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A simple generalization of the energy gap law for nonradiative processes

Seogjoo J. Jang

TL;DR

This paper reexamines the energy-gap law for radiationless transitions, clarifying that the classic EG expression is a particular stationary-phase approximation to the Fermi Golden Rule within a spin–boson framework. It introduces a generalized energy-gap (GEG) law that incorporates finite-temperature effects and an Ohmic low-frequency environment, using a tau_s-based prescription to capture nontrivial bath contributions. Numerical tests against exact FGR calculations demonstrate that GEG substantially improves upon EG, while an SPI regime (ΔE/λ < 2) and a global interpolation (k_GI) provide practical tools for broad parameter ranges. The work offers a more robust, broadly applicable framework for modeling nonradiative decay in molecules and materials, with implications for ET, exciton transfer, and related phenomena, and outlines directions for including more structured baths and non-Condon effects.

Abstract

For more than 50 years, an elegant energy gap (EG) law developed by Englman and Jortner [Mol. Phys. {\bf 18}, 145 (1970)] has served as a key theory to understand and model nearly exponential dependence of nonradiative transition rates on the difference of energy between the initial and final states. This work revisits the theory, clarifies key assumptions involved in the rate expression, and provides a generalization for the cases where the effects of temperature dependence and low frequency modes cannot be ignored. For a specific example where the low frequency vibrational and/or solvation responses can be modeled as an Ohmic spectral density, a simple generalization of the EG law is provided. Test calculations demonstrate that this generalized EG law brings significant improvement over the original EG law. Both the original and generalized EG laws are also compared with stationary phase approximations developed for electron transfer theory, which suggests the possibility of a simple interpolation formula valid for any value of EG.

A simple generalization of the energy gap law for nonradiative processes

TL;DR

This paper reexamines the energy-gap law for radiationless transitions, clarifying that the classic EG expression is a particular stationary-phase approximation to the Fermi Golden Rule within a spin–boson framework. It introduces a generalized energy-gap (GEG) law that incorporates finite-temperature effects and an Ohmic low-frequency environment, using a tau_s-based prescription to capture nontrivial bath contributions. Numerical tests against exact FGR calculations demonstrate that GEG substantially improves upon EG, while an SPI regime (ΔE/λ < 2) and a global interpolation (k_GI) provide practical tools for broad parameter ranges. The work offers a more robust, broadly applicable framework for modeling nonradiative decay in molecules and materials, with implications for ET, exciton transfer, and related phenomena, and outlines directions for including more structured baths and non-Condon effects.

Abstract

For more than 50 years, an elegant energy gap (EG) law developed by Englman and Jortner [Mol. Phys. {\bf 18}, 145 (1970)] has served as a key theory to understand and model nearly exponential dependence of nonradiative transition rates on the difference of energy between the initial and final states. This work revisits the theory, clarifies key assumptions involved in the rate expression, and provides a generalization for the cases where the effects of temperature dependence and low frequency modes cannot be ignored. For a specific example where the low frequency vibrational and/or solvation responses can be modeled as an Ohmic spectral density, a simple generalization of the EG law is provided. Test calculations demonstrate that this generalized EG law brings significant improvement over the original EG law. Both the original and generalized EG laws are also compared with stationary phase approximations developed for electron transfer theory, which suggests the possibility of a simple interpolation formula valid for any value of EG.

Paper Structure

This paper contains 9 sections, 45 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Energy Gap law: Stationary Phase Approximation for High Frequency Single Vibrational Mode
  3. Generalization of the EG law
  4. General case
  5. Ohmic spectral density with exponential cut-off for low frequency modes
  6. Numerical results and discussion
  7. Conclusion
  8. First order correction for $\tau_s^0$ for general case
  9. Stationary phase approximations for $|\Delta \bar{E}| \sim \lambda$

Figures (4)

  • Figure 1: Logarithms of dimensionless rate $\kappa$, which is obtained by multiplying an actual rate $k$ with $\hbar\sqrt{k_BT\lambda}/(\sqrt{\pi}J^2)$ for Case I of Table \ref{['table2']}. The upper panel (High Temp.) is for $k_BT=\hbar\omega_l$ and the lower panel (Low Temp.) is for $k_BT=\hbar\omega_l/2$. Exact results are those obtained by exact numerical evaluation of the FGR rate expression. EG, GEG, SC, and SPI respectively represents rates of the EG law, GEG law, semiclassical approximation, and the stationary phase approximation with interpolation.
  • Figure 2: Logarithms of dimensionless rate $\kappa$ for Case II of Table \ref{['table2']}. All other conventions are the same as Fig. 1.
  • Figure 3: Logarithms of dimensionless rate $\kappa$ for Case III of Table \ref{['table2']}. All other conventions are the same as Fig. 1.
  • Figure 4: Logarithms of dimensionless rate $\kappa$ for Case IV of Table \ref{['table2']}. All other conventions are the same as Fig. 1.