Electronic Band Structure

In a normal cluster calculation, the Fock matrix is diagonalized to yield eigenvalues corresponding to various points in the Brillouin zone. For m unit cells, the points generated are 0, 1/m, 2/m, ...up to 1/2. If m is odd, the upper bound becomes (m-1)/(2m). No other points in the Brillouin zone can be generated by diagonalization.

In order to represent a general point, k, in the Brillouin zone, a complex secular determinant, F_k, of size n must be constructed. The elements of this matrix are

$\begin{displaymath}F_{k}(\lambda,\sigma) = \sum_{r=-\infty}^{r= \infty}E(\lambda,\sigma+nr)e^{-ikr2\pi}. \end{displaymath}$

Because interactions between atomic orbitals fall off rapidly with distance, the limits of r can be truncated to include all non-vanishing elements of E, for the sake of convenience. However, these elements are precisely those which were used in the construction of the Fock matrix. Using this, and the fact that periodic boundary conditions were employed in the construction of the Fock matrix, this summation can be simplified to

$\begin{displaymath}F_{k}(\lambda,\sigma) = \sum_{r=0}^{r=m-1}E(\lambda,\sigma+nr)exp(-ikr'2\pi), \end{displaymath}$

where r', the index of the unit cell, equals r while r is less than m/2, otherwise r' = m-r. Band structures can then readily be constructed by varying the wave-vector k over the range 0-0.5. Units of k are 2π/a, where a is the fundamental unit cell repeat distance. The band structure is then constructed by simply joining the points in the order in which they are generated. Within band structures, bands of different symmetry are allowed to cross. Simply joining the points does not allow for band crossing. However, when the resulting bands are represented graphically, visual inspection readily reveals which bands should, in fact, cross.