Efficient implementation of the $GW$ approximation within the all-electron FLAPW method

We present an implementation of the $GW$ approximation for the electronic self-energy within the full-potential linearized augmented-plane-wave (FLAPW) method. The algorithm uses an all-electron mixed product basis for the representation of response matrices and related quantities. This basis is derived from the FLAPW basis and is exact for wave-function products. The correlation part of the self-energy is calculated on the imaginary-frequency axis with a subsequent analytic continuation to the real axis. As an alternative we can perform the frequency convolution of the Green function $G$ and the dynamically screened Coulomb interaction $W$ explicitly by a contour integration. The singularity of the bare and screened interaction potentials gives rise to a numerically important self-energy contribution, which we treat analytically to achieve good convergence with respect to the $\mathbf{k}$-point sampling. As numerical realizations of the $GW$ approximation typically suffer from the high computational expense required for the evaluation of the nonlocal and frequency-dependent self-energy, we demonstrate how the algorithm can be made very efficient by exploiting spatial and time-reversal symmetry as well as by applying an optimization of the mixed product basis that retains only the numerically important contributions of the electron-electron interaction. This optimization step reduces the basis size without compromising the accuracy and accelerates the code considerably. Furthermore, we demonstrate that one can employ an extrapolar approximation for high-lying states to reduce the number of empty states that must be taken into account explicitly in the construction of the polarization function and the self-energy. We show convergence tests, CPU timings, and results for prototype semiconductors and insulators as well as ferromagnetic nickel.

Figure 1

Convergence of (a) the ${\Gamma }_{25^{\prime }\text{v}}\to {\Gamma }_{15\text{c}}$ and ${\Gamma }_{25^{\prime }\text{v}}\to {\text{X}}_{1\text{c}}$ gaps of Si and (b) the ${\Gamma }_{15\text{v}}\to {\Gamma }_{25^{\prime }\text{c}}$ and ${\text{R}}_{15^{\prime }\text{v}}\to {\Gamma }_{25^{\prime }\text{c}}$ gaps of ${\text{SrTiO}}_3$ as a function of the MPB cutoff parameter $L_{\text{max}}$ for the angular quantum number.

Figure 2

Convergence of (a) the ${\Gamma }_{25^{\prime }\text{v}}\to {\Gamma }_{15\text{c}}$ and ${\Gamma }_{25^{\prime }\text{v}}\to {\text{X}}_{1\text{c}}$ gaps of Si and (b) the ${\Gamma }_{15\text{v}}\to {\Gamma }_{25^{\prime }\text{c}}$ and ${\text{R}}_{15^{\prime }\text{v}}\to {\Gamma }_{25^{\prime }\text{c}}$ gaps of ${\text{SrTiO}}_3$ as a function of the MPB cutoff parameter $G_{\text{max}}^{\prime }$ for the IPWs.

Figure 3

Convergence of (a) the ${\Gamma }_{25^{\prime }\text{v}}\to {\Gamma }_{15\text{c}}$ and ${\Gamma }_{25^{\prime }\text{v}}\to {\text{X}}_{1\text{c}}$ gaps of Si and (b) the ${\Gamma }_{15\text{v}}\to {\Gamma }_{25^{\prime }\text{c}}$ and ${\text{R}}_{\text{v}}\to {\Gamma }_{25^{\prime }\text{c}}$ gaps of ${\text{SrTiO}}_3$ as a function of $\sqrt{4\pi /v_{\text{min}}}$.

Figure 4

Rank of the matrix $W$ for the screened interaction, which equals the number of basis functions after the optimization, for Si and ${\text{SrTiO}}_3$ as a function of $\sqrt{4\pi /v_{\text{min}}}$.

Figure 5

Convergence of the ${\Gamma }_{25^{\prime }\text{v}}\to {\text{X}}_{1\text{c}}$ gap of Si with respect to the number of bands used in the construction of the polarization function as well as the self-energy with (solid line) and without the extrapolar correction (dashed line).

Figure 6

Band structure of Ni calculated by the LSDA (dashed lines) and the $GW$ approximation (solid lines) for majority (left panel) and minority spin (right panel). The Fermi energy is set to zero.