Q-Chem can deal with spin-orbit effects by using a perturbative approach in which one first computes non-relativistic zero-order states (by CIS, EOM-CCSD, ADC, etc) and then computes matrix elements of the spin-orbit part of the Breit-Pauli Hamiltonian. This matrix (expressed in the basis of zero-order states) can then be treated as a perturbation that mixes non-relativistic states, giving rise to the spin-orbit perturbed states. Because there are different experimental manifestations of spin-orbit effects (i.e., spectroscopic splittings, intensity borrowing, inter-system crossing, magnetic anisotropies, to name a few), there are different ways in which spin-orbit couplings (SOCs) can be related to the experimental observables. Hence, the user should know what is it they want to compute and carry out the post-processing of the raw data (zero-order non-relativistic states, their energies and properties, and SOC matrix) accordingly.

Here are some pointers to help to navigate through the relativistic calculations. The key point to remember is that “coupling” and “splitting” are not the same. These two terms have distinct meanings. Couplings are the matrix elements of spin-orbit operator between non-relativistic states: <SM| Hso | S’M’>. This is precisely what is computed and printed in Q-Chem’s output. The definition of the splitting depends on a particular type of a transition; hence, the calculations of splittings are system-dependent and should be carried as a post-processing step.

Since spin-orbit operator depends on spin, the values of the matrix elements depend on the spin projection. The values of the individual matrix elements also depend on the orientation of coordinate system — this is perfectly normal.

In order to connect to the experimental observables, such as SOCC (spin-orbit coupling constant), one needs to have the SOC matrix elements for the entire multiplet. For example, when computing couplings between the singlet and triplet states, one needs to to obtain the couplings for all three components of the triplet (Ms=-1, 0, 1). These components are degenerate in non-relativistic calculations and usually only one component is computed (Ms=0).

In the EOM-CC implementation of SOC, the algorithm computes one spin-projection in a given calculation. Then it generates matrix elements for all other spin projections by using Wigner-Eckart’s theorem. The details of the theory are described here:

http://iopenshell.usc.edu/pubs/pdf/jcp-151-034106.pdf

SOCC stands for spin-orbit coupling constant, which is defined as the sum of the squared matrix elements between the states of all possible spin projections in a given transition A->B:

SOCC = \sqrt(\sum_{M, M’} |<B, S, M | H_so | A, S, M>|^2).

In contrast to SOC, SOCC is independent of molecular orientation. It is related to ISC rates via Fermi’s golden rule.

Breit-Pauli spin-orbit operator has one- and two-electron parts. The full two electron part is expensive, but it can be very accurately approximated by the spin-orbit mean-field (SOMF) treatment. This is what is implemented for EOM-EE. The output contains both one-electron part of SOCs and the full SOCs with two-electron part evaluated with SOMF approximation.