Electron Transfer, Diabatic Couplings and Vibronic Energy Gaps in a Phase Space Framework

TL;DR

This work compares phase-space (PS) electronic structure with traditional Born-Huang (BH) theory for vibronic energies in the Shin-Metiu electron-transfer model. By parameterizing electronic states with nuclear position and momentum and applying an ADT to form PS diabats, the authors compute vibronic gaps and show PS yields substantially lower relative errors than BH in moderate nonadiabaticity, often by up to two orders of magnitude, while BH remains reliable only in weakly coupled regimes. PS succeeds by effectively incorporating contributions from higher-lying states and maintaining total momentum conservation, though it struggles in strongly nonadiabatic limits due to a simplistic choice of the generator $\hat{\Gamma}$; improvements like localized, multi-nuclear partitions of unity are proposed. The study suggests PS diabats could extend to spin-dependent ET and phenomena like chiral-induced spin selectivity (CISS), offering a path to more accurate and physically faithful simulations of correlated electron-nuclear dynamics. Overall, PS provides a promising framework for vibronic structure and ET dynamics beyond conventional BH methods, with practical impact for modeling spin effects and multistate crossings in molecular systems.

Abstract

We investigate the well-known Shin-Metiu model for an electronic crossing, using both a standard Born-Huang (BH) framework and a novel phase space (PS) electronic Hamiltonian framework. We show that as long as we are not in the strongly nonadiabatic region, a phase space framework can obtain a relative error in vibrational energy gap which is consistently one order of magnitude smaller than what is found within a BH framework. In line with recent results showing that dynamics on one phase space surface can outperform dynamics on one Born-Oppenheimer surface, our results indicate that the same advantages should largely hold for curve crossings and dynamics on two or a handful of electronic surfaces, from which several implications can be surmised as far as the possibility of spin-dependent electron transfer dynamics.

Figures (6)

• Figure 1: Diabatic (Left) and adiabatic (Right) Marcus parabolas with relevant quantities for nonadiabatic electron transfer. For the adiabats, diabatic character is labeled by color.
• Figure 2: Born-Oppenheimer Adiabatic (Left), 2-State Boys (Center), and 3-State Boys (Right) surfaces for various external ion screening constants, denoted on the right of each row. Boys Surfaces are colored based on diabat (Left is Red, Blue is right, Green is Center). For 2-State Boys, diabatic coupling is shown in grey, and couplings are not shown for the 3-State Boys for clarity. Note that, due to the third state becoming an intruder at C=3, two-state Boys becomes unstable. Three state boys alleviates the instability at intermediate $C$ but becomes unstable when the third state becomes well separated as seen in the C=2 case.
• Figure 3: Lowest vibrational energy gap on a linear scale (top), on a log scale (middle), and the relative absolute error of the vibrational energy gap on a log scale (bottom) as a function of fixed ion screening constant $C$ (bottom axis) or log of the nonadiabaticity parameter (top axes). We plot results for electron-mobile ion mass ratios of 10 (left) and 200 (right). Note that the dips present in PS results indicate when the calculated-energy crosses the exact energy. The point at which PS outperforms BH is indicated by the gray vertical dashed line. All PS calculations in these plots use $\hat{\Gamma}=\hat{p}/i\hbar$. In general, for $C > 3.5$ a.u., one finds that a Boys-PS strongly outperforms all other results.
• Figure 4: (Top) The vibrational energy gap with or without including nonadiabatic couplings in the diabatized subspace. (Bottom) The relative absolute error in the vibrational energy gap for the same methods as above. Note the minor instability in Boys-2-NAC calculations (blue, bottom right) near the vertical grey dashed line; this instability occurs due to the numerical instability of a 2-state model in this region caused by the presence of an intruder state. Overall, from this data, one can infer that a Boys-PS approach performs well by including the effects of higher lying states--rather than by addressing the minimal residual derivative couplings between the two or three diabatic states included.
• Figure 5: Ground (blue) and 1st excited (red) adiabatic electronic wavefunction probability distributions a function of electronic position $r$ for various nuclear positions $R$ and screening constants $C$. The adiabatic potential felt by the electron is shown in black and the position of the mobile nucleus is shown with the gray vertical dashed line. Here, we plot data for $C=2$ a.u. (top), $C = 4$ a.u. (middle) and $C = 10$ a.u. (bottom) in order to model the transition from nonadiabatic to adiabatic ET. Note that, for the middle and lower panels (where PS performs quite well), there is some electronic density at $r=0$ for $R=0$. However, no such density is present in the nonadiabatic limit, $C= 2$ a.u. Note that, in this same limit, for $R=0$, the ground and excited adiabatic surfaces are near-identical, with density split across both fixed ions.