Efficient calculation of the exact exchange energy in periodic systems using a truncated Coulomb potential

We present a scheme for calculating the exact exchange energy in periodic solids within a Kohn–Sham or Hartree–Fock approach, which removes the need to treat the integrable singularities via an auxiliary function. In the exchange integrals, we use a modified Coulomb potential, which tends to the exact potential as the number of $\mathbf{k}$ points increases yet has no singularities, and which is also very simple to implement. We compare this approach to the auxiliary function scheme for diamond, graphite, and two allotropes of silicon carbide and show that it converges more rapidly with the number of wave vectors.

Figure 1

(Color online) EXX for $\alpha$-silicon carbide using the attenuated scheme and the auxiliary functions proposed by Carrier et al. (Ref. 17) and Wenzien et al. (Ref. 14). The dashed and solid lines guide the eyes for all $\mathbf{k}$ point meshes and symmetric $\mathbf{k}$ point meshes, respectively. The $\mathbf{k}$ point meshes used are indicated by parentheses.

Figure 2

(Color online) EXX for $\beta$-silicon carbide using the attenuated scheme and the auxiliary functions proposed by Carrier et al. (Ref. 17) and Gygi and Baldereschi (Ref. 13). The dashed and solid lines guide the eyes for all $\mathbf{k}$ point meshes and symmetric $\mathbf{k}$ point meshes, respectively, and the $\mathbf{k}$ point meshes used are indicated by parentheses.

Figure 3

(Color online) EXX for diamond using the attenuated scheme and the auxiliary functions proposed by Carrier et al. (Ref. 17) and Gygi and Baldereschi (Ref. 13). The dashed and solid lines guide the eyes for all $\mathbf{k}$ point meshes and symmetric $\mathbf{k}$ point meshes, respectively, and the $\mathbf{k}$ point meshes used are indicated by parentheses.

Figure 4

(Color online) EXX for graphite using the attenuated scheme and the auxiliary functions proposed by Carrier et al. (Ref. 17) and Wenzien et al. (Ref. 14). The dashed and solid lines guide the eyes for all $\mathbf{k}$ point meshes and symmetric $\mathbf{k}$ point meshes, respectively, and the $\mathbf{k}$ point meshes used are indicated by parentheses. The attenuated scheme behaves poorly for very unsymmetric meshes, in particular the 422 and 844 meshes. These meshes do not, however, model the physical system well.

Figure 5

(Color online) Contributions to EXX for $\alpha$-silicon carbide for both the attenuated scheme and the singularity correction approach by using the Carrier et al. (Ref. 17) auxiliary function.