Nonlocal exchange correlation in screened-exchange density functional methods

We present a systematic study on the exchange-correlation effects in screened-exchange local density functional method. To investigate the effects of the screened-exchange potential in the band gap correction, we have compared the exchange-correlation potential term in the screened-exchange local density approximation (sX-LDA) formalism with the self-energy term in the GW approximation. It is found that the band gap correction of the sX-LDA method primarily comes from the downshift of valence band states, resulting from the enhancement of bonding and the increase of ionization energy. The band gap correction in the GW method, on the contrary, comes in a large part from the increase of the conduction band energies. We also studied the effects of the screened-exchange potential in the total energy by investigating the exchange-correlation hole in comparison with quantum Monte Carlo calculations. When the Thomas-Fermi screening is used, the sX-LDA method overestimates (underestimates) the exchange-correlation hole in short (long) range. From the exchange-correlation energy analysis, we found that the LDA method yields better absolute total energy than the sX-LDA method.

Figure 1

(Color online) LDA (blue open squares) and sX-LDA (red filled circles) band gap vs the experimental value for several semiconductors in the zinc-blende structure.

Figure 2

The eigenvalue difference between sX-LDA and LDA, ${ϵ}_{n\mathbf{k}}^{\mathrm{sX}\text{−}\mathrm{LDA}}-{ϵ}_{n\mathbf{k}}^{\mathrm{LDA}}$, is plotted against the sX-LDA eigenvalue for several symmetric Brillouin zone $k$ points: (a) diamond, (b) Si, (c) Ge, and (d) GaAs. The zero of the eigenvalue is set at the top of the valence band maximum.

Figure 3

(Color online) Matrix elements of exchange-correlation potential $V_{\mathrm{xc}}$ evaluated using the LDA wave functions. The matrix elements of the conduction band minimum and the valence band maximum are shown.

Figure 4

(Color online) $V_{\mathrm{xc}}(\mathbf{r},R)R^2$. One electron is fixed at (a) bond center, (b) atom, and (c) the interstitial site. The solid (black), dashed (red), and dashed-dotted (blue) lines correspond to sX-LDA, GW, and HF results. The raw GW data show oscillation owing to the finite wave vector cutoff used in our calculations. The data in the current figure were obtained by taking the median values of the oscillating curves. For HF results, we used sX-LDA wave functions. Note that the figure shows only the nonlocal part. For sX-LDA, there is a sharp peak in $V_{\mathrm{xc}}(\mathbf{r},R)$ at $R=0$ due to the local part of the exchange-correlation potentials. This delta peak is not shown in the figure when $V_{\mathrm{xc}}(\mathbf{r},R)$ is multiplied by $R^2$.

Figure 5

(Color online) The spherically averaged screened (black solid line) and Hartree-Fock (red dashed line) exchange holes of bulk silicon are shown with one electron fixed at three different spatial points: (a) bond center, (b) atom, and (c) interstitial site. The exchange holes were calculated using LDA wave functions.

Figure 6

(Color online) The spherically averaged sX-LDA exchange-correlation hole of bulk silicon in the screened-exchange method (black solid lines) is shown with one electron fixed at three different spatial points: (a) bond center, (b) atom, and (c) interstitial site. For comparison purposes, we also show the HF (green dotted lines), LDA (red dashed lines), and coupling-strength-averaged VMC (blue dashed-dotted lines) exchange-correlation hole. The LDA and HF exchange-correlation holes were calculated using sX-LDA wave functions. The VMC results are from Ref. 16.

Figure 7

(Color online) The distance weighted exchange-correlation hole of bulk silicon in the screened-exchange method (black solid lines) is shown with one electron fixed at three different spatial points: (a) bond center, (b) atom, and (c) interstitial site. The LDA (red dashed lines) and HF (green dotted lines) exchange-correlation holes were calculated using sX-LDA wave functions for comparison purpose. The VMC (blue dashed-dotted lines) results are from Ref. 16.