Spin-Orbit Coupling in All-Electron Mixed Basis Approach

In the evaluation of the spin-orbit coupling (SOC), the use of the L center dot S formula is invalid in the interatomic region where the effective potential is not spherically symmetric. This problem occurs in the LCAO, LMTO, and APW methods, while the plane-wave pseudopotential and PAW methods cannot treat the spin-orbit splitting (SOS) of core orbitals. To avoid these problems, the all-electron mixed basis approach is adopted, which uses both plane waves (PWs) and atomic orbitals (AOs) as a basis set. The general form S center dot backward difference V x p can be used for PWs, while the standard form L center dot S can be used for AOs, which are well localized inside the non-overlapping atomic sphere in the spherical potential region and composed of the numerical radial function on a logarithmic radial mesh and analytic cubic harmonics. The explicit formula of the AO-AO, PW-AO, and PW-PW matrix elements of the SOC for spin-polarized systems is presented. In particular, the AO-AO matrices are explicitly derived for p, d, and f orbitals. The method is applied to the SOS of core and valence levels in X-ray photoelectron spectra. The results are in excellent agreement with the available experimental data, which suggests the validity of the present method.