Effective spin-orbit models using correlated first-principles wave functions

Diffusion Monte Carlo using continuous real-space wave functions is one of the most accurate scalable many-body methods for solid-state systems. However, to date, spin-orbit interactions have not been incorporated into large-scale calculations at a first-principles level, only having been applied to small systems. In this technique, we use explicitly correlated first-principles diffusion Monte Carlo calculations to derive an effective spin-orbit model Hamiltonian. The simplified model Hamiltonian is then solved to obtain the energetics of the system. To demonstrate this method, benchmark studies are performed in main-group atoms and monolayer tungsten disulfide, where high accuracy is obtained.

Figure 1

Calculated spin-orbit interaction energies vs SOI descriptors $\langle {\sum }_{iI}{\widehat{\mathbf{L}}}_{iI}·{\widehat{\mathbf{S}}}_i\rangle$ for 20 wave-function samples for the Ga atom. The Dolg-Stoll small-core ECP is used. The blue circles (fixed-phase DMC) are the fixed-phase DMC extrapolated estimators. The fitted ${\lambda }_{\mathrm{SOI}\left(\mathrm{DMC}\right)}$ is 0.058(1) eV.

Figure 2

Fitted $\lambda$ using the results obtained from fixed-phase DMC (circles), VMC with Slater determinants (upper triangles), and Slater-Jastrow wave functions (lower triangles). The experimental estimates are labeled with short horizontal lines. For Sn and Ge, experimental $\lambda \mathrm{s}$ are evaluated as the average of the values obtained from $J=1$ and $J=2$ SOI splittings.

Figure 3

(a). Spin-orbit splittings of the ground-state electron configurations of Ga, In, Ge, Sn, Br, and I atoms. (b) Fixed-phase DMC extrapolated estimators (the mixed estimators are extrapolated to the zero-time-step limit) vs the experimental values of the fine-structure splittings for the chosen main-group atoms. Different colors and shapes of markers denote calculated values using different ECPs.

Figure 4

Band structure of monolayer ${\mathrm{WS}}_2$ calculated on the unit cell, using the Perdew-Burke-Ernzerhof exchange-correlation functional [45] without spin-orbit interaction. The red dashed line represents the Fermi level. The conduction-band minimum and valence-band maximum at $K$ are labeled $A_1^{\prime }$ and $A^{\prime \prime }$. The corresponding single-particle orbitals are plotted on the right using the visual molecular dynamics software [50], with orbital shape derived from the isosurface of the magnitude, colored by the phase.

Figure 5

Calculated spin-orbit interaction energies vs SOI descriptors $\langle {\sum }_{iI}{\widehat{\mathbf{L}}}_{iI}·{\widehat{\mathbf{S}}}_i\rangle$ for 24 fixed-phase wave functions constructed on a manifold that has degenerate scalar-relativistic energy at $K$ for monolayer ${\mathrm{WS}}_2$. The circles are the fixed-phase DMC extrapolated estimators, which are fitted to the solid line. The fitted slope ${\lambda }_{\mathrm{SOI}\left(\mathrm{DMC}\right)}$ is 0.159(3) eV.

Figure 6

(a) Calculated spin-orbit interaction energies vs descriptors for the Sn atom (Stuttgart-Stoll small-core ECP) using VMC, with single Slater determinants (HF) and Slater determinants multiplied by the same three-body Jastrow factor (VMC-SJ), and fixed-phase DMC. (b) The differences between the energies calculated using fixed-phase DMC and VMC(HF) and VMC(SJ).

Figure 7

Normalized distribution of the fixed-phase DMC extrapolated estimators (the DMC mixed estimators are extrapolated to the zero-time-step limit) calculated with available basis sets using the Dolg-Stoll small-core ECPs.

Figure 8

CPU times needed for performing fixed-phase DMC on 1 core (16 nodes) for the same number of blocks, with and without the SOI part computed.