Charge Transfer with a Spin. I: A Generalized CASSCF Framework for Investigating Charge Transfer in the Presence of Spin-Orbit Coupling

We present a generalized extension of the recently developed electron/hole-transfer Dynamically-weighted State-Averaged Constrained CASSCF (eDSC/hDSC) method to model charge transfer in the presence of spin-orbit coupling (SOC) for systems containing an odd number of electrons. Our approach incorporates complex-valued spinor orbitals and incorporates four electronic configurations in describing ground-excited state curve crossings between Kramers-restricted doublet states. The method achieves smooth potential energy surfaces and rapid self-consistent field (SCF) convergence across a wide range of spin-orbit coupling strengths, providing an efficient framework for investigating charge transfer processes in the presence of nontrivial spin degrees of freedom.

💡 Research Summary

In this work the authors present a significant extension of the recently introduced electron‑/hole‑transfer Dynamically‑weighted State‑Averaged Constrained CASSCF (eDSC/hDSC) methodology to treat charge‑transfer (CT) processes in open‑shell systems where spin‑orbit coupling (SOC) cannot be ignored. The key innovation lies in moving from the conventional real‑valued orbital framework to a fully complex‑valued spinor‑based description that naturally incorporates SOC at the self‑consistent‑field (SCF) level. By employing a Kramers‑restricted open‑shell formalism, each electronic configuration (ground and excited) is represented as a pair of time‑reversal‑related spinors, guaranteeing that the two members of a Kramers doublet remain degenerate in the absence of external fields.

The method defines two electronic configurations per state: for electron‑transfer (eDSC) the active space consists of a minimal CASSCF(1,2) where the unpaired electron is promoted from orbital N+1 to N+2; for hole‑transfer (hDSC) a CASSCF(3,2) active space is used, moving an electron from a doubly‑occupied orbital to the active pair. Dynamical weights w₁ and w₂ are assigned to the two configurations based on a temperature‑like parameter T and the energy gap ΔE = E₂ – E₁, following a Boltzmann‑type expression. This weighting ensures that, as the system approaches the CT region, the ground‑ and excited‑state contributions are smoothly balanced, avoiding abrupt changes in the potential energy surface.

A crucial physical constraint is imposed on the active orbitals: the projection of the active‑space density onto the donor fragment (P_L) must equal that onto the acceptor fragment (P_R). Mathematically this is expressed as Tr(P_L P_active − P_R P_active) = 0 and enforced via a Lagrange multiplier λ in the overall Lagrangian L = w₁E₁ + w₂E₂ − λ Tr