Brillouin Zone: Generation of Band Structures

Using a modified cluster technique, band structures of polymers can readily be calculated. When a sufficiently large repeat unit is used, errors introduced due to the methodology of the cluster procedure become vanishingly small. Even for delocalized π systems, such as polyacetylene, accurate band structures can be generated using repeat units of about 20Å. For less highly conjugated systems, a shorter cluster length should be sufficient. In contrast to earlier oligomer work, no allowance need be made for end-effects. In addition, the set of points in the BZ to be used is determined explicitly by the step-size.

The technique outlined here is very fast compared to earlier methods [70].

Geometry optimization of clusters of the size reported here (i.e., having translation vectors of about 25Å) require only a little more time than molecules of similar size, the extra time being used to calculate the inter-unit cell interactions. Band structure calculations are also very fast. The time required depends on the size of the fundamental unit cell. For polyacetylene, this amounted to 3% of the time for a single self consistent field calculation of the cluster.

Band structures calculated using the program BZ are accurate in the sense that any errors are due to the Hamiltonian used. A more accurate method, for example a large basis set ab initio calculation, should yield highly accurate band structures. In addition, limited use of symmetry in the construction of the cluster secular determinant and in the geometry optimization should increase the speed of such a calculation considerably. Electrical conductivity in semiconductors is caused by holes in the valence band and electrons in the conduction band. Conductivity also depends on the hole and electron effective masses, which are readily calculable from the second derivative of the energy of the band with respect to wave-vector. Band structures for linear polymers, calculated using semiempirical methods, should be suitable for calculation of effective masses, and consequently electrical conductivity. Unfortunately, NDDO type semiemp

As generated by MOPAC, the Fock matrix is unsuitable for band-structure work. First, the matrix represents the cluster, not the unit cell, and second, the Fock matrix will not exhibit the high symmetry of the associated space-group. The perturbation is small, but fortunately it can readily be eliminated.

The steps involved in converting the MOPAC Fock matrix into one suitable for band-structure work are as follows:

Generation of solid-state Fock matrix

BZ assumes that the unit cells used in constructing the MOPAC cluster were supplied in the order defined in MAKPOL. Based on this assumption, the first unit cell will have the index (0,0,0). If there are N atomic functions in a unit cell, then the first N rows of the MOPAC Fock matrix will correspond to the central unit cell (CUC). Of all the unit cells, this one is the only one for which the entire Fock matrix is not present; instead only the lower-half triangle is available. However, since the CUC is symmetric, the missing data are generated by forming the transpose, i.e., H_i,j = H_j,i.

The Fock matrix representing the interaction of the CUC with the next unit cell, (0,0,1), or (0,0,2) if BCC is specified, is then extracted, as are all the small Fock matrices. Each Fock matrix, representing the CUC interacting with each unit cell, is stored in a large array, of size N² times the number of unit cells used. As phrases of the type "The Fock matrix representing the interaction of the CUC with unit cell (i,j,k)" are cumbersome, from here on, the term "unit cell (i,j,k)" should be understood as having the same meaning.

The indices of each unit cell is also generated and stored. However, the cluster theory assumes that the interaction matrix relating two unit cells which are separated by more than half the distance of the translation vector does not represent that interaction. Rather, it represents the interaction of two unit cells which are separated by less that half the translation vector distance. In order to conform with this definition, all unit cell indices more that half of the number of unit cells specified by the MERS keyword are changed. For example, if MERS(4,4,4) is used, the unit cells (0,1,1) and (2,2,2) would be unchanged, but unit cells(0,1,3) and (0,0,4) would become (0,1,-2) and (0,0,-1), respectively.

As a result of this operation, most of the unit cells surrounding the CUC are generated. The next step is to symmetrize the Fock matrices so that they have the symmetry of the space group. Note that if symmetrization is not done, the band-structures generated would be almost, but not quite, identical to those which use symmetrized Fock matrices.

Symmetrization of Fock matrices

From group-theory we know that if a matrix is operated on by every operation of a group exactly once, the resulting matrix will have the symmetry of that group. In other words,

$begin{displaymath}F_{sym} = frac{1}{M}sum_{i=1}^M<R_ivert F_{unsym}vert R_i^$ .

The index i covers all operations of the group, including the identity.