Periodicity

For periodic systems the case of solving one problem for an infinitely large system is transposed to infinitely many problems, each one for a small system. The `infinitely many' part is of course interpolated by a finite number of sampling points.

The $H$ and $S$ are Bloch transformed[5] for each point $k$ of the Brillouin zone:

$$H^k_{RLR'L'} = \sum_T H_{(R+T)LR'L'} e^{ik T}$$ (27)

where $T$ is the translation vector.

The density matrix and other quantities are integrated over the $k$ index to yield the real valued matrices, as described. Each $k$ index is entirely independent from all others in TB and this is a great opportunity for efficient and simple parallelisation. Furthermore the number of $k$-points can be reduced by a significant factor if symmetry is present. This is employed by providing symmetry weight factors for the eventual summation.

The practical complication is that matrices turn from real symmetric to complex Hermitian and the transposition becomes a conjugate transposition.

Figure 1: General diagram of the workflow in a typical TB calculation

To summarise, a general schematic view of the workflow in a typical TB calculation is shown in Fig 1.