Coulomb singularity

From VASP Wiki
Revision as of 12:11, 10 May 2022 by Ftran (talk | contribs)

The bare Coulomb operator

in the unscreened HF exchange has a representation in the reciprocal space that is given by

It has a singularity at , and to alleviate this issue and to improve the convergence of the exact exchange with respect to the supercell size (or the k-point mesh density) different methods have been proposed: the auxiliary function methods[1], probe-charge Ewald [2] (HFALPHA), and Coulomb truncation methods[3] (HFRCUT). These mostly involve modifying the Coulomb Kernel in a way that yields the same result as the unmodified kernel in the limit of large supercell sizes. These methods are described below.

Probe-charge Ewald method

Auxiliary function methods

Truncation methods

In this method the bare Coulomb operator is truncated by multiplying it by the step function , and in the reciprocal this leads to

whose value at is finite and is given by . The screened Coulomb operators

and

have representations in the reciprocal space that are given by

and

respectively. Thus, the screened potentials have no singularity at . Nevertheless, it is still beneficial for accelerating the convergence with respect to the number of k-points to multiply these screened operators by , which in the reciprocal space gives

and

respectively, with the following values at :

and