Perturbation Theory Treatment of Spin–Orbit Coupling. III: Coupled Perturbed Method for Solids

Abstract

A previously proposed noncanonical coupled-perturbed Kohn−Sham density functional theory (KS-DFT)/Hartree− Fock (HF) treatment for spin−orbit coupling is here generalized to infinite periodic systems. The scalar-relativistic periodic KS-DFT/ HF solution, obtained with a relativistic effective core potential, is taken as the zeroth-order approximation. Explicit expressions are given for the total energy through third-order, which satisfy the 2N + 1 rule (i.e., requiring only the first-order perturbed wave function for determining the energy through third-order). Expressions for additional second-order corrections to the perturbed wave function (as well as related one-electron properties) are worked out at the uncoupled-perturbed level of theory. The approach is implemented in the CRYSTAL program and validated with calculations of the total energy, electronic band structure, and density variables of spin-current DFT on the tungsten dichalcogenide hexagonal bilayer series (i.e., WSe 2 , WTe 2 , WPo 2 , WLv 2 ), including 6p and 7p elements as a stress test. The computed properties through second-or thirdorder match well with those from reference two-component self-consistent field (2c-SCF) calculations. For total energies, E (3) was found to consistently improve the agreement against the 2c-SCF reference values. For electronic band structures, visible differences w.r.t. 2c-SCF remained through second-order in only the single-most difficult case of WLv 2 . As for density variables of spin-current DFT, the perturbed electron density, being vanishing in first-order, is the most challenging for the perturbation theory approach. The visible differences in the electron densities are, however, largest close to the core region of atoms and smaller in the valence region. Perturbed spin-current densities, on the other hand, are well reproduced in all tested cases.

Unveiling the Role of Spin Currents on the Giant Rashba Splitting in Single-Layer WSe2

“…Crystalline orbitals are expressed as linear combinations of atomic orbitals . We use the ECP60MDF and ECP28MDF effective-core potentials by Dolg and co-workers, for W and Se, respectively, derived from multiconfigurational four-component Dirac–Coulomb–Breit calculations. , The valence basis sets are those from ref . We use the PBE generalized-gradient approximation, GGA, xc functional (plain and hybridized with a fraction α of exact Fock exchange), as well as the r 2 SCAN meta-generalized-gradient, meta-GGA, xc functional, − in its original (SU(2) gauge-dependent) version and in its (SU(2) gauge-invariant), current-dependent version J-r 2 SCAN .…”

“…We use the PBE generalized-gradient approximation, GGA, xc functional (plain and hybridized with a fraction α of exact Fock exchange), as well as the r 2 SCAN meta-generalized-gradient, meta-GGA, xc functional, − in its original (SU(2) gauge-dependent) version and in its (SU(2) gauge-invariant), current-dependent version J-r 2 SCAN . SOC integrals are computed analytically for both the energy and forces, and SOC is treated self-consistently. ,,,− …”

“…In conclusion, we stress that the key to these SCDFT results is the possibility to account for SOC self-consistently while employing energy functionals and effective potentials that depend (implicitly or explicitly) on spin currents. Indeed, it is this dependence on J⃗ that unlocks a full SOC-induced orbital relaxation. , Without spin currents in the xc potential, any SOC-including calculation with a (semi)­local DFA reduces to a second-variational post-self-consistent approach, for all practical purposes. Figure C further shows this by comparing, on the same scale, the spin current J z in the xy plane close to a Se atom computed without (top) and with (bottom) inclusion of J⃗ in the xc potential.…”