Accuracy of generalized gradient approximation functionals for density-functional perturbation theory calculations

We assess the validity of various exchange-correlation functionals for computing the structural, vibrational, dielectric, and thermodynamical properties of materials in the framework of density-functional perturbation theory (DFPT). We consider five generalized-gradient approximation (GGA) functionals (PBE, PBEsol, WC, AM05, and HTBS) as well as the local density approximation (LDA) functional. We investigate a wide variety of materials including a semiconductor (silicon), a metal (copper), and various insulators (SiO${}_2$ $\alpha$-quartz and stishovite, ZrSiO${}_4$ zircon, and MgO periclase). For the structural properties, we find that PBEsol and WC are the closest to the experiments and AM05 performs only slightly worse. All three functionals actually improve over LDA and PBE in contrast with HTBS, which is shown to fail dramatically for $\alpha$-quartz. For the vibrational and thermodynamical properties, LDA performs surprisingly very well. In the majority of the test cases, it outperforms PBE significantly and also the WC, PBEsol and AM05 functionals though by a smaller margin (and to the detriment of structural parameters). On the other hand, HTBS performs also poorly for vibrational quantities. For the dielectric properties, none of the functionals can be put forward. They all (i) fail to reproduce the electronic dielectric constant due to the well-known band gap problem and (ii) tend to overestimate the oscillator strengths (and hence the static dielectric constant).

Figure 1

Energy for various quartz structures along the path from $\alpha$- to $\beta$-quartz for the different XC functionals: LDA in solid black with circles, PBE in solid red (gray) with circles, PBEsol in dashed black with squares, AM05 in dashed red (gray) with squares, WC in dotted black with triangles, and HTBS in dotted red (gray) with triangles. HTBS fails to predict $\alpha$-quartz as the stable phase.

Figure 2

Relative error with respect to the experimental frequencies of zircon for the different XC functionals: LDA (black circles), PBE [red (gray) circles], PBEsol (black squares), AM05 [red (gray) squares], WC (black triangles), and HTBS [red (gray) triangles].

Figure 3

Phonon band structure and density of states of silicon computed using the different XC functionals: LDA in solid black, PBE in solid red (gray), PBEsol in dashed black, AM05 in dashed red (gray), WC in dotted black, and HTBS in dotted red (gray). The experimental data from Refs. [31, 38] are also reported as blue (black) circles.

Figure 4

Phonon band structure and density of states of quartz computed using the different XC functionals: LDA in solid black, PBE in solid red (gray), PBEsol in dashed black, AM05 in dashed red (gray), WC in dotted black, and HTBS in dotted red (gray). The experimental data from Refs. [39, 40, 41] are also reported as blue (black) circles.

Figure 5

Phonon band structure and density of states of stishovite computed using the different XC functionals: LDA in solid black, PBE in solid red (gray), PBEsol in dashed black, AM05 in dashed red (gray), WC in dotted black, and HTBS in dotted red (gray). The experimental data from Ref. [42] are also reported as blue (black) circles.

Figure 6

Phonon band structure and density of states of zircon computed using the different XC functionals: LDA in solid black, PBE in solid red (gray), PBEsol in dashed black, AM05 in dashed red (gray), WC in dotted black, and HTBS in dotted red (gray). The experimental data from Refs. [43, 44, 45] are also reported as blue (black) circles.

Figure 7

Phonon band structure and density of states of periclase computed using the different XC functionals: LDA in solid black, PBE in solid red (gray), PBEsol in dashed black, AM05 in dashed red (gray), WC in dotted black, and HTBS in dotted red (gray). The experimental data from Ref. [37] are also reported as blue (black) circles.

Figure 8

Phonon band structure and density of states of copper computed using the different XC functionals: LDA in solid black, PBE in solid red (gray), PBEsol in dashed black, AM05 in dashed red (gray), WC in dotted black, and HTBS in dotted red (gray). The experimental data (with error bars) from Ref. [46] are also reported as blue (black) circles.

Figure 9

Variation of the average frequency $\overline{\omega }$ (in cm${}^{-1}$) of (a) periclase and (b) copper as a function of the lattice constant $a$ (in Å) computed using the different XC functionals: LDA in solid black, PBE in solid red (gray), PBEsol in dashed black, AM05 in dashed red (gray), WC in dotted black, and HTBS in dotted red (gray). The vertical lines indicate the experimental lattice constant taken from Refs. [27, 28], and the two extreme PBE and LDA lattice constants for periclase and copper, respectively.

Figure 10

Variation of (a) the linear thermal expansion $\varepsilon$, (b) the coefficient of linear thermal expansion $\alpha$, and (c) the bulk modulus $B_0$ of periclase as a function of the temperature computed using the different XC functionals: LDA in solid black, PBE in solid red (gray), PBEsol in dashed black, AM05 in dashed red (gray), WC in dotted black, and HTBS in dotted red (gray). Relevant experimental data are also reported. The labels correspond to the first author and year of publication of Refs. [60, 61, 62, 63, 64].

Figure 11

Variation of (a) the linear thermal expansion $\varepsilon$, (b) the coefficient of linear thermal expansion $\alpha$, and (c) the bulk modulus $B_0$ of copper as a function of the temperature computed using the different XC functionals: LDA in solid black, PBE in solid red (gray), PBEsol in dashed black, AM05 in dashed red (gray), WC in dotted black, and HTBS in dotted red (gray). Relevant experimental data are reported. The labels correspond to the first author and year of publication of Refs. [65, 66].