Fermi's golden rule rate expression for transitions due to nonadiabatic derivative couplings in the adiabatic basis

TL;DR

This work develops a comprehensive Fermi's golden rule formulation for nonadiabatic transitions between adiabatic states that accounts for nonadiabatic derivative couplings (NDC) and second-order terms. A quasi-adiabatic (crude adiabatic) approximation yields a general, well-defined FGR expression that includes non-Condon momentum contributions and can be evaluated with standard ab initio and dynamics data; for linearly coupled harmonic baths, a closed-form rate expressed via bath spectral densities $\mathcal{J}(\omega)$, $\mathcal{J}_D(\omega)$, and $\mathcal{J}_F(\omega)$ is obtained, enabling efficient Fourier-transform based calculations. Model studies show that NDC terms can substantially boost nonadiabatic rates, with a high-frequency mode and bath broadening enhancing the effect, while applications to azulene indicate that environmental broadening helps but conical-intersection pathways likely dominate fast S1→S0 decay. Overall, the framework provides a practical, ab initio-accessible route to quantify nonadiabatic rates beyond the Condon approximation and to probe the balance between NDC effects and more complex dynamical bottlenecks in molecular systems.

Abstract

Starting from a general molecular Hamiltonian expressed in the basis of adiabatic electronic and nuclear position states, where a compact and complete expression for nonadiabatic derivative coupling (NDC) Hamiltonian term is obtained, we provide a general analysis of the Fermi's golden rule (FGR) rate expression for nonadiabatic transitions between adiabatic states. We then consider a quasi-adiabatic approximation that uses crude adiabatic states evaluated at the minimum potential energy configuration of the initial adiabatic state as the basis for the zeroth order adiabatic and NDC coupling terms of the Hamiltonian. Although application of this approximation is rather limited, it allows deriving a general FGR rate expression without further approximation and still accounts for non-Condon effect arising from momentum operators of NDC terms and its coupling with vibronic displacements. For a generic and widely used model where all nuclear degrees of freedom and environmental effects are represented as linearly coupled harmonic oscillators, we derive a closed form FGR rate expression that requires only Fourier transform. The resulting rate expression includes quadratic contributions of NDC terms and their couplings to Franck-Condon modes, which require evaluation of two additional bath spectral densities in addition to conventional one that appears in a typical FGR rate theory based on the Condon approximation. Model calculations for the case where nuclear vibrations consist of both a sharp high frequency mode and an Ohmic bath spectral density illustrate new features and implications of the rate expression. We then apply our theoretical expression to the nonradiative decay from the first excited singlet state of azulene, which illustrates the utility and implications of our theoretical results.

Figures (11)

• Figure 1: Natural logarithms of scaled rates $\kappa=\sqrt{k_BT \lambda/\pi} k_{FGR}$ (in the units where $k_B=\hbar=\omega_c=1$) for case I-A (upper panel) and cases I-B and I-C (lower panel), are compared with reference results of a Condon approximation, eq. \ref{['eq:k-app']}, for the same cases.
• Figure 2: Natural logarithms of scaled rates $\kappa=\sqrt{k_BT \lambda/\pi} k_{FGR}$ (in the units where $k_B=\hbar=\omega_c=1$) for case II-A (upper panel) and cases II-B and II-C (lower panel), are compared with reference results of a Condon approximation, eq. \ref{['eq:k-app']}, for the same cases.
• Figure 3: Histograms of $g_j$, which is dimensionless, and ${\rm Im}\ \tilde{F}_j$ (in atomic units) versus the wavenumber of normal modes for azulene.
• Figure 4: Plots of ${\mathcal{K}}^*(t)$, $F_s(t)=F(t)/\hbar$ (in the unit of ${\rm ps^{-1}}$), and $D_s(t)=D(t)/\hbar^2$ (in the unit of ${\rm ps^{-2}}$) versus time (in the unit of ${\rm ps}$) for azulene. Black solid lines are real parts and red dashed lines are imaginary parts.
• Figure 5: FGR rates of nonradiative decay from ${\rm S_1}$ to ${\rm S_0}$ versus the energy gap $E_1-E_0$ of azulene for five different bath model parameters provided in Table \ref{['table-bath']}. The two vertical lines represent the minimum and maximum values of the energy gap values in solution.