Coulomb singularity

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In the unscreened HF exchange, the bare Coulomb operator

is singular in the reciprocal space at :

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.

Truncation methods

The potential is truncated by multiplying it by the step function , and in the reciprocal this leads to

which has no singularity at , but the value


Auxiliary function methods