Coulomb singularity: Difference between revisions

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=== Probe-charge Ewald ===
=== Probe-charge Ewald ===


=== Truncation ===
=== Spherical truncation ===


In this method{{cite|spenceralavi:prb:08}} the bare Coulomb operator <math>V(\vert\mathbf{r}-\mathbf{r}'\vert)</math> is truncated by multiplying it by the step function <math>\theta(R_{\text{c}}-\left\vert\mathbf{r}-\mathbf{r}'\right\vert)</math>, and in the reciprocal this leads to
In this method{{cite|spenceralavi:prb:08}} the bare Coulomb operator <math>V(\vert\mathbf{r}-\mathbf{r}'\vert)</math> is spherically truncated by multiplying it by the step function <math>\theta(R_{\text{c}}-\left\vert\mathbf{r}-\mathbf{r}'\right\vert)</math>, and in the reciprocal this leads to
:<math>
:<math>
V(q)=\frac{4\pi}{q^{2}}\left(1-\cos(q R_{\text{c}})\right)
V(q)=\frac{4\pi}{q^{2}}\left(1-\cos(q R_{\text{c}})\right)
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</math>
</math>


Note that the truncation method described above works very well in the case of 3D systems. However, it is not recommended for systems with a lower dimensionality {{cite|sundararamanarias:prb:13}}.
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{{cite|sundararamanarias:prb:13}}. For such systems, the approach proposed in ref. {{cite|sundararamanarias:prb:13}} (not implemented in VASP) is more adapted since the truncation is done according to the Wigner-Seitz cell.
 
 


== Related tags and articles ==
== Related tags and articles ==

Revision as of 13:04, 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 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 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

Spherical truncation

In this method[3] the bare Coulomb operator is spherically 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

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[4]. For such systems, the approach proposed in ref. [4] (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

References