Making Symmetry (.sym) Files
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All the.SYM files for the non-crystallographic point groups in the Point-Group Symmetry dialog are in the /SYM directory or folder, which should be directly beneath (or inside) the directory or folder containing the VIBRATZ.EXE file. These files may be consulted as examples.

File Format

All data is read in a free-form manner (no restriction of the fields to particular columns), the various fields being separated by spaces. The following descriptions gives the input in terms of lines, but in fact line breaks are disregarded.

The first line contains four fields; the point-group label; the centric/non-centric flag; the number of Cartesian symmetry matrices; and the number of symmetry species. The label should contain no blanks. The centric flag is 0 if the group is centric and non-zero if non-centric. If the group is centric, the center should not be present in the matrices - that is, only half of the total number of matrices should be given.

Following this are the Cartesian symmetry matrices, nine floating-point numbers per matrix. For clarity it is best to enter these three per line, with blank lines between matrices. These can be taken directly from the output of the SYMGRP program (see below).

Next come two lines which indicate which species contain the infrared and Raman polarization components.

The line for infrared has 3 pairs of numbers, the first number of each pair indicating the number of the species, and the second number indicating the x, y or z polarization components, with numbers 1, 2 or 3 respectively.

Likewise the line for Raman has 6 pairs of numbers, the first number of each pair indicating the number of the species, and the second number, from 1 to 9, indicating the Raman polarization component; the choices are xx, yy, zz, xy, xz, yz, xx+yy, xx-yy, xx+xx+zz and 2zz-xx-yy respectively. Of course components such as "xx" are often given as "x2".

Now come the data for each species in succession, with four or more lines for each species.

The first line for each species contains four fields, the species label, the degeneracy, the number of basis functions and the total number of terms in the basis functions. Degeneracy is 3 for T species, 2 for E species with non-complex characters, and 1 for "doubly-degenerate" E species with complex characters and all A and B species. The number of functions should be equal to the true degeneracy of the species, which is 2 for all E species.

The second line for each species contains flags for pure translation and rotation, in the order x, y, z translation, x, y, z rotation (6 numbers total).

The third line for each species consists of triplets of integer numbers giving the exponents for x, y and z in each term of the basis functions - these exponents apply to all the basis functions. There should be as many triplets as there are numbers of terms specified in species line 1 above. The individual exponents can be as large as 8, but the sum of exponents in each term is also limited to 8 (i.e. maximum is 8th order).

Finally, the fourth set of lines for each species contains the coefficients for each term, in each basis function. There should be as many lines as there are basis functions specified in species line 1 above, and each line should have as many coefficients are there are total terms.

The arrangement of terms in the basis functions is important, because the matrix of coefficients must be inverted, as a square matrix. If there are n functions and m terms, and m is larger than n, only the leftmost n terms in each function are used in this inversion (although all terms must always be entered in this input). The terms should be arranged so as to give a non-singular n by n matrix on the left. To do this it is usually sufficient to make sure that diagonal terms are non-zero. For example, consider the pair of basis functions xx-yy and 2xy, which are common for E species. Lines 3 and 4 could be entered as

2 0 0 0 2 0 1 1 0
1 -1 0
0 0 2

but disregarding the last term and taking the square matrix of coefficients on the left gives a singular matrix, since there is no longer any information from the second function. The terms in this case can be arranged as follows (although other arrangements are possible):

2 0 0 1 1 0 0 2 0
1 0 -1
0 2 0

The completed.sym file should be placed in the /SYM directory or folder, which should be directly beneath (or inside) the directory or folder containing the VIBRATZ.EXE file. This folder contains all the files for the non-crystallographic point groups in the Point-Group Symmetry dialog, and these files may be consulted as examples.

Derivation of Symmetry Matrices.

The easiest way to provide Cartesian symmetry matrices is to use the program SYMGRP, provided with VIBRATZ (also with ATOMS and SHAPE Professional). In principle, the symmetry matrices can have any orientation, provided the basis functions for the species (next section) are consistent with this orientation. In practice there may be difficulties with non-standard orientations, such as the fact that VIBRATZ is not set up for any arbitrary mixtures of Cartesian coordinates - for example only certain Raman polarization components may be listed. It is best to conform to the standards given in the Orientation section.

Polynomial Basis Functions for Species.

Provided a character table for the desired group is available, it is usually possible to specify the first and second order basis functions which are the infrared and Raman polarization components, since these are a standard part of the tables. Since only the species which contain these polarization components are active in the infrared and Raman, this information suffices to calculate the modes which can be observed in the spectra. The character tables in the book Symmetry and Spectroscopy (D. C. Harris and M. D. Bertolucci, 1978, Dover) also gives the third-order functions representing the f atomic orbitals (except for icosahedral groups). This book also gives character tables for groups with axes up to order 10. Note that these functions often assume a certain orientation of the symmetry elements with respect to the reference axes - see the previous section.

The basis functions beyond third order for the non-crystallographic point groups in the Point-Group Symmetry dialog were obtained with a program which steps sytematically through possible basis-function terms up to the desired order (VIBRATZ currently accepts 8th order), and derives the complete functions with the projection-operator technique. This program requires as input the complete characters for the group, as well as the symmetry matrices in the same order as in the character table.