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

Abstract

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.

Figures (6)

• Figure 1: Schematic illustration of crossing between potential energy surfaces (PESs) associated with charge-transfer doublet states in a molecule containing an odd number of electrons. The two diabatic surfaces, correspond to the electron localization of transferring electron on the left or right fragment. The doubly-degenerate nature of each diabat is indicated by colored dashed lines {yellow,green} and {blue, red}.
• Figure 2: Schematic representation of the active spaces (highlighted in magenta boxes) used to model electron transfer (left) and hole transfer (right) in an open-shell molecular system containing an odd number of electrons. For electron transfer, the active orbitals involve excitation of the unpaired electron, represented by a minimal CASSCF(1,2) active space. The hole-transfer case involves excitation from a doubly occupied orbital, corresponding to a CASSCF(3,2) active space with three electrons distributed over two orbitals.
• Figure 3: Molecular orbital (MO) density matrices for the (A) ground ($\bm{K_g}$) and (B) excited electronic ($\bm{K_e}$) states. The diagonal elements indicate the occupancies of individual MOs, with pairs of degenerate (Kramers-restricted) orbitals forming two corresponding diagonal sub-blocks (green $(1)$ and yellow $(\bar{1})$). The magenta columns highlight the active-space orbitals involved in the electronic excitation in the spin-up space. Columns to the left (right) of the active space correspond to core (virtual) orbitals in each half of the matrix.
• Figure 4: Potential energy curves for a bridging hydrogen transferring between two symmetrically located fragments. (A) Ground and excited energy surfaces shown in blue and orange, respectively. (B) Molecular geometry of the phenoxyl–phenol (ph–ph) system, illustrating the hydrogen transfer pathway (black arrow) between the two ring fragments highlighted in red and blue. The reaction coordinate $R$ represents the displacement of the transferring hydrogen relative to the midpoint between the two oxygen atoms. The internuclear distance between oxygen atoms is 2.459 Å. See Supplementary Information for Cartesian coordinates of molecular geometry.
• Figure 5: Effect of increasing spin-orbit coupling strength. (A) Ground and excited state PES. The energies are shifted to set zero as mean of two surfaces in the middle. The SOC term in Eq. \ref{['eq:h_so']} is scaled by a factor $\eta$ for all geometries along the reaction coordinate. Each color represents the two PES for $\eta$ = [1, 50, 100, 200]. (B) PES near avoided crossing region. (C) Energy gap between ground and excited state at $R = 0.0\ \text{\AA}$ for different values of $\eta$.