Coulomb singularity: Difference between revisions

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respectively. Thus, the screened potentials have no singularity at <math>q=0</math>. Nevertheless, it is still beneficial for accelerating the convergence with respect to the number of k-points to multiply these screened operators by <math>\theta(R_{\text{c}}-\left\vert\mathbf{r}-\mathbf{r}'\right\vert)</math>, which in the reciprocal space gives
respectively. Thus, the screened potentials have no singularity at <math>q=0</math>. Nevertheless, it is still beneficial for accelerating the convergence with respect to the number of k-points to multiply these screened operators by <math>\theta(R_{\text{c}}-\left\vert\mathbf{r}-\mathbf{r}'\right\vert)</math>, which in the reciprocal space gives
:<math>
:<math>
\frac{4\pi}{\left\vert\mathbf{q}\right\vert^{2}+\lambda^{2}}
\frac{4\pi}{q^{2}+\lambda^{2}}
\left(
\left(
1-e^{-\lambda R_{\text{c}}}\left(\frac{\lambda}{\left\vert\mathbf{q}\right\vert}
1-e^{-\lambda R_{\text{c}}}\left(\frac{\lambda}{q}
\sin\left(\left\vert\mathbf{q}\right\vert R_{\text{c}}\right) +
\sin\left(qR_{\text{c}}\right) +
\cos\left(\left\vert\mathbf{q}\right\vert R_{\text{c}}\right)\right)\right)
\cos\left(qR_{\text{c}}\right)\right)\right)
</math>
</math>
and
and

Revision as of 11:57, 10 May 2022

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.

Truncation methods

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.

Auxiliary function methods