Coulomb singularity

The bare Coulomb operator

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

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

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

It has an (integrable) singularity at $q=\vert\mathbf{k}'-\mathbf{k}+\mathbf{G}\vert=0$ 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 different methods have been proposed: the auxiliary function , probe-charge Ewald (HFALPHA), and Coulomb truncation 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.

Spherical truncation

In this method the bare Coulomb operator $V(\vert\mathbf{r}-\mathbf{r}'\vert)$ is spherically 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

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

whose value at $q=0$ is finite and is given by $V(q=0)=2\pi R_{\text{c}}^{2}$, where the truncation radius $R_{\text{c}}$ is chosen as $R_{\text{c}}=\left(3/\left(4\pi\right)N_{\mathbf{k}}\Omega\right)^{1/3}$ with $N_{\mathbf{k}}$ being the number of $k$-points in the full Brillouin zone.

The screened Coulomb operators

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

and

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

have representations in the reciprocal space that are given by

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

and

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

respectively. Thus, the screened potentials have no singularity at $q=0$. Nevertheless, it is still beneficial for accelerating the convergence with respect to the number of k-points to multiply these screened operators by $\theta(R_{\text{c}}-\left\vert\mathbf{r}-\mathbf{r}'\right\vert)$, which in the reciprocal space gives

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

and

$V(q)=\frac{4\pi}{q^{2}} \left( 1-\cos(qR_{\text{c}})\text{erfc}\left(\lambda R_{\text{c}}\right) - e^{-q^{2}/\left(4\lambda^2\right)} \Re\left({\text{erf}\left(\lambda R_{\text{c}} + \text{i}\frac{q}{2\lambda}\right)}\right)\right)$

respectively, with the following values at $q=0$:

$V(q=0)=\frac{4\pi}{\lambda^{2}}\left(1-e^{-\lambda R_{\text{c}}}\left(\lambda R_{\text{c}} + 1\right)\right)$

and

$V(q=0)=2\pi\left(R_{\text{c}}^{2}\text{erfc}(\lambda R_{\text{c}}) - \frac{R_{\text{c}}e^{-\lambda^{2}R_{\text{c}}^{2}}}{\sqrt{\pi}\lambda} + \frac{\text{erf}(\lambda R_{\text{c}})}{2\lambda^{2}}\right)$

Note that the truncation method described above works very well in the case of 3D systems. However, it is not recommended for systems with lower a dimensionality. For such systems, the approach proposed in ref. (not implemented in VASP) is more adapted since the truncation is done according to the Wigner-Seitz cell.

Related tags and articles

HFRCUT, Hybrid_functionals: formalism, Downsampling_of_the_Hartree-Fock_operator

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