Coulomb singularity

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The bare Coulomb operator

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

It has an (integrable) singularity at that leads to a very slow convergence of the results with respect to the cell size or number of k-points. In order 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 [1], probe-charge Ewald [2] (HFALPHA), and Coulomb truncation[3] methods (selected with 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, which can also be applied to screened Coulomb operators, are described below.

Auxiliary function

Probe-charge Ewald

Truncation

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 , where the truncation radius is chosen as with being the number of -points in the full Brillouin zone.

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

Related tags and articles

HFRCUT, Hybrid_functionals: formalism, Downsampling_of_the_Hartree-Fock_operator

References