Imaginary part of spin-orbit coupling in ricc2

[lbertram12]

Hi,

After running some SHARC/RICC2 simulations with ADC(2) using spin-orbit coupling, I noticed that the Hamiltonian listed in QM.out/output.dat is entirely real. This is opposed to e.g. CASSCF calculations on the same system, which have large imaginary parts to the spin-orbit coupling Hamiltonian off-diagonal elements. I couldn't find anything in the manual about only including the real parts so I'm wondering why this is the case?

Thanks, Lauren

[maisebastian]

Dear Lauren, this is due to different conventions used in the definition of the triplet spin functions in ricc2 versus the CASSCF codes interfaced to SHARC (MOLPRO, OpenMolcas). Essentially, every triplet consists of three degenerate eigenstates, which are typically indexed as M_S=+1,0,-1. This choice diagonalizes the S_z operator, which is the reason why its typically used. However, as the triplets are degenerate, one could form any linear combinations of those and they would still be eigenfunctions of the spin-free Hamiltonian.

The choice of the triplet spin functions thus is arbitrary, but it affects how the SOC matrix looks like. As described in DOI: 10.1021/jp9901242 (around equations 46 and 47), for even-electron systems one can choose the triplet spin functions such that all SOC matrix elements become real-valued. For odd-electron systems, this is not possible. This real-valued choice is used by ricc2 (and also by COLUMBUS, see DOI: 10.1063/1.4892060 section IIIB), in order to not deal with complex numbers.

Note that this is not an approximation, as diagonalization would produce the same eigenvalues of the H_SOC operator. However, if you inspect the individual populations of the tripet components in a SHARC trajectory, the interpretation will be slightly different. With some interfaces, you will see the populations of T+, T0, and T-, whereas in others you will basically see Tx, Ty, and Tz.

Best, Sebastian