Theory and Implementation
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The basic theory and methodology used in VIBRATZ have been described before (Dowty, Phys. Chem. Minerals, 14:67, 1987). There have been no fundamental changes in basic methods described in that paper, but many additions and extensions. The following discusses some special topics which may be important for proper use of VIBRATZ.

VIBRATZ uses the general method of calculation described by E. B. Wilson, J. C. Decius and P. C. Cross for molecules (Molecular Vibrations, Dover 1955 - hereafter abbreviated WDC). Using this method for crystals requires only extending the continuity of the structure across unit-cell boundaries, which is done in VIBRATZ by using secondary or dummy atoms when necessary. These dummy atoms are used in calculating changes in bond lengths and angles, for example, but not in the calculation of frequencies, which is done on the basis of atoms in the unit cell only.

Although forces are entered in the form of internal coordinates or bonds and angles, these are transformed to Cartesian coordinates at an early stage. The solution of the secular equation is done with Wilson's fg method (it could be done in other ways), but this is applied to the Cartesian coordinates rather than internal coordinates. Symmetry coordinates are also worked out on a Cartesian basis.

In addition to facilitating computer programming, conversion to Cartesian coordinates has the advantage that there is no necessity for conforming the forces to the mathematical requirements of the secular equation. Thus it is not necessary to limit the number of internal coordinates or force constants to the number of normal modes (3n-3 or 3n-6), or to provide redundancy conditions if the number is excessive. This facilitates changes in the forces used, and transferral and comparison of force constants between compounds which are chemically similar but geometrically and/or symmetrically different. If an insufficient number of force constants is specified, there will be some zero frequencies, which is harmless.

Derivation of symmetry coordinates, which has often been done on a somewhat ad hoc basis, is systematized by using the projection/transfer operator technique.

Symmetry - Basis Functions

When symmetry is present, VIBRATZ requires a set of symmetry matrices, and a polynomial basis function for each species.

Symmetry matrices are obtained in two ways. For space groups and crystallographic point groups, they are generated using licensed excerpts from the program SgInfo by Ralf Grosse-Kunstleve. The matrices for the entire group are obtained by intermultiplying a small number of basis matrices. The information for all this is stored internally in VIBRATZ.

For non-crystallographic point groups, the symmetry matrices are read in from a file, since there is theoretically an infinite number of such point groups. The files for the most common point groups are supplied. The matrices for any point group, in any orientation, may be generated with the auxiliary program SYMGRP. This program also uses a small number of basis operators or matrices to generate the entire group.

Symmetry matrices are used in two ways in VIBRATZ; to generate all the atoms in a molecule or unit cell, starting from a symmetry-unique set; and to derive the irreducible representations and symmetry coordinates which are used to factor the large matrix of the secular equation into smaller matrices for each species. For the first use, lattice translations are included in the case of crystals and polymers, but lattice translations are ignored for the second use.

The polynomial basis functions for the space groups and crystallographic point groups are stored internally in VIBRATZ, since there is a finite number of them. In fact, only 32 sets of functions are required since translations are irrelevant in this context.

The basis functions for non-crystallographic point groups may be obtained in several ways. Firstly, infrared-active species always have first-order basis functions of the type x, y, z, and Raman-active species have second-order basis functions which are the components of the polarizability tensor. These functions are almost always listed in character tables. Thus the actual observed infrared and Raman species of a point group can always be calculated if a character table is available The third-order functions which are the representations of the f atomic orbitals are also sometimes listed in character tables (e.g. D.C. Harris and M. D. Bertolucci, Symmetry and Spectroscopy, Dover 1978). The book by M. Lax () lists many basis functions for space groups. Finally, a program to find basis functions was written. This program steps through all products of x, y and z up to a given order, and derives the correct basis function(s) for a given species using the projection operator technique. It requires a valid character table for the group. Polynomial basis functions have thus been determined up to 8th order, and with a few exceptions (species xx of group xx, and optically inactive species of the icosahedral groups) this is sufficient for all the groups listed in the Point Group Symmetry dialog.

Note that published character tables may contain errors. All published character tables for group S8 so far examined show rotations on x and y in species E1, whereas test calculations indicate that they really belong in species E3 (also note that x and y rotations follow the xz and yz components in all other groups).

Orientation and Rotations.

Orientation of symmetry elements with respect to coordinate axes is critical for several reasons. Unfortunately there are no universally accepted conventions for such orientation, and structure descriptions do not always specify orientation completely.

VIBRATZ uses the following conventions for orientation of symmetry elements. Any unique symmetry axis is parallel to z. In Cnv groups, a vertical mirror plane is perpendicular to the x axis (not the y axis if there is a difference). In groups which have both horizontal 2-fold axes and vertical mirror planes, a 2-fold axis is oriented parallel to the x axis and mirror planes are oriented accordingly. The icosahedral groups are oriented with a 5-fold axis parallel to z, and an adjacent 3-fold axis in the x-z plane. These orientations have been used for the basis functions in all cases, and for the symmetry matrices for non-crystallographic symmetry generated by the program SYMGRP. Note that trigonal and hexagonal crystals are oriented by VIBRATZ with the a2 axis parallel to y, so that a1 does not coincide with x (the crystal/Cartesian conversion is z = c, x = a*).

VIBRATZ has provisions for two different kinds of rotation to correct any non-correspondence in orientation. Note that there is also a third type of rotation, which is applied in the Rotation menu and the Dialog Bar of the Atoms Window. This third type of rotation is applied only to the image of the structure and the atomic displacement vectors, and has nothing to do with symmetry analysis.

Firstly, the atomic coordinates must correspond with the orientation of symmetry operators, for proper generation of all equivalent atoms. In the case of space groups, several settings may be possible, but all the common possibilities are listed in the Space Group Symmetry dialog, and it is usually only necessary to select the setting which corresponds to your atomic coordinates. In the rare case that the list is not adequate, it may be necessary to use the Custom Symmetry option.

In the case of molecules belonging to a point group, you may have a) atomic positions which do not correspond to the above standard orientation, or b) a complete set of coordinates for all atoms in the structure, rather than just a symmetry-unique set.

There are two solutions to situation a). The easiest is probably to rotate the actual atomic coordinates, using the Rotate option in the Input Atoms dialog - this is the first type of rotation mentioned above. Enter the original coordinates, then apply the appropriate rotation on the z-axis. The second option is to supply symmetry matrices in the appropriate orientation for your atomic coordinates. Such matrices can be generated with SYMGRP. It is then necessary to write a.sym file for input in the Point-Group Symmetry dialog, or to modify one of the existing files (see Making Symmetry Files for preparation of.sym symmetry input files). If the point group is one of those listed, the standard basis functions may no longer be valid for your specific orientation. In this case you can either supply new basis functions, or use the Pre-Calculation Rotations option (described below) to rotate both atoms and symmetry matrices to the standard orientation.

If you have set of atomic coordinates which includes symmetry equivalents, you can simply enter the entire set and VIBRATZ will identify the redundant atoms and delete them at your option. Of course this assumes that the atomic coordinates are consistent with the symmetry matrices - if they are not, refer to the previous paragraph.

Assuming that the input atomic coordinates are consistent with the symmetry matrices, a second type of rotation may be necessary to bring both the atoms and the symmetry matrices into the orientation assumed by the basis functions. This is done with the Pre-Calculation Rotations option in the Input menu. This type of rotation is required for certain space groups, and may be required if you use your own .sym file in the Point-Group Symmetry option.

Most space groups are oriented in the standard way described above, but in some cases, namely point or factor groups D3, D3d, D2d, D3h and C3v, the International Tables for X-Ray Crystallography uses two different standard orientations, only one of which is consistent with the above conventions. Also, monoclinic crystals are usually given an orientation with the unique axis as b or y, instead of z. When a primitive cell is used for rhombohedral crystals instead of the non-primitive hexagonal cell, rotations are also required. In all such cases, VIBRATZ should be capable of recognizing the misorientation, and should supply appropriate rotations which appear in the Pre-Calculation Rotations dialog.

Note that results such as atomic displacements are given in terms of the rotated structure, not the input structure, since displacements should correspond to the allowed polarization components which are determined by the basis functions.

Least Squares

Least-squares adjustment of the force constants in the analyical method (see Control Window) is based on the changes in the internal coordinates calculated from the atomic motions. The total energy of each mode is a linear function of the squares of these changes. Several aspects of this process should be noted.

It assumes in the first place that all the assignments are correct. If an observed mode is known to be missing it can be assigned observed frequency zero and it will be skipped, but the sequence of observed frequencies in each species must be correct, and each observed frequency must be matched exactly with the corresponding calculated frequency. If only two calculated modes are switched in order with respect to the observed list, which can easily happen with slight inaccuracies in force constants, least-squares adjustment of any force constants involved in these modes will probably fail.

Secondly, the directions of atomic motions are assumed to be correct for each mode. Of course this is not correct unless the calculated frequencies already match the observed - obviously refinement will work only if the deviations of calculated from observed frequencies are small. This requirement becomes more stringent as the possible vibrations become more complex, i.e. as the number of modes in each species increases. If the average deviation of the calculated from observed is not much smaller than the average difference between observed modes in a given species, it is quite likely that the model is at least partially incorrect and least-squares will not be reliable.

Thirdly, there may be strong correlations between force constants because of geometric redundancies and pseudo-symmetry. Because forces are converted to Cartesian, and the calculation is not directly based on internal coordinates, there is nothing to prevent duplicate or redundant forces. There are unavoidable correlations in some cases, for example between Urey-Bradley X-X repulsive forces and X-C-X angles.

In the step-by-step method (see Control Window) the second point is not strictly applicable. Redundant forces will not halt the refinement. But succes is still not to be expected unless the model is basically correct to begin with.

In summary, least-squares is appropriate only for special conditions, typically when there is a relatively small number of modes in each species, or when observed and calculated modes can be correlated precisely for other reasons. It is often necessary to refine only some force constants at a time, perhaps selecting only some of the modes which are known (positively) to be influenced by these force constants.

Intensities

Infrared intensities are simply derived from the change in dipole moment, using the atomic charges in the Atom Types list.

Raman Polarizability and Intensities. The "differential bond-polarizability tensor" rho is now defined as

rho(ij) = [sum over all k bonds] Pk db(i) b(j) / |b|

where Pk is the polarizability factor (Bond Forces dialog), b is the bond vector and db is the vector giving the change in the bond. This is a change from the original paper, which has little effect on the spatially-averaged intensities, but does improve the agreement of individual polarization components with observations. The spatially averaged intensities are determined by the standard formulas (WDC, Ch. 3 - equations 23 and 24), and the factors alpha and beta in these formulas are adjustable (Basic Parameters dialog). A value of 0.3 for beta instead of 4-7 has been found necessary to give qualitative agreement with spectra of silicates.

The depolarization ratio, which is given for each mode along with the spatially averaged intensity, is calculated with WDC Ch. 3, equation 26. The lower curve in the calculated Raman spectrum (average) represents the calculated intensity multiplied by the depolarization factor - thus the two curves are; upper, parallel and lower, perpendicular orientation of polars.

These very simple intensity calculations give surprisingly good qualitative agreement with powder spectra of silicate minerals or with spectra of silicate glasses, at least for modes involving significant stretching of Si-O or other strongly polarizable bonds. The agreement for individual polarized spectra is much poorer. Although much more elaborate approaches to intensity have been used, it should be kept in mind that calculating intensities of individual vibrational modes depends on near perfect description of the atomic motions, and that in complex structures, different vibrational components may intermix in complex ways in each mode. Changing force constants slightly may result in different admixtures of these components in modes which are close in frequency. For example, if bonds X-Y and Z-Y are fairly similar, but not identical, they may contribute to two or more individual modes in a fairly narrow frequency range. The exact relative intensities of these modes, and their polarizations, may not be reliable because of variable contributions of X-Y and Z-Y stretching to each mode, which is difficult to reproduce in a calculation. However, in spatially averaged (powder or glass) spectra an envelope containing all the modes with X-Y and Z-Y stretching may be reproduced fairly well even with rather approximate force constants.

Changes from the Original Release of VIBRAT

2-Coordinate Angle Forces. This option, for the bending forces involved in 3-atom valence angles, has been refined and corrected. The original option (if selected) always constructed two coordinates for each angle (in-plane and out-of-plane) and assigned a force constant to the out-of-plane bending based on the value of the angle. Now, when two coordinates are assigned to each angle with the same input specification the force constants are the same - this is intended for 180 degree angles. Separate in-plane and out-of-plane coordinates, with separate force constants, may also be defined for non-180-degree angles. The calculation of changes in the out-of-plane coordinates has also been corrected.

Calculation of Doubly-Degenerate E Species. VIBRATZ actually calculates both the frequency-redundant vibrations of each pair in these species (to avoid using complex numbers). The original version simply listed both vibrations, but VIBRATZ now screens out half the frequencies.

Additions and Extensions. Symmetry is now integral with the program for all space groups and all common point groups. Force constants may be refined by least-squares. Complete Urey-Bradley forces according to the Shimanouchi transformation have been added since the initial release. Tau (torsion) and psi (bond-plane) 4-atom angle coordinates have been added. The possible types of interaction between bonds and 3-atom angles sharing one or more atoms, located with the Automatic option, have been extended, and with the Manual option it is now possible to specify interactions between any internal coordinates even if there are no shared atoms. Angle forces may now be specified in units either of md-A or md/A.