Calculations on Crystals using CHARMM The crystal section within CHARMM allows calculations on crystals to be performed. It is possible to build a crystal with any space group symmetry, to optimise its lattice parameters and molecular coordinates and to carry out a vibrational analysis using the options. * Menu: * Syntax:: Syntax of the CRYSTAL command * Function:: A brief description of each command * Examples:: Sample testcases * Implementation:: Background and implementation

[Syntax CRYStal command] CRYStal [BUILd_crystal] [CUTOff real] [NOPErations int] [DEFIne xtltyp a b c alpha beta gamma] [FREE] [PHONon] [NKPOints int] [KVECtor real real real TO real real real] [VIBRation] [READ] [CARD UNIT int] [PHONons UNIT int] [PRINt] [PRINt] [PHONons] [FACT real] [MODE int THRU int] [KPTS int TO int] [WRITe] [CARD UNIT int] [PHONons UNIT int] [VIBRations] [MODE int THRU int] [UNIT int] xtltyp ::= { CUBIc } { TETRagonal } { ORTHorhombic } { MONOclinic } { TRIClinic } { HEXAgonal } { RHOMohedral } { OCTAhedral/trnc} { RHDO } a b c alpha beta gamma ::= (six real numbers) The crystal module is an extension of the image facility within the CHARMM program. All crystal commands are invoked by the keyword CRYStal. The next word on the command line can be one of the following : Build - builds a crystal. Define - defines the lattice type and constants of the crystal to be studied. Free - clear the crystal and image facility. Phonon - calculates the crystal frequencies for a single value or a range of values of the wave vector, KVEC. Print - prints various crystal information. Read - reads the crystal image file. Vibration - calculates the harmonic crystal frequencies when the wave vector is the zero vector. Write - writes out to file various crystal information.

A brief description of each command follows. 1. Crystal Build. A crystal of any desired symmetry can be constructed by repeatedly applying a small number of transformations to an asymmetric collection of atoms (called here the primary atoms). The transformations include the primitive lattice translations A, B and C which are common to all crystals and a set of additional transformations, {T}, which determines the space group symmetry. The Build command will generate, given {T}, a data structure of all those transformations which produce images lying within a user-specified cutoff distance of the primary atoms. The data structure can then be used by CHARMM to represent the complete crystal of the system in subsequent calculations. The symmetry operations, {T}, are read from the lines following the Crystal Build command. The syntax of the commmand is : Crystal Build Cutoff <real> Noperations <int> ... <int> lines defining the symmmetry operations. The Cutoff parameter is used to determine the images which are included in the transformation list. All those images which are within the cutoff distance are included in the list. There is no limit to the number of transformations included in the lists as they are allocated dynamically. The crystal symmetry operations are input in standard crystallographic notation. The identity is assumed to be present so that (X,Y,Z) need not be specified (in fact, it is an error to do so). For example, a P1 crystal is defined by the identity operation and so the input would be Crystal Build .... Noper 0 whilst a P21 crystal would need the following input lines : Crystal Build .... Noper 1 (-X,Y+1/2,-Z) A P212121 crystal is specified by Noper 3 (X+1/2,-Y+1/2,-Z) (-X,Y+1/2,-Z+1/2) (-X+1/2,-Y,Z+1/2) It should be noted that in those cases where the atoms in the asymmetric unit have internal symmetry or in which a molecule is sited upon a symmetry point within the unit cell not all symmetry transformations for the crystal need to be input. Some will be redundant. It is up to the user to check for these cases and modify the input accordingly. 2. Crystal Define. The define command defines the crystal-type on which calculations are to be performed. It is usually the first crystal command that is specified in any job using the crystal facility. It has the format : Define lattice-type a b c alpha beta gamma The input lattice parameters are checked against the lattice-type to ensure that they are compatible. Nine lattice types are permitted. They are listed below along with any restrictions on the lattice parameters : CUBIc - a = b = c and alpha = beta = gamma = 90.0 degrees. (example: 50.0 50.0 50.0 90.0 90.0 90.0 ) (volume = a**3) (degrees of freedom = 1) TETRagonal - a = b and alpha = beta = gamma = 90.0 degrees. (example: 50.0 50.0 40.0 90.0 90.0 90.0 ) (volume = c*a**2) (degrees of freedom = 2) ORTHorhombic - alpha = beta = gamma = 90.0 degrees. (example: 50.0 40.0 30.0 90.0 90.0 90.0 ) (volume = c*b*a) (degrees of freedom = 3) MONOclinic - alpha = gamma = 90.0 degrees. (example: 50.0 40.0 30.0 90.0 70.0 90.0 ) (volume = c*b*a*sin(beta) ) (degrees of freedom = 4) TRIClinic - no restrictions on a, b, c, alpha, beta or gamma. (example: 50.0 40.0 30.0 60.0 70.0 80.0 ) (volume = c*b*a*sqrt(1.0 - cos(alpha)**2 - cos(beta)**2 - cos(gamma)**2 + 2.0*cos(alpha)*cos(beta)*cos(gamma)) ) (degrees of freedom = 6) HEXAgonal - a = b, alpha = beta = 90.0 degrees and gamma = 120.0 (example: 40.0 40.0 120.0 90.0 90.0 120.0 ) (volume = sqrt(0.75)*c*a**2 ) (degrees of freedom = 2) RHOMbohedral - a = b = c ; alpha=beta=gamma<120 (trigonal) (example: 40.0 40.0 40.0 67.0 67.0 67.0 ) (volume = a**3*(1.0-cos(alpha))*sqrt(1.0+2.0*cos(alpha)) ) (degrees of freedom = 2) OCTAhedral - a = b = c, alpha = beta = gamma = 109.4712206344907 (a.k.a truncated octahedron) (example: 40.0 40.0 40.0 109.471220634 109.471220634 109.471220634 ) (volume = 4*sqrt(3))/9 * a**3 ) (truncated cube length = a * sqrt(4/3) ) (degrees of freedom = 1) RHDO (Rhombic Dodecahedron) - a = b = c, alpha = gamma = 60.0 and beta = 90.0 (example: 40.0 40.0 40.0 60.0 90.0 60.0 ) (volume = sqrt(0.5) * a**3 ) (truncated cube length = a * sqrt(2) ) (degrees of freedom = 1) It is up to the user to ensure that the lattice parameters have the desired values for the system at all times. The values are stored by the program but, at present, there is no way of transmitting this information between jobs. For example, if the lattice parameters have been changed during a lattice optimisation then the new parameters, which are printed out at the end of the minimization, must be input here at the beginning of the next CHARMM run. Lattice parameters are stored in dynamic trajectory and restart files. 3. Crystal Phonon. Phonon calculates the dispersion curves for a crystal. Any value of the wavevector can be used (although, in practice, each component of KVEC is normally limited to the range -0.5 to +0.5). The dynamical matrix and normal mode eigenvectors determined in the phonon calculation are complex although the eigenvalues remain real. The syntax for the command is : Crystal Phonon Nkpoints <i> Kvector <f> <f> <f> To <f> <f> <f> Nkpoints tells the program the number of points at which the derivative matrices must be built and diagonalised whilst the Kvector ... To ... clause determines the values of KVEC for each calculation. Thus, Kvector 0.0 0.0 0.0 To 0.5 0.5 0.5 Nkpoints 3 would solve for the crystal frequencies at the points, KVEC=(0.0,0.0,0.0), (0.25,0.25,0.25) and (0.5,0.5,0.5). If it is desirable, point calculations can be carried out by omitting the To statement and putting Nkpoints 1. For single calculations at KVEC=(0.0,0.0,0.0) the Crystal Vibration command is faster. The eigenvalues and eigenvectors at each value of the wave vector from the phonon calculation are saved and they can be written out to a file using the Crystal Write Phonon command. No analysis facilities exist within CHARMM for the phonon data structure as the eigenvectors are complex. It is to be noted that phonon and vibration calculations can only be performed on crystals of P1 symmetry. No information about the symmetry operations is used when generating the dynamical matrix. 4. Crystal Print. Two options exist with the Print command. If no keyword is given then the crystal image file is printed out. The Crystal Print Phonon command performs a similar function to the Print Normal_Modes command in the vibrational analysis facility. Selected frequencies and eigenvectors for a range of values of the wave vector can be printed out. The syntax is : Crystal Print Phonon Kpoints <i> To <i> Modes <i> Thru <i> Factor <f> The Kpoints .. To .. clause determines the wave-vectors at which the modes are to be printed, the Modes .. Thru .. gives the range of the eigenvectors and the Factor command gives the scale factor to multiply each normal mode by. 5. Crystal Read. The Crystal Read command reads in a crystal image file. The file has the same output as produced by the Crystal Print or Crystal Write commands. The command is useful if a crystal image file was produced using the Crystal Build command and saved using the Crystal Write command in a previous job and it is desired to reuse the same transformation file for analysis or comparison purposes. The command can also be used to read in limited sets of transformations if specific crystal interactions need to be investigated. The transformation file is formatted so the Card keyword needs to be specified and the unit number must be given after the Unit keyword. 6. Crystal Vibration. For a free molecule with N atoms the dynamical equations have 3N-6 non-zero eigenvalues. This is no longer so for a crystal. If a crystal is made up of L unit cells each containing Z molecules with N atoms, the dynamical equations would have a dimension of 3NZL. However, using the symmetry properties of the lattice it is possible to factor the equations into L sets each with a dimension of 3NZ and each depending upon a vector, KVEC, which labels the irreducible representation of the translation group to which the set belongs. The force constant matrix is complex. Its form may be found in the references given at the end of the documentation. Vibration solves the dynamical equations for the case where the wave-vector is zero, i.e. when the equations are real. The procedure is invoked by the Crystal Vibration command. The syntax is : Crystal Vibration 7. Crystal Write. There are three Crystal Write options. If no keyword is given the crystal image file is written out, in card format, to the specified unit. The CARD and UNIT keywords are required. The Crystal Write Phonon command writes out the phonons from a phonon calculation. All the eigenvalues and eigenvectors for all values of the wavevector that are stored are written automatically. The Crystal Write Vibration command writes out the eigenvalues and eigenvectors from a vibration calculation. The modes to be written are given by the Mode .. Thru .. clause. All Write commands require that the Fortran stream number be given after the Unit keyword and a CHARMM title may be specified on the following lines. The structure of the phonon and vibration files for a crystal may be found by looking at the routines WRITDC and XFRQW2 respectively in the file [.IMAGE]XTLFRQ.SRC. The vibration modes are written in the same form as a for VIBRAN normal mode file and may be read in using the appropriate VIBRAN commands. Unfortunately no analysis facilities exist for complex eigenvectors within CHARMM and so users will have to write their own if they want to perform phonon calculations. 8. Crystal Minimization. It is possible to perform a lattice minimization using the normal CHARMM MINImize command and the ABNR minimizer. Two extra keywords have been introduced. If none of them is present then a coordinate minimization is performed as usual. If LATTICE is specified then the LATTice parameters and the atomic coordinates are minimized together. If NOCOoordinates is given with the keyword LATTice then only the lattice parameters are optimised. Specifying NOCOordinates by itself is an error. It should be noted that when the lattice is being optimised the crystal symmetry is maintained. A cubic crystal will remain cubic, etc.

Examples of input may be found in the test directory. All crystal files are prefixed by the string "xtl_". All the jobs involve L-Alanine. Briefly the jobs are : 1. XTL_ALA1.INP. The crystallographic fractional coordinates are read in and converted to real space coordinates using the CHARMM COORdinate CONVert command and the experimental values for the lattice parameters. 2. XTL_ALA2.INP. A crystal image file is generated for the crystal using a value of 10.0 Angstroms for the crystal cutoff. 3. XTL_ALA3.INP. A coordinate and lattice minimization are performed for the crystal. The crystal image file from the previous job is used and the optimised coordinates are saved. The main point to note is that before using the crystal package for energy calculations and other manipulations that involve the image non-bond lists an image update must be performed. For safety always do an update after building or reading in the crystal. Note too that the new, optimised lattice parameters are used in the all the subsequent input files. 4. XTL_ALA4.INP. For subsequent calculations a coordinate file that contains the coordinates of all atoms (four molecules of L-alanine) is generated. A crystal image file suitable to do this is read in directly from the input stream. It contains 6 transformations (not 3 as might be expected) because the CHARMM image facility requires that the inverses of all transformations be present. The first three are the ones needed and the last three are their inverses. An update is needed after reading the file to make known to the program the coordinates of the atoms in the first transformation of all the inverse pairs in the image list. The Print Coor Image file will then print out the coordinates of the atoms in the original asymmetric unit and the first three of the images. If the coordinates of the atoms in all the images are required then the keyword NOINV in the UPDATE command must be used (check IMAGE.DOC). 5. XTL_ALA5.INP. The same job as the second except that the crystal is generated for a whole unit cell (i.e. the system generated in the fourth job). The same value of the crystal cutoff is used. An energy is calculated too. The energy and its RMS coordinate derivative should be exactly four times (apart from a small round-off error) the value obtained for an energy calculation on a single asymmetric unit with the same lattice parameters and crystal cutoff (see job 3). 6. XTL_ALA6.INP. Peform a crystal vibration and phonon calculation for the optimised structure of the L-alanine crystal. The vibrational and phonon modes are written out to files and components of the first 24 phonon normal modes for the three values of the wavevector that were calculated are printed. To do the same for the vibrations it would be necessary to use the appropriate VIBRAN commands in another job.

Background and Implementation. The Crystal options and their commands were described above. The present section discusses relevant background material and briefly reviews the methods used in the implementation. Some technical points are also made. The crystal option is an extension to the CHARMM program. The source code is in the directory [.IMAGE] whilst the crystal data structure is in the file IMAGE.FCM. Two additional source code files have been added - CRYSTL.SRC and XTLFRQ.SRC. Small modifications have been made to the files ENERGY.SRC and EIMAGE.SRC. CHARMM Images and the Crystal Image Data Structure. As outlined above a crystal structure can be specified entirely by the action of the primitive translations A, B and C, and a small set of transformations, {T} (which themselves are functions of A, B and C), on an asymmetric group of atoms. In CHARMM the calculation of the energy assumes that there exists a cutoff distance beyond which all interactions between particles are neglected so that when performing calculations on supposedly infinite crystals only a limited portion of that crystal, i.e. that portion containing those atoms within the cutoff distance of the primary atoms, need be considered. The CHARMM image option, of course, already enables the energies of crystals to be calculated but the input required to use it to do so is cumbersome and time consuming. It is a great simplification to include an extra data structure that defines the crystal in terms of A, B and C and {T}. There are a number of advantages: 1. A crystal is regular so that its generation can be automated. All that needs to be done is to systematically transform the primary atoms by one of the set {T} and a linear combination of A, B and C. The result is obviously best stored in terms of A, B, and C rather than as absolute numerical values of the transformations. 2. It is essential to define a CHARMM crystal by A, B and C and {T} if the lattice parameters a, b, c, alpha, beta and gamma are to be varied because the coordinates of all the image atoms within the crystal will change during successive cycles of the optimisation as a, b, c, alpha, beta and gamma themselves change. 3. When constructing the dynamical matrix for a non-zero wave-vector it is necessary to know the unit cell to which a particular atom belongs in order to evaluate the exponential factor in the expression. Although the crystal data structure and the values of the lattice parameters define the crystal the individual transformations have to be worked out explicitly in order to determine energies, harmonic frequencies and so on. In the present version of the program the IMAGE facility is used, so that a new set of IMAGE transformations are calculated from the crystal data structure as soon as a crystal is built or every time the lattice parameters are changed. The use of the IMAGE facility means that the number of transformations that can be used is determined by the dimension of the IMAGE arrays (MAXTRN in DIMENS.FCM). Crystal and Image Patching. Crystal image patching is unavailable in the present version of the program so that bonds between images are not permitted. Similarly hydrogen-bond interactions described by an explicit hydrogen-bond function are also forbidden. The only forces that can be calculated between primary and image atoms are non-bonded ones. The Lattice Coordinate System. WARNING: If your system is not properly rotated, there will usually be bad contacts. If you have bad contacts, check the alignment. The convention used by CHARMM for orientating the crystal in real space involves the use of a symmetric transformation (h) matrix. For non-orthorhombic systems, these coordinates are different (rotated) from the aligned conventioned used by PDB and others. The conversion is performed by the COOR CONVert command, see *note convert:(corman.doc)Syntax. The Structure of the Crystal File. The crystal file is divided into three parts. A standard CHARMM title. A symmetry operation declaration section headed by the word Symmetry and terminated by an End. The transformations are written in the same way as for the Crystal Build command except that the identity transformation has to be explicitly listed. An image section headed by Images and terminated by an End. Here the images are defined in terms of the symmetry transformations and the lattice translations A, B and C. The comment line shows the column labelling. Sometimes it is useful to write one's own crystal files without recourse to the Crystal Build option. In this case the symmetry and image blocks can be put in any order (although only one of each is allowed) and there is no restriction on the positioning of blank and comment lines. Two examples of a crystal file are: * Crystal file for a P1bar crystal. * Symmetry (X,Y,Z) (-X,-Y,-Z) End Images ! Operation a b c 1 0 0 -1 1 0 0 1 2 0 0 0 End * Crystal file for a P212121 crystal. * Symmetry (X,Y,Z) (X+1/2,-Y+1/2,-Z) (-X,Y+1/2,-Z+1/2) (-X+1/2,-Y,Z+1/2) End Images ! Operation a b c 2 0 0 0 3 0 0 0 4 0 0 0 2 -1 0 0 3 0 -1 0 4 0 0 -1 End Second Derivative Calculations and the Use of Symmetry. Consider a crystal with a unit cell in which there is more than one asymmetric unit (i.e. all space groups other than P1). The dynamical matrix then takes a blocked form, with Z**2 blocks if Z is the number of asymmetric units. Each block is of dimension 3N x 3N and contains the sum over all unit cells of the second derivative interaction elements between the Mth and Nth asymmetric units. It is possible to calculate only the Z blocks (11), (12), ..., (1M), ..., (1Z) and then transform them to produce the full matrix. In the present program, however, it is necessary to perform vibration calculations on entire unit cells. It should be emphasised that while this symmetry transformation can be used for calculations of the normal mode eigenvectors and frequencies for the zero wavevector it does not hold at other values for all additional values. Therefore, simple symmetry arguments such as these do not hold for phonon calculations. Symmetry can also be used to block the dynamical matrix into several smaller matrices each corresponding to a different symmetry species, thereby greatly reducing the time needed for diagonalisation and automatically helping to identify the normal modes. Symmetry blocking is not coded at the moment. References. Lattice Dynamics of Molecular Crystals", Lecture Notes in Chemistry 26, S.Califano, V.Schettino and N.Neto (1981), Springer-Verlag, Berlin, Heidelberg and New York. A comprehensive monograph with good sections on the theory of lattice vibrations and normal mode symmetries. A.Warshel and S.Lifson, J.Chem.Phys. (1970), 53, 582. The original CFF paper on crystal calculations. It describes the theory behind crystal optimisations and vibrational calculations. E.Huler and A.Warshel, Acta Cryst. (1974), B30, 1822. An extension of the work in reference 2. "Infrared and Raman Spectra of Crystals", G.Turrell (1972), Academic Press, London and New York. A nice clear introduction to the subject.

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