Spin-Orbit Coupling

Spin–orbit coupling (SOC) is the spin-dependent relativistic effect that mixes electronic states of different spin multiplicity. It is the complement to the scalar (spin-free) corrections provided by X2C, and it is what makes otherwise spin-forbidden processes — intersystem crossing, phosphorescence, zero-field splitting, and fine structure — possible.¹

TeraChem computes SOC perturbatively by state interaction: it first solves for non-relativistic, spin-pure states of each multiplicity, then builds the spin–orbit matrix between them and diagonalizes it to obtain spin-mixed states and the coupling matrix elements. The operator is an effective one-electron Breit–Pauli spin–orbit operator (using the true nuclear charges), and the \(M_S\) sublevels are handled with the appropriate Clebsch–Gordan algebra. Coupling matrix elements are reported in cm⁻¹.

Because SOC is evaluated by mixing states of different multiplicity, a SOC calculation must request more than one spin multiplicity in the same run.

Two routes

SOC can be computed on top of either linear-response excited states or multireference CAS states:

Keyword	Reference states	Where to read more
`cisspinorbit yes`	CIS / TDA-DFT singlets and triplets	CIS / TDDFT
`cassoc yes`	State-averaged CASSCF / CASCI states spanning several multiplicities	CASSCF

Both use run energy.

Example 1: SOC at the CIS / TDDFT level

Request singlet and triplet excited states together (singlets via cisnumstates with cismult 1, triplets via cisnumtriplet), then turn on cisspinorbit. This computes the SOC between the singlet and triplet manifolds (and the ground state). The reference can be Hartree–Fock (CIS) or a DFT functional (TDA-DFT); here B3LYP is used:

soc_cis.in

run            energy
coordinates    geom.xyz
charge         0
spinmult       1
method         b3lyp
basis          sto-3g

cis            yes
cismult        1
cisnumstates   4        # singlet excited states
cisnumtriplet  6        # triplet excited states
cisspinorbit   yes
cismaxiter     50
cisprintthresh 0.01

threall        1e-14
convthre       1e-08
precision      mixed
purify         no
end

Example 2: SOC at the CASSCF level

For multireference systems, state-average a CASSCF (or CASCI) calculation over the multiplicities of interest and add cassoc yes. This acrolein example averages over two singlets, two triplets, and two quintets:

soc_casscf.in

run            energy
coordinates    geom.xyz
charge         0
spinmult       1
method         hf
basis          3-21g
guess          hcore

casscf         yes
closed         12
active         5
cassinglets    2
castriplets    2
casquintets    2
cassoc         yes

casscfmaxiter        100
casscfconvthre       1e-5
casscfenergyconvthre 1e-8

precision      double
threall        1.0e-16
purify         no
end

For an open-shell reference, average over the appropriate odd-electron multiplicities instead — e.g. a Cr⁺ calculation would use casdoublets and casquartets with the relevant castargetmult/casweights. See the CASSCF section for the state-averaging keywords.

Notes

SOC requires states of more than one multiplicity in the same run; with only a single multiplicity there is nothing to couple.
The treatment is perturbative (state interaction) using an effective one-electron Breit–Pauli spin–orbit operator; coupling elements are printed in cm⁻¹.
SOC is the spin-dependent counterpart to the scalar-relativistic X2C corrections; the two address different relativistic effects and can be used together.

Summary of keywords

Activating SOC

Keyword	Type	Default	Description
`cisspinorbit`	bool	no	Compute SOC among CIS / TDA-DFT states
`cassoc`	bool	no	Compute SOC among CASSCF / CASCI states

Companion state-count keywords

Keyword	Used with	Description
`cismult`	`cisspinorbit`	Multiplicity of the primary CIS manifold (`1` for singlets)
`cisnumstates`	`cisspinorbit`	Number of states in the primary manifold
`cisnumtriplet`	`cisspinorbit`	Number of triplet states to add (with `cismult 1`)
`cassinglets` / `castriplets` / `casquintets`	`cassoc`	Number of state-averaged states per (even-electron) multiplicity
`casdoublets` / `casquartets`	`cassoc`	Number of state-averaged states per (odd-electron) multiplicity

References

C. M. Marian, Spin–orbit coupling and intersystem crossing in molecules, WIREs Comput. Mol. Sci. 2, 187 (2012). ↩