How to incorporate the effect of spin-orbit coupling in electronic structure calculation

Since the effect of spin-orbit coupling plays an important role in many transition metal complexes, what are the common methods to incorporate the effect of spin-orbit coupling?

Basically there are two kinds of approaches which may be found in many text books, L-S coupling and j-j coupling.

L-S coupling means that scalar electronic states (e.g. atomic L-S states and linear molecular Lambda-S states) are calculated first, and then the SO matrix is constructed with the help of 1-e (and optional 2-e) SO integrals. After diagonalizaion, energies of spinor states (atomic J levels and linear Omega states) may be obtained. Most of Q.C. programs like Molpro, Molcas, Orca, and Gamess do SOC in this way.

In j-j coupling, orbitals and spins are combined into spinors (atomic j and linear omega) at the very beginning, so there are no orbitals and scalar electronic states any more. The most representative program is Dirac. In addition, some Q.C. programs can do two-component HF/DFT only, including NWchem (sodft), Turbomole, Gaussian (int=dkh4), and so on. ADF can do j-j coupling DFT also (spinorbit zora), whereas L-S coupling DFT has to be performed through TDDFT.

From the perspective of methodology, in addition to L-S coupling and j-j coupling, there are also some intermediate approaches, which do scalar SCF/MCSCF calculatons first but SOC calculations at the post-HF/MCSCF stage. The programs I can think of are Columbus (soci), Cfour (ccsdso), and (maybe) Nooijen's STEOM-CC which is integrated in Orca.

The inclusion of spin-orbit coupling in electronic structure calculations is done by including the interaction between the electron-spin and the orbital angular momentum in the Hamiltonian. Such interaction is described according to the spin-orbit Hamiltonian defined as follows,

$$\hat{H}_\mathrm{SO} = \frac{1}{2m_\mathrm{e}\mathrm{c}^2} \frac{1}{r} \left( \frac{\partial{V}}{\partial{r}} \right) \hat{L} \cdot \hat{S},$$

where $$V$$ is the Coulombic potential of the electron in the field of the atom.

Spin-orbit coupling effect can be included in electronic structure calculations using variational methods such as fully relativistic four-component Dirac HF or KS approach, or two-component approaches such as ZORA, X2C, DKH etc. Spin-orbit coupling effect can also be included as a perturbative correction.

• +1. While short, this answer does actually provide a lot of new information about methods that can be used. Commented Jun 9, 2020 at 6:24