Divide-and-Conquer Method for Instanton Rate Theory

Ring-polymer instanton theory has been developed to simulate the quantum dynamics of molecular systems at low temperatures. Chemical reaction rates can be obtained by locating the dominant tunneling pathway and analyzing fluctuations around it. In the standard method, calculating the fluctuation terms involves the diagonalization of a large matrix, which can be unfeasible for large systems with a high number of ring-polymer beads. Here we present a method for computing the instanton fluctuations with a large reduction in computational scaling. This method is applied to three reactions described by fitted, analytic and on-the-fly ab initio potential-energy surfaces and is shown to be numerically stable for the calculation of thermal reaction rates even at very low temperature

Semiclassical instanton formulation of Marcus-Levich-Jortner theory

Marcus-Levich-Jortner (MLJ) theory is one of the most commonly used methods for including nuclear quantum effects into the calculation of electron-transfer rates and for interpreting experimental data. It divides the molecular problem into a subsystem treated quantum-mechanically by Fermi's golden rule and a solvent bath treated by classical Marcus theory. As an extension of this idea, we here present a "reduced" semiclassical instanton theory, which is a multiscale method for simulating quantum tunnelling of the subsystem in molecular detail in the presence of a harmonic bath. We demonstrate that instanton theory is typically significantly more accurate than the cumulant expansion or the semiclassical Franck-Condon sum, which can give orders-of-magnitude errors and in general do not obey detailed balance. As opposed to MLJ theory, which is based on wavefunctions, instanton theory is based on path integrals and thus does not require solutions of the Schr\"odinger equation, nor even global knowledge of the ground- and excited-state potentials within the subsystem. It can thus be efficiently applied to complex, anharmonic multidimensional subsystems without making further approximations. In addition to predicting accurate rates, instanton theory gives a high level of insight into the reaction mechanism by locating the dominant tunnelling pathway as well as providing information on the reactant and product vibrational states involved in the reaction and the activation energy in the bath similarly to what would be found with MLJ theory.Comment: 21 pages, 4 figure

Microcanonical and thermal instanton rate theory for chemical reactions at all temperatures

Semiclassical instanton theory is used to study the quantum effects of tunnelling and delocalization in molecular systems. An analysis of the approximations involved in the method is presented based on a recent first-principles derivation of instanton rate theory [J. Chem. Phys., 2016, 144, 114106]. It is known that the standard instanton method is unable to accurately compute thermal rates near the crossover temperature. The causes of this problem are identified and an improved method is proposed, whereby an instanton approximation to the microcanonical rate is defined which is integrated numerically to obtain a thermal rate at any temperature. No new computational algorithms are required, but only data analysis of a number of standard instanton calculations

Nonadiabatic quantum transition-state theory in the golden-rule limit. I. Theory and application to model systems

We propose a new quantum transition-state theory for calculating Fermi's golden-rule rates in complex multidimensional systems. This method is able to account for the nuclear quantum effects of delocalization, zero-point energy and tunnelling in an electron-transfer reaction. It is related to instanton theory but can be computed by path-integral sampling and is thus applicable to treat molecular reactions in solution. A constraint functional based on energy conservation is introduced which ensures that the dominant paths contributing to the reaction rate are sampled. We prove that the theory gives exact results for a system of crossed linear potentials and also the correct classical limit for any system. In numerical tests, the new method is also seen to be accurate for anharmonic systems, and even gives good predictions for rates in the Marcus inverted regime.Comment: 18 pages and 6 figure

Elucidating the NuclearQuantum Dynamics of Intramolecular Double Hydrogen Transfer in Porphycene

We address the double hydrogen transfer (DHT) dynamics of the porphycene molecule: A complex paradigmatic system where the making and breaking of H-bonds in a highly anharmonic potential energy surface requires a quantum mechanical treatment not only of the electrons, but also of the nuclei. We combine density-functional theory calculations, employing hybrid functionals and van der Waals corrections, with recently proposed and optimized path-integral ring-polymer methods for the approximation of quantum vibrational spectra and reaction rates. Our full-dimensional ring-polymer instanton simulations show that below 100 K the concerted DHT tunneling pathway dominates, but between 100 K and 300 K there is a competition between concerted and stepwise pathways when nuclear quantum effects are included. We obtain ground-state reaction rates of $2.19 \times 10^{11} \mathrm{s}^{-1}$ at 150 K and $0.63 \times 10^{11} \mathrm{s}^{-1}$ at 100 K, in good agreement with experiment. We also reproduce the puzzling N-H stretching band of porphycene with very good accuracy from thermostatted ring-polymer molecular dynamics simulations. The position and lineshape of this peak, centered at around 2600 cm$^{-1}$ and spanning 750 cm$^{-1}$, stems from a combination of very strong H-bonds, the coupling to low-frequency modes, and the access to $cis$-like isomeric conformations, which cannot be appropriately captured with classical-nuclei dynamics. These results verify the appropriateness of our general theoretical approach and provide a framework for a deeper physical understanding of hydrogen transfer dynamics in complex systems

Nonadiabatic instanton rate theory beyond the golden-rule limit

Fermi's golden rule describes the leading-order behaviour of the reaction rate as a function of the diabatic coupling. Its asymptotic $(\hbar \rightarrow 0)$ limit is the semiclassical golden-rule instanton rate theory, which rigorously approximates nuclear quantum effects, lends itself to efficient numerical computation and gives physical insight into reaction mechanisms. However the golden rule by itself becomes insufficient as the strength of the diabatic coupling increases, so higher-order terms must be additionally considered. In this work we give a first-principles derivation of the next-order term beyond the golden rule, represented as a sum of three components. Two of them lead to new instanton pathways that extend the golden-rule case and, among other factors, account for the effects of recrossing on the full rate. The remaining component derives from the equilibrium partition function and accounts for changes in potential energy around the reactant and product wells due to diabatic coupling. The new semiclassical theory demands little computational effort beyond a golden-rule instanton calculation. It makes it possible to rigorously assess the accuracy of the golden-rule approximation and sets the stage for future work on general semiclassical nonadiabatic rate theories

Generalized spin mapping for quantum-classical dynamics

We recently derived a spin-mapping approach for treating the nonadiabatic dynamics of a two-level system in a classical environment [J. Chem. Phys. 151, 044119 (2019)] based on the well-known quantum equivalence between a two-level system and a spin-1/2 particle. In the present paper, we generalize this method to describe the dynamics of $N$-level systems. This is done via a mapping to a classical phase space that preserves the $SU(N)$-symmetry of the original quantum problem. The theory reproduces the standard Meyer--Miller--Stock--Thoss Hamiltonian without invoking an extended phase space, and we thus avoid leakage from the physical subspace. In contrast with the standard derivation of this Hamiltonian, the generalized spin mapping leads to an $N$-dependent value of the zero-point energy parameter that is uniquely determined by the Casimir invariant of the $N$-level system. Based on this mapping, we derive a simple way to approximate correlation functions in complex nonadiabatic molecular systems via classical trajectories, and present benchmark calculations on the seven-state Fenna--Matthews--Olson complex. The results are significantly more accurate than conventional Ehrenfest dynamics, at a comparable computational cost, and can compete in accuracy with other state-of-the-art mapping approaches.Comment: 22 pages, 14 figure

Using Instanton Theory to Study Quantum Effects in Photosensitization

Electronic excitation is usually accomplished using light (photoexcitation) and is a key step in a vast number of important physical and biological processes. However, in instances where photoexcitation is not possible, a photosensitizer can excite the target molecule in a process called photosensitization. Unfortunately, full details of its mechanism are still unknown. This perspective gives an overview of the current understanding of photosensitization and describes how instanton theory can be used to fill the gaps, especially with regard tothe importance of quantum tunnelling effects

Instanton theory of tunnelling in molecules with asymmetric isotopic substitutions

We consider quantum tunnelling in asymmetric double-well systems for which the local minima in the two wells have the same energy, but the frequencies differ slightly. We derive a generalization of instanton theory for these asymmetric systems, leading to a semiclassical expression for the tunnelling matrix element and hence the energy level splitting. We benchmark the method using a set of one- and two-dimensional models, for which the results compare favourably with numerically exact quantum calculations. Using the ring-polymer instanton approach, we apply the method to compute the level splittings in various isotopic-substituted versions of malonaldehyde in full dimensionality and analyse the relative contributions from the zero-point energy difference and tunnelling effects.Comment: 13 pages, 4 figure