next up previous
Next: Charge Symmeterisation Up: k-point Sampling Previous: Discretisation of k-space

Special k-point Sets

The k-point sampling approximation is used by evaluating Equation (3.18) with


where v is the unit cell volume,



and tex2html_wrap_inline5608 is the Fourier transform of tex2html_wrap_inline5610 , that is, the region of space which is occupied by electrons given by the contribution of k-point k. Here k runs over the a set of k-points and i labels the occupied states and the integral over the Brillouin zone is replaced by a sum over the discrete k-point set. This shows that a calculation to obtain the charge density is required at each k-point in the set. To optimize any total energy calculation it is necessary to compute the charge density at a carefully chosen set of k-points which characterises the shape of the reciprocal space of the cell. For example, the space group operations of a given cell will map some k-point onto others - these will have a related reciprocal space charge density[42], therefore only a single computation is necessary.

This fact is used by Chadi and Cohen to generate a special set of k-points at which the reciprocal space should be sampled. The k-point set is generated as follows: picking two starting k-points tex2html_wrap_inline5636 and tex2html_wrap_inline5638 satisfying certain uniqueness conditions[40] in the reciprocal space of the cell with point group operations tex2html_wrap_inline5640 then a new set of points can be generated by


The new set of points generated in this way can then be used in a similar process to generate larger sets.

This procedure does not generate k-points that are necessarily unique in the irreducible part of the Brillouin zone. Through the symmetry operations, T, the new set of k-points can be `folded back' into the irreducible part. A normalised weighting factor can then be associated with each point with their ratios indicating the number of times that each point in the irreducible zone has been generated. The generation procedure can be repeated until a sufficiently dense set of k-points is achieved to give as small an error required by the approximation given by Equation (3.27).

The Monkhorst-Pack scheme uses an alternate approach which gives an identical set of k-points to Chadi and Cohen. It is based on generating periodic functions based on the point group symmetries on a regular grid of points in the Brillouin zone.

next up previous
Next: Charge Symmeterisation Up: k-point Sampling Previous: Discretisation of k-space

Stewart Clark
Thu Oct 31 19:32:00 GMT 1996