Marcus Theory and The Condon Approximation Revisited II: The Horror of Triplet Energy Transfer

TL;DR

This work investigates the applicability of the Condon approximation in ET and TET, using a 1D four-electron model and adiabatic-to-diabatic transformations to isolate the origin of their distinct distance dependences. It demonstrates that TET couplings arise from small, geometry-sensitive tails of diabatic states, yielding a scaling where $\beta_{TET} \approx 2\beta_{ET}$ under the model, and that non-Condon fluctuations can significantly affect TET, breaking the simple rate relation in certain Closs derivatives. The study combines high-level electronic-structure methods (ATD-based diabatization, BoysOV) with constrained SCF approaches (eDSC/hDSC) to accurately represent ET/HT and TET states, validating the approach against experimental Closs data. The results highlight the importance of many-body treatments and diabatization in predicting nonadiabatic transfer rates and suggest that TET behavior is more susceptible to environmental and geometric fluctuations, with conical intersections further undermining simple scaling in some systems.

Abstract

We investigate the applicability of the Condon approximation (i.e. the notion that the diabatic coupling is invariant to geometry) in the context of both electron transfer (ET) and triplet energy transfer (TET) and compare the two cases. Although it is well appreciated that diabatic couplings usually arise from the interactions of electronic wavefunction tails, we show that ET tails are very different from TET tails. Using a simple model problem, our analysis explains in detail why the rates of TET decays with twice the rate of ET, while also leading to the hypothesis that the smaller diabatic couplings found for TET (versus ET) should imply more sensitivity to non-Condon fluctuations. As an example, for the classic sets of molecules investigated by Closs, we show that the Condon approximation is indeed less applicable for TET than for ET.

Figures (9)

• Figure 1: Schematic MO diagram of two molecules (donor, blue; acceptor, pink) containing two relevant orbitals (horizontal lines) with electrons (up/down arrows signifying spin) moving (thick red arrow) for electron transfer (ET), hole transfer (HT), and triplet energy transfer (TET) from donor to acceptor.
• Figure 2: The Closs donor--bridge--acceptor molecules; The donor (D) molecules from top to bottom: benzaldehyde (TET calculations), benzophenone (TET experiments), and biphenyl (ET/HT). The acceptor molecule (A) is naphthalene. The two bridge molecules: decalin (D-##xx) and cyclohexane (C-##xx). The D and A are bound at either the 1,3 or 1,4 position of the cyclohexane bridge (C-13xx/C-14xx) and the 2,6 and 2,7 position of the decalin bridge (D-26xx and D-27xx). The D and A are covalently bonded to the bridge either equatorially (e) or axially (a).
• Figure 3: (a-c) Semi-logarithmic plots of experimental rates as measured from Ref. closs1988intramolecular and closs1988EET for the 'ee' Closs molecules versus number of sigma bonds separating the D-A moieties. Note that the rate of triplet transfer falls off exponentially with only small differences. (d-f) The computationally predicted squares of the diabatic coupling (Hartrees). The data for TET in Fig. (d) is from Ref. subotnik2010closs. We calculate the ET (e) and HT (f) using (eDSC/hDSC) see Sec \ref{['sec:discussion-calculations']}. All theoretical calculated used the optimized geometries from a Hartree-Fock singlet ground state calculation of the neutral species. Note that theory matches experiment closely and also falls off exponentially.
• Figure 4: 1D model of a 4-electron system using (a) $V_{ext}$ external potential as an approximate nuclear-electron attraction. The e-e repulsion term in the Hamiltonian is $c_{ee} / \sqrt{|x_{12}|^2+a_h^2}$ where $a_h = 0.01 \, a_0$ and $c_{ee} = 2.5e-5$. (b) Occupied canonical (green; orange) and (c) localized (blue; red) orbitals from HF.
• Figure 5: A log-log plot of the magnitude of the EET coupling of the 4-electron system from CIS boysOV diabats (black circles) and the HT coupling of the 3-electron system from hDSC (blue circles) with linear fit plotted in dashed lines. Note that the HT slope of -5.48 is approximately half that of the TET curve (-12.60).