Instanton Theory for Nonadiabatic Tunneling through Near-Barrier Crossings

TL;DR

The paper resolves a gap in nonadiabatic rate theory by extending semiclassical instanton theory to the non-convex regime, where the crossing occurs near a diabatic barrier. It introduces an analytic continuation scheme to handle sign changes in the instanton prefactor, yielding rates that agree with fully quantum FGR across deep-tunneling and classical limits. Benchmarking on model systems reveals two distinct nonadiabatic tunneling pathways—round-top (concerted) and cusped-top (sequential)—and demonstrates how concerted tunneling can dominate at low temperatures, while sequential pathways prevail at higher temperatures. The HON isomerization example confirms real-system applicability, offering a computationally efficient tool to study competing sequential and concerted nonadiabatic tunneling in near-barrier crossings across multidimensional molecular systems.

Abstract

Many reactions in chemistry and biology involve multiple electronic states, rendering them nonadiabatic in nature. These reactions can be formally described using Fermi's golden rule (FGR) in the weak-coupling limit. Nonadiabatic instanton theory presents a semiclassical approximation to FGR, which is directly applicable to molecular systems. However, there are cases where the theory has not yet been formulated. For instance, in many real-world reactions including spin-crossover or proton-coupled electron transfer, the crossing occurs near a barrier on a diabatic state. This scenario gives rise to competing nonadiabatic reaction pathways, some of which involve tunneling through a diabatic barrier while simultaneously switching electronic states. To date, no rate theory is available for describing tunneling via these unconventional pathways. Here we extend instanton theory to model this class of processes, which we term the ``non-convex'' regime. Benchmark tests on model systems show that the rates predicted by instanton theory are in excellent agreement with quantum-mechanical FGR calculations. Furthermore, the method offers new insights into multi-step tunneling reactions and the competition between sequential and concerted nonadiabatic tunneling pathways.

Figures (10)

• Figure 1: Schematic diagrams of crossing diabatic states (0 and 1) in the convex (a) and non-convex (b) regimes along with the instantons involved. (a) A system without any diabatic barrier. The only possible reaction is the transition from reactants (R) to products (P). The instanton consists of tunneling paths on either state (in blue and red) and changes electronic state at the hopping point (purple dot). (b) A system with a near-barrier crossing. Both potentials exhibit barriers, which lie in the vicinity of the crossing point. Hence, there exist multiple competing instanton pathways.
• Figure 2: Illustration of inverse temperature $\beta$ (imaginary time) of (a) cusped-top instantons (upper path in Figure \ref{['fig:schematic']}b) and (b) round-top instantons (lower path in Figure \ref{['fig:schematic']}b) as a function of tunneling energy $E$. The solid line is for the normal regime and the dashed line is for the inverted regime. Both regimes have the same crossover point. The crossing point energy $V^\ddagger$ and the crossover inverse temperature $\beta_\mathrm{c}$ of round-top instantons for each case are marked on the axes. $\beta_\mathrm{c}$ corresponds to the highest temperature where a round-top instanton can exist, and the superscripts n and i stands for the normal and inverted regimes respectively. The wrong-order instantons have an overall minus sign in their prefactors, leading to unphysical imaginary rates.
• Figure 3: Illustration of an analytic continuation scheme from a convex regime instanton to a non-convex regime instanton. The tunneling paths on either state (in blue and red) form the instantons. The top row is for the normal regime, and the bottom row is for the inverted regime. From left to right, we deform the potentials from a 1D convex model to a linear model, and then introduce negative curvature in $V_0$ and $V_1$.
• Figure 4: Diabatic potentials defined in Eq. \ref{['equ:modelpots']}, where the instanton describing concerted tunneling is in the normal regime for model A (a) and in the inverted regime for model B (b). All instantons displayed are optimized at $\beta=1000$. Panels c-f show the instantons in phase space. (c) Instanton 1 in Panel a (non-convex). (d) Instanton 1 in Panel b (non-convex). (e) Instanton 2 in Panel a (convex). (f) Instanton 2 in Panel b (non-convex). Note that there is no fundamental difference in the shapes of the convex and non-convex instantons in phase space.
• Figure 5: Rate constants for the reaction from state 0 to state 1 $k_{0\to1}$ as a function of inverse temperature $\beta$ for model A (a) and model B (b). SCI 1 and 2 represent the rates given by instantons 1 and 2 in Figure \ref{['fig:models']}. SCI 1+2 stands for the sum of the two rates. For FGR, the exact reactant partition function is used, while the remaining methods employ the harmonic approximation to $Z_0$ evaluated at the minimum of the left well.