Accuracy of the pseudopotential approximation in ab initio theoretical spectroscopies

A large number of today’s ab initio calculations, in particular in solid-state physics, are based on density-functional theory using first-principles pseudopotentials. This approach, initially developed for the ground state, is nowadays widely used as a starting point for the calculation of excited-state properties, as, for instance, those involved in optical spectroscopy. In this paper we investigate the validity and the accuracy of the pseudopotential approximation, analyzing how different choices within the latter can influence the calculated electronic response of silicon and silicon carbide. We consider norm-conserving first-principles pseudopotentials, both in the fully nonlocal (Kleinman-Bylander) and the semilocal forms, with different choices for the reference (local) component. The effects of the inclusion of outer-core states in the valence shell are analyzed in order to obtain a detailed comparison with all-electron calculations. We present accurate results for different pseudopotential descriptions of Kohn-Sham and quasiparticle band structures and of many spectroscopic quantities in the linear and the nonlinear response regimes for different momentum transfers $\mathbf{Q}$. Moreover, the effects of the pseudopotential nonlocality have been analyzed for electron-energy-loss spectra in the limit of vanishing momentum transfer. Our results show that the pseudopotential approximation can be quite safely applied to excited-state calculations, even when they involve Kohn-Sham eigenvalues and eigenvectors several tens of eV above the Fermi energy.

Figure 1

Logarithmic derivatives of the $s$, $p$, and $d$ components of our standard and outer-core pseudopotentials, in both their semilocal (dashed lines) and separable (KB) (dot-dashed lines) forms compared with their all-electron counterparts (solid lines). Results for different reference components ($l_{\text{ref}}=0$ and $l_{\text{ref}}=2$) are reported in the top and bottom parts of the figure, respectively. Note the large energy scale, ranging from $-5.0$ to 5.0 hartree.

Figure 2

Effects of the pseudopotential choice on individual KS-LDA eigenvalues of bulk Si at the $\Gamma$ point in the high-energy region (60 eV above the Fermi level). Top panel: Influence of a change in the local reference component for the case of the standard pseudopotential used in its fully separable form. Bottom panel: Same as top panel, but for the outer-core pseudopotential. Results for $l_{\text{ref}}=0$ (points) and $l_{\text{ref}}=2$ (squares) are shown in both panels.

Figure 3

Top panel: High-energy KS-LDA eigenvalues for bulk Si at the $\Gamma$ point for the standard pseudopotential in semilocal (square) and separable forms (crosses), compared with results using the outer-core pseudopotential in its separable form (circle). $l_{\text{ref}}=0$ has been used throughout. Central panel: Same as top panel, but using $l_{\text{ref}}=2$ throughout. Bottom panel: Same as the central panel, but showing eigenvalues at a lower-symmetry $\mathbf{k}$ point [Baldereschi point (Ref. 45)].

Figure 4

Comparison between the LDA Kohn-Sham silicon DOSs calculated with a standard pseudopotential using its semilocal form (dashed line) and using its separable (KB) form (solid line). The reference component is taken as $l_{\text{ref}}=0$ in both cases.

Figure 5

Same as Fig. 4, but using $l_{\text{ref}}=2$ in both the semilocal and separable forms of our standard pseudopotential.

Figure 6

Matrix elements entering ${\Sigma }_x$ at the $X$ point. The matrix elements have been calculated between the top valence (TV) [bottom valence (BV)] and the top conduction (TC) bands, and are shown as a function of the $\mathbf{G}$ vectors. The outer-core and the standard PPs are in their separable form.

Figure 7

Effects of the pseudopotential choice on the calculated imaginary part of the finite-$\mathbf{Q}$ macroscopic dielectric function of bulk silicon. Results obtained with the standard pseudopotential used in the semilocal and separable forms (stars and light solid line, respectively) and for the outer-core pseudopotential (solid line) with $l_{\text{ref}}=0$ are reported. In the inset, $l_{\text{ref}}=2$ is shown for the standard pseudopotential. The $\mathbf{Q}$ value is $\mathbf{Q}=1.86\text{a}\text{.u}\text{.}$, and a grid of 32 (shifted) $\mathbf{k}$ points has been employed.

Figure 8

Effects of the pseudopotential choice on the calculated finite-$\mathbf{Q}$ loss function of bulk silicon. Results obtained with the standard and the outer-core PPs used in their separable form are compared for different choices of $l_{\text{ref}}$. Standard (outer-core): $l_{\text{ref}}=0$ solid line (dashed line) and $l_{\text{ref}}=2$ light solid line (crosses). The transferred momentum $\mathbf{Q}$ is 0.53 a.u.

Figure 9

Bulk silicon loss function for $\mathbf{Q}=1.86\text{a}\text{.u}\text{.}$, calculated using the outer-core (solid line) and the standard (dashed line) pseudopotentials with $l_{\text{ref}}=0$.

Figure 10

Influence of the inclusion of the $\left[V_{\text{nl}},\mathbf{r}\right]$ commutator on the computed loss function of bulk silicon at $\mathbf{Q}=0$, for both the standard pseudopotential (dashed line) and the outer-core pseudopotential (solid line). Dark (light) lines refer to calculations with (without) including the commutator. $l_{\text{ref}}=0$ has been used throughout. The standard and the outer-core PPs are in their separable form.

Figure 11

Influence of the inclusion of the $\left[V_{\text{nl}},\mathbf{r}\right]$ commutator on the dielectric function of bulk silicon at $\mathbf{Q}=0$, for both a standard pseudopotential (dotted line) and an outer-core pseudopotential (solid line). Dark (light) lines refer to calculations with (without) including the commutator. $l_{\text{ref}}=0$ has been used throughout. The standard and the outer-core PPs are in their separable form.

Figure 12

Effects of the inclusion of the $\left[V_{\text{nl}},\mathbf{r}\right]$ commutator on the calculated second-order susceptibility of bulk SiC, using a standard pseudopotential in its separable form and choosing $l_{\text{ref}}=0$.