Which value to choose when calculating SOC?

Questions
#1

Hello everyone!
I’m calculating SOC.
Should the final SOC be the sum of One-electron SO and Mean-field SO（0.027879 + 0.027482） or Mean-field SO（0.027482）?

The output：
State A: Ground state
State B: T2
Analysing Sz ans S^2 of the pair of states…
Ket state: Computed S^2 = 0.000000 will be treated as 0.000000 Sz = 0.000000
Bra state: Computed S^2 = 2.000000 will be treated as 2.000000 Sz = 0.000000
Clebsh-Gordan coefficient: <0.000,0.000;1.000,0.000|1.000,0.000> = 1.000

One-electron SO (cm-1)
Reduced matrix elements:
<S|| Hso(L-) ||S’> = (-0.000000,-0.000000)
<S|| Hso(L0) ||S’> = (0.000000,0.027879)
<S|| Hso(L+) ||S’> = (-0.000000,0.000000)

SOCC = 0.027879

Actual matrix elements:
|Sz=0.00>
<Sz=-1.00|(-0.000000,0.000000)
<Sz=0.00|(0.000000,0.027879)
<Sz=1.00|(-0.000000,-0.000000)

Mean-field SO (cm-1)
Reduced matrix elements:
<S|| Hso(L-) ||S’> = (-0.000000,-0.000000)
<S|| Hso(L0) ||S’> = (0.000000,-0.027482)
<S|| Hso(L+) ||S’> = (-0.000000,0.000000)

Singlet part of <S|| Hso(L0) ||S’> = (-0.000000,0.000000) (excluded from all matrix elements)
L-/L+ Averaged reduced matrix elements:
<S|| Hso(L-) ||S’> = (-0.000000,-0.000000)
<S|| Hso(L+) ||S’> = (-0.000000,0.000000)

SOCC = 0.027482

The input is as follows:

$rem
METHOD          B3LYP
BASIS           6-31G*
CIS_N_ROOTS     5
CIS_TRIPLETS    TRUE
CALC_SOC        2
$end
#2

The SOC operator has one- and two-electron contributions. The latter is sometimes neglected (although you can see it’s not small), and in Q-Chem it is computed within a mean-field approximation. So the total SOC should include both.

In this paper we report a Python script (obtainable from a link in the paper) to extract these quantities from a Q-Chem TD-DFT calculation, then use them to construct and diagonalize the Breit-Pauli Hamiltonian to get the coupled states:
Computing L- and M-edge spectra using the DFT/CIS method with spin–orbit coupling - Physical Chemistry Chemical Physics (RSC Publishing)

#3

Thanks! If I use the total SOC to substitute into the Marcus formula for calculation, should the value be the sum of One-electron SO and Mean-field SO (0.027879 + 0.027482), or use the matrix processed by ext_soc.py, and then take the root of the sum of the squares of each value?
Processed by ext_soc.py:

    "S0,T1": [
        [
            5.362911704952321e-05,
            8.361494340096e-07
        ],
        [
            0.0,
            0.0
        ],
        [
            5.362911704952321e-05,
            -8.361494340096e-07
        ]
    ],
#4

If you’re just trying to do a state-to-state Marcus theory rate, e.g., < S0 | H_SOC | T1>, then use the sum of 1-electron and (mean-field) 2-electron SOC contributions, because there your basis states |S0> and |T1> are uncoupled ones. The script that’s in my paper is for spectroscopy - if you wanted to compute energies and intensities of the coupled states that emerge from the BP Hamiltonian.