Relativistic self-consistent $GW$: Exact two-component formalism with one-electron approximation for solids

We present a formulation of relativistic self-consistent $GW$ for solids based on the exact two-component formalism with one-electron approximation (X2C1e) and nonrelativistic Coulomb interactions. Our theory allows us to study scalar relativistic effects, spin-orbit coupling, and the interplay of relativistic effects with electron correlation without adjustable parameters. Our all-electron implementation is fully ab initio and does not require a pseudopotential constructed from atomic calculations. We examine the effect of the X2C1e approximation by comparison to the established four-component formalism and reach an excellent agreement. The simplicity of X2C1e enables the construction of higher order theories, such as embedding theories, on top of perturbative calculations.

Figure 1

The DFT band structure of $\mathrm{AgI}$ calculated using the PBE functional with various Hamiltonians. The left panel is obtained from the nonrelativistic Hamiltonian ($a_0=6.280$ Å). The middle and the right panel show the result from the sfX2C1e-Coulomb and the X2C1e-Coulomb Hamiltonian ($a_0=6.169$ Å). The coloring of bands is employed to highlight orbitals close to the Fermi level.

Figure 2

The DFT band structure of $\mathrm{AgI}$ calculated using the PBE functional in the Gaussian gth-tzvp-molopt-sr basis [93] and the gth-pbe pseudopotential [94]. $a_0=6.169$ Å. The coloring of bands is employed to highlight orbitals close to the Fermi level.

Figure 3

The $\mathrm{sc}GWk$-resolved spectral functions of $\mathrm{AgCl}$ (top row), $\mathrm{AgBr}$ (middle row), and $\mathrm{AgI}$ (bottom row) calculating from the nonrelativistic (left), the sfX2C1e-Coulomb (middle), and the X2C1e-Coulomb (right) Hamiltonian ($a_0$ = 5.550, 5.774, and 6.169 Å, respectively).

Figure 4

The orbital-resolved $k$-dependent spectral functions of $\mathrm{AgI}$ at the $\mathrm{\Gamma }, X$, and $L$ point, calculated using $\mathrm{sc}GW$ based on the X2C1e-Coulomb Hamiltonian. $a_0=6.169$ Å.

Figure 5

$\mathrm{sc}GW$ band gaps of AgBr calculated using the sfX2C1e-Coulomb (top) and the X2C1e-Coulomb (bottom) Hamiltonian as a function of $N_k^{-1/3}$. Both the $\mathrm{sc}GW$ band gaps with ($GW$-corr) and without ($GW$-uncorr) the head corrections to the dynamic part of the $GW$ self-energy as shown.