Spin‐Orbit Coupling in All‐Electron Mixed Basis Approach

In the evaluation of the spin‐orbit coupling (SOC), the use of the L · 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 · ∇ V × p can be used for PWs, while the standard form L · 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.