Perspective of Fermi's golden rule and its generalizations in chemical physics

Abstract

This perspective provides a succinct history of Fermi's golden rule (FGR), overview of its derivation, assumptions, and representative forms. Major applications of FGR, mostly in the field of chemical physics, are reviewed. These illustrate the broad applicability and success of FGR. Ambiguities and open issues encountered in practical applications of FGR are clarified. Recent advances in generalizations of FGR and computational methods for practical applications are addressed.

Figures (6)

• Figure 1: Some milestones in the development and applications of FGR.
• Figure 2: (A) The real (black) and imaginary (red) parts of the time-dependent GF in Eq. (\ref{['eq:kF_GF']}) for a system with single vibrational mode and two diabatic electronic states. The dashed lines indicate Gaussian apodization functions with two different widths. (B) The rate spectrum with narrower apodization function, and (C) the rate spectrum with the wider one. In the rate spectra, vertical dashed lines represent the locations of adiabatic gaps for upward and downward transitions, with the gap denoted as $E_{ad}$. Reproduced from Ref. landi-jpcc128 with permission from the American Chemical Society.
• Figure 3: Dimensionless rates $\kappa= \hbar\sqrt{k_BT\lambda}/(\sqrt{\pi}J^2)k_{_{\rm F}}$ (in natural logarithmic scale) versus -$\Delta G/\lambda$ for the case where the spectral density of Eq. (\ref{['eq:bath-sp']}) is given by an Ohmic form with exponential cutoff, ${\mathcal{J}}(\omega)=\pi\hbar \eta \omega e^{-\omega/\omega_c}$ with $\eta=1$ and $\hbar\omega_c=5k_BT$. $\lambda=5k_BT$ for this case. Results based on Eqs. (\ref{['eq:kf-m']}) (Marcus), (\ref{['eq:kf-2-sca-1']}) (SC), and (\ref{['eq:kf-2-spi']}) (SPI) are compared with exact numerical evaluation of ${\rm k_{_F}}$.
• Figure 4: Dimensionless rates $\kappa= \hbar\sqrt{k_BT\lambda}/(\sqrt{\pi}J^2)k_{_{\rm F}}$ (in natural logarithmic scale) versus -$\Delta G/\lambda$ for the case where the spectral density of Eq. (\ref{['eq:bath-sp']}) is given by ${\mathcal{J}}(\omega)=\pi\hbar \eta \omega e^{-\omega/\omega_c}+\pi \hbar s_h \omega_h^2 \delta (\omega-\omega_h)$ with $\eta=1$, $\hbar\omega_c=5k_BT$, $s=0.1$, and $\omega_h=5\omega_c$. Rates calculated by EJ-EG law,englman-mp18 GEG law,jang-jcp155-1 and Eq. (\ref{['eq:kf-2-spi']}) (SPI) are compared with exact numerical evaluation of ${\rm k_{_F}}$.
• Figure 5: The structure of PPBA (left) and its FGR-based kinetic model (right). The black arrows represent non-radiative transitions with FGR--based rates, while the blue arrow shows the emission with an experimentally determined rate. Rates are given in the unit of s$^{-1}$. Adapted from Ref. min-jpca129 with permission from the American Chemical Society.