Charge Transfer with a Spin. II: A Framework for Diabatization which Localizes Charge and Spin

We investigate a diabatization procedure that localizes charges (in real space) and localizes spins (in spin space) for open-shell systems that exhibit charge transfer in the presence of spin-orbit coupling. The procedure is applied to a two-state crossing between pairs of Kramers-restricted doublet states (which can also be considered effectively a four-state crossing). To generate the relevant electronic states, we employ the recently developed electron/hole-transfer Dynamically-weighted State-Averaged Constrained CASSCF (eDSC/hDSC) method that treats systems with an odd number of electrons. To generate the relevant diabatic states, we employ a two-step optimization over complex-unitary rotations that sequentially maximizes dipole and spin moments through iterative Jacobi sweeps; the resulting update rules are effectively equivalent to those of approximate joint diagonalization (AJD) applied to charge and spin. The method converges rapidly and yields smooth diabatic potential energy surfaces that preserve dipole and spin properties (e.g., a slowly varying pseudospin texture) along the reaction coordinate while maintaining time-reversal symmetry.

💡 Research Summary

This paper presents a comprehensive diabatization framework that simultaneously localizes charge in real space and spin in spin space for open‑shell systems where charge transfer occurs in the presence of strong spin‑orbit coupling (SOC). The authors focus on a prototypical four‑state problem that can be viewed as two Kramers‑restricted doublets (effectively a four‑state crossing) and develop a two‑step optimization scheme based on complex‑unitary rotations.

First, the electronic states required for the study are generated with the recently introduced electron/hole‑transfer Dynamically‑weighted State‑Averaged Constrained CASSCF (eDSC/hDSC) method. eDSC/hDSC imposes separate constraints on the electron and the hole, thereby producing adiabatic states that already reflect the charge‑transfer character of the system.

The core of the diabatization is an adiabatic‑to‑diabatic transformation matrix U, which is a general complex‑unitary matrix rather than a real orthogonal one because SOC mixes spin components. U is constructed as a product of sequential Jacobi rotations, each acting on a low‑dimensional subspace of the adiabatic basis. The authors enforce two physical constraints on U: (i) preservation of time‑reversal symmetry (TRS) so that Kramers pairs remain Kramers pairs after the transformation, and (ii) a gauge choice that yields a smoothly varying spin‑quantization axis (pseudospin texture) along the reaction coordinate.

The two‑step optimization proceeds as follows. In the first step the dipole‑moment expectation values ⟨ξ_i|μ̂_α|ξ_i⟩ (α = x, y, z) are maximized across all four states. This is equivalent to the Boys localization criterion for charge and can be written as a sum of squares of the diagonal elements of the dipole matrices in the diabatic basis. Maximizing this sum is mathematically identical to an approximate joint diagonalization (AJD) of the three dipole matrices. In the second step, for each Kramers pair (i, \bar{i}) the spin‑expectation values ⟨ξ_i|Ŝ_α|ξ_i⟩ and ⟨\bar{ξ}_i|Ŝ_α|\bar{ξ}_i⟩ are maximized, which corresponds to an SU(2) rotation within the pair. The rotation matrix for a pair has the form

Q(γ, ϕ) =