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User Manual v2.1 - Manchester Molecular Magnetism Group

1. 2 re 15 F idi A 1 10 gg en A Ansnesa canes 5L es x or Dd E 2 A AAA 2 E c 40 4 45 4 20 e 95 paxil LTD lLIL 3 30 1 L L L d 0 1 2 3 4 5 Field T 20 T T T T L OS J 10 A e e ER 5r S n OF 4 E ge D S oue RR a oe 4 2 E c woof 4 15 4 20 7 Field T Figure 4 3 1 Zeeman plots using a full calculation top and the approximation method bottom Survey block To perform a parameter sweep the Survey block is utilized The first line of this block specifies what the user wishes to survey For example if the first line is Residual the output will be the residual error between the calculation and experiment against the parameters in the survey Other options include M i j S CG j or HG j which represent the value of the magnetization susceptibility MCE or heat capcity respectively for the i field and the j temperature Following the first line this block is internally delimited into sections which belong to the same variable by The start and end values for the parameter and the number of steps required are first specified followed by the properties that they control In the following example the exchange coupling parameter is varied between 10 and 10 cm lin 20 steps and the isotropic g factors of sites 1 and 4 are varied from 1
2. User Manual v2 1 Copyright 2011 2015 Nicholas F Chilton Contents TAOS IS Ecco aces cath ol un tni ee ne een iv Acknowledgements En Re cites ARS a pp Puts A medo A ns vi T Introduction enis eee a 1 2 Theoretical Background ii nt nt is 2 2 1 NOTATIONS A und A a as ossi A NA a 2 X2 CO T 2 Spiarbit COUPLING aoc sa fa ces ian ds a eros tf E do 3 PERG Ha SCE UIT Ss sarl ES ne ed Oe ertet M ae ae 4 Cry Stale Mela potential usse a O ahve Ed Coke eat ed es 5 Zeeman Effecta eto oi tees dtu eof dto bte l val iore te tic fun 9 Orbital reduction factor uiii e edicto ins di 9 MASASHUE pEODOITIeS S oerte breeds as ada xum commute na pef beer 9 Accuracy and approximations att aed as era speed do dada das audes 11 Poivderantesratlolg SRE ba aiat tbe t e bud e dr ee 13 Pseudo EAN cu sodes tectis fei o bie herd to aee ea tbe iode D deter 13 Transition probabiles eosam oo A I 14 TAMIL E 14 NoOn ecol me ATIEY cise cee ane a RE aa dae a er di 14 TIP intermolecular interactions and magnetic impurities eene 15 Error residuals ss ooo ee eed ea s v eus 16 Electron Paramagnetic Resonadlegz ossa a a 16 3 Code Description siess eteesi C 18 o US 18 SCIUNT 19 2 ser CSUN Ghia el ns ne a qae 20 4 1 Binaries ANG compilation arias 20 Howto use the Maketle nd A ae ti a 20 4 2 PT OSTA IS
3. 10 0B 0Bg The entropy change associated with the application and removal of a magnetic field is the quantity associated with the Magnetocaloric Effect MCE The magnetic entropy change is easily calculated for isotropic or anisotropic systems through Equation 2 2 23 Note that M is the molecular mass of the complex and the entropy change is in units of J kg K Ba AB pese NOR Ma 3 2 2 23 T M OT a a Ba 0 The low temperature heat capacity of a paramagnetic system can be very sensitive to the magnetic interactions PHI is equipped to calculate the magnetic heat capacity through Equation 2 2 24 which includes a phenomenological term to capture the effect of the lattice heat capacity The heat capacity is given in units of R Naks where Tp is the Debye temperature and a is the lattice exponent Ei E 2 ist 5 eat _ zere zl 5 4234 Tp 2 2 24 kg T222 10 Accuracy and approximations Originally in PHI the magnetic properties were determined using Equations 2 2 18 and 2 2 21 directly where the derivatives were calculated using the finite difference method This method was prone to some instability due to level crossings where the eigenvalues changed index as well as hypersensitivity to Hamiltonian parameters when numerical limitations of double precision arithmetic became significant From version 1 8 PHI now uses Equations 2 2 19 and 2 2 22 which prove to be more numerically stable and
4. Zeeman block is used to specify the options for the Zeeman calculation The possible keywords are given in Table 4 3 7 Table 4 3 7 Zeeman options Parameter Options Syntax Comments Magnetic field direction Field STR Selection of magnetic field STR is either x y or z for single directions or xyz for principal Field Vector X Y Z axes integration If STR is Vector an arbitrary single direction is given If STR is Field Angles 0 y Angles an arbitrary single direction is given in polar coordinates Default Field z Magnetic field sweep Sweep Low High N Sets the magnetic field range in Tesla and number of points Default Sweep 0 7 250 Either the full calculation or the approximation scheme in cases where applicable may be used to perform such a sweep If using the full calculation method please note that due to the convention of matrix diagonalization routines the eigenvalues are returned in ascending order thus presenting artefacts at level crossings which appear like avoided crossings This is demonstrated in Figure 4 3 1 with a simple isotropic case using the full calculation top and the approximation method bottom both with 10 steps It is clearly seen that the level crossings are not correctly displayed in the top figure due to the width of the steps and the eigenvalue re ordering This can be corrected visually by increasing the number of steps 30
5. 2 3 g E 57 111 7 at 5024 11 302 so L 1 0i ur 15 112 38 15 12 38 _ 65 L5 17 t 1 1 1 7 1 Table 2 2 3 Operator equivalent factors for the lanthanides in the J my basis Ion Multiplet 2 Rank 4 Rank 6 Rank ce Esp 2 35 2 315 0 prt Ha 52 2475 4 5445 272 4459455 Nd Hoy 7 1089 136 467181 1615 42513471 Pm y 14 1815 952 2335905 2584 42513471 Sm Hsp 13 315 26 10395 0 Eu To 0 0 0 Ga SS 0 0 0 Tp Es 1 99 2 16335 1 891891 Dy gt lites 2 315 8 135135 4 3864861 Ho je 1 450 1 30030 5 3864861 Er alee 4 1575 2 45045 8 3864861 Tm He 1 99 8 49005 5 891891 yp ENT 2 63 2 1155 4 27027 Table 2 2 4 Operator equivalent factors for the lanthanides in the L m S ms basis Ion Term 2 Rank 4 Rank 6 Rank ce E 2 45 2 495 4 3861 Pr H 2 135 4 10395 2 81081 Na T 2 495 2 16335 10 891891 Pm I 2 495 2 16335 10 891891 Sm H 2 135 4 10395 2 81081 Eu TE 2 45 2 495 4 3861 Gus 0 0 0 Tp E 2 45 2 495 4 3861 Dy opes 2 135 4 10395 2 81081 Ho SI 2 495 2 16335 10 891891 Er a 2 495 2 16335 10 891891 Tm 7H 2 135 4 10395 2 81081 Yb E 2 45 2 495 4 3861 Note that only 2 and 4 rank operators are required to describe CFs for d block
6. Ref 1 228 2 16 0 0368 0 0 Opt 241 2 34 0 0315 0 000743 0 00000883 Eu Ref 1 214 3 82 0 147 0 0 0 Opt 230 3 28 0 269 0 000715 0 00164 0 000144 TbU Ref 1 252 4 50 0 267 0 0 0 Opt 260 0 997 0 223 0 0402 0 00685 0 000267 Dy Ref 1 357 4 40 0 121 0 0 E Opt 362 2 73 0 221 0 00655 0 000110 E Ho Ref 1 497 7 06 0 139 0 S E Opt 515 7 83 0 121 0 00629 s Er Ref 1 629 18 2 0 517 P E Opt 572 12 6 1 85 E z E Tm Ref 1 875 123 7 z Opt 684 177 3 3 Yb Ref 1 2910 E E n z Opt 2957 Exchange coupling For both spin only and orbitally degenerate cases the exchange Hamiltonian Equation 2 2 7 is parameterized with the complete J5 tensor Equation 2 2 8 In many cases this can be separated into two components the an isotropic exchange Equation 2 2 9 and the antisymmetric exchange Equation 2 2 10 in which case pm takes the form of Equation 2 2 11 While such an approach is commonplace in spin only situations the subject of magnetic exchange between orbitally degenerate ions is non trivial and a number of attempts have been made to determine an effective operator for such cases Currently in PHI the exchange interaction for orbitally degenerate ions follows the treatment of Lines which includes only the spin spin interaction between the true spins in the Li ML Si Mg basis In
7. gt New or File gt Open 4 3 Input files and syntax Input to the program is via plain text input and data files The job name used to launch the program Section 4 2 defines the name of the associated input and data files PHI will look for It will look for files in the directory that the program was launched from the current working directory For the above example PHI will look for test job input in the current directory This input file contains all the instructions that PHI needs to perform calculations The other data files required vary based on the type of calculation specified by the input file A total list of input and data files is given below using the example job name test job input Contains all input specifications and parameters test job_mag exp Contains experimental magnetization data test job_sus exp Contains experimental susceptibility data test job_levels exp Contains experimental energy levels test job_G exp Contains experimental g tensors test job_mce exp Contains experimental MCE data test job_epr exp Contains experimental EPR data test job_heat exp Contains experimental heat capacity data Please note that when running PHI on a Windows or Unix machine the end of line characters for the data files must be in DOS or Unix format and that data files prepared on Macintosh computers may have to be converted before they will work This can be accomplished with the free utility flip https ccrma stanford ed
8. sites 2 and 3 Note that handedness is preserved with PHI so that switching the order of the interacting pair is equivalent to negating the antisymmetric exchange vector Antisymmetric 2 3 0 1 0 1 1 5 25 Interaction block To define completely asymmetric exchange interactions between centres the ke Tnteraction block is used The interactions are all zero by default so only the required interactions should be listed The syntax is similar to that for the Fit and ee Survey blocks see below The first line for each exchange pair gives the site indices followed by three lines for each row of the interaction tensor The final line must be which signifies the end of the interaction tensor Note that handedness is preserved with PHI so that switching the order of the interacting pair will imply usage of the transpose of the given exchange tensor INCIENSO 2 3 JXX JXy JXZ JyX Jyy Jyz JZX JZy JZZ SOCoupling block To define or modify the SO Coupling parameters the SOCoupling block is used The syntax requires the site index followed by up to six values representing the first to sixth order SO Coupling parameters in wavenumbers This example sets the parameters for the sites 1 and 5 only where site 5 has only a first order component while site 1 has both first and second order HAS O SOU PINO dl AVAL O ES 5 1 65 0 OReduction block The combined orbital reduction pa
9. 0 O symmetry strong CF Fe IIDTd w CA 1 5 2 0 Ta symmetry weak CF Fe II FI ES 520 9 Spherical symmetry Fe IDOh w Ty 2 1 100 0 1 0 O symmetry weak CF Fe ID Td w E 2 0 Ta symmetry weak CF Fe IDFI 5D 2 2 100 0 1 0 Spherical symmetry CodIDOh w Tag 2 1 145 0 1 0 O symmetry weak CF Co Td w E 2 0 Ta symmetry weak CF Co III FI SB 2 2 145 0 1 0 Spherical symmetry Co IDOh w Tie 3 2 1 171 5 1 5 O symmetry weak CF Co IDOh s E 12 0 i E Co ID Td A 32 0 Ta symmetry Co ID FI F J2 3 171 5 1 0 Spherical symmetry Ni IIDOh w Tig 3 2 1 235 0 1 5 O symmetry weak CF Ni IIDOh s E W2 O Oy symmetry strong CF Ni III Td FAS 32 0 Ta symmetry Ni IID FI F W2 3 235 0 1 0 Spherical symmetry Ni IDOh Aog 1 0 E On symmetry Ni ID Td S 1 1 315 0 5 Ta symmetry Ni ID FI F 1 3 315 0 1 0 Spherical symmetry Cu IDOh AES 12 0 E z On symmetry Cu IDTd T2 1 2 I 830 0 0 Ta symmetry Cu ID FI zB A 830 0 1 0 Spherical symmetry Ce J Esp 5 2 0 Spherical symmetry Ce LS F W C Fable 2 251 120 Spherical symmetry Pr J Ha 4 0 Spherical symmetry Pr LS 7H 1 5 able 2 2 1 10 Spherical symmetry Nd J Toy 9 2 0 Spherical symmetry Nd LS I 325 265 able 2 29 0 eO Spherical symmetry Pm J L 4 0 Spherical symmetry Pm LS SI E 6 Table2 2 1 1 0 Spherical symmetry Sm J E
10. 0 0 Monomeric impurity Fit block To fit experimental data the Fit block must be detailed This block is very similar in syntax to the Survey block however in place of the start finish and number of steps either the starting value for the parameter or the lower limit starting value and upper limit is required Also before the beginning of the variable sub blocks the first line is either 32 Powell or Simplex specifying the fitting algorithm to be used The example below would fit the isotropic exchange coupling between sites 1 and 2 and the isotropic g factor for site 1 limited between 1 9 and 2 1 using the Simplex method The user should be reminded that the residual in fitting modes is not an absolute reference and will vary dramatically from problem to problem Also it is advisable to always visually check the results of the minima obtained from fits to aid in the determination of the global minimum Note that while some fits may be numerically better than others it does not necessarily mean that they are actually better this must be visually confirmed Note the ordering of parameters will not affect a Simplex minimization however it will affect a Powell minimization The Simplex method is often more useful than the Powell method when approximate values for the parameters are already known Fit Simplex ORTO EX il LD 2 0 241 GF 1 4 Params block Finally the Params bl
11. 7 to 2 3 in 10 steps 31 SUR Residual Table 4 3 8 lists the syntax for different properties note that the dummy integers zeros must be present Table 4 3 8 Fit and Survey block syntax Syntax Comment EX SiteA SiteB 1 2 3 4 5 6 7 Exchange coupling third integer represents x y Z isotropic antisymmetric x antisymmetric y or antisymmetric z IN SiteA SiteB 1 2 3 4 5 6 7 8 9 Interaction tensor third integer represents Jxx Jxy Jxz Jyx Jyy Jyz Jzx Jzy or Jzz SO Site 1 2 3 4 5 6 0 Spin orbit coupling second integer is the order GF Site 1 2 3 4 0 G factor second integer represents x y Z or isotropic CF Site Rank Order Crystal field parameter RC Site 128 0 Reference frame rotation second integer represents a p or y RE SiteA SiteB 1 2 3 Exchange frame rotation third integer represents a p or y OR Site 0 0 Orbital reduction parameter LW Freq 0 0 EPR linewidth second integer selects corresponding frequency where O implies all frequencies VO Freq 0 0 EPR pseudo voigt parameter second integer selects corresponding frequency where 0 implies all frequencies MO Freq 0 0 EPR mosacity second integer selects corresponding frequency where O implies all frequencies TI 0 0 0 Temperature Independent Paramagnetism DT 0 0 0 Debye temperature DA 0 0 0 Debye exponent ZJ 0 0 0 Mean field intermolecular interaction IM 0
12. Art of Parallel Scientific Computing Cambridge University Press 2nd edn 1996 P D Stevenson Comput Phys Commun 2002 147 853 858 S K Langley L Ungur N F Chilton B Moubaraki L F Chibotaru and K S Murray Chem Eur J 2011 17 9209 9218 S K Langley N F Chilton B Moubaraki and K S Murray Dalton Trans 2011 41 1033 1046 M R Razali N F Chilton A Urbatsch B Moubaraki S K Langley K S Murray G B Deacon and S R Batten Polyhedron 2013 52 797 803 S K Langley N F Chilton B Moubaraki and K S Murray Dalton Trans 2012 41 9789 9796 S K Langley N F Chilton L Ungur B Moubaraki L F Chibotaru and K S Murray Inorg Chem 2012 51 11873 11881 S K Langley N F Chilton I A Gass B Moubaraki and K S Murray Dalton Trans 2011 40 12656 12659 S K Langley N F Chilton B Moubaraki and K S Murray Dalton Trans 2011 40 12201 12209 A S R Chesman D R Turner K J Berry N F Chilton B Moubaraki K S Murray G B Deacon and S R Batten Dalton Trans 2012 41 11402 11412 M Nematirad W J Gee S K Langley N F Chilton B Moubaraki K S Murray and S R Batten Dalton Trans 2012 41 13711 M R Razali A S R Chesman N F Chilton S K Langley B Moubaraki K S Murray G B Deacon and S R Batten Dalton Trans 2012 42 1400 1405 B Bleaney and K D Bowers Proc R Soc Math
13. PHI however the interaction can also be anisotropic and or antisymmetric thus is much more general than the original Lines model The exchange coupling using the Lines approach may also be calculated in the my basis when used in conjunction with the lanthanide ions in the simple input method see section 4 3 utilizing a Clebsch Gordan decomposition By default the reference frame of the exchange matrix is coincident with the global coordinate system however this can be rotated such that the anisotropic and antisymmetric interactions can be described in simple local reference frames Note that upon swapping the site indices of the exchange Hamiltonian the exchange tensor T becomes its transpose 1 e 3 Ju Jy 5p LJEN mrad gt Se 2 2 7 i lt j Js os Jizz Reis A 2 2 8 Hise duos die LJEN His 2 gt Ji Sus tJ SiS tJ iz Si 2 2 9 i lt j i lt j LjeN 2 2 dij SiS 55 tdg Ssi dij 25 3 i lt j 2 2 10 Ju gs Cu Juj d Ju du 2 2 11 d UJ US Crystal field potential The CF potential is constructed from spherical harmonics to represent the environment in which the spin carrier resides While twenty seven terms exist in the full expansion the number required may be reduced as the CF Hamiltonian must be invariant under the operations of the point group of the molecule see below for a brief outline of the rules for non zero parameters Many approac
14. Phys Eng Sci 1952 214 451 465 M Gerloch and J H Harding Proc R Soc Math Phys Eng Sci 1978 360 211 227 44
15. and the CF Hamiltonian The diagonal g tensor is easily rotated into the local frame using the Z Y Z convention according to Equations 2 2 41 2 2 44 The rotation of the CFPs is performed according to Mulak and Mulak s convention with a slight modification The rotation of a set of CFPs of a given rank in Wybourne notation is given by Equation 2 2 45 where the elements of the unitary rotation matrix D are given by Equation 2 2 46 The symbols in brackets in Equation 2 2 46 are binomial coefficients The rotation convention in PHI is different to that of EasySpin the rotation matrices are the transpose of each other therefore RP a B y REASYSPR y p a 14 cos sinO 0 R 0 sinO cos 0 2 2 41 0 0 1 coso 0 sin Ry 0 0 1 0 2 2 42 sinO 0 cosO0 RPH a B y Rz Ry B Rz y 2 2 43 wa RPH a B y G RPH a p y 2 2 44 BY D a B y Bk 2 2 45 D n a B y snem EE mn O 2 2 46 2k 2p m n TIP intermolecular interactions and magnetic impurities A Temperature Independent Paramagnetic TIP component can be added to the calculated magnetic susceptibility directly in units of cm mol Equation 2 2 47 Intermolecular interactions between spin systems can be modelled using the mean field approximation Equation 2 2 48 this expression changed as of version 2 0 to allow its use in anisotropic systems Magnetic impurities are included employin
16. have the added benefit of being 1 5 and 1 67 times faster overall respectively Whilst the general method for the calculation of the magnetic properties of arbitrary systems has been given above a useful simplification of the method is possible when considering magnetically isotropic spin only compounds Taking advantage of the spherical symmetry of the Hamiltonian in conjunction with first order approximation methods can lead to a substantial reduction in the computational demands of the problem While the uncoupled basis is most useful for anisotropic systems easily allowing formulation of the SO and CF Hamiltonians isotropic systems requiring only the isotropic exchange Hamiltonian are block diagonal in a total spin basis In this case the problem can be solved by considering each block independently greatly reducing the dimension of the problem and speeding up the calculation The matrix elements can be calculated using Irreducible Tensor Operators ITOs and the Wigner Ekhart theorem and while the literature is well established the necessary equations and procedures are presented to clarify frequent typographical errors and to present a consistent notation In this example the coupled basis is formed by first coupling S4 and Sz to make S or followed by coupling to 4 to make 5454 or etc to the final total spin S expressed in bra ket notation in Equation 2 2 25 Recall that these are vector sums such that Equation 2 2 26 must
17. in Table 2 2 2 The operator equivalents themselves are polynomials of angular momentum operators derived from the tesseral harmonics and PHI includes all even odd positive and negative orders q for the gd 4 and 6 rank k operators The rank k is restricted to k 2 k 4 and k 6 as only the ground terms of the ground configuration are considered The use of the negative q operators is equivalent to the sine type operators of Hutchings and the imaginary CFPs in Wybourne notation The method relies on the use of the operator equivalent factors 04 to relate the total angular momentum matrix elements to the single electron matrix elements These factors have been tabulated for the ground multiplets for all lanthanides but not as far as the author is aware for the ground terms of the lanthanides these are now presented in Table 2 2 4 In PHI the CF Hamiltonian is applied to either the orbital or the total angular momentum components of a given centre That is if the centre possesses a non zero orbital moment the CF Hamiltonian directly acts on the orbital component as a true CF However if the centre does not possess an orbital moment the CF Hamiltonian acts on the effective spin or total angular momentum depending on one s interpretation of the assigned spin Note that the orbital reduction parameter o is only relevant when the CF Hamiltonian is applied to an orbital moment directly Ais y D c BA 0 08
18. lt gz Transition probabilities For anisotropic systems the zero field average transition probability between states u and v is calculated through Equation 2 2 40 using the expectation values of the three Cartesian magnetic moment operators Equation 2 2 38 The transition probabilities are in units of squared Bohr magnetons Ug uv T 3 v J mixing a X Y Z For calculations on single lanthanide ions in the Li m Si ms basis the wavefunction is N 2 gt cili Sd Si gta Iiza 4 2 2 40 i 1 expressed also in the Ji my basis though a Clebsch Gordan decomposition This provides a means of investigating the extent of J mixing by the CF Non collinearity For single magnetic centres the orientation of the reference frame is always an arbitrary choice and any symmetry elements that may be identified by crystallography or other means can be related to this axis When considering multiple magnetic sites in a single compound while the global reference frame is still arbitrary the individual reference frames which may possess defined symmetry elements may not be coincident and in which case it would not be ideal to enforce the global frame upon all sites Therefore PHI allows users to rotate individual reference frames of the magnetic centres to allow for a description of each centre in its own most convenient reference frame The two sources of magnetic anisotropy in PHI are the anisotropic g tensor
19. of orbital angular momentum The intra atomic coulomb interaction is treated with the Russell Saunders or LS formalism such that only the total spin and the total orbital moments of the ground term are employed Whilst designed for anisotropic calculations PHI is also optimized for calculations involving magnetically isotropic or spin only systems Another major design feature was to employ the Zeeman term in the Hamiltonian such that non perturbative field dependent magnetic properties could be calculated This also facilitates the calculation of field dependent properties such as Electron Paramagnetic Resonance EPR and Zeeman spectra One of the main goals is for the program to be approachable by non experts a goal that has been facilitated though the use of plain text input files and the provision of pre compiled binaries for common operating systems A Graphical User Interface GUI is also available to aid running calculations and provide real time visualization of data such that the program is even more accessible to beginners This feature makes the program perfect for use as a teaching aid for magneto chemical studies 2 Theoretical Background 2 1 Notation This manual uses the following notation for common mathematical quantities Table 2 1 1 Mathematical notation Quantity Symbol Scalar A Vector A Vector component ak Matrix A Matrix component Aag Operator Vector operator A Vector
20. simulation of EPR spectra is not a simple task Due to the field swept nature of the experiment the action of the magnetic field on the sample must be accounted for and generally cannot be treated as a perturbation Therefore evaluation of the field dependent wavefunctions is required Many approaches for this task have been employed using various approximations most of which involve searching the energy manifolds for transitions In PHI the EPR spectrum is calculated via a brute force approach which considers the transition probability for every pair of states at each field point explicitly While this approach is very computationally intensive it does not rely on any approximations and includes all transitions whether deemed to be allowed or forbidden as well as looping transitions The EPR absorption as a function of field is calculated through Equation 2 2 53 or 2 2 54 for perpendicular or parallel mode respectively There is also the possibility in PHI to calculate the EPR spectrum using an infinite order perturbation theory approach The structure of the routine is almost identical to that for the full method however in place of diagonalization of the full Hamiltonian the exchange and Zeeman components are treated as perturbations to the zeroth order wavefunctions which are the eigenfunctions of the SO and CF Hamiltonians This method therefore assumes knowledge of the single site properties for each ion which can then b
21. symmetry V IDOh Ax 32 0 On symmetry V IDTd w T 3 2 1 56 5 1 5 Tasymmetry weak CF V IDTd s E 12 0 Ta symmetry strong CF V IDFI F s 56 5 1 0 Spherical symmetry Cr IIDOh Ax 32 0 p On symmetry Cr IID Td w E ya i 91 5 1 5 Ta symmetry weak CF Cr IIDTd s E 12 0 Ta symmetry strong CF Cr IID FI F S PADS 91 5 1 0 Spherical symmetry Cr IDOh w E 2 0 O symmetry weak CF Cr IDTd w Um 2 i 57 5 1 0 Ta symmetry weak CF Cr ID FI SD 2 2 57 5 1 0 Spherical symmetry Mn VDOh A eA 540 0 1 0 On symmetry Mn VI Td E 12 0 Ta symmetry Mn VI FI D VE 540 0 1 0 Spherical symmetry Mn IV Oh Ax 32 0 On symmetry Mn V Td w T gem 138 5 L5 Tasymmetry weak CF Mn IV Td s E 12 0 Ta symmetry strong CF Mn IV FI F W2 3 138 5 1 0 Spherical symmetry Mn IDOh w E 2 0 O symmetry weak CF MndIDTd w T 2 i 89 0 1 0 Ta symmetry weak CF Mn IID FI SD 2 2 89 0 1 0 Spherical symmetry Mn IDOh w A 52 0 E On symmetry weak CF Mn IDOh s US 1 2 1 300 0 1 0 O symmetry strong CF MndDTdw A 5 2 0 z E Ta symmetry weak CF Mn IDFI S 5 2 0 Spherical symmetry 23 g On symmetry strong CF Fe VDOh TE 1 1 332 5 EE On symmetry Fe VDTd FA 1 0 Ta symmetry Fe VDFI SF il 3 332 5 1 0 Spherical symmetry FedINOh w 52 0 O symmetry weak CF Fe IIDOh s um 1 2 1 460 0 1
22. two times the orbital moment 2L If this block is omitted the orbital moments are all assumed to be zero This example assigns L4 Lz L4 0 and L 5 corresponding to the spins above O Ts o ilio Method 2 Ion block The above two blocks can be efficiently replaced in the case of common situations by using the simple input method To use the simple input method the first line must be Ion and the subsequent lines define the centres in a standard notation The example below would make exactly the same assignments as specified in the examples above under certain assumptions 22 TIon Mn IIT Oh w III Oh w h w Fe Fe III O Dy LS The possible keywords for the Ions block are given in Table 4 3 1 with the specifications that they designate Table 4 3 1 Ion types Keyword Term S L cem o Comment Ee ne Radical Ti IIDOh Tag 1 2 1 155 0 1 0 O symmetry Ti IIDTd E 12 0 Ta symmetry Ti IIDFI D 1 2 2 155 0 1 0 Spherical symmetry Ti IDOh ZI 1 1 61 5 n5 On symmetry Ti DTd FA 1 0 Ta symmetry Ti ID FI F 1 3 61 5 1 0 Spherical symmetry V IV Oh p 1 2 1 250 0 1 0 O symmetry V IV Td E 125150 Ta symmetry V IV FI D 1 2 2 250 0 1 0 Spherical symmetry V IIDOh E 1 1 105 0 2175 On symmetry V IIDTd A 1 0 Ta symmetry V IIDFI SF 1 3 105 0 1 0 Spherical
23. work amongst an arbitrary number of processes connected by a network Multi dimensional non linear optimization is a difficult problem often requiring in depth parameter space analysis to determine the global minimum for a given problem For this reason PHI contains two internal fitting algorithms Powell s method and the Simplex method which have been implemented based on those described in Numerical Recipes for Fortran The Simplex method is well suited to optimizing nearby minima while Powell s method is often useful in situations where a good initial guess is not available PHI contains several functions from the Fortran version of Stevenson s anglib library modified versions of the functions cleb sixj binom and angdelta are contained within the source 3 2 GUI The GUI is written in Python using the PyGTK Matplotlib and ReportLab libraries and has been designed in Glade The GUI is provided as a standalone executable for Windows while MacOS and Linux users must run the GUI as a python script 19 4 User Guide 4 1 Binaries and compilation PHI is available as both pre compiled statically linked binaries and as source code There are eight available binaries as listed below Windows 32 bit phi_vx x_windows32 exe Windows 32 bit OpenMP phi_vx x_windows32omp exe Windows 64 bit phi_vx x_windows64 exe Windows 64 bit OpenMP phi_vx x_windows64omp exe MacOS 64 bit phi_vx x_mac64 x Ma
24. 2 2 12 i 1 k 2 4 6 q k where o are the orbital reduction parameters Bi are the CFPs arar in Steven s notation 9 are the operator equivalent factors g are operator equivalents Table 2 2 2 Definition of the Stevens operators Operator 4 09 31 P MEE NE es EA Os 3 UL L_ L _ E H JS Rae Peper 2 7L 7L 312 2 _ L 72 stt 1 2 6 CL P 5 1 1 1 1 7 78 7 P 5 2 09 352 30121 250 31 61 Op 7 312 1 1 L 1 727 3121 1 67 01 P 5 2 1 1 1 2 78 7 P 5 OP EE cene en ELL 4 i EN JRE e le IS T 1 15 Qi 3pIp 59r a pr E He EEE IEEE A Ly on 3722 59 aies ES 29 0 a i cate Sys OR NL ane a 102 4 332 OL CE UE VERS TUS DOS 102 hy 1 7 nk SS a A en A m 61 i 1 331 30072 15L 5I L 1012L 121 332 _ 30772 152 522 L 10220 12152 List 02 231 31522 7352 10522 E 525072 2942 5f2 408 601 PETT AR Ost L 1 832 3002 152 SIL 10121 121 3817 30820 150 512 L E 4 125 Ltt ae vee ECE 7 6 2 15 33L 18121 1232 I 101 102 33 181 L 1232 2 1012 102 L
25. Az the CF interaction Hcp and the Zeeman effect Azer H Aso Hey Aer Azer 2 2 4 Spin orbit coupling The SO coupling operator is usually given as Equation 2 2 5 however this first order model results in the SO multiplets following the Land interval rule This is correctly obeyed by ions of low atomic mass such as the 3d ions however deviations from the Land interval rule for heavy ions due to term mixing by SO coupling are significant and must be accounted for Thus in PHI the SO operator is expanded as a power series following the parameterization of Karayianis Equation 2 2 6 The sum extends to order 25 where 5 is the total spin of the term in question The coefficients A1 A2 and 43 were tabulated for the tripositive lanthanides by Karayianis however we have optimized these and included higher orders where required Table 2 2 1 N Bc YA ai 2 2 5 i 1 N 28 P D 2 by il amp y 2 2 6 i 1 j 1 where j are the SO coupling constants g are the orbital reduction parameters Table 2 2 1 Optimized spin orbit parameters for the triply ionized rare earths Ion A cem Az cem 23 cem 44 cm As cm A6 cm Ce Ref 1 640 7 3 Opt 691 Pr Ref 1 390 4 63 Opt 421 5 78 Nd Ref 1 299 2 48 0 0475 Opt 326 2 66 0 0247 Pm Ref 1 251 1 99 0 0239 0 Opt 269 1 85 0 00977 0 000920 Sm
26. G tensors D G tensors with directions C H E Z MCE Heat capacity EPR Zeeman The number of directions used with the ZCW integration scheme is not a linear trend with ZCW level Table 4 3 12 shows the number of directions for each ZCW level up to ZCW 20 Table 4 3 12 Number of directions in ZCW integration ZCW Number 0 21 1 34 2 55 3 89 4 144 5 233 6 377 7 610 8 987 9 1597 10 2584 11 4181 12 6765 13 10946 14 17711 15 28657 16 46368 17 75025 18 121393 19 196418 35 20 317811 The options for the residual types can be used to favour the better fitting of particular regions of data Table 4 3 13 Residual types Residual type string Comments LowT Low temperature bias HighT High temperature bias LowB Low field bias HighB High field bias LowT LowB Low temperature and low field bias LowT HighB Low temperature and high field bias HighT LowB High temperature and low field bias HighT HighB High temperature and high field bias LowE Low energy bias HighE High Energy bias sus exp specification This file is used to define the experimental data for fitting purposes The file is plain text composed of floating point numbers where the first column represents the temperature in K and the subsequent columns represent the experimental data yy7 in cm mol K for
27. LS 5 2 0 Spherical symmetry Sm LS CH Sees Table 2210 MC Spherical symmetry Eu LS 7E 3 3 Table2 2 1 1 0 Spherical symmetry Gd IID 8S 7 2 0 Spherical symmetry Tb J Fg 6 0 Spherical symmetry Tb LS TF 3 3 Table2 2 1 1 0 Spherical symmetry Dy J His 15 2 O Spherical symmetry Dy LS CH W2 9 Fable 2 2 1 00 Spherical symmetry Ho J Is 8 0 Spherical symmetry Ho LS I 2 6 Table2 2 1 1 0 Spherical symmetry Er J lis 15 2 0 Spherical symmetry Er LS I 3 2 6 Table 2 2 1 1 0 Spherical symmetry Tm J Ho 6 0 Spherical symmetry 24 Tm LS H 1 Ss able 22s ESTO Spherical symmetry Yb J o 7 2 0 Spherical symmetry Yb LS F W To lanez 00 Spherical symmetry Note that for all d block ions and LS type f block ions the isotropic electronic spin g factor is set to 2 0 For J type lanthanides the isotropic g factor is set to the appropriate gy value For the d block free ions FI and LS type f block ions the appropriate operator equivalent factors are automatically included Other blocks Gfactors block To specify the spin g factors for the centres the Gfactors block is used Unless specified all g factors are taken to be 2 0 by default The syntax requires the site index followed by one or three values indicating either an isotropic or anisotropic spin g factor gx Jy 9z In the following example the second centre is given an a
28. Me CUT oi ze asics canes Sn coo bdo ES 21 A Input Tiles amd Sta ease ee Salas i on ida 21 Input specification E 22 Other blocks e nes nad dtes iO eee esce Luce nn 25 S s exp specificato ET MR SRE Rea Sie MARIN ESAE Rr Sa EAD Mu DRM 36 HAS EXD doceo CAMION DEN P Wet T EE We 36 ilice exp specification oai conten dics deat hie ion a sd cate me ruota ss dt uos 36 heat exp speclfICatiOf iei eo tor Pe YES NO 37 DREXD Speer ation sedet en aie adieu tiene ld ne nt 37 levels expapectHic alonso so e e SN A Ne ere 37 Ero SP CITICAHOM sn dedans 37 4 4 Output files and interpretation sanitaires annees 38 sus res mag res mce res heat res epr res levels res and G res specification 38 TEMA TES SDOCITICatOTi us Cos A Ru Ae i e eere 38 SULVEY ES specification on antenne ananas ab rude ondes ede e de 38 States res specifiCatioTi cei e ii R EOD e lists 38 4 Use of tlie CUT auis noto sott o Sese be andit shots stalim tales a ess 39 TE AIAN IO NA 40 AT Testing na an nn Mae a Oa a a NOS EE EEE 41 S Bugs and Feedback tia 42 11 6 References 111 License PHI Copyright 2011 2015 Nicholas F Chilton email nfchilton E gmail com This document is part of PHI Any results obtained through the use of PHI that are published in any form must be accompanied by the following reference N F Chilton R P Anderson L D Turner A Soncini and K S Murr
29. RABCD Sets the frequency ies for the simulation in GHz Default FEPR 9 5 35 94 Spectrum type Type N Selects whether the absorption N 0 first derivative N 1 second derivative N 2 or integrated absorption N 1 spectrum is to be calculated Default Type 1 Parallel mode Parallel Selects parallel mode Default Off Magnetic field sweep Sweep Low High N Sets the magnetic field range in Tesla and number of spectrum points Default Sweep 0 1 6 250 Linewidth Linewidth ABCD Sets the base FWHM isotropic pseudo voigt linewidth in cm for each frequency If only one linewidth is given and multiple frequencies are to be simulated then all frequencies have the same linewidth Default Linewidth 0 27 Pseudo voigt parameter Voigt ABCD Sets the pseudo voigt parameter for each frequency If only one parameter is given and multiple frequencies are to be simulated then 29 all frequencies have the same parameter Default Voigt 0 Gaussian Mosaicity Subspace perturbation Zeeman block Mosaic AB C D Subspace N Sets the mosacity parameter for each frequency If only one parameter is given and multiple frequencies are to be simulated then all frequencies have the same parameter Default Mosaic 0 Turns on the subspace perturbation method and selects how many states to include Default Off The
30. an be calculated according to Equations 2 2 36 and 2 2 37 where the cg factors are the components of the eigenvector describing state Wa With the g factors known the magnetic properties are calculated by considering the first order Zeeman perturbation to the m states 2 2 36 N 1 ON Sb aa S S D 25 1 aie bai Y Y casrtast S S I8I 8 5 S X Y Casitas QR 1 08 1 28 1 2 S2 kz S 1 ky Six ud p N N 1 I os uso Ek 1 25 1 25 1 4 Sis Sui ga i 1 i 2 Si amp kh 2 2 37 Powder integration For powder measurements on anisotropic systems the calculations must be integrated over many orientations to accurately reflect the experiment While poly crystalline samples contain a finite number of crystallites with discrete orientations it is usually assumed that the size of the crystals is small enough that it is closely representative of a powder sample with an infinite number of orientations evenly distributed on the sphere For the magnetic susceptibility it is sufficient to use the xyz integration scheme as this is exact for a second rank tensor property For the magnetization however the xyz scheme is inadequate and should not be used A number of orientation integration schemes are possible PHI uses the Zaremba Conroy Wolfsberg ZCW scheme as presented by Levitt The implementation in PHI samples the magnetic properties over a hemisphere as magnetic properties are invar
31. ande Grandjean Dr Lincoln Turner Dr Russell Anderson Dr Alessandro Soncini Dr Angus Gray Weale Dr David Paganin Dr Stuart Langley Dr Willem van den Heuvel Dr Marta Estrader Dr James Walsh Mr Chris Billington Mr Philip Chan School of Chemistry Monash University Clayton Victoria Australia P School of Chemistry The University of Manchester Manchester United Kingdom Institut de Physique Universit de Li ge Belgium Department of Chemistry Missouri S amp T United States of America d School of Physics Monash University Clayton Victoria Australia School of Chemistry University of Melbourne Victoria Australia f School of Chemistry Universitat de Barcelona Catalonia Spain Monash eResearch Centre Monash University Clayton Victoria Australia vi vii 1 Introduction PHI is a computer package designed for the calculation and interpretation of the magnetic properties of paramagnetic compounds While the use of phenomenological Hamiltonians is not at all a new concept the program was conceived as an update to older methods while adding new functionality new approaches and access increased computational power The program was designed primarily for the treatment of systems containing orbitally degenerate and strongly anisotropic ions through the inclusion of Spin Orbit SO coupling and Crystal Field CF effects Thus PHI was written with the explicit inclusion
32. ay J Comput Chem 2013 34 1164 1175 Redistributions of source code must retain the above copyright notice this list of conditions and the following disclaimer Redistributions in binary form must reproduce the above copyright notice this list of conditions and the following disclaimer in the documentation and or other materials provided with the distribution THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS AS IS AND ANY EXPRESS OR IMPLIED WARRANTIES INCLUDING BUT NOT LIMITED TO THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT INDIRECT INCIDENTAL SPECIAL EXEMPLARY OR CONSEQUENTIAL DAMAGES INCLUDING BUT NOT LIMITED TO PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES LOSS OF USE DATA OR PROFITS OR BUSINESS INTERRUPTION HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY WHETHER IN CONTRACT STRICT LIABILITY OR TORT INCLUDING NEGLIGENCE OR OTHERWISE ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE By downloading and or using this software you agree to the terms of this license iv Acknowledgements The author wishes to acknowledge advice assistance and contributions from the following people Prof Keith Murray Prof Stuart Batten Prof Richard Winpenny Prof Eric McInnes Prof David Collison Prof Fern
33. be satisfied for all coupling steps iS fig CSS Ms 2 2 25 S Si lt Sk Si S 2 2 26 The isotropic exchange Hamiltonian can be represented by use of a 0 rank tensor operator Equation 2 2 27 The matrix elements of spherical tensor operators are evaluated by applying the Wigner Ekhart theorem followed by a decoupling procedure to calculate the reduced matrix elements as expressed in Equation 2 2 28 where the numerator of the fraction is a Clebsch Gordan coefficient and the quantities in braces are Wigner 9j symbols see below for simplifications The remaining reduced matrix element can be easily calculated see below Note that the k and k values are the ranks of the component and intermediate spins respectively and can be easily determined using simple rules i jEN i jeN frx 2 2 J S 2V3 b3 JT 2 2 27 i lt j i lt j 11 Es ms P S ms 1 1 k aN TUM MS eis 1 28 1 28 1 SS 2573 E i kia Fosse ns Ji IK 2k 1 2 1 25 1 Sui Si Jua UR 2 2 28 As the tensor is rank zero the Clebsch Gordan coefficient is equivalent to two Kronecker delta functions Equation 2 2 29 While this simplifies the calculations it also implies something much more meaningful there is no dependence on the magnetic quantum number at all such that it may be excluded from the basis and the dimensionality of the Hamiltonian matrix further reduced Coupled with block diagona
34. cOS 64 bit OpenMP phi_vx x_mac64omp x Linux 64 bit phi_vx x_linux64 x Linux 64 bit OpenMP phi_vx x_linux64omp x These binaries have been compiled using the Intel compiler suite and the Intel MKL To compile PHI from source the supplied Makefile must be tailored to the specifications of the system at hand a Fortran 95 compiler must be available and the appropriate libraries need to be compiled and in known locations The source files must be listed in the order shown in the Makefile or errors will be encountered PHI must be compiled using a C pre processor which provides a means for compiling different versions of the code from the same source files Table 4 1 1 shows the compile time options Even without SMP or SPMD activated the C pre processor must still be invoked If compiling for SPMD it is recommended to use your MPI library s wrapper compiler eg mpifort with the additional libraries and options required by PHI It is recommended that PHI be compiled with the highest level of compiler optimization and inter procedural optimization Note that the source code is written in free form Fortran95 and therefore compilers such as gfortran may need the ffree form flag or similar Table 4 1 1 Compile time options Option C pre Other required Additional Forbidden processor flag flags libraries flags SMP OpenMP Domp openmp Dmpi SPMD MPI Dmpi MPI Domp How to use the Makefile The sup
35. calculation phi py s 200 B parvzoom Completed Mag Reduced Mag MCE Cp x x Ly Zeeman cpr States Figure 4 5 1 Screenshot of the PHI GUI in operation on a Debian Linux platform 39 When the execute button is clicked green arrow on top toolbar the text in the Input tab is written to a file with the name given in the Job Name Input File box and PHI is executed The PHI output is re routed to the Output tab in the left pane while simultaneously the plotting routine initiates polling the data files for new data This updating feature is designed to reflect the progress of a fitting routine however also plots static data The routine ends when the calculation is complete The Reset zoom on update checkbox controls whether the plot auto scales with each new refresh and can be unchecked to allow the observation of a particular region during the fitting process The Plot Update Interval field specifies the time in seconds between each update and can be changed to suit the computational demand of the calculations The Output Line Buffer field specifies how many lines are buffered from PHI before they are printed into the Output tab and should be increased if the calculations are very quick and producing a copious amount of output If performing a fit where the plotting routine is constantly plotting new data the Pause button can be used to freeze the plot t
36. e magnitude of the largest peak positive or negative at the first temperature in the input file Note that the temperature 1s the inner loop and varies first followed by the frequency 3 P2 EL 102 12 192 13 3 mL 2 02 wa MES 3 m 102 12 9259 levels exp specification This file defines the experimental energy levels which are only required if a fit or survey with respect to the energy levels is required The format for this file is one floating point value per line for each energy level given in wavenumbers It is possible to specify unknown energy levels in the file using a question mark see example below which will not be used when calculating error residuals Note that there should be no blank lines at the end of the file G exp specification This file is used to define the experimental g tensors used for the fitting and survey modes The file consists of lines with three or twelve values defining either gx gy and gz or gx i j pk pi j pk pi gy 9z gt Dg Dg Dox Day Do Days Paz D DD is the unit vector denoting the direction of g Each line represents the diagonalized g tensor for a pseudo spin 1 2 Kramers doublet Therefore the number of g D DE for each Kramers doublet where 37 tensors must be less than or equal to half the total dimension of the problem Note that there should be no blank lines at the end of the file D gyl Digy1 DE AI Dred Diei Diori D gy2 D gy2 D
37. e perturbed by the exchange interaction and magnetic field Of course this is inappropriate in large magnetic fields or if the exchange interactions are stronger than the SO or CF terms In both cases the linewidth function assumes a pseudo voigt profile Equation 2255 which has shown to be required in certain applications The linewidth is treated in frequency space and therefore no frequency field conversion factor commonly referred to as 16 1 Eoo the factor 1s required 4 Note that the x and y directions are determined as mutually orthogonal to the main magnetic field B while the z direction is parallel to it E Ej i jedim P 2 emt erat A B gt I Hes i j Healy as J rtm i lt j 2 2 53 Ei Ej i jedim E em ett A B y li Accel li V AE njo 2 2 54 i lt j i where i and j are two eigenstates evaluated at B AE E Ej Euw Nij is the linewidth v is the voigt parameter 2vIn 2 V AE m v Wn EET 1 mE 2 2 55 nn 4 s Jaye D As with the calculation of powder thermodynamic magnetic properties of anisotropic systems the EPR absorption signal must be integrated over all possible orientations of the magnetic field the ZCW scheme as discussed above is used for this purpose After the absorption spectrum is calculated it is normalized and if requested the first or second derivative or the integration is taken via finite differe
38. ea D g 2 D gz2 D g 2 Eo Dea aD ga Do3 MD de Did 4 4 Output files and interpretation PHI outputs information regarding the operation of the program and the type of calculations 1t is performing to stdout shell command prompt or terminal This is redirected to the GUI output panel however when running PHI without a GUI this can be directed to a specified output file by appending for example gt test job out to the execution command so that it would read on Linux phi_vx x_linux64 x test job gt test job out PHI writes all calculated data to files in the working directory of the job in the files described below Note that the naming of the files is identical to that of the exp input data files the job name is appended with an underscore to the following output files for example test job_mag res sus res mag res mce res heat res epr res levels res and G res specification Data is written to this file in exactly the same format as the input exp files zeeman res specification This file contains the results from a calculation of a Zeeman plot The file consists of dim 1 columns where dim is the dimension of the total Hilbert space of the system The first column contains the magnetic field strength B in Tesla followed by the corresponding energy for each state in the system in wavenumbers survey res specification This file contains a number of columns one for each operator defined in the Su
39. efunction is also transformed into the J m Jd basis and is printed in the same manner as that for the regular wavefunction If the g tensor calculation is appropriate the diagonal g tensors are printed along with their directions in the internal coordinate system In the approximation mode a list of the intermediate and final spin states along with their energies are provided 4 5 Use of the GUI The GUI runs completely independently from PHI and is provided as a visualization aid as well as a tool to abstract the command line from users It is therefore the perfect platform for teaching the fundamentals of magneto chemistry in an interactive environment Figure 4 5 1 shows a screenshot of the GUI in operation under Debian Linux The interface is divided into three sections The main left pane is for PHI input and output the right pane is for displaying the results and the bottom left section is the control panel To begin an input file can be typed into the Input tab in the main left pane or a file opened using the File gt Open menu If a new input is typed into the Input tab then a name must be given to the new job in the Job Name Input File box The current working directory can be changed from that where the GUI was launched through the File gt New menu which also clears the Input box The box to the left of the Pause button allows the user to specify the version of PHI to use for the
40. es 0 y direction is given in polar coordinates Default Field Z isotropic Field Powder 3 anisotropic BHeat ABCD Selects the magnetic field s in Tesla for the given If STR is Angles an arbitrary single Magnetic field Temperature sweep Sweep Low High N calculation Any number of fields can be listed on the same line Default BHeat 0 1 Sets the temperature range in Kelvin and number of points Temperatures are on a base 10 logarithmic scale Default Sweep 0 5 20 250 Debye lattice contribution Debye Tp a Sets the Debye temperature in Kelvin and exponent Default Debye 0 0 EPR block The EPR block is used to specify the options for the EPR calculation The possible keywords are given in Table 4 3 6 Table 4 3 6 EPR block options Parameter Options Syntax Comments Magnetic field direction Field STR Selection of magnetic field STR is either x y integration or z for single directions or xyz for principal Field Powder N axes integration If STR is Powder then ZCW integration is used where N gt 0 If STR Field Vector X Y Z 1s Vector an arbitrary single direction is given If STR is Angles an arbitrary single Field Angles 0 y direction is given in polar coordinates Default Field Powder 6 Temperature TEPRABCD Sets the temperature s for the simulation in Kelvin Default TEPR 5 Frequency FEP
41. from anti ferromagnetic superexchange between the Cu II ions leading to an S 0 ground state To investigate the magnitude of the superexchange interaction a fit of the ymT vs T data to a single J isotropic HDVV spin Hamiltonian with a variable g factor in the Zeeman Hamiltonian was performed The entire input file required to perform this calculation with PHI is presented to highlight the simplicity of such operations Figure 4 6 1 inset This analysis found a very good fit to the experimental data Figure 4 6 1 with a coupling constant of J 144 6 cm and g 2 12 40 0 5 KKTONS Cu II Oh Cu II Oh a Simplex 50 EX 124 0 4 0 3 2 00 G 140 GF 240 KKESUS Bsus 1 Params OpMode Fit S 0 2 mT cm mol K 0 1 0 50 100 150 200 250 300 Temperature K Figure 4 6 1 Magnetic susceptibility of the Cu ID OAc dimer in a field of 1 T the solid line is a fit to the data using the parameters in the text Inset Entire PHI input file required to perform the calculation 4 7 Testing The Coupling Report Operation Mode is provided to inform the user of the block diagonal structure of the HDVV Hamiltonian matrix in a coupled total spin basis without performing any demanding calculations It is useful to check to see the requirements of large problems and determine whether it can be solved on the available hardware 41 5 Bugs and Feedback The development of PHI is an ongoi
42. g analytical expressions for the field and temperature dependent magnetization and magnetic susceptibility for pure spin centres with g 2 0 As of version 2 0 the impurity value represents the fraction of the system Equations 2 2 49 and 2 2 50 These effects are included in the order of TIP zJ magnetic impurity giving the final expression for the magnetic susceptibility Equation 2 2 51 Xrip Xcaic TIP 2 2 47 where TIP is the temperature independent paramagnetism s XTIP Xz a Bea 2 2 48 Wa TUE where zJ is the intermolecular interaction parameter 15 M 1 IMP Mcaie UMP M yp 2 2 50 where Ximp is the field and temperature dependent magnetic susceptibility of the impurity Mymp is the field and temperature dependent magnetization of the impurity IMP is the fraction of magnetic impurity Xcalc TIP zJ 1 nz ene TIP x 1 IMP UMP Xime 2 2 51 Error residuals In all cases the error for a particular data set is calculated following the sum of squares approach Equation 2 2 52 as an example for magnetization When calculating the total error for a simultaneous comparison to multiple data sets the total residual is calculated as the product of the individual sum of squares errors for each data set In this way different error scales of the individual data sets will not obscure any features points 2 Residual gt Mexp Maz 2 2 52 i 1 Electron Paramagnetic Resonance The
43. hes have been attempted over the years to determine Crystal Field Parameters CFPs such as the Point Charge PCM Angular Overlap AOM and Superposition SM models however all have fallen short of consistently predicting these parameters This is because in reality the electrostatic CF is inadequate due to the overlooked contributions from covalency non orthogonality screening and polarization of the orbitals In spite of these criticisms the CF model succeeds in describing experimental results when it is considered a phenomenological Hamiltonian where the resultant parameters have no direct physical interpretation Given this interpretation and the close similarity of the operators see below the CF Hamiltonian is also used in PHI to model Zero Field Splittings ZFS of effective spins There are numerous parameterization schemes for the effective CF Hamiltonian and care must be taken to avoid confusion For a good grounding see Mulak and Gajek Hutchings and Rudowicz As PHI constructs the Hamiltonian within a total spin orbit basis the operator equivalent technique of Stevens ef al was chosen as the most efficient method for the evaluation of matrix elements even though the notation is less transparent than others Equation 2 2 12 Here the definitions of the a operators are consistent with Hutchings Mulak and Gajek and Stevens however for clarity definitions of all the positive and negative operators are given below
44. iant under inversion of the magnetic field Pseudo g tensors For calculations involving anisotropic ions which give rise to doublet states pseudo g tensors may be calculated within the basis of each doublet This is equivalent to treating each doublet as a pseudo spin 1 2 state whose magnetic anisotropy is given by the g tensor For Kramers systems these doublets are related by time inversion symmetry and the treatment is rigorous however for non Kramers systems the g tensors for pseudo doublets are only approximate and only gz is non zero due to vanishing off diagonal elements between the conjugate states The theory is well established but a brief overview of the method will be given Note that PHI does not currently support the g tensor calculation in bases of other values of pseudo spin 13 For a given system the expectation values of the three Cartesian magnetic moment operators are evaluated in the basis of the zero field wavefunction Equation 2 2 38 The g tensor is then constructed for each doublet through Equation 2 2 39 where w and w are the wavefunctions of the doublet and a B x y z wt Hzgg e cem 2 2 38 Gage gt D Mosa 2 2 39 uc papr v wpapr This g tensor is then diagonalized to yield three principle values and their corresponding directions leading to the definition of the anisotropic g tensor for each pseudo spin doublet By convention the directions are defined such that gy lt gy
45. ion 2 2 15 It is this response to the magnetic field which is responsible for the observable magnetic properties such as magnetization and magnetic susceptibility N Acer s cil T 4 S i i 1 2 2 15 S AE tul where T is the identity matrix g is the g tensor Orbital reduction factor Note that in all Hamiltonians above the o parameter has been included with all orbital operators This is the combined o A k factor required when using the T P equivalence for orbital triplet terms A is required when making the T P substitution and takes the value of 1 0 when representing a T term and 1 5 when representing a T term or k in some texts is the orbital reduction factor which is an empirical constant 0 lt k lt 1 and accounts for a reduction in the effective orbital angular momentum due to covalency or low symmetry effects It can be effectively removed setting o to unity default Note that for the SO and CF Hamiltonians the orbital reduction factor is included as 6 0 0 0 0 or oj for first second third fourth fifth and sixth rank respectively where required Magnetic properties The inclusion of the Zeeman Hamiltonian allows the magnetic properties to be calculated from first principles without resorting to perturbation theory Thus full mixing of all states by the magnetic field is implicitly included The fundamental definitions for the magnetic properties are expressed in Equation
46. ions whereas the 6 rank is also generally required for f block ions Of course however higher rank operators may be required to accurately describe ZFS effects The second order CF operators are intimately related to those of the standard ZFS Spin Hamiltonian and using the definitions of the CF operators as in Table 2 2 2 the relationships between the ZFS parameters and the CFPs are therefore expressed in Equations 2 2 13 and 2 2 14 D 3B20 2 2 13 E B20 2 2 14 The non zero CFPs are determined solely by the point group of the ion in question Often the assumed point group symmetry does not include the entire molecule but only the first coordination sphere of the paramagnetic ion as this is the largest contribution to the perturbation Often idealized symmetry may be used initially followed by small corrections to allow for distortions of lower symmetry For a full C representation all 27 CFPs are required If the group is not C then only CFPs with even q are required If a C axis is present only CFPs with q jn where j is an integer are required Only the following groups need negative q CFPs C1 C S5 C3 Cs Sg Ca S4 and Cs A comprehensive list of non zero CFPs for all point group symmetries can be found in Gorller Walrand and Binnemans chapter n Zeeman Effect The Zeeman Effect is the stabilization and destabillization of angular momentum projections parallel and anti parallel to a magnetic field Equat
47. is a string of x y z ora combination thereof Default GDir xyz Fitting algorithm vigour FitVigour X Sets how vigourous the fitting algorithm starts as a parameter percentage Default FitVigour 10 Fitting algorithm limiting FitLimit X Sets how strongly the fitting algorithm enforces parameter limits Limiting function is eXIAl where A is difference between the fitting parameter and its limit Default FitLimit 12 Table 4 3 10 Operation modes OpMode Comments Sim STR2 Simulation STR2 is a string see below and Table 4 3 11 Fit STR2 Fit STR2 is a string see below and Table 4 3 11 Sur STR2 Survey STR2 is a string see below and Table 4 3 11 Coupling Report Reports the block diagonal structure of the matrix Matrix Elements Prints Hamiltonian matrix 34 The second required string is composed of letters representing the calculations to be performed For example MS would represent Magnetization and Susceptibility whilst LSG would represent energy Levels Susceptibility and G tensors The possible letters are given in Table 4 3 12 Note that a Simulation involving L energy levels will result in the printing of the wavefunction in states res that G and D are mutually exclusive and that the letter codes may be in any order i e LMSG MGLS etc Table 4 3 11 Operation mode STR2 STR2 Comments E Energy levels M Magnetization S Susceptibility G
48. ler angles a f and y given in degrees 23 Default Off equivalent factors Number of CPU cores MaxCPU N Sets the upper limit of CPU cores available N is an integer Default MaxCPU 1 Fitting algorithm display NoPrint Turns off the printing of fit progress to the terminal and intermediate results to disk Default Off Full wavefunction printing FullWF Prints the full wavefunction in the states res file Default Off Save survey calculations Save Saves a file for each step of the survey calculation Default Off Disable operator NoOEF Disables the Operator Equivalent Factors such that CF input values are assumed to contain Ok Default Off G tensor multiplets Mults NABCD Gives the multiplicities of the multiplets for the calculations of pseudo spin gt states N gives the number of multiplets followed by N integers giving the multiplicity Default Off Single crystal experiment Single Circumvents checking for the need to integrate magnetic properties 1 e requested single field direction is allowed Default Off Survey percentage Percent Prints survey percentage completion Default completion Off Residual type Residual STR Selects residual calculation method STR is a string see Table 4 3 13 Default Off G tensor direction residual GDir STR Selects which directions to include in the residual calculation for g tensors and directions STR
49. lity Read Input Files A gt Magnetization Construct Basis Evaluate Matrix Diagonalize Matrix _Mognetization Elements Energy Levels H Residual Simulation Print Results Figure 3 1 1 Operational schematic of PHI Powder Average 18 The heart of the program is the construction and diagonalization of the Hamiltonian matrix which is required for any calculation PHI relies on the use of external linear algebra routines BLAS and LAPACK routines for matrix diagonalization and multiplication PHI has been written to take advantage of multiple processor cores now common in consumer and specialized machines There are two models of parallelism currently supported by PHI Symmetric Multi Processing SMP and Single Process Multiple Data SPMD which use fundamentally different ideas to perform tasks more efficiently compared to a sequential model The simplest approach to increase computational efficiency is to employ multiple cores on a shared memory machine SMP model to perform multiple diagonalizations simultaneously which is achieved in PHI using OpenMP threads to distribute work However the SMP model is clearly limited by the size of the machine both the number of cores and available memory For this reason the SPMD model is one of the most common parallel strategies due to the cost effectiveness of multiple smaller machines PHI uses the MPI standard to distribute
50. lization of the matrix this leads to a tremendous reduction in the computational effort required for the problem S ms 0 015 ms Os 516 2 2 29 msm The occurrence of Wigner 9j symbols in every matrix element is unfortunate due to their computational complexity however in this case there are only four possible 9j symbols which are easily simplified They are presented below in Equations 2 2 30 2 2 3 1 where the quantities in the braces on the right hand side of the equations are Wigner 6j symbols E y 8a p0c 40 c d 0142 Af 2 2 30 e f 0 2a 1 2c 1 2e 1 a b 1 SS Lot RE b d 0 2 2 31 E f V3 2c 1 t a b 1 8 1 5 e e 1 d 1 A e 2232 f 3 e 1 4 ere a b 0 a d e 1 Os p 1 f d 1 2 2 2 33 k f j y3Qa 1 The reduced matrix elements remaining in Equation 2 2 28 are easily calculated depending on the rank of the particular spin operator Sill 11S 25 1 Sill SD 115 J SCS DS 1 2 2 34 2 2 35 12 Once the matrix elements have been calculated the matrix is diagonalized to determine the eigenvalues and eigenvectors of the coupled states To evaluate the magnetic properties it is necessary to determine the effective g factors for the coupled spin multiplets In general the spin multiplets Ya originate from a mixture of the different CS Sa coupled basis states necessitating further ITO algebra The g factors for each multiplet c
51. lock provides all the options for the calculation of magnetization Table 4 3 3 gives all the available options for this block Table 4 3 3 Mag block options Parameter Options Syntax Comments Magnetic field direction Field STR Selection of magnetic field STR is either x y integration or z for single directions or xyz for principal Field Powder N axes integration If STR is Powder then ZCW integration is used where N gt 0 If STR Field Vector X Y Z is Vector an arbitrary single direction is 27 given If STR is Angles an arbitrary single Field Angles 0 y direction is given in polar coordinates Default Field z isotropic Field Powder 3 anisotropic Temperature TMag ABCD Selects the temperature s in Kelvin for the calculation Any number of temperatures can be listed on the same line Default TMag 2 4 10 20 Magnetic field sweep Sweep Low High N Sets the magnetic field range in Tesla and number of points Default Sweep 0 7 10 x x MCE block This block provides all the options for the calculation of the magnetocaloric effect Table 4 3 4 gives all the available options for this block Table 4 3 4 MCE block options Parameter Options Syntax Comments Magnetic field direction Field STR Selection of magnetic field STR is either x y integration or z for single directions or xyz for principal Field Powder N axe
52. nces The EPR linewidth can be augmented to include the effects of crystal mosacity The linewidth is augmented for each orientation independently as in Equation 2 2 56 2 2 56 where w is the mosacity parameter 17 3 Code Description 3 1 PHI PHI is written entirely in Fortran95 and split into six modules for easy maintenance These modules are data f90 ang_mom f90 powder f90 props f90 fitting f90 and phi f90 data f90 contains all the explicit variable declarations for global variables and arrays It also contains a number of subroutines which initialize constants read input files write output files and perform diagnostics ang_mom f90 contains all the Hamiltonian operators and tools required for matrix operations powder f90 contains the routines required for powder integration procedures props f90 contains the subroutines for the calculation of the magnetic properties fitting f90 contains the subroutines necessary to perform surveys and fits containing residual calculation routines and fitting algorithms phi f90 is the main program which controls what calculations are to be performed The program can be well understood by means of an operational schematic Figure 3 1 1 Note the boxes are only representative of the flow of the program and do not necessarily correspond to individual subroutines or functions Heat Capacity Electron Paramagnetic Resonance Magnetocaloric Effect Susceptibi
53. nd B Ng Rep Prog Phys 1989 52 699 762 J Mulak and Z Gajek The Effective Crystal Field Potential Elsevier 2000 C Rudowicz J Phys C Solid State Phys 1985 18 1415 1430 B Bleaney and K W H Stevens Rep Prog Phys 1953 16 108 159 K W H Stevens Proc Phys Soc Sect A 1952 65 209 215 A Abragam and B Bleaney Electron Paramagnetic Resonance of Transition Ions Oxford University Press 1970 C Gorller Walrand and K Binnemans in Handbook on the Physics and Chemistry of Rare Earths Elsevier 1996 vol 23 F Lloret M Julve J Cano R Ruiz Garc a and E Pardo Inorganica Chim Acta 2008 361 3432 3445 R Bo a Theoretical Foundations of Molecular Magnetism Elsevier 1999 M Evangelisti and E K Brechin Dalton Trans 2010 39 4672 M Evangelisti F Luis L J de Jongh and M Affronte J Mater Chem 2006 16 2534 C Voglis P E Hadjidoukas I E Lagaris and D G Papageorgiou Comput Phys Commun 2009 180 1404 1415 M Abramowitz and I A Stegun Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables U S Department of Commerce Washington D C 10th edn 1972 J J Borr s Almenar J M Clemente Juan E Coronado and B S Tsukerblat J Comput Chem 2001 22 985 991 J J Sakurai and S F Tuan Modern quantum mechanics Addison Wesley Longman 2010 J J Borr s Almenar J M Clemente Juan E Coronado and B S Tsukerblat Ino
54. ng process which the author hopes will continue with the advent of new technologies and or interfaces which may enhance the computational power available to the user The author welcomes any bug reports feature requests comments suggestions or queries about the code Please address all correspondence to nfchilton E gmail com Please keep in mind that the code is continually under development and bugs may still be present Updated source code and binaries are uploaded to nfchilton com phi regularly 42 No Am 18 19 20 21 22 23 24 25 26 24 28 29 30 31 32 33 34 6 References N Karayianis J Chem Phys 1970 53 2460 2469 N F Chilton R P Anderson L D Turner A Soncini and K S Murray J Comput Chem 2013 34 1164 1175 R J Elliott and M F Thorpe J Appl Phys 1968 39 802 807 O Kahn Mol Phys 1975 29 1039 1051 J J Borr s Almenar J M Clemente Juan E Coronado A V Palii and B S Tsukerblat Chem Phys 2001 274 131 144 W P Wolf J Phys Colloq 1971 32 C1 26 C1 33 A Palii B Tsukerblat J M Clemente Juan and E Coronado Int Rev Phys Chem 2010 29 135 230 M E Lines J Chem Phys 1971 55 2977 2984 M Hutchings in Solid State Physics Elsevier Amsterdam 1964 vol 16 pp 227 273 C E Schaffer and C K Jorgensen Mol Phys 1965 9 401 412 W Urland Chem Phys 1976 14 393 401 D J Newman a
55. nisotropic spin g factor Of 9x gy 1 9 and g 2 2 the third centre is also given an anisotropic spin g factor of 9x 0 1 gy 2 5 and g 11 9 and the fourth centre is given an isotropic spin g factor of 1 98 usus ES EEES 2 We LS 2 2 3 2525 Lie Exchange block To define an isotropic exchange coupling interactions between the centres the eK x change block is used The interactions are all zero by default so only the required interactions should be listed This is done on one line by specifying the index of the first site followed by the index of the second site followed by the isotropic or anisotropic exchange values Only one or three values should be given indicating either an isotropic exchange or the three diagonal components of the exchange matrix Jxx Jyy and J The following example specifies three coupling pathways between sites 1 and 2 2 and 3 and 1 and 3 where the exchange involving site 1 is axially anisotropic Exchange Antisymmetric block To define the antisymmetric components of exchange interactions between centres the a Antisymmetric block is used The interactions are all zero by default so only the required interactions should be listed This is done on one line by specifying the index of the first site followed by the index of the second site followed by the three components of the antisymmetric exchange The following example specifies antisymmetric exchange between
56. o inspect the plot without it changing Note that this does not pause the calculation by PHI it only pauses the plotting loop which resumes upon disengaging the Pause button The calculation can be aborted by clicking the Abort button red cross on top toolbar killing the PHI instance The plot legend can be dragged to the desired location when a fitting is not taking place Plots can be saved from the plotting section however the results of the calculation remain in the project directory and can be plotted or examined outside the GUI if desired A report of the calculation can be generated by clicking the report button page icon on top toolbar which produces a pdf file containing the input output and any plots that are available The most common causes of errors encountered in the GUI are incorrect user input output files in use by other programs Keep in mind when using the GUI that any errors arising from PHI or the GUI will be printed to the terminal shell command prompt etc and this information should be supplied if reporting a bug 4 6 Example A number of examples utilizing PHI can be found in the literature however a brief example is provided to show how to write a simple input file The classic Cu IDo OAc 4 dimer originally investigated by Bleaney and Bowers and subsequently by Gerloch et al shows a strong decrease in the yyT vs T data with a reduction in temperature Such behaviour originates
57. ock is used to choose the operation mode and other calculation options Table 4 3 9 gives the options available in this section Table 4 3 9 Params block options Parameter Options Syntax Comments Operation mode OpMode STR1 STR2 Selection of operation mode STR1 and STR2 are strings see Table 4 3 10 Must be present Magnetism approximation Approx Turns on the block diagonal approximation for isotropic systems Default Off Monomeric impurity Zero field splitting IMPNx ZFSIJK Adds a monomer impurity of spin S N 2 with fraction x i e x 2 1 for one uncoupled spin Default Off Alters the convention of Bg such that it equals D Any number of sites can be listed on the same line Default Off Cubic crystal field Cubic IJ K Forces cubic CFP ratios for B and B based on B and B2 Any number of sites can be listed on the same line Default Off Static magnetic field StaticB IBI X Y Z StaticB B 0 q Includes the presence of a static magnetic field of magnitude IBI Tesla with vector X Y Z or polar coordinates 0 9 Default Off Rotate reference frame Rotate Na B y Rotates the reference frame CFPs and or g for site N through the Euler angles a p and y given in degrees Default Off Rotate exchange frame EXRotateN Ma 6 y Rotates the exchange frame for the exchange defined between sites N and M through the Eu
58. operator component A 7 2 2 Theory For systems in thermodynamic equilibrium the underlying postulate is the solution of the time independent Schr dinger equation Equation 2 2 1 The action of the Hamiltonian operator on the wavefunction Y gives the energy of the state E The wavefunction is usually separated into radial and angular parts and in the domain of spin Hamiltonians the angular part 1s solved explicitly while the radial integrals essentially become parameters to be determined For a given problem the Hilbert space is constructed from N sites with angular momentum basis states of either Si ms JJ my Or Li my Si Ms where i N Note only a single term is used to describe each ion i e S J or L and S are fixed The total uncoupled basis of the system is the direct product of all the individual basis states Equation 2 2 2 This system is solved by evaluating the matrix elements of the Hamiltonian over the basis states and diagonalizing the Hamiltonian matrix The dimension of the Hilbert space and therefore the Hamiltonian matrix is given by Equation 2 2 3 AW EW 2 2 1 L S m Ms La S mi Ms e Lz 5 Mi ms Q Lay 55 mss ms EN 2 2 2 N dim a 4 Gs 1 2 2 3 The Hamiltonian is composed of operators which act on the angular momentum basis functions to yield the matrix elements The Hamiltonian is split into four components the SO coupling Aso the exchange coupling
59. plied Makefile provides a skeleton to set up a custom compilation of PHI The variables at the top of the file must be set in order to compile the program COMPILER is your Fortran 95 compiler e g ifort or gfortran MPI COMPILER is your wrapper MPI compiler e g mpifort FLAGS and MPI FLAGS can be adjusted as the user pleases LAPACK must be set to the appropriate library destination and contain links to lapack and blas SOURCES is the list of source files for PHI and must be in the default order The flags in the Makefile may be specific to the ifort compiler and therefore must be 20 substituted for their equivalent flag for your compiler e g openmp becomes fopenmp for gfortran 4 2 Program execution To run PHI it is as simple as launching the executable on the command line from the working directory containing the input file e g phi_vx x_linux64 x test job where test job is the name of the input file The standalone GUI is launched by simply running the gui executable whereas the GUI script is run with Python as python phi py The PHI executable s should also be in the same directory as the GUI files allowing the GUI to automatically select the PHI executable This selection can be changed by altering the executable name in the GUI or by placing only the desired PHI executable in the program directory The default job directory is the same as that containing the GUI files however this can be changed by selecting File
60. ral fields are described in all cases with a positive Bi parameter and tetrahedral fields with a negative B parameter LUST Cronica Jal SILC al 2 0 0 001 0 0006 0 5 25930 2 6 3 3 4 Sus block This block provides all the options for the calculation of magnetic susceptibility Table 4 3 2 gives all the available options for this block Table 4 3 2 Sus block options Parameter Options Syntax Comments Magnetic field direction Field STR Selection of magnetic field STR is either x y integration or z for single directions or xyz for principal Field Powder N axes integration If STR is Powder then ZCW integration is used where N gt 0 If STR Field Vector X Y Z is Vector an arbitrary single direction is given If STR is Angles an arbitrary single Field Angles 0 y direction is given in polar coordinates Default Field Z isotropic Field xyz anisotropic Magnetic field BSus ABCD Selects the magnetic field s in Tesla for the calculation Any number of fields can be listed on the same line Default BSus 0 01 Temperature sweep Sweep Low High N Sets the temperature range in Kelvin and number of points Default Sweep 1 8 300 250 Temperature Independent TIP X Sets a TIP in cm mol Default TIP 0 Paramagnetism Intermolecular interaction zJ X Sets the mean field zJ parameter in cm Default zJ 0 Mag block This b
61. rameters can be set through the use of this block These are specified by the index of the site followed by the value of the total orbital reduction parameter Note that the value for the parameter should be negative A value of 1 0 removes the feature default i e no orbital reduction In this example site 5 is given a total orbital reduction parameter of 1 35 be sure not to confuse the terminology and sign of the parameter HAS OREAUCIEALON 5 SIS CrystalField block A CF may be specified by using the CrystalField block The syntax requires the site index followed by the rank order and then value of the parameter The site index rank and order must be integers while the CFPs must be real numbers If the full input method was used operator equivalent factors are not included by PHI automatically and they must be included in the input parameters explicitly if required However if the simple input method was used the operator equivalent factors for the lanthanides and d block free ions are automatically included by PHI The following example specifies the B0 parameter equal to 0 001 for site 1 the B 0 parameter equal to 0 0006 for site 2 and the B 0 parameter equal to 0 230 26 To describe a weak cubic field for the d block free ions it is recommended to use the Cubic N parameter see Table 4 3 3 As the operator equivalent factors are taken into account using the simple input method octahed
62. rg Chem 1999 38 6081 6088 D Gatteschi and L Pardi Gazzetta Chim Ital 1993 123 231 240 B W Shore and D H Menzel Principles of atomic spectra Wiley 1967 A Bencini and D Gatteschi EPR of Exchange Coupled Systems Springer Verlag M Eden and M H Levitt J Magn Reson 1998 132 220 239 M Gerloch and R F McMeeking J Chem Soc Dalton Trans 1975 2443 2451 H Bolvin ChemPhysChem 2006 7 1575 1589 J Mulak and M Mulak Phys Status Solidi B 2006 243 2796 2810 43 35 36 37 38 39 40 41 42 43 44 45 46 4T 48 49 50 51 22 33 54 55 56 57 G R Hanson K E Gates C J Noble M Griffin A Mitchell and S Benson J Inorg Biochem 2004 98 903 916 S Stoll and A Schweiger J Magn Reson 2006 178 42 55 S D Bruce J Higinbotham I Marshall and P H Beswick J Magn Reson 2000 142 57 63 H Husein Mor H Weihe and J Bendix J Magn Reson 2010 207 283 286 G van Veen J Magn Reson 1969 1978 30 91 109 R Aasa and tore Vanngard J Magn Reson 1969 1975 19 308 315 S K Klitgaard F Galsb l and H Weihe Spectrochim Acta A Mol Biomol Spectrosc 2006 63 836 839 M J D Powell Comput J 1964 7 155 162 J A Nelder and R Mead Comput J 1965 7 308 313 W H Press S A Teukolsky W T Vetterling and B P Flannery Numerical Recipes in Fortran 90 The
63. rvey block and one more the final column which represents the residual for the parameter set defined by the row noting that the columns are ordered as the variables in the Survey block states res specification This file is produced when a simulation of the energy levels is requested and contains information regarding the wavefunction and energy levels and transition probabilities J mixing and g tensors if applicable In the full calculation case the wavefunction is printed in a matrix type manner with row and column headers The first column contains the row headers which are the basis elements in which the Hamiltonian was constructed i e the single ion states The subsequent columns are the different eigenstates of the system with the column headers displaying the energies in wavenumbers The columns show the expansion coefficients for the basis states that comprise the given eigenstate Unless FullWF is 38 selected coefficients below 1x10 are not printed In the case of wavefunctions with imaginary components two matrices are printed the top one being the real part of the coefficients and the bottom one being the imaginary components Below this are shown the percentage contribution of each basis state to the wavefunction For anisotropic systems the transition probabilities between the states are printed note that this matrix 1s symmetric For single lanthanide ions calculated under the simple input method the wav
64. s 2 2 16 and 2 2 17 OE Max 2 2 16 OM The molar magnetization is the sum of the magnetization of each state weighted by its Boltzmann population Equation 2 2 18 where Z is the partition function Equation 2 2 20 giving the magnetization for a single Cartesian direction a x y z in Bohr Magnetons per mole ug mol Equivalently the Magnetization can be calculated using Equation 2 2 19 dim ES 9E E 2 2 18 B La Zus a OB i 1 _ ksT lnZ Gu NU Hg 0B id T eke 2 2 20 Following Equation 2 2 17 the molar magnetic susceptibility is the first derivative of Equation 2 2 18 resulting in Equation 2 2 21 which contains terms that depend on the first and second derivatives of the eigenvalues with respect to the magnetic field As there are two derivative steps there are nine possible combinations of the Cartesian directions f X y Z leading to the definition of the 3 X 3 magnetic susceptibility tensor Equation 2 2 21 reduces to the traditional vanVleck formula in the limit of zero magnetic field however the numerical method employed here is capable of accurately determining the susceptibility in the presence of non zero fields as used in experiment Following Equation 2 2 19 Equation 2 2 22 is entirely equivalent to Equation 2 2 21 dim dim 0M Na OE OE eek 02E El CB QBg 10kgTZ 4 OB OB 4 OBa0Bp WS QE CEA ND OE Z5 Zi 0B Li 0Bg 2 2 21 NakgT 0 lnZ ok 2 2 22
65. s integration If STR is Powder then ZCW integration is used where N gt 0 If STR Field Vector X Y Z is Vector an arbitrary single direction is given If STR is Angles an arbitrary single Field Angles 8 direction is given in polar coordinates Default Field z isotropic Field Powder 3 anisotropic Magnetic field BMCE ABCD Selects the magnetic field s in Tesla for the calculation Any number of fields can be listed on the same line Default BMCE 7 Temperature sweep Sweep Low High N Sets the temperature range in Kelvin and number of points Default Sweep 1 8 50 250 Molecular mass Mass X Sets the molecular mass for the sample in g mol Default Mass 2000 Integration Integrate N Sets the number of magnetic field integration points Default Integrate 50 x Heot block This block provides all the options for the calculation of the heat capacity Table 4 3 5 gives all the available options for this block Table 4 3 5 Heat block options Parameter Options Syntax Comments Magnetic field direction Field STR Selection of magnetic field STR is either x y integration or z for single directions or xyz for principal Field Powder N axes integration If STR is Powder then ZCW integration is used where N gt 0 If STR Field Vector X Y Z is Vector an arbitrary single direction is 28 Field Angl
66. the different fields as defined in the input file for example 0 01 0 1 and 1 T respectively Note that there should be no blank lines at the end of the file mag exp specification This file is very similar to the sus exp file however the first column represents the magnetic field in T and the subsequent columns represent the experimental data M in Bohr Magnetons per mole N ug for the different temperatures as defined in the input file mce exp specification This file is very similar to the sus exp file however the first column represents the temperature points in K and the subsequent columns represent the experimental data 45 in J kg K for the different magnetic fields Note that the appropriate molecular mass must be defined in the Params block 36 heat exp specification This file is very similar to the sus exp file however the first column represents the temperature points in K and the subsequent columns represent the experimental data C in units of R J mol Ko for the different magnetic fields epr exp specification This file is very similar to the sus exp file however the first column represents the magnetic field points in T and the subsequent columns represent the experimental data either integrated absorbance absorbance first derivative or second derivative for the different frequencies and temperatures The data should be normalized to th
67. u craig utility flip 21 input specification This file is delimited into blocks by headers signified by four asterisks The first block must be the Spin or on block see below and the input file is terminated by ae Find The input file is not case sensitive despite the notation given in this manual for clarity After the End termination line the input file is not read by PHI and so may contain descriptions other input specifications or comments Also any line that begins with is interpreted as a comment and not read by PHI The first block which must be the Spin or lon block specifies the number and type of magnetic centres in the problem and this can be done in one of two ways Method 1 Spin block In the first method the full input method the first line must be Spin and the subsequent lines denote the spin angular momentum of the centres The spins are entered as two times the spin 25 or the number of unpaired electrons Note that these spins may be real spins or pseudo spins In the following example three paramagnetic centres are declared with spins 4 2 S2 4 S4 5 2 Spin 4 S 5 5 Orbit block In the full input method the Orbit block is also used which details the corresponding orbital angular momentum of each site declared in the Spin block Like the Spin block the orbital moments must be entered as

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