Marcus Theory and The Condon Approximation Revisited I: E-SHAKE and Seam Sampling

TL;DR

This work addresses systematic failures of Marcus theory for floppy donor–bridge–acceptor systems by developing E-SHAKE to sample the diabatic seam where E_D(R)=E_A(R) and by constructing localized diabats to compute $H_{DA}$. The approach combines Boys/BoysOV diabatic localization with an approximate strictly diabatic gradient and a constrained MD scheme (E-SHAKE) to explore the seam beyond minimum-energy crossings. The key finding is that C-13-ae traverses a conical intersection along the seam, driving $H_{DA}$ to near zero, while C-13-ea/ee maintain nonzero, relatively steady couplings, signaling non-Condon dynamics and breakdown of the Marcus framework for the ae variant; an isotopic substitution is predicted to produce a measurable rate difference. These results establish a practical, generalizable method for probing nonadiabatic transitions in complex systems and have implications for designing experiments and understanding TET beyond traditional high-temperature, two-state models.

Abstract

Marcus theory is the workhorse of theoretical chemistry for predicting the rates of charge and energy transfer. Marcus theory overwhelmingly agrees with experiment -- both in terms of electron transfer and triplet energy transfer -- for the famous set of naphthalene-bridge-biphenyl and naphthalene-bridge-benzophenone systems studied by Piotrowiak, Miller, and Closs. That being said, the agreement is not perfect, and in this manuscript, we revisit one key point of disagreement: the molecule C-13-ae ([3,equatorial]-naphthalene-cyclohexane-[1,axial]-benzophenone). To better understand the theory-experiment disagreement, we introduce and employ a novel scheme to sample the seam between two diabatic electronic states (E-SHAKE) through which we reveal the breakdown of the Condon approximation and the presence of a conical intersection for the C-13-ae molecule; we also predict an isotopic effect on the rate of triplet-triplet energy transfer.

Figures (7)

• Figure 1: Donor (D) and acceptor (A) locations for C-13-ae and C-13-ea Closs molecules.
• Figure 2: Comparison between left: theoretically predicted couplings (ref. subotnik2010closs and this work, 'ae' systems) and right: experimentally determined rates (refs. closs1988EETcloss1989dexter). Slopes to linear fits give inferred values for $\alpha$ in Eq. \ref{['eq:coupling-bonds']}. The upper panels show the 'ee' systems (black squares), which have excellent agreement between the the calculated couplings and measured rates. The lower panels show the 'ae', red triangle-down, and 'ea', blue triangle-up, systems. Note that Closs and co-workers did not identify whether they had the 'ae' or 'ea' variant of the C-13 bridge (purple diamond). For the 'ae' and 'ea' systems, the correlation between the rates and number of bonds is weaker than for the 'ee' systems; note the log scale on the y-axis.
• Figure 3: As in Fig. \ref{['fig:exptheory']}, a comparison between computed diabatic coupling and number of bonds for 'ae' and 'ea' systems. Couplings are computed in the vicinity of the $\textrm{T}_1$/$\textrm{T}_2$ crossing and are largely the same as at the ground state minimum. However, here we also plot the 'ae' variants of the C-13 and C-14 bridges. While C-14-ae is relatively similar to C-14-ea, C-13-ae represents a particularly striking outlier that we will return to below.
• Figure 4: Coupling between donor and acceptor diabats as a function of energy within the seam above the minimum for C-13-ae. Note that for this molecule, the diabatic coupling is centered around zero, indicating the presence of a CI.
• Figure 5: Coupling between donor and acceptor diabats as a function of energy within the seam above the minimum for C-13-ea. Note that for this molecule, the diabatic coupling is not centered at zero and is reasonably constant, reflecting a reasonable adherence to the Condon approximation.