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]
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,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. Seven lattice types are permitted. They
are listed below along with any restrictions on the lattice parameters :
Triclinic - no restrictions on a, b, c, alpha, beta or gamma.
Monoclinic - alpha = gamma = 90.0 degrees.
Orthorhombic - alpha = beta = gamma = 90.0 degrees.
Rhombohedral - a = b = c ; alpha=beta=gamma<120 (!=90) (trigonal)
Tetragonal - a = b and alpha = beta = gamma = 90.0 degrees.
Hexagonal - a = b, alpha = beta = 90.0 degrees and gamma = 120.0
degrees.
Cubic - a = b = c and alpha = beta = gamma = 90.0 degrees.
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.
The convention used by CHARMM for orientating the crystal in real space
may be found in the routine CONCOR in [.MANIP]CORMAN2.SRC.
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.
NIH/DCRT/Laboratory for Structural Biology
FDA/CBER/OVRR Biophysics Laboratory