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

Abstract

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.

Figures (8)

• Figure 1: Schematic illustration of charge transfer and pseudospin texture in an odd-electron system. (A) Doubly degenerate diabatic potential energy surfaces corresponding to charge being localized on left (or right fragment) as shown on yellow (magenta) balls. (B) Multi-dimensional nuclear configuration space. A nuclear trajectory from configuration P to Q (orange curve) defines an effective reaction coordinate; along this path, the spin-quantization axis of a Kramers doublet (blue arrows) can rotate, generating a non-trivial pseudospin texture.
• Figure 2: Potential energy curves for charge transfer along the reaction coordinate $R$ corresponding to a proton coupled electron transfer for a phenoxy-phenol (ph-ph) complex. (A) A visualization of the system whereby the bridging hydrogen is transfers between the two symmetrically located fragments. The internuclear distance between oxygen atoms is 2.459 Å. The numerical value on the x-axis represents the displacement of the hydrogen from the midpoint between the two oxygens. (B) Adiabatic surfaces. (C) The left (L) and right (R) diabatic surfaces. The charge transfer diabatic states are doubly degenerate (just like the adiabatic states).
• Figure 3: Diabatic PESs for different SOC values with scaling factor $\eta$ = [1, 50, 100, 200]. Details about the form of spin-orbit coupling are in Paper 1. Note that increasing the SOC can drastically change the surfaces.
• Figure 4: A comparison of the active orbital electron densities for two charge transfer diabatic states at different SOC coupling strengths. Plotted at R = [-0.12 Å, 0.0 Å, 0.12 Å]
• Figure 5: Spin expectation values $\langle \hat{S}_{\alpha}\rangle$ ($\alpha=x,y,z$) along the reaction coordinate for SOC strengths (A) $\eta=1$ and (B) $\eta=200$. Diabats L and R are shown in blue and orange, respectively. Solid, dashed, and dotted lines denote $\langle \hat{S}_x\rangle$, $\langle \hat{S}_y\rangle$, and $\langle \hat{S}_z\rangle$. For each diabat, the two time-reversal partners exhibit equal-and-opposite spin expectation values. See Fig. \ref{['fig:mol_axis']} for a definition of the $xyz$ axes.