scattering Another
27. are given by Xa a 1 N The corresponding positions in the reference structure are denoted as xo a 1 N For both the given structure and the reference structure we introduce the yet undeter mined centroids X and X respectively and define the deviation A D q ES z xo xa X 4 21 Here D q is a rotation matrix which depends on also yet undetermined an gular coordinates which we chose to be quaternion parameters abbreviated as vector q q0 q1 42 43 The quaternion parameters fulfill the normalization condition q q 1 26 The target function to be minimized is now defined as m q X X Y walAl2 4 22 65 The wa are normalized positive weights 37 wa 1 The minimization with respect to the centroids is decoupled from the minimization with respect to the quaternion parameters and yields X Y ex 4 23 XO Y wax 4 24 We are now left with a minimization problem for the rotational part which can be written as m q Y wa D a r ra eaten 4 25 The relative position vectors Ta Xa X 4 26 ro x _ xO 4 27 are fixed and the rotation matrix reads q q q2 q3 2 qoq3 q1q2 2 q0g2 193 D q 2 gog3 gige 4 2 q0q1 9293 2 q092 q1q43 2 qog1 gags q 4 28 Quaternions and rotations The rotational minimization problem can be elegantly solved by using quaternion algebra Quaternions are so called hyper complex numbers having a real u
28. cement trajectory home vania trajectories spce300K_lbar_100ps log file msd_spce300K_lbar_100ps 1og time_info 0 9990 1 atoms Water Hydrogen projection_vector None weights incoherent units length Units nm time_steps 2000 Save Run Figure 2 7 Almost Done window quantities relevant to the calculation such as a possible projection vector the weights the length units the time steps in output the name and the location of the output file edit the corresponding python script close the window to come back to the start up window of nMoldyn see Fig 2 8 To calculate the VACF with respect to a given axis the user can enter in the Projection Vector field the components of the corresponding unit vector in the form z y z Otherwise writing None the VACF of each atom in the selected system will be calculated taking v over all directions In the panel Weights the user can select how to weight the different con tributions to the VACF coming from all atoms of the system The weights are normalized to one see Section 2 2 The units length for the VACF and the number of time steps in the output file can be selected in the panels Length units and Time Steps in output respectively The output file is a file ascii plot containing the VACF of the selected system as a function of the time in ps Since the VACF is symmetric in time it is stored only for values on the positive
29. cients of the series on the lhs and the rhs of Eq 1 57 To construct a numerical method we replace the series by polynomials of order N where t N At defines the time window for the memory function to be computed After this first step a polynomial division is performed on the rhs of Eq 1 57 and after a subsequent multiplication of both sides with z one obtains the time dependent memory function j by comparison of coefficients ee ie ee eg AP Seer m TE 2 2 Cpe R Y El 27 4 58 Within nMoldyn w n is replaced by the autocorrelation function calculated in the framework of the autoregressive model 47 n as in Eqs 50land 51 The coefficients c are obtained by polynomial division and R is a rest which does not contain information on the memory function within the time interval te 0 NAt The discrete memory function is therefore given by j cn A remark concerning the discretization scheme 4 54 is in place here The discrete convolution sum is effectively a first order approximation of the convolu tion integral More sophisticated approximations could be used but they would lead to less convenient expressions upon z transformation Correspondingly we have chosen a first order approximation for the differentiation on the left hand side of 3 60 In this way the first order integro differential equation 3 60 is transformed into the first order difference equation 4 54 However this simple
30. d trajectory and 1 This means that the program takes by default the first and the last point of the input MD trajectory as first and last time step for the calculation of the required quantity and that every time step is extracted from the input trajectory i e all of the time steps of the input trajectory are taken into account skip 1 Any other choice can be made by replacing the above default values by new integer values and then pressing Enter in every text field The corresponding time values expressed in units of ps are shown on the right side of the text fields 2 2 3 Input Atom Selection Group Selection Once the trajectory of the system has been selected the buttons Atom selec tion allows one to select all or part s of the full system that will be used in a particular calculation and to specifying which parts of the system should be deuterated The selection will be used to calculate Dynamics e mean square displacement e velocity autocorrelation function e density of states Scattering e coherent scattering function e coherent scattering AR analysis e incoherent scattering function Gaussian approximation e incoherent scattering function e incoherent scattering AR analysis e EISF 15 Pressing the Atom Selection button a window nemed Atom selection pops up This window contains a input field named Read from PDB file having an auxiliary button Browse and a button named my system A detailed selection of t
31. df guide 12 html M Abramowitz I A Stegun Handbook of Mathematical Functions Dover New York 1972 p 914 http sourcesup cru fr projects mmtk 73 17 R Rew G Davis S Emmerson H Davies Netcd 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 fUser sGuide for c an interface for Data Access version 3 http www unidata ucar edu packages netcdf guidec 1997 K Hinsen ScientificPython http dirac cnrs orleans fr K Hinsen J Comp Chem 2000 21 79 OpenSource web site http www opensource org 1997 L van Hove Phys Rev 95 1 249 1954 P Schofield Phys Rev Letters 4 5 239 1960 A Rahman K S Singwi and A Sj lander Phys Rev 126 986 1962 J P Boon and S Yip Molecular Hydrodynamics McGraw Hill New York 1980 J P Ryckaert G Ciccotti and H J C Berendsen J Comput Phys 23 327 1977 S L Altmann Rotations Quaternions and Double Groups Clarendon Press Oxford 1986 G R Kneller and A Geiger Molecular Simulation 3 5 6 283 300 1989 R M Lynden Bell and A J Stone Molecular Simulation 3 271 81 1989 M P Allen and D J Tildesley Computer Simulation of Liquids Oxford University Press Oxford 1987 A R Edmonds Angular momentum in Quantum Mechanics Princeton University Press Princeton New Jersey 1957 D M Brink and G R Satchler Angular Momentum Clarend
32. discretization scheme together with the use of the one sided z transform leads to a significant error in 0 It is clear from Eq that due to the symmetry of the autocorrelation function w t t the derivative d dt should vanish at t 0 However in the discretized version it is approximated by a forward difference that is always negative A higher order calculation shows that the estimate for 0 that results from the procedure described above should be doubled 70 MSD within the AR model The relation between the VACF and the MSD reads MSD t x t 2 0 2 2 dr t T Cov t 4 59 By discretizing the above equation one obtains MSD n 2 gt At n k Cwy k 4 60 k 0 Using fi n O n nfa n O n Cuu n 4 61 into Eq E60 gives 00 MSD n 2 Y At fi n k folk 4 62 k oo Making use of the one side z transform equivalent to the Laplace transform for discrete functions we obtain MSD 2 2F Fy At 4 63 where gt F Goi 4 64 Introducing Eq 64 into Eq 1 64 yields MSD 2 2 te 4 65 and its inverse z transform reads 1 MSD n 2At a per 2mi Using the expression of the non normalized Cn z obtained in the framework of the AR model p ee 1 j zZ uP 4 67 Z Zj one finds the expression of MSD within the same AR model P 1 ga 1 j 1 j n 2 ER 2At v D o z aay a j 1 J J J 71 In the limit of n 00 the
33. e ee Oe es 5 AR PU DS a a ee a 6 8 E AA de ed ed en ie ot 8 dd ee DANSE ces ee 10 Gaines Susan a te 13 luput First Step Last Step Skin 14 a oe aie tec aca 15 Seta Re a ee eee 18 Outp t gt LOS LS da roue 30 ion Vi 46 47 2 3 Command Line Interface pMoldyn 47 51 51 52 53 54 55 56 57 i 3 57 ey ee rere a ree ee ere 60 62 4 1 omputation of time correlation functions 62 4 2 Mean square displacementg 64 4 Rigid body ftg 2 44 65 4 4 Autoregressive AR procesd 68 List of Figures ada ae 11 2 2 file director selection windo eds aa E 14 15 16 17 a Ht a eae A 20 Tarma a set Dee 21 ACF from velocitied 22 ACF from velocities Python script 23 eee BE ee ee ee ee a 24 PRESTEN AR 25 eH Hee ts Rubias oe 26 26 27 30 32 34 35 g 3 37 2 20 herent Scatt ring AR ana l is P thon scrip 39 ee 40 EEE WEINE 43 u a feu eee ee oe ae 49 hd a oe oe eh re ee De D eet ie 50 Introduction Although Molecular Dynamics simulation techniques are widely used in physics chemistry and biology their possibilities are often not fully exploited because of the lack of easy to use analysis tools This is especially true for very complex systems which force most computational scientists to use standard program packages cont
34. elocities are computed by numerical differentiation of the coordinate trajectories correcting first for possible jumps due to periodic 19 Mean Square Displacement lt 2 gt O El Projection vector Weights none incoherent mass coherent Length units 4 nm v Time steps in output 2000 Output file text MSD_spce300K_1bar_11 Browse OK Cancel Figure 2 6 Mean Square Displacement window boundary conditions In both cases the program computes the discrete velocity autocorrelation function Cyy k At Colk At Y woCriaa k At k 0 N 1 2 6 a 2 7 where N is the total number of time steps The velocity autocorrelation functions Cuv aa k At is computed with the FCA algorithm described in Section LI Cyy k At can be computed either for the isotropic case or with respect to a user defined axis see Section B J The quantities wa are user defined weights which are normalized to one gt wa 1 The VACF is normalized such that Cyv 0 1 From Velocities Pressing the button velocities will open the window Veloc ity Autorrelation Function from velocities from which the user can enter some 20 Ds o Br A Almost done Your settings will stored in an input file whose contents are shown below You can Save them and run the calculations later or Run the calculations immediately from MMTK import title Mean Square Displa
35. elocity Autocorrelation Function e Density of States e Autoregressive Model e Angular Trajectory e Rigid Body Trajectory e Rigid Body Rotation Trajectory e Digital Filter e Angular Velocity Autocorrelation Function e Spectrum of angular VACF e Reorientational Correlation Function In this section we give a description of the dynamical quantities calculated by nMoldyn For a description of the theoretical background see Chapter B 18 Mean Square Displacement MSD nMoldyn allows one to computes the average mean square displacement of atoms see Section B 5 A k At waAl k At k 0 N 1 2 4 N is the total number of time steps in the coordinate time series The wa are user defined weights which are normalized to one see Section 2 2 One can also compute the mean square displacement with respect to a user defined axis A k At n waA2 k Atin k 0 N 1 2 5 In both cases the algorithm described in Section 2 is applied nMoldyn cor rects the atomic input trajectories for jumps due to periodic boundary condi tions The above calculation can be carried out through the graphical interface pressing the button Mean Square Displacement from the drop down menu Dynamics Pressing this button will open the window Mean Squre Displace ment see Fig 2 0 The user can calculate the mean square displacement with respect to a given axis by entering in the Projection Vector field the components of the corre
36. enght of the trajectory The units of the q vectors can be selcted in the panel q units The number N of points in the output file containing the intermediate incoherent scattering function and the number of frequency points in the output file containing its fourier spectrum can be entered in the panels time steps in output and Points in spectrum respectively The output files are two netCDF files containing the incoherent interme diate scattering functions as a function of the time in ps for all q values and their Fourier spectra as a function of the frequency in THz respectively Since F ams k At is symmetric in time it is stored only for the positive time axis and its Fourier spectrum is stored only for values on the positive fre quency axis The default names of the two output files contain the strings ISFG and ISFG_SPECT respectively followed by the name of the trajectory The user can then change the names and or the destination directory through the Browse buttons Pressing the OK button switches to the Almost Done window containing all settings in python language These settings can be saved in a python script file via the Save button and then run later using the command line interface of nMoldyn or run immediately throgh the button Run Pressing the button cancel closes the window without saving settings Incoherent Scattering Function Option Incoherent scattering Function in the drop down menu dynamics allows one to c
37. es the program calculates a set of P complex coefficients an for the AR model averaging aver all atoms of the system and over all cartesian components The correlation functions and their Fourier spectra are then computed according to the algorithm de scribed in Section L Starting from the discretized memory function equation whitch relates the time evolution of the correlation function to its memory func tion see Section LA and using the correlation function calculated by the AR model the program computes for each q value the discretized memory function The program performs the above calculations isotropically Pressing the but ton Incoherent Scattering AR analysis the relevant window pops up see Fig 23 Here the user can enter the order poles number of the Autoregressive model in the field Model Order see Section 4 A priori the autocorrelation function and its power spectrum can be approximated to almost arbitrary preci sion by increasing the order of the autoregressive model In practice it has been proven that reliably computation can be carried out up to P of the order of 1000 poles Since the program performs the calculations in high precision by default type None in the text field Extended precision memory function The user can enter the moduli of the required q values through the text field named Q values using the form qmin Aq qmaz In this way all of the functions will be calcultad for qm values defined as qm
38. ese settings can be saved in a python script file via the Save button and then run later using the command line interface of nMoldyn or run immediately through the button Run Pressing the button cancel closes the window without saving settings 38 gt Almost done Your settings will stored in an input file whose contents are shown below You can Save them and run the calculations later or Run the calculations immediately trajectory home vania trajectories spce300K_1bar_100ps log_file arcsf_spce300K_lbac_100ps log time_info 0 9990 1 atoms Water deuter None ar_order 50 ar_precision None q vector_set_ani q vector_set_exp q vector_set 0 2 0 4 0 8 0 8 0 10 0 12 0 14 0 7 Save Run Cancel 0 0 0 Figure 2 20 Coherent Scattering AR analysis Python script Incoherent Scattering Function Gaussian Approx nMoldyn allows one to compute the incoherent intermediate scattering function in the Gaussian approximation see Eq 3 35 Fran PO Y su k 0 N 1 m 0 Nq 1 2 15 N is the total number of time steps in the coordinate time series and N is a user defined number of q shells The q t grid is the same as for the calculation of the intermediate incoherent scatering function see Paragraph2 2 5 The wa are user defined weights which are normalized to one gt wa 1 such that F q 0 1 The atomic intermediate scattering funct
39. gorithm 61 Chapter 4 Algorithms 4 1 Computation of time correlation functions In Section Blit has been outlined that most of the quantities which can be ex tracted from MD simulations are time correlation functions Correlation func tions of discrete time series can be efficiently calculated by using the Fast Fourier Transform FFT 40 The so called Fast Correlation Algorithm FCA al lows the number of multiplications complexity to be reduced from N to x N loga N In nMOLDYN all time correlation functions are computed using the FCA method which will be outlined in the following We will also briefly comment on spectral smoothing of Fourier transformed correlation functions We consider two time series alk At 0 k At k 0 M 1 4 1 of length T N 1 At which are to be correlated In the following the shorthands a k and b k will be used The discrete correlation function of a k and b k is defined as won Lato a k b k m m 0 N 1 Cab M Waa Erin O HG Im m M 1 1 4 2 The prefactors in front of the sums ensure the proper normalization of the individual channels m N 1 N 1 The asterisk denotes a complex conjugate According to 1 2 cau m has 2N 1 data points and obeys the symmetry relation Calm Cia mM 4 3 In case that a k and b k are identical the corresponding correlation func tion Caa m is called an autocorrelation function
40. he Display button in the View menu The option write PDB file allows a PDB file to be written starting from a netCDF trajectory The positions of all of the atoms of the system at a given time step of the simulated trajectory can be stored in a PDB file At first the trajectory has to be selected via the Open Trajectory or the Open Trajec tory set buttons The user can then select the desired time step by inserting the corresponding integer value in the First Step field and then pressing enter Finally the corresponding PDB file can be created via the write PDB file button Once a calculation is running the user can check its progress by selecting theShow running calculations option A window showing in real time job progress time and date will pops up see Fig 23 The Quit button will terminate the program 2 2 2 Input First Step Last Step Skip Once the trajectory has been selected the user can to specify the starting time step the last time step and the number of time steps which have to be skipped from the input MD trajectory to perform the calculation The above values 14 Progress indicator 33 Module Job ID Job Owner Started Progress in msd 6264 vania Fri Nov 17 13 00 14 2006 Figure 2 3 Job progress have to be entered in the text fields named First Step Last Step and Skip respectively The default values are 1 N with N equal to the total number of time steps contained the selecte
41. he system can be made by marking the atoms to be used in a PDB file which will be imported via the Browse button The selection is made by replacing the atom label in the last column of the PDB file by The user can specify in the same PDB file possible deuteration of part s of the system or of the overall system Alternatively most common selections can be made via the button my system In case of trajectories of simple atomic or molecular systems the button my system allows one to take into account all of the atoms or molecules of the simulation while in case of more complex systems containing for example proteins it allows the single macromolecule to be selected see Fig2 4Jand Fig 2 5 here the button and the window my system become respectively Water and Protein 0 Pressing the my system button a new window my system pops up allowing the user to specify which atoms or groups have to be taken into account for the calculations and whethever the system or part of it should be deuterated The first column lists the atoms or the groups which can be selected and the second column lists the atoms or the groups which can be deuterated Fig24 and FigB 5 show as an example the possible choices listed in the window my system in the case of water and in the case of a protein X Atom selection Read from PDB file D Browse 256 Molecules 768 Atoms Water 256 molecules Douter Sslschen All ox a No
42. hm emploied for the computation of the discrete memory function of the VACF Since the memory function q t depends on q as well as on time for each g vector one obtains a set of P complex coefficients c 72 Bibliography 1 S W Lovesey Theory of Neutron Scattering from Condensed Matter Vol 2 ao N 10 11 12 13 14 15 16 1 Clarendon Press Oxford 1984 M Bee Quasielastic Neutron Scattering Principles and Applications in Solid State Chemistry Biology and Materials Science Adam Hilger Bris tol 1988 G R Kneller Mol Phys in press G R Kneller and A Geiger Molecular Physics 68 2 487 498 1989 G R Kneller and A Geiger Molecular Physics 70 3 465 483 1990 H J Boehm and R Ahlrichs Mol Phys 54 6 1261 1274 1985 J Anderson J J Ullo S Yip J Chem Phys 86 7 1987 G R Kneller J C Smith S Cusack and W Doster J Chem Phys 97 12 8864 1992 A J Dianoux G R Kneller J L Sauvageol and J C Smith J Chem Phys 99 7 5586 1993 G van Rossum et al The Python web site http www python org D Ascher P Dubois K Hinsen J Huguin T Oliphant Numerical Python Technical report UCRL MA 128569 Lawrence Livermoore National Li brary http numpy sourceforge net The FFTW library Technical Report http fftw org net http my unidata ucar edu content software netcdf software html http www unidata ucar edu software netc
43. http pythonmac org packages ScientificPython 2 4 9 MMTK 2 4 2 N D A A WwW nMOLDYN 3 0 1 You then have a clickable application called nMOLDYN in your Applica tions folder and the various command line tools pMoldyn xMoldyn ded_to_nc nc_to_ded dlpoly_to_nc usr local bin If you use these command line tools make sure that usr local bin is in your shell s search path It is of course also possible to install nMOLDYN from the source code dis tribution as for other Unix systems To download all of the packages above see the site http dirac cnrs orleans fr nMOLDYN download html Chapter 2 Using nMOLDYN nMoldyn provides two user interface to accomodate a wide range of users e Graphical Interface xMoldyn e Command Line Interface pMoldyn and expects input trajectories to be in netCDF format 2 1 Trajectories nMOLDYN expects trajectories to be in netCDF format and follow the conven tions of the Molecular Modeling Toolkit MMTK Trajectories that have not been produced with MMTK or MMTK based programs must be converted to MMTK format before they can be analyzed with nMOLDYN This conversion is necessary because no other common trajectory format permits efficient ac cess both to conformations at a given time and to one atom trajectories for all times In addition to providing such an access the netCDF format has several advantages that make it particularly suitable for archiving trajectories e compac
44. ich the EISF is computed should be long enough to allow for a representative sampling of the conformational space 3 4 Velocity correlation functions and Density of States The velocity autocorrelation function VACF of atom a in an atomic or molec ular system is usually defined as Crnaalt z va 0 valt 3 24 94 In some cases e g for non isotropic systems it is useful to define VACFS along a given axis Coviaa t n va 0 n ua t n 3 25 where va t n is given by Va t n n va t 3 26 The vector n is a unit vector defining a space fixed axis The VACFs of the particles in a many body system can be related to the incoherent dynamic structure factor It is easy to show that 2 limg 0 75 9 4 GW 3 27 where G w is the density of states DOS For an isotropic system it reads Gl X b inc oviaalw 3 28 z 1 00 Covjaalw a dt expl iwt Cyv aa t 3 29 For non isotropic systems relation 8 27 holds if the DOS is computed from the atomic velocity autocorrelation functions C v aa t ng where ny is the unit vector in the direction of q 3 5 Mean square displacements Another important quantity describing the dynamics of an atomic or molecular system is the mean square displacement MSD of a particle Defining da t R t Ra 0 3 30 the MSD of particle a is given by A t da2 t 3 31 As for the VACF one can introduce a mean square displacement with respect to a given a
45. ilarly to the translational velocity autocorrelation functions introduced in Section 22 4 one can define angular velocity autocorrelation functions to char acterize the angular motion of molecules In general the angular velocity is referred to an orthonormal body fixed coordinate system Usually this is the principal axis system in which the tensor of inertia is diagonal Depending on its geometry a molecule will behave differently with respect to rotational motion about different body fixed axes The autocorrelation function for the angular velocity components w is defined as Colt w Wi 0 3 41 The prime indicates a body fixed coordinate system The components w are related to the quaternion parameters describing the orientation of the molecule and their time derivatives 8 29 0 do QU Q2 4 do ws _o a do Ba q 3 42 Wry 43 q da w q43 2 q 4 d3 Here the quaternion parameters describe the rotation of the space fixed coordi nate system into the body fixed coordinate system The corresponding rotation matrix is explicitly given in Eq 4228 The components of the angular velocity may be used to define rotation angles describing rotations about the body fixed axes 8 ex f drwi r 3 43 3 8 Reorientational correlation functions The molecular reorientational correlation function is defined as the conditional probability to find a molecule with orientation Q at time t given it had the orientati
46. in the AR model Evaluating V 4 2 as given by 48 at z exp iwAt yields the density of states within the AR model Ather 2 M GUR w YP expliwAt 4 53 Here M is the mass of the tagged atom kg is the Boltzmann constant and T the temperature Note that the VACF and the density of states within the AR model are entirely determined by the coefficients al Discrete memory function of the VACF within the AR model Starting from the memory function equation of the VACF Eq B60 the first step towards a numerical computation of the memory function consists in discretizing Eq B 60 pin 1 y _ E 2 r 2 At n k b k 4 54 Eq 4 54 is now subjected to a one sided z transform Using that Z gt f n 1 f n 2F gt 2 z 0 4 55 for any discrete function f n whose one sided z transform exists one obtains from 454 1 z 7 1 4 56 ol 4 56 69 using that 0 1 The one sided z transform of an arbitrary discrete function f n is defined as F gt 2 Xpo f n z Here it has been used that the one sided z transform of the discrete convolution integral is just the product Es z Ws z Inserting the definition of the one sided z transform for z and Y z this equation can be rewritten as D gpa L FE WO 9G 1 27 TA ONES a J Note that the term proportional to z cancels out The time dependent memory function is in principle obtained by comparing the coeffi
47. ions are given by 2 FY eam k At exp ET ao isotropic system 2 16 2 FincalQmk At exp ET At n non isotropic syst 2ul 7 The quantities A t and A2 t n are the mean square displacements defined in Equations 3 31 and 3 32 respectively They are computed by using the algorithm described in Section 2 nMoldyn corrects the atomic input tra jectories for jumps due to periodic boundary conditions It should be noted that the computation of the intermediate scattering function in the Gaussian approximation is much cheaper than the computation of the full intermedi ate scattering function Finc q t since no averaging over different g vectors needs to be performed It is sufficient to compute a single mean square dis placement per atom Pressing the button Incoherent Scattering Function 39 Gaussian Approx in the graphical interface the window Incoherent Scat tering Function Gaussian Approx pops up The user can enter the moduli y A A A Incoherent Scattering Function Gaussian approx gt Q values Q shell width Vectors per shell Q direction 1 0 0 Weights none incoherent x mass coherent Window width for FFT f 0 of trajectory length Q units 4 1 nm v M Time steps in output z000 Points in spectrum 2000 Output file netCDF ISFG_spce300K_1bar_1C Browse Spectrum output file output file netCDF IISFG_SPECT_spce300K Browse
48. la tion function Moreover it can compute several quantities related to neutron scattering the coherent and incoherent intermediate scattering functions with their Fourier transforms and their memory functions and the elastic incoherent structure factor Additionally the nMOLDYN package allows one to construct modified trajectories from an input trajectory rigid body trajectories in which the internal motions of the molecules or parts thereof are eliminated angu lar trajectories which describe rigid body motions by center of mass and ori entational quaternion coordinates frequency filtered trajectories from which motions outside a specified frequency interval are eliminated As nMOLDYN 2 0 nMOLDYN 3 0 use the Molecular Modelling Toolkit MMTK an Open Source program library for molecular simulation applications and expects tra jectories to be in MMTK format netCDF files Trajectory converters for CHARMM X PLOR NAMD DLPOLY and AMBER trajectories are included in the distribution of nMOLDYN All nMOLDYN calculations can be applied to a whole system or to arbitrary subsets The most common subsets can be selected in the graphical interface from a list less common selections can be specified by Python code via the command line interface nMOLDYN 2 0 was written by Tomasz Rer Krzysztof Murzyn and Konrad Hinsen and the upgrade nMOLDYN 3 0 by Paolo Calligari Contents 3 5 1 1 Linux sau en De E
49. lues using the form Qmin AQ maz In this way the intermediate scattering function will be calcultad for q values defined as qm Qmin tm Aq with m running from 0 to Ng 1 and qmax N 1 Ag Through the text field Vectors per shell the user can enter the number of g vectors N that have to be used in the average of Eq 2 21 The expected value is un integer The required tolerance on the g moduli can be entered via the text field Q shell width The program looks for N q vectors whose moduli deviate from the grid of the selcted values within the given tolerance in order to carry out the average of Eq 2 21 The Q shell width fix the g resolution In the panel Weights the user can select how to weight the different contri butions to the Intermediate coherent scattering function coming from all atoms of the system see Section 2 2 The units of the q vectors can be selcted in the panel q units 45 Here the output files is a netCDF files containing the EISF as a function of q in the selcted units The default names of the output file contains the strings EISF followed by the name of the trajectory The user can then change the names and or the destination directory through the Browse buttons Pressing the OK button switches to the Almost Done window containing all settings in python language These settings can be saved in a python script file via the Save button and then run later using the command line interface of nMoldy
50. memory function that can be calculated by nMOLDYN is the memory function related to the coherent intermediate scattering function It is defined through the cor responding memory function equation t OrFeon q t dr amp q t T Fcon q T 3 61 0 The memory function q t which depends on time as well as on q permits the analysis of memory effects on different length scales From a numerical point 60 of view the calculation of the memory function equation relevant to the coherent intermediate scattering function is completely analogous to the case of the VACF memory function the discrete time signal being here N 5 ba coh expl iq Ra t 3 62 a 1 See section 4 4 for more details on the numerical algorithm Memory function equation of Incoherent scattering nMOLDYN allows one to calculate the memory function related to the incoherent intermediate scattering function as well It is defined through the corresponding memory function equation E f CN 3 63 The memory function q t which depends on q as well as on time permits the analysis of memory effects on different length scales As in the previuos cases the numerical calculation of the memory function equation relevant to the incoherent intermediate scattering function is based on the Autoregressive model the discrete time signal being here N XC ba ine exp iq Ra t 3 64 a 1 See section 41 4 for more details on the numerical al
51. n a python script file via the Save button and then run later using the command line interface of nMoldyn or run immediately through the button Run Pressing the button cancel closes the window without saving settings Anisotropic Through this option the average of Eq 2 19 will be carried out taking the q vectors lying on a given plane or along a given direction whose moduli deviate within the prescribed tolerance from the required values Press ing this button the window Incoherent Scattering Function Anisotropic pops up The input parameters to be specified in order to carry out the compu tation the coherent intermediate scattering function and its Fourier spectrum are the same as for the Isotropic option In addition here one has to specify the plane or the direction on which the q vectors have to be taken This can be done via the text field Q direction or plane entering either a unit vector in 42 Incoherent Scattering Function Isotropic 3 Q values 0 2 100 Q shell width ft Vectors per shell 50 Weights none incoherent x mass coherent Window width for FFT 10 of trajectory length Q units 4 1 nm v M Time steps in output 2000 Points in spectrum 2000 Output file netCDF SF_spce300K_1bar_100 Browse Spectrum output file output file netCDF ISF_SPECT_spce300K_1 Browse ox Figure 2 22 Incoherent Sca
52. n be written as 00 See 3 5 00 51 F q t is called the intermediate scattering function and is defined as Flat Dd Taslexp iq Ra 0 explig Ra 3 6 in Tas L Babs CES 3 7 The operators R t in Eq are the position operators of the nuclei in the sample The brackets denote a quantum thermal average and the time dependence of the position operators is defined by the Heisenberg picture The quantities ba are the scattering lengths of the nuclei which depend on the isotope and the relative orientation of the spin of the neutron and the spin of the scattering nucleus If the spins of the nuclei and the neutron are not prepared in a special orientation one can assume a random relative orientation and that and position of the nuclei are uncorrelated The symbol appearing in Pag denotes an average over isotopes and relative spin Aai of neutron and nucleus Usually one splits the intermediate scattering function and the dynamic structure factor into their coherent and incoherent parts which describe collective and single particle motions respectively Defining bacon ba 3 8 2 da inc b2 ba gt 3 9 the coherent and incoherent intermediate scattering functions can be cast in the form wd bacon 6 con exp iq Ra 0 expliq Rg t 3 10 Feoh q t A Finc a t HL Panelesnl ia Ro Oplia RaO 3 11 The corresponding dynamic structure factors are obtai
53. n or run immediately throgh the button Run Pressing the button cancel closes the window without saving settings 2 2 6 Output Input Inspection View Pressing the button View brings up a drop down menu from which it is possible to choose the following options e Trajectory information e View variables e Animation Display Display 3D e Reciprocal basis e Direct basis In this section we give a description of the above options Trajectory information Pressing this button one can visualize the trajectory information e name of the trajectory file and location directory e simulated system molecules number atoms number and time steps e creation date Press the Close button to come back to the main start up window of nMoldyn View variables Pressing the button View variables a drop down menu from which one can select the variable to be inspected brings up The option termodynamics shows the evolution of the temperature in Kelvin during the simulation while the option energy shows the evolution of the kinetic energy kJ mol 1 46 Animation Animation of the imported trajectoty can be displayed through this option Pressing button Animations a window trajectory animation pops up From this window the user can enter the first and the last step of the trajectory to be precessed in the text fields first step and last step respectively and the number of time steps to be skipped in the text field skip Display
54. n the quaternion parameters j _ GM Gin GP Diana IP mn IO 3 54 x go ig TP go iqa T q2 igi PT q2 iqi P Here the quaternion parameters describe the rotation of the space fixed coordi nate system into the body fixed coordinate system The corresponding rotation matrix is given in Eq 4 28 According to Eq 3 53 the spherical harmonics are just special cases of the Wigner functions Yi m JH G m 5 m 25 a ae CD rot 3 55 x go iqa TP qo iqa q2 iqi q2 iqu P Using the normalization of the spherical harmonics and Eq 8 52 one arrives at the following expression for the p coefficients Phan t 4r Yinla t n a 0 3 56 The following relations for the p coefficients hold Pan Gani 3 57 Prin t P m nlt Phm t 3 58 The coefficients 6 are the components of the 27 1 x 2j 1 unit matrix The initial value of the the p coefficients is an immediate consequence of definition 59 3 46 and p Q 0 0 0 6 Q Eq 3 58 follows from the symmetry of the Wigner functions and the symmetry of classical time correlation functions Since measurable quantities must be real it follows from that only p coefficients with m n 0 can be directly measured pjg t is measured by infrared spectroscopy dipole dipole correlation function and p2 t by re laxation NMR experiments Here one measures in most cases the integr
55. nMOLDYN User s Guide Vania Calandrini Paolo Calligari Konrad Hinsen Gerald R Kneller Institut Laue Langevin 6 rue Jules Horowitz BP 156 38042 Grenoble Cedex 9 France Centre de Biophysique Moleculaire CNRS Rue Charles Sadron 45071 Orleans Cedex 02 France Synchrotron Soleil Division Experiences Saint Aubin BP 48 91192 Gif sur Yvette Cedex France December 19 2006 Corresponding author Electronic mail kneller cnrs orleans fr Abstract nMOLDYN is a modular program package for the analysis of Molecular Dy namics trajectories especially designed for the computation and decomposi tion of neutron scattering spectra The current release 3 0 of nMOLDYN is an upgrade of the version nMOLDYN 2 0 which extends the functionality of the original version nMOLDYN 1 0 G R Kneller V Keiner M Kneller M Schiller Comp Phys Comm 91 191 214 1995 and provides a much more convenient user interface both graphical interactive and batch and can be used as a tool set for implementing new analysis modules This was made pos sible by the use of a high level language Python and of modern object oriented programming techniques nMOLDYN allows one to calculate the mean square displacement the velocity autocorrelation function as well as its Fourier Trans form the density of states and its memory functions the angular velocity autocorrelation function and its Fourier transform the reorientational corre
56. nction output file output file netCDF AR CSF_spce300K_1bar Browse Structure factor output file output file netCDF AR CSF_SPECT_spce30 Browse Memory function output file output file netCDF AR CSF_Memory_spced Browse J ok Cancel Figure 2 19 Coherent Scattering AR analysis 37 Here the user can enter the order poles number of the Autoregressive model in the field Model Order see Section Z4 A priori the autocorrelation function and its power spectrum can be approximated to almost arbitrary preci sion by increasing the order of the autoregressive model In practice it has been proven that reliably computation can be carried out up to P of the order of 1000 poles Since the program performs the calculations in high precision by default type None in the text field Extended precision memory function The user can enter the moduli of the required q values through the text field named Q values using the form qmin AQ Gmax In this way all of the functions will be calcultad for qm values defined as qm Qmin m Aq with m running from 0 to N 1 and qmars N 1 Ag Through the text field Vectors per shell the user can enter the number of g vectors N oriented isotropically that have to be used in the average The expected value is un integer The required tolerance on the g moduli can be entered via the text field Q shell width The prgram
57. ndow in FigB 15 pops up The user can enter the frequency interval in THz units in the fields Filter window The output file is a file netCDF containing only motions in the chosen interval The default name of the output trajec tory contains the string AT followed by the name of the trajectory The user can then change the name and or select the destination directory through the Browse button in the Angular trajectory window Pressing the OK button switches to the Almost Done window containing all input settings in python language These settings can be saved in a python script file via the Save but ton and then run later using the command line interface of nMoldyn or run immediately through the button Run Pressing the button cancel closes the window without saving settings 2 2 5 Output Scattering Pressing the button Scattering brings up a drop down menu from which it is possible to choose the following options e Coherent Scattering Function e Coherent Scattering AR analysis e Incoherent Scattering Function Gaussian Approx e Incoherent Scattering Function e Incoherent Scattering AR analysis 30 e EISF In this section we give a description of the scattering quantities calculated by nMoldyn For a description of the theoretical background see Chapter B Coherent Scattering Function nMoldyn computes the coherent intermediate scattering function on a rectangu lar grid of equidistantly spaced points along the time and the q axi
58. ne 1 Hydrogen Oxygen Figure 2 4 Atom selection in water Pressing the close button the choice is confirmed and the window my system is closed Pressing the OK button in the my system window the user confirms the choice of the system To analyze the dynamics of complex molecular systems it is often desirable to consider the overall motion of molecules or molecular subunits i e their 16 Atom selection x Protein 0 Read from PDB file Protein 0 sn Browse 129 Residues 1960 Atoms Protein 0 129 residues Deer Scie OK i a D J ci Ju 2 af a Figure 2 5 Atom selection in lysozyme rigid body motion The Group Selection button allows the user to select within the system already selected by the atom selection button a particular atomic group that will be used to calculate some functions describing its rigid body dynamics Dynamics angular trajectory rigid body trajectry rigid body rotation trajectory angular velocity autocorrelation function VACF spectrum of angular VACF reorientational correlation function The functions describing the rigid body dynamics have not been fully tested yet but they will be soon available At present the guide 17 contains only a description of the theoretical background and of the relevant algorithms Once the trajectory has been imported pressing the Group Selection a window named Groups listing all the possible group
59. ne or along a given direction whose moduli deviate within the prescribed tolerance from the required values Press ing this button the window Coherent Scattering Function Anisotropic pops up see Fig 2 17 The input parameters to be specified in order to carry out the computation the coherent intermediate scattering function and its Fourier spectrum are the same as for the Isotropic option In addition here one has to specify the plane or the direction on which the q vectors have to be taken This can be done via the text field Q direction or plane entering ei ther a unit vector in the form x y 2 or two vectors in the form x y z x y 2 defining a plane 33 26 Coherent Scattering Function Anisotropic Q values 022 400 0 Q shell width 1 Vectors per shell EC Q direction or plane PEAR Weights none incoherent Mass coherent Window width for FFT hi 0 of trajectory length Q units 4 1 nm v M Time steps in output 2000 Points in spectrum 2000 Output file netCDF CSF_spce300K_1har 10 Browse Output file for Dynamic Structure Factor output file netCDF CSF_SPECT_spce300K_ Browse ox Figure 2 17 Coherent Scattering Function Anisotropic Explicit List Through this option the average of Eq 2 10 Jwill be carried out taking the g vectors whose moduli deviate within the prescribed tolerance from the
60. ned by performing the Fourier transformation defined in Eq BD An important quantity describing structural properties of liquids is the static structure factor which is defined as 00 S q l dw Scon q w Feon 4 0 3 12 00 If the sums over a and 8 in 3 10 are extended over subsets of respectively equivalent atoms one obtains the partial static structure factors 3 2 Classical framework In the classical framework the intermediate scattering functions are interpreted as classical time correlation functions The position operators are replaced by 52 time dependent vector functions and quantum thermal averages are replaced by classical ensemble averages It is well known that this procedure leads to a loss of the universal detailed balance relation S q w exp Ghw S q w 3 13 and also to a loss of all odd moments 00 Grr dw w 1 S q w Dy Derg 3 14 The odd moments vanish since the classical dynamic structure factor is even in w assuming invariance of the scattering process with respect to reflections in space The first moment is also universal For an atomic liquid containing only one sort of atoms it reads _ ha w 2M where M is the mass of the atoms Formula shows that the first moment is given by the average kinetic energy in units of of a particle which receives a momentum transfer q Therefore w is called the recoil moment A number of recipes has been suggested to c
61. nit 1 and three imaginary units I J and K Since IJ K cyclic quaternion multiplication is not commutative A possible matrix representation of an arbitrary quaternion A ad0 1 01 14 02 J 03 K 4 29 reads ag a aQ dz A Se 2 4 30 a2 43 ao a3 a2 ay ao The components a are real numbers Similarly as normal complex numbers allow one to represent rotations in a plane quaternions allow one to represent rotations in space Consider the quaternion representation of a vector r which is given by R x I4 y J 2 K 4 31 and perform the operation R QRQ 4 32 66 where Q is a normalized quaternion 1 IQP d af 92 di t Q Q i 4 33 The symbol tr stands for trace We note that a normalized quaternion is represented by an orthogonal 4 x 4 matrix R may then be written as R x I y J 7 K 4 34 where the components x y z abbreviated as r are given by r D q r 4 35 The matrix D q is the rotation matrix defined in 4 28 Solution of the minimization problem In quaternion algebra the rota tional minimization problem may now be phrased as follows m q Y wall QR Q Rall Min 4 36 Since the matrix Q representing a normalized quaternion is orthogonal this may also be written as m q Su QRO R Q Min 4 37 This follows from the simple fact that A AQ if Q is normalized Eq 4 37 shows that the targe
62. ns can be obtained The second generation nMoldyn presented here offers an interactive graphi cal user interface for standard calculations highly flexible script based process ing for non standard applications and a machine independent compact binary file format These improvement were made possible by the use of 1 the object oriented high level Python TO 2 the fast array package Numerical Python L the efficient FFT implementation FFTW 12 the portable binary file format netCDF and the corresponding library 17 the scientific computing library Scientific Python TS me aie io the molecular simulation library MMTK 19 All of these packages are developped and distributed following the Open Source principles 20 anyone can use and improve them without being hindered by licensing restricitons All the time consuming algorithms use efficient imple mentations in C or Fortran which are provided by other packages Numerical Python FFTW and MMTK together with a Python interface while nMoldyn itself contains only Python modules This NOTE is organized as follows Section I contains a description of the requirements for nMOLDYN installation and a description of the installation procedure Section B contains a description of the different options and quanti ties that can be calculated with nMOLDYN described in the order in which they appear within the graphical interface Section 3 contains the definitions and the the
63. o the lenght of the trajectory The units of the q vectors can be selcted in the panel q units The number N of points in the output file containing the intermediate coherent scattering functions and the number of frequency points in the output file containing their fourier spectra can be entered in the panels time steps in output and Points in spectrum respectively The output files are two netCDF files containing the Coherent intermediate scattering functions as a function of the time in ps for all q values and their Fourier spectra as a function of the frequency in THz respectively Since Feoh dm k At is symmetric in time it is stored only for the positive time axis and its Fourier spectrum is stored only for values on the positive frequency axis The default names of the two output files contain the strings CSF_ and CSF_SPECT respectively followed by the name of the trajectory The user can then change the names and or the destination directory through the Browse buttons Pressing the OK button switches to the Almost Done window containing all settings in python language These settings can be saved in a python script file via the Save button and then run later using the command line interface of nMoldyn or run immediately through the button Run Pressing the button cancel closes the window without saving settings Anisotropic Through this option the average of Eq 2 10 will be carried out taking the q vectors lying on a given pla
64. ompute the incoherent intermediate scattering function on a rectangular grid of equidistantly spaced points along the time and the q axis repectively Finc Qm k At 5 Wa Finc a Gm k At 2 18 q Fincalqm k At exp iq Ra 0 expliq R t 2 19 The indices k and m run from 0 to N 1 and from 0 to N 1 respectively N is the total number of time steps in the coordinate time series and N is a user defined number of q shells The values for qm are defined as qm Qmin m Ag The program corrects the atomic input trajectories for jumps due to periodic boundary conditions The symbol 7 in denotes an average over q vectors having the same modulus qm For each q value and each atom an average over intermediate scattering functions Finc a q k At for a fixed number of 41 vectors q is performed The q vectors can be isotropically distributed in the space in a plane or along a direction The correlation functions Finca Qi k At are computed by using the FCA algorithm described in Section LI The quantities Wa are user defined weights which are normalized to one Xa wa 1 such that Fine q 0 1 Although the efficient FCA technique is used to compute the atomic time correlation functions the program may consume a considerable amount of CPU time since the number of time correlation functions to be computed equals the number of atoms times the total number of q vectors The above calculations ma
65. on Press Oxford 1968 M E Rose Elementary Theory of Angular Momentum John Wiley New York 1957 G R Kneller K Hinsen J Chem Phys 2001 115 11097 http www cecill info http www python org download http www unidata ucar edu software netcdf http sourcesup cru fr projects scientific py F J Harris Proc IEEE 66 1 51 1978 74 39 40 41 42 43 44 45 46 47 T Rog K Murzin K Hinsen G R Kneller J Comp Chem 2002 24 657 E O Brigham The Fast Fourier Transfrom Prentice Hall Englewood Cliffs NJ USA 1974 A Papoulis Signal Analysis McGraw Hill Singapore 1984 G R Kneller Technical Report J l 2215 Forschungszentrum J lich ISSN 0366 0885 ZB Forschungszentrum J lich D 52425 J lich Germany G R Kneller Molecular Simulation 7 113 1991 A Papoulis Probability random variables and Stochastic Processe McGraw Hill New York 1991 3rd ed Makhoul J Proc EEE 1975 63 561 J Burg Maximum Entropy Spectral Analysis PhD thesis Stanford Uni versity Stanford CA 1975 Makhoul J IEEE Transactions on Acoustics Speech and Signal Process ing ASSP 25 1977 423 75
66. on Qo at time to In the following this probability will be denoted by p Q t1 Qo to Here N denotes a set of angular coordinates as Euler angles or 97 quaternion parameters The joint probability p Q1 t1 Qo to which gives the probablity to find a molecule with orientation Qo at time to and with orientation Q at time ti can be expressed as p Q ti No to p Q t Qo to p Qo to Here p Mo to is the probability to find a molecule with orientation Qo at time to If we consider an isotropic system in thermal equilibrium the reorientational correlation function depends only on the time difference t t1 to and the change in orientation Q i e p Q1 t1 Mo to p Q t 0 0 In addition we have p Qo to 1 877 where 87 is the volume of the angular space The reorientational correlation function may now be expanded in Wigner rotation matrices 28 which form a complete set of basis functions in Q BABI p t10 0 DI DE 3 44 872 mn mn jmn In the following the coefficients p7 t are called p coefficients Using the or thogonality of the Wigner functions i 8r dQ DH MD Q Sisi mm Onn an Di OD 0 Gr m n 3 45 Eq 8 44 can be inverted to give Phin t I d9 p Q t0 0 D3 2 3 46 Writing the reorientational correlation function as p 9 10 0 FILE H 3 47 a where Qa t is the orientation of molecule a with respect to its initial orientation and is
67. onality of nMoldyn The only limitation concerns the selection of the part s of the full system that will be used in a particular calculation Ac tually only the most common selections can be made small molecules can be selected collectively e g all water molecules and macromolecules can be se lected individually allowing in addition the restriction to certain subparts e g only the backbone The same options exist to specifying which parts of the system should be deuterated Alternatively a detailed selection can be made by marking the atoms to be used in a PDB file Moreover from the graphical user interface it is possible to create an input file for the command line interface Such input file provides a convenient starting point for customization To run the interactive program based on the graphical interface type xMoldyn The start up window of nMoldyn will pop up see FigB T This window con tains five drop down menu buttons e File e Dynamics e Scattering e View e Help three input text fields 10 NN ER ue nMOLDYN File Dynamics Scattering View Help First step 0 0 000000 ps Last step I 0 001000 ps Skip fi 0 001000 ps Atom Selection Group Selection Figure 2 1 Start up window of nMoldyn e First Step e Last step e Skip and two buttons e Atom selection e Group Selection All inputs concerning the system are set via the drop down menu button File the three te
68. oretical background of the various quantities provided by nNMOLDYN In Section QZ the algorithms for the numerical calculation of the various quantities are discussed Chapter 1 Installing n MOLDYN The current version of nMOLDYN is 3 0 1 The nMOLDYN distribution con tains the full Python source code which is covered by the CeCILL License 34 1 1 Linux users If your Linux distribution uses the RPM package manager use the source RPM file to build a binary RPM file compatible with your system The command is rpm rebuild nMOLDYN 3 0 1 1 src rpm Then install the resulting binary RPM file The source RPM file of nMOLDYN can be downloaded from the site http dirac cnrs orleans fr nMOLDYN download htmi Before compiling and installing nMOLDYN please make sure the following packages are installed and working correctly 1 the netCDF library 2 the Scientific Python the MMTK library the FFTW library oa Ae o the Python interface to FFTW If you don t have them you can get the corresponding source RPM file from the site http dirac enrs orleans fr nMOLDYN download html although these are not necessarily the most recent versions available For installation compile and install the packages one by one in the order given above followed by the source RPM for nMOLDYN It is important to install the binary RPM for each package before building the binary RPM for the following one If your Linux distribution does not
69. orrect classical dynamic structure factors for detailed balance and to describe recoil effects in an approximate way The most popular one has been suggested by SCHOFIELD 22 3 15 S q w palco 3 16 One can easily verify that the resulting dynamic structure factor fulfills the relation of detailed balance Formally the correction 3 16 is correct to first order in A Therefore it cannot be used for large q values which correspond to large momentum transfers hg This is actually true for all correction methods which have suggested so far For more details we refer to 3 3 3 Elastic incoherent structure factor The elastic incoherent structure factor EISF is defined as the limit of the incoherent intermediate scattering function for infinite time EISF q lim Finc g 1 3 17 Using the above definition of the EISF one can decompose the incoherent inter mediate scattering function as follows where Fj q t decays to zero for infinite time Taking now the Fourier trans form it follows immediately that Sinc q w EIS F q 6 w Si q w 3 19 53 The EISF appears as the amplitude of the elastic line in the neutron scattering spectrum Elastic scattering is only present for sytems in which the atomic motion is confined in space as for solids To understand which information is contained in the EISF we consider for simplicity a system where only one sort of atoms is visible to the neutrons To a very good appr
70. oximation this is the case for all systems containing a large amount of hydrogen atoms as biological systems Incoherent scattering from hydrogen dominates by far all other contri butions Using the definition of the van Hove self correlation function Gs r t I red ba Galr t aa f exp iq r Finc q t 3 20 which can be interpreted as the conditional probability to find a tagged particle at the position r at time t given it started at r 0 one can write EISF q Be f dr expliq r Gs r t oo 3 21 The EISF gives the sampling distribution of the points in space in the limit of infinite time In a real experiment this means times longer than the time which is observable with a given instrument The EISF vanishes for all systems in which the particles can access an infinite volume since G r t approaches 1 V for large times This is the case for molecules in liquids and gases For computational purposes it is convenient to use the following representa tion of the EISF 8 EISF a Y 82 ino lexplia Rell 3 22 This expression is derived from definition 8 17 of the EISF and expression ETT for the intermediate scattering function using that for infinite time the relation exp iq Ra 0 expliq Ra t explia Ra 3 23 holds In this way the computation of the EISF is reduced to the computation of a static thermal average We remark at this point that the length of the MD trajectory from wh
71. s repectively Feon qm k At o q Opla k AD k 0 N 1 m 0 N 1 2 10 N is the total number of time steps in the coordinate time series and N is a user defined number of q shells p q k At is the Fourier transformed particle density pla k At Y wa expliq Ra k At 2 11 The symbol 7 in denotes an average over q vectors having approxi mately the same modulus qm dmin mM Aq The particle density must not change if jumps in the particle trajectories due to periodic boundary conditions occcur In addition the average particle density N V must not change This can be achieved by choosing q vectors on a lattice which is reciprocal to the lattice defined by the MD box Let b b2 b3 be the basis vectors which span the MD cell Any position vector in the MD cell can be written as R x by y ba z b3 2 12 with x y 2 having values between 0 and 1 The primes indicate that the coordinates are box coordinates jump due to periodic bounday conditions causes x y z to jump by 1 The set of dual basis vectors b b b is defined by the relation bib 2 13 If the g vectors are now chosen as q 2r kb Ib mb 2 14 where k l m are integer numbers jumps in the particle trajectories produce phase changes of multiples of 27 in the Fourier transformed particle density i e leave it unchanged One can define a grid of q shells or a grid of g vectors along a gi
72. s spce300K_lbar_100ps log_file dos vel_spce300K_lbar_100ps log time info 0 9990 1 atoms Water Hydrogen projection_vector None weights incoherent ft_window 10 units length Units nm Save Run 2 Density Of states from coordinates gt Projection vector None Weights none incoherent x mass coherent Differentiation fast Window width for FFT Fi 0 of trajectory length Length units 4 nm Ww A Frequency units 4 1 ps x Vlicm Points in spectrum 2000 Output file text DOS_spce300k_1 bar_1 Browse ox Figure 2 13 DOS from coordinates 26 _ Autoregressive Model from velocities ut Model order 50 Extended precision memory function None Projection vector None Weights gt none incoherent mass coherent Length units 4 nm v Frequency units 4 1 ps licm Time steps in output 2000 Points in spectrum 2000 Memory function output file output file text AR Memory_spce300K_1 Browse VACF output file output file text AR VACF_spce300K_1b Browse DOS function output file output file text AR DOS_spce300K_1ba Browse MSD function output file output file text AR MSD_spce300K_1ba Browse AR parameter output file output file text
73. s which can be taken into ac count to perform the above calculations pops up For example small molecules can be selected collectively e g all water molecules and macromolecules can be selected individually This option is very useful when one deals with com plex systems containing for example proteins In this case a drop down menu button relevant to the individual macromolecule appears wthin the Groups win dow This button allows the user to select some standard groups For example in the case of a protein we have the overall macromolecule the sideChain the backbone the methyl groups For each group the user can select within the window named selection wherever it has to be taken into account to perform the calculation In addition within the SideChain group selection the user can selectively mark the residues to be used lysine histidine alanine etc Once a group has been selected in the selection window the two buttons on the right become active The first button allows one to select the Reference file PDB containing the group reference position which has to be used to calculate one of the above quantities The second button allows one to display some information about the reference file Pressing the close button the choice is confirmed and the window is closed 2 2 4 Output Dynamics Pressing the button Dynamics brings up a menu from which it is possible to choose the following options e Mean Square Displacement e V
74. simulated and measured neutron scattering intensities for classical systems if recoil effects in the scat tering process are not dominant 3 The experimental data can be used to test the quality of the MD force field which is the central input for the simula tions 5 6 7 Conversely the simulated intensities allow a detailed analysis of the dynamical and structural behaviour of the system under consideration Sl O The latter is particularly important for complex systems for which an interpretation of the measured intensities in terms of simple analytical models is difficult if not impossible The program package nMOLDYN allows neutron scattering intensities to be efficiently calculated from MD simulations The calculation of various space and time correlation functions permits a detailed analysis of the structure and dynamics of the system under consideration nMOLDYN contains modules for the calculation of incoherent and coherent dynamic structure factors elastic incoherent structure factors EISFs mean square displacements translational velocity autocorrelation functions and memory functions In addition rigid body trajectories of subunits of the system can be extracted from molecular dy namics trajectory files These subunits can be arbitrarily defined their size can range from a few atoms to a whole domain in a macromolecule From the rigid body trajectories angular correlation functions and reorientational correlation functio
75. solution This occurs tipically when one need to perform a large number of similar calculations or when high flexibility is required in the selection of 47 the atoms and groups that are to be used for a given calculation For these situations nMoldyn provides a command line interface that reads all input in formation from a single specification file The specification files are Python scripts In the following we show as an example the specification file for a sim ple mean square displacement calculated over all of the atoms of a protein from MMTK import trajectory lysozyme nc output files msd msd plot title Mean Square Displacement time_info 0 None 1 weights mass atoms Protein 0 The command line interface provides high flexibility for advanced users For example if the standard atom selections offered by the graphical interface are insufficients it is possible to customize the atom selection by few lines of Python in the nMoldyn specification file The following file specifies the calculation of the mean square displacement for only the sidechains of residues 5 to 25 from MMTK import trajectory lysozyme nc output files msd msd plot title Mean Square Displacement time_info 0 None 1 weights mass def atoms_code trajectory retrieve all proteins from
76. sponding unit vector in the form z y z Otherwise writing None the MSD will be calculated by averaging over all the directions In the panel Weights the user can select how to weight the different contributions to the MSD coming from all atoms of the system The weights are normalized to one see Section The units length for the Mean Square Displacement and the number of time steps in the output file can be selected in the panel Length units and Time Steps in output respectively The output file is a file ascii plot containing the MSD of the selected system as a function of the time in ps The default name of the output file contains the string MSD followed by the name of the trajectory The user can then change the name and or select the destination directory through the Browse button Pressing the OK button switches to the Almost Done window containing all settings in python language see Fig 2 7 These settings can be saved in a python script file via the Save button and then run later using the command line interface of nMoldyn or run immediately through the button Run Pressing the button cancel closes the window without saving settings Velocity Autocorrelation Function VACF Pressing the button Velocity Autocorrelation Function will open a drop down menu button from which the user can select how to perform the calculation of the velocity autorrelation function directly from the velocities or from the coordinates In this case the v
77. t files binary storage e machine independent format e fully self contained complete information about the system is stored in the trajectory file The conversion of the trajetories from the DLPOLY format to the MMTK format can be made directly via the nMOLDYN graphical interface see para graph B 2 T Otherwise nMOLDYN comes with two converters for foreign for mats e dcd_to_nc converts from the DCD format used by the programs CHARMM X PLOR and NAMD In addition to the DCD file a compatible PDB is required as a system definition Note that DCD files are machine dependent the converter accepts only files that are compatible with the machine it runs on There is also an inverse converter nc_to_dcd e dlpoly_to_nc converts from the DLPOLY format It reads the DLPOLY files FIELD and HISTORY in order to produce a trajectory file in netCDF format Run any of these converters without input arguments in order to get usage instructions An elaborate converter for AMBER trajectory files is available separately Many netCDF utilities and various visualization and analysis packages to access netCDF data are currently available For an up to date list of freely available and commercial software that can access or manipulate netCDF data see the NetCDF Software list in Ref I3 Here we recall the ncdump and ncgen utilities These two tools convert between binary netCDF files and a text representation of netCDF files ncdump generates an
78. t function to be minimized can be written as a simple quadratic form in the quaternion parameters 43 m q q Ma 4 38 M Y waMa 4 39 The matrices Ma are positive semi definite matrices depending on the positions Yq and ro Mani ne es z2 Loa Yo a 2LaL0a 2YaY0a 220200 Ma 12 2 Ya20a ZaYoa Ma 13 2 LaZ0a ar ZaXLoa Ma a 2 LaYoa YoX0a Ma22 Da T y a alr Lda T You T Zda 2La Toa 2YaYoa 224200 Ma 23 2 LaY0a T Ya Too Ma 24 2 LaZ00 Zu ZaT0a Mass 22 y2 22 28 Ya La La Loa 2YaYoa 220200 Ma aa 2 YaZ0a ZaYoa Masa a y Z T Ta Ya 2a 2 aToa 2YaY0a 224200 4 40 67 The rotational fit is now reduced to the problem of finding the minimum of a quadratic form with the constraint that the quaternion to be determined must be normalized Using the method of Lagrange multipliers to account for the normalization constraint we have m q A a Mq q q 1 Min 4 41 This leads immediately to the eigenvalue problem Ma q 4 42 q q 1 4 43 Now any normalized eigenvector q fulfills the relation q Mq m q Therefore the eigenvector belonging to the smallest eigenvalue Amin is the desired solution At the same time Amin gives the average error per atom 4 4 Autoregressive AR process To compute the memory function t from a discrete signal x n x nAt the latter is modelled by an a
79. time axis The default name of the output file contains the string VACF followed by the name of the trajectory The user can then change the name and or select the destination directory through the Browse button Pressing the OK button switches to the Almost Done window containing all settings in python language see Fig 9 These settings can be saved in a python script file via the Save button and then run later using the command line interface of nMoldyn or run immediately throgh the button Run Pressing the button cancel closes the window without saving settings 21 Velocity Autocorrelation Function from velo s Projection vector None Weights none incoherent w mass coherent Length units 4 nm vA Time steps in output 2000 Output file text VACF_spce300K_1bar_1 Browse OK Cancel Figure 2 8 VACF from velocities FromCoordinates Pressing the button coordinates will open the window Velocity Autorrelation Function from coordinates From this window it is possible to enter the same quantities as for the calculation from velocities In addition the paneldifferentiation allows one to select how to perform the differentiation of the coordinate trajectories to obtain the velocities trajectory see FigB T0 Refer to Section 2 2 for more details about the differentiation schemes Density of States DOS The program calculates the discrete density
80. tory window Pressing the OK button switches to the Almost Done window containing all input settings in python language These settings can be saved in a python script file via the Save button and then run later using the com mand line interface of nMoldyn or run immediately through the button Run Pressing the button cancel closes the window without saving settings Digital filtering It is often of interest to restrict attention to motions in a specific frequency interval both for quantitative analysis and for visualization by animated dis play this is particularly useful to study low frequency motions without being distracted by the high frequency noise nMoldyn can apply frequency filter to the atomic trajectories either of the whole system or of a user defined subset The result is stored in a new trajectory file that contains only motions in the chosen interval Frequency filtering uses a straightforward algorithm take the discrete Fourier transform of each particle coordinate set the Fourier coefficients outside the filtering interval to zero and transform back to the time domain This corresponds to applying a rectangular window in the frquency domain 29 Digital Filter 85 Filter window from to Output file netCDF DF_lysozyme_1atm_300 Browse Cancel Figure 2 15 Digital Filtering window Pressing the button Digital Filtering from the menu Dynamics of nMoldyn the wi
81. tral smoothing procedure since the VACF is weighted by this window Gaussian function nMoldyn computes the density of states starting from both atomic velocities and atomic coordinates In this case the velocities are computed by numerical differentiation of the coordinate trajecto ries correcting first for possible jumps due to periodic boundary conditions From velocity Pressing the button velocity will open the window Density of states from velocities see ZII To calculate the DOS with respect to a given axis the user can enter in the Projection Vector field the components of the corresponding unit vector in the form z y z Otherwise writing None the DOS will be isotropically calculated In the panel Weights the user can select how to weight the different con tributions to the DOS coming from all atoms of the system see Section 2 2 The panel window width for FFT percento of trajectory lenght allows one to select the width of the Gaussian function to be used in the smoothing procedure for the calculation of the DOS see Section kI The width is defined with respect to the lenght of the trajectory The length units for the DOS the frequency units and the number of points in the output spectrum can be selected in the panels Length units Frequency units and Points in spectrum respectively 23 Velocity Autocorrelation Function from coore Projection vector None Weights none w incoherent mass coherent
82. ttering Function Isotropic the form x y z or two vectors in the form x y z 2 y 2 defining a plane Explicit List Through this option the average of Eq ZIJ will be carried out taking the q vectors whose moduli deviate within the prescribed tolerance from the required values and lying along explicitly listed directions Pressing this button the window Incoherent Scattering Function Explicit List pops up Here the input parameters to be specified in order to carry out the computation the incoherent intermediate scattering function and its Fourier spectrum are the same as for the Isotropic option In addition here one has to specify the directions along which the q vectors have to be taken This can be done via the text field Q List entering a list of unit vectors in the form L Y 230 Y 23 43 Incoherent Scattering AR analysis In the framework of the Autoregressive model nMoldyn allows the intermediate coherent scattering function its Fourier spectrum the incoherent dynamical structure factor and its memory function to be computed on a rectangular grid of equidistantly spaced points along the time and the q axis repectively The user is referred to Section 5 9 for more theoretical details The dynamical variable of the correlation function under consideration sn Da ine EXP iq Ra nAt is considered as a discrete signal which is modeled by an autore gressive stochastic process of order P For each q valu
83. uded in the output This is often the most suitable option to use for a brief look at the structure and contents of the netCDF file h Show only the header information in the output that is output only the dec larations for the netCDF dimensions variables and attributes of the input file but no data values for any variables The output is identical to using the c option except that the values of coordinate variables are not included At most one of c or h options may be present v varl The output will include data values for the specified variables in addition to the declarations of all dimensions variables and attributes One or more variables must be specified by name in the comma delimited list following this option The list must be a single argument to the command hence cannot contain blanks or other white space characters The named variables must be valid netCDF variables in the input file The default without this option and in the absence of the c or h options is to include data values for all variables in the output 2 2 Graphical Interface xMoldyn Through the nMoldyn graphical interface the user first open a trajectory then specifies the parameters for the calculation he wishes to perform and finally starts the calculation itself Moreover he can check the status of running cal culations and inspect their results The graphical interface gives access to most of the functi
84. ues on the positive frequency axis The default names of the output files contain the strings AR ISF AR ISF_SPECT and AR ISF Memory followed by the name of the trajectory The user can then change the names and or the destination directory through the Browse buttons Pressing the OK button switches to the Almost Done window containing all settings in python language These settings can be saved in a python script file via the Save button and then run later using the command line interface of nMoldyn or run immediately through the button Run Pressing the button cancel closes the window without saving settings EISF nMoldyn allows one to compute the elastic incoherent structure factor see Section B 3 on a grid of equidistantly spaced points along the q axis EISF qm Y WaEISFa m m 0 Nq 1 2 20 N is a user defined number of q shells The values for gm are defined as qm Gmin M Aq The atomic EISFs are computed as EISFa dm lexplia Rall 2 21 Here the symbol denotes an average over the q vectors having the same modulus qm The program corrects the atomic input trajectories for jumps due to periodic boundary conditions The quantities wa are user defined weights which are normalized to one 7 Wa 1 such that ETS F 0 1 Pressing the button EISF the corresponding graphical interface pops up see Fig 2 24 The user can enter the moduli of the required q vectors through the text field named Q va
85. utoregressive stochastic process of order P 14 45 P a t Sal a t nAt ep t 4 44 Here ep t is white noise with zero mean and amplitude op The coeffients Lay are fitted to the discrete signal using Burg s algorithm EG 47 and op is given by P o r 0 Sal nd 4 45 n 1 where r t is the autocorrelation function of x t r t z t x 0 4 46 In all following calculations nMOLDYN works with a set of coefficients an which has been averaged over all selected atoms and the three Cartesian coor dinates VACF within the AR model The autocorrelation function r t introduced in the previous Section is here the normalized velocity autocorrelation function VACF t hence r 0 0 1 Within the AR model the z transform of the VACF has the form oe y E 4 48 Lie e 35 68 Here the zz are the zeros of P plz 5 aP zP 4 49 k 1 We recall that the z transform of an arbitrary discrete function f n is given by F z DIZ f n 2 and the inverse transform by f n 34 o dzz F 2 n co Applying the inverse z transform to 48 yields P PAP n Y Bel 4 50 j l where the coefficients 5 are given by 1 a 02 py 4 51 aP Tiiz 2 Mi 25 H Note that y 4 n has a multiexponential form and that the stability criterion ie j 1 P 4 52 must be fulfilled This is guaranteed by the Burg algorithm 46 47 Density of states with
86. ven direction or on a given plane giving in addition a tolerance for q nMoldyn looks then for q vectors of the form 2 14 whose moduli deviate within the prescribed tolerance from the equidistant q grid From these q vectors only a maximum number per grid point called generically q shell also in the anisotropic case is kept The weights wa of which the square roots appear in 2 11 are normalized to one X Wa 1 The density density correlation is computed via the FCA technique described in Section I 31 The above calculations may be run via the graphical interface Pressing the button Coherent Scattering Function brings up a drop down menu from which the user can choose wether to calculate Fron Gm k At isotropically Isotropic or on a given plane gt Anisotropic or along a given direction Explicit List Isotropic Through this option the average of Eq B10 will be carried out taking on a q shell the q vectors whose moduli deviate within the prescribed tolerance from the required values Pressing this button the window Coherent Scattering Function Isotropic pops up see Fig 2 T6 The user can enter Coherent Scattering Function Isotropic Q values 0 2 100 Q shell width 1 Vectors per shell 50 Weights none gt incoherent y mass coherent Window width for FFT fi 0 of trajectory length Q units 4 1 nm v V Time steps in output 2000 Points in spectrum 2000
87. wn in the file name field The selection is confirmed by pressing the button Open of the file directory selection window By doing this the file directory selection window disappears The Open trajectory set option allows a trajectory set file to be selected as input file The trajectory set file permits to treat a sequence of trajectory files like a single trajectory for reading data The trajectory files must all contain data for the same system The variables stored on the individual files need not be the same but only variables common to all files can be accessed see pag 141 of the MMTK manual T6 for more details about the construction of these files The selection of a Trajectory Set file can be made via a file directory selection window like that of Fig 2 2 13 Open trajectory Directory Momeivaniatrajectories El lysozyme_1atm_300K_filtered_full nc El spce300K_1bar_100ps nc We Pd fue name en Files of type netCDF files nc Cancel Figure 2 2 file directory selection window TheOpen DLPOLY trajectories button allows a DLPOLY trajectory to be selected and converted in netCDF format The selection is made through a file directory selection window like that of Fig 2 2 The new netCDF file of the trajectory can then be selected as input file from the Open trajectory button The option Open Sacttering Function allows one to open an output netCDF file previously calculated in order to display it via t
88. xis A t n d t n 3 32 with dalt n n d t 3 33 The calculation of mean square displacements is the standard way to obtain diffusion coefficients from MD simulations Assuming Einstein diffusion in the long time limit one has for isotropic systems 1 Da lim Al t Jim At 3 34 55 The mean square displacement can be related to the incoherent intermediate scattering function via the cumulant expansion 231 24 1 Finca F 2 ine expl 0 pa 1 t 9 pa 2 t F 3 35 The cumulants Pa x t are defined as paal deny 3 36 paalt 2 Takt 3 d2 t5m0 3 37 The vector ny is the unit vector in the direction of q In the Gaussian approx imation the above expansion is truncated after the q term For certain model systems like the ideal gas the harmonic oscillator and a particle undergoing Einstein diffusion this is exact For these systems the incoherent intermediate scattering function is completely determined by the mean square displacement There exists also a well known relation between the mean square displace ment and the velocity autocorrelation function Writing da t de dT va T in Eq 31 one can show see e g PA that t A2 t 6 A oem co 3 38 0 Using now the definition 8 34 of the diffusion coefficient one obtains the rela tion t Da dr Cuviaa T 3 39 0 With Eq this can also be written as Da Couiaa 0 3 40 3 6 Rigid body motions
89. xt fields First Step Last Step Skip and the two auxil iary buttons Atom Selection and Group Selection The drop down menu buttons Dynamics and Scattering allow one to start the calculation of the quantities one is interested in The View and Help buttons allow one to in spect input output data and ask for help respectively All of these functions will be described below The following notation is used throughout this section N is the number of selected atoms in the system or in the subsystem for which the analysis is performed The atoms are labelled by Greek indices a 8 Their positions are denoted by Ra and their velocities by va The coherent and incoherent scattering lengths are defined by 11 be coh ba 2 1 da inc b2 g ba 2 2 respectively where the symbol denotes an average over isotopes and relative spin orientations of neutrons and nucleus In quantities that are averages over all atoms wa denotes a set of weights nMoldyn implements different weighting schemes for the atoms e Equal weighting 1 Wa N e Incoherent neutron scattering 2 DS ane Wa a a b2 inc e Coherent neutron scattering ba coh VWa SE Dea b2 con e Mass weighting where eae We 1 For the calculation of the velocity autocorrelation function starting from the atomic coordinates and for the calculation of the angular autocorrelation function starting from the quaternion parameters nMoldyn performs a numer
90. y be run via the graphical interface Pressing the button textbfIncoherent Scattering Function brings up a drop down menu from which the user can choose whether to calculate Finc qm k At isotropically gt Isotropic or on a given plane gt Anisotropic or along a given direction Explicit List Isotropic Through this option the average of Eq 2 19 will be carried out taking isotropically the q vectors whose moduli deviate within the prescribed tolerance from the required values Pressing this button the window Incoher ent Scattering Function Anisotropic pops up The input quantities to be entered are the same as for the isotropic calcula tion of the Coherent scattering Here the output files are two netCDF files containing the Incoherent inter mediate scattering functions as a function of the time in ps for all q values and their Fourier spectra as a function of the frequency in THz respectively Since Finc m k At is symmetric in time it is stored only for the positive time axis and its Fourier spectrum is stored only for values on the positive fre quency axis The default names of the two output files contain the strings ISF and ISF_SPECT respectively followed by the name of the trajectory The user can then change the names and or the destination directory through the Browse buttons Pressing the OK button switches to the Almost Done window containing all settings in python language These settings can be saved i

nMOLDYN: User's Guide - Theoretical Biophysics, Molecular

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