Coulomb singularity

In the unscreened HF exchange, the bare Coulomb operator

$V(\vert \mathbf {r} -\mathbf {r} '\vert )={\frac {1}{\vert \mathbf {r} -\mathbf {r} '\vert }}$

is singular in the reciprocal space at $q=\vert \mathbf {k} '-\mathbf {k} +\mathbf {G} \vert =0$:

$V(q)={\frac {4\pi }{q^{2}}}$

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, probe-charge Ewald (HFALPHA), and Coulomb truncation methods (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 $V(\vert \mathbf {r} -\mathbf {r} '\vert )$ is truncated by multiplying it by the step function $\theta (R_{\text{c}}-\left\vert \mathbf {r} -\mathbf {r} '\right\vert )$, and in the reciprocal this leads to

$V(q)={\frac {4\pi }{q^{2}}}\left(1-\cos(qR_{\text{c}})\right)$

which has no singularity at $q=0$, but the value

$V(0)=2\pi R_{\text{c}}^{2}$

The screened potentials

$V(\vert \mathbf {r} -\mathbf {r} '\vert )={\frac {e^{-\lambda \left\vert \mathbf {r} -\mathbf {r} '\right\vert }}{\left\vert \mathbf {r} -\mathbf {r} '\right\vert }}$

$V(\vert \mathbf {r} -\mathbf {r} '\vert )={\frac {{\text{erfc}}\left({-\lambda \left\vert \mathbf {r} -\mathbf {r} '\right\vert }\right)}{\left\vert \mathbf {r} -\mathbf {r} '\right\vert }}$

have representations in the reciprocal space that are given by

${\frac {4\pi }{q^{2}+\lambda ^{2}}}$

and

${\frac {4\pi }{q^{2}}}\left(1-e^{-q^{2}/\left(4\lambda ^{2}\right)}\right)$

respectively. Thus, the screened potentials have no singularity at $q=0$.