Instanton theory for Fermi’s golden rule and beyond

Abstract

Instanton theory provides a semiclassical approximation for computing quantum tunnelling effects in complex molecular systems. It is typically applied to proton-transfer reactions for which the Born–Oppenheimer approximation is valid. However, many processes in physics, chemistry and biology, such as electron transfers, are non-adiabatic and are correctly described instead using Fermi’s golden rule. In this work, we discuss how instanton theory can be generalized to treat these reactions in the golden-rule limit. We then extend the theory to treat fourth-order processes such as bridge-mediated electron transfer and apply the method to simulate an electron moving through a model system of three coupled quantum dots. By comparison with benchmark quantum calculations, we demonstrate that the instanton results are much more reliable than alternative approximations based on superexchange-mediated effective coupling or a classical sequential mechanism. This article is part of the theme issue ‘Chemistry without the Born–Oppenheimer approximation’.

Heavy-Atom Quantum Tunnelling in Spin Crossovers of Nitrenes

“…68,69 Like the WC method, it is also based on Fermi's golden rule, but importantly treats the fulldimensional potential-energy surfaces in an ab initio manner. This golden-rule instanton theory has been shown to be in excellent agreement with exact quantum-mechanical rates across the whole temperature range [69][70][71][72] and can even be extended beyond the golden-rule limit. 73 Unlike full quantum-mechanical solutions, which are intractable for all but the simplest systems, instanton theory is computationally amenable to on-the-fly simulations of sizeable molecular systems.…”

“…We can go beyond the WC approximation of the rate constant using semiclassical goldenrule instanton theory, which provides an accurate description of nuclear quantum effects such as zero-point energy and multidimensional tunnelling. 68,69,72 In contrast to Fermi's golden rule, which formally requires the computation of wavefunction overlaps between vibronic states, 87 instanton theory is rooted in the pathintegral formulation of quantum mechanics. 88 Here, the path integral is dominated by two classical trajectories travelling on the triplet and singlet surfaces, which join smoothly together at the so-called "hopping point".…”

“…The key computational step is the optimization of this instanton pathway (including finding the optimal τ ), which is facilitated by discretizing the trajectories in the form of a ring polymer. 72,90 In this way, the problem can be turned into a standard saddle-point search of the ring polymer, based on potentials, gradients and Hessians computed on-the-fly with doublehybrid DFT.…”

“…75 The convergence of the instanton method becomes slightly more challenging under these conditions, but similar to a tunnelling-splitting calculation, which is also defined in the low-temperature limit, [76][77][78][79][80][81][82] the theory is in principle able to capture this important physical case. 72 In this work we will investigate nonadiabatic tunnelling in spin-crossover processes by considering two specific examples from nitrene chemistry. Although heavy-atom tunnelling is conventionally thought of as being restricted to cryogenic temperatures, we here unveil the significance of such effects even at room temperature, implying that heavy-atom tunnelling may surprisingly be relevant under typical reaction conditions of synthetic chemistry.…”

Chemistry without the Born–Oppenheimer approximation

“…chemically intuitive yet non-Cartesian coordinates. Richardson et al [11] discuss in their work the importance of quantum-mechanical effects on the calculation of rate constants in tunnelling regime and beyond the Fermi Golden rule, focusing on the semiclassical instanton theory and its extension beyond the Born-Oppenheimer approximation.…”