Mapping variable ring polymer molecular dynamics: A path-integral based method for nonadiabatic processes

Quasi-Diabatic Scheme for Nonadiabatic On-the-Fly Simulations

We use the quasi-diabatic (QD) propagation scheme to perform on-the-fly nonadiabatic simulations of the photodynamics of ethylene. The QD scheme enables a seamless interface between accurate diabatic-based quantum dynamics approaches and adiabatic electronic structure calculations, explicitly avoiding any efforts to construct global diabatic states or reformulate the diabatic dynamics approach to the adiabatic representation. Using the partial linearized path-integral approach and the symmetrical quasi-classical approach as the diabatic dynamics methods, the QD propagation scheme enables direct nonadiabatic simulation with complete active space self-consistent field on-the-fly electronic structure calculations. The population dynamics obtained from both approaches are in a close agreement with the quantum wavepacket-based method and outperform the widely used trajectory surface-hopping approach. Further analysis of the ethylene photodeactivation pathways demonstrates the correct predictions of competing processes of nonradiative relaxation mechanism through various conical intersections. This work provides the foundation of using accurate diabatic dynamics approaches and on-the-fly adiabatic electronic structure information to perform ab initio nonadiabatic simulation.

Probing photodissociation dynamics using ring polymer molecular dynamics

The performance of the ring polymer molecular dynamics (RPMD) approach to simulate typical photodissociation processes is assessed. The correct description of photodissociation requires the calculation of correlation functions or expectation values associated with non-equilibrium initial conditions, which was shown to be possible with RPMD very recently [J. Chem. Phys. 145, 204118 (2016)]. This approach is combined with treatment of the nonadiabatic dynamics employing the ring polymer surface hopping approach (RPSH), which is based on Tully's fewest switches surface hopping (FSSH) approach. The performance is tested using one-dimensional photodissociation models. It is found that RPSH with non-equilibrium initial conditions can well reproduce the time-dependent dissociation probability, and adiabatic and diabatic populations for cases where the crossing point is below and above the Franck-Condon point, respectively, while standard FSSH fails to reproduce the exact quantum dynamics in the latter case. Thus, it is shown that RPSH is an efficient and accurate alternative to standard FSSH, which is one of the most widely employed approaches to study photochemistry.

Stochastic Transition State Theory

Kramers's original paper on the diffusion model of chemical reactions was based on the consideration that only the barrier region determines the outcome of transmission over a barrier. Subsequently it became understood that Kramers's approach was identical to variational transition state theory (VTST) and as such used only thermodynamic information. Here, using Kramers's philosophy in conjunction with perturbation theory and the realization that the dynamics which is rate-determining usually occurs in the vicinity of the transition state leads to a novel stochastic rate theory in which the momentum change induced by the medium is the stochastic variable. A first successful application of the theory is to the old and challenging problem of motion over a cusped barrier. This has implications for the study of transition path time distributions as well as the theory of tunneling via nonadiabatic coupling.

On the exact continuous mapping of fermions

We derive a rigorous, quantum mechanical map of fermionic creation and annihilation operators to continuous Cartesian variables that exactly reproduces the matrix structure of the many-fermion problem. We show how our scheme can be used to map a general many-fermion Hamiltonian and then consider two specific models that encode the fundamental physics of many fermionic systems, the Anderson impurity and Hubbard models. We use these models to demonstrate how efficient mappings of these Hamiltonians can be constructed using a judicious choice of index ordering of the fermions. This development provides an alternative exact route to calculate the static and dynamical properties of fermionic systems and sets the stage to exploit the quantum-classical and semiclassical hierarchies to systematically derive methods offering a range of accuracies, thus enabling the study of problems where the fermionic degrees of freedom are coupled to complex anharmonic nuclear motion and spins which lie beyond the reach of most currently available methods.

Inclusion of nuclear quantum effects for simulations of nonlinear spectroscopy

The computation and interpretation of nonlinear vibrational spectroscopy is of vital importance for understanding a wide range of dynamical processes in molecular systems. Here, we introduce an approach to evaluate multi-time response functions in terms of multi-time double symmetrized Kubo transformed thermal correlation functions. Furthermore, we introduce a multi-time extension of ring polymer molecular dynamics to evaluate these Kubo transforms. Benchmark calculations show that the approximations are useful for short times even for nonlinear operators, providing a consistent improvement over classical simulations of multi-time correlation functions. The introduced methodology thus provides a practical way of including nuclear quantum effects in multi-time response functions of non-linear optical spectroscopy.

Harmonic-phase path-integral approximation of thermal quantum correlation functions

We present an approximation to the thermal symmetric form of the quantum time-correlation function in the standard position path-integral representation. By transforming to a sum-and-difference position representation and then Taylor-expanding the potential energy surface of the system to second order, the resulting expression provides a harmonic weighting function that approximately recovers the contribution of the phase to the time-correlation function. This method is readily implemented in a Monte Carlo sampling scheme and provides exact results for harmonic potentials (for both linear and non-linear operators) and near-quantitative results for anharmonic systems for low temperatures and times that are likely to be relevant to condensed phase experiments. This article focuses on one-dimensional examples to provide insights into convergence and sampling properties, and we also discuss how this approximation method may be extended to many-dimensional systems.

Effects of tunnelling and asymmetry for system-bath models of electron transfer

We apply the newly derived nonadiabatic golden-rule instanton theory to asymmetric models describing electron-transfer in solution. The models go beyond the usual spin-boson description and have anharmonic free-energy surfaces with different values for the reactant and product reorganization energies. The instanton method gives an excellent description of the behaviour of the rate constant with respect to asymmetry for the whole range studied. We derive a general formula for an asymmetric version of the Marcus theory based on the classical limit of the instanton and find that this gives significant corrections to the standard Marcus theory. A scheme is given to compute this rate based only on equilibrium simulations. We also compare the rate constants obtained by the instanton method with its classical limit to study the effect of tunnelling and other quantum nuclear effects. These quantum effects can increase the rate constant by orders of magnitude.

Photoexcited Ag ejection from a low-temperature He cluster: a simulation study by nonadiabatic Ehrenfest ring-polymer molecular dynamics

Nonadiabatic ring-polymer molecular dynamics simulations were performed to understand the photoexcitation dynamics of a low-temperature Ag·He₅₀₀ cluster.

Dynamically consistent method for mixed quantum-classical simulations: A semiclassical approach

We introduce a new semiclassical (SC) framework, the Mixed Quantum-Classical Initial Value Representation (MQC-IVR), that can be tuned to reproduce existing quantum-limit and classical-limit SC approximations to quantum real-time correlation functions. Applying a modified Filinov transformation to a quantum-limit SC formulation leads to the association of a Filinov parameter with each degree of freedom in the system; varying this parameter from zero to infinity controls the extent of quantization of the corresponding mode. The resulting MQC-IVR expression provides a consistent dynamic framework for mixed quantum-classical simulations and we demonstrate its numerical accuracy in the calculation of real-time correlation functions for a model 1D system and a model 2D system over the full range of quantum- to classical-limit behaviors.

Kinetically-constrained ring-polymer molecular dynamics for non-adiabatic chemistries involving solvent and donor–acceptor dynamical effects

We investigate the performance of the recently developed kinetically-constrained ring polymer molecular dynamics (KC-RPMD) method for the description of model condensed-phase electron transfer (ET) reactions in which solvent and donor–acceptor dynamics play an important role. Comparison of KC-RPMD with results from Golden-Rule rate theories and numerically exact quantum dynamics calculations demonstrates that KC-RPMD accurately captures the combination of electronic- and nuclear-dynamical effects throughout the Marcus (intermediate solvent friction) and Zusman (large solvent friction) regimes of ET. It is also demonstrated that KC-RPMD accurately describes systems in which the magnitude of the diabatic coupling depends on the position of a dynamical donor–acceptor mode. In addition to these successes, however, we present an unsurprising failure of KC-RPMD to capture the enhancement of the ET rate in the low solvent friction regime associated with nuclear coherence effects. In this analysis, we re-visit several aspects of the original KC-RPMD formulation, including the form of the kinetic constraint and the choice of the mass of the auxiliary electronic variable. In particular, we introduce a Langevin bath for the auxiliary electronic variable to correct for its unphysically low coupling with the nuclear degrees of freedom. This work demonstrates that the KC-RPMD method is well suited for the direct simulation of non-adiabatic donor–acceptor chemistries associated with many complex systems, including those for which solvent dynamics plays an important role in the reaction mechanism.

Deriving the exact nonadiabatic quantum propagator in the mapping variable representation

We derive an exact quantum propagator for nonadiabatic dynamics in multi-state systems using the mapping variable representation, where classical-like Cartesian variables are used to represent both continuous nuclear degrees of freedom and discrete electronic states. The resulting Liouvillian is a Moyal series that, when suitably approximated, can allow for the use of classical dynamics to efficiently model large systems. We demonstrate that different truncations of the exact Liouvillian lead to existing approximate semiclassical and mixed quantum–classical methods and we derive an associated error term for each method. Furthermore, by combining the imaginary-time path-integral representation of the Boltzmann operator with the exact Liouvillian, we obtain an analytic expression for thermal quantum real-time correlation functions. These results provide a rigorous theoretical foundation for the development of accurate and efficient classical-like dynamics to compute observables such as electron transfer reaction rates in complex quantized systems.

Direct simulation of electron transfer in the cobalt hexammine(ii/iii) self-exchange reaction

We present an atomistic simulation of electron transfer in a transition metal complex system using path integral methods.

Mean field ring polymer molecular dynamics for electronically nonadiabatic reaction rates

We present a mean field ring polymer molecular dynamics method to calculate the rate of electron transfer (ET) in multi-state, multi-electron condensed-phase processes. Our approach involves calculating a transition state theory (TST) estimate to the rate using an exact path integral in discrete electronic states and continuous Cartesian nuclear coordinates. A dynamic recrossing correction to the TST rate is then obtained from real-time dynamics simulations using mean field ring polymer molecular dynamics. We employ two different reaction coordinates in our simulations and show that, despite the use of mean field dynamics, the use of an accurate dividing surface to compute TST rates allows us to achieve remarkable agreement with Fermi's golden rule rates for nonadiabatic ET in the normal regime of Marcus theory. Further, we show that using a reaction coordinate based on electronic state populations allows us to capture the turnover in rates for ET in the Marcus inverted regime.

Nonadiabatic Dynamics in Atomistic Environments: Harnessing Quantum-Classical Theory with Generalized Quantum Master Equations

The development of methods that can efficiently and accurately treat nonadiabatic dynamics in quantum systems coupled to arbitrary atomistic environments remains a significant challenge in problems ranging from exciton transport in photovoltaic materials to electron and proton transfer in catalysis. Here we show that our recently introduced MF-GQME approach, which combines Ehrenfest mean field theory with the generalized quantum master equation framework, is able to yield quantitative accuracy over a wide range of charge-transfer regimes in fully atomistic environments. This is accompanied by computational speed-ups of up to 3 orders of magnitude over a direct application of Ehrenfest theory. This development offers the opportunity to efficiently investigate the atomistic details of nonadiabatic quantum relaxation processes in regimes where obtaining accurate results has previously been elusive.

Simulating Excited State Dynamics in Systems with Multiple Avoided Crossings Using Mapping Variable Ring Polymer Molecular Dynamics

Mapping variable ring polymer molecular dynamics (MV-RPMD) is an approximate quantum dynamics method based on imaginary-time path integrals for simulating electronically nonadiabatic photochemical processes. By employing a mapping protocol to transform from a discrete electronic state basis to continuous Cartesian phase-space variables, the method captures electronic state transitions coupled to nuclear motion using only classical MD trajectories. In this work, we extend the applicability of MV-RPMD to simulations of photoinduced excited electronic state dynamics in nonadiabatic systems with multiple avoided crossings. We achieve this by deriving a new electronic state population estimator in the phase space of electronic variables that is exact at equilibrium and numerically accurate in real time. Further, we introduce an efficient constraint protocol to initialize an MV-RPMD simulation to a particular electronic state. We numerically demonstrate the accuracy of this estimator and constraint technique in describing electronic state dynamics from an initial nonequilibrium state in six model systems, three of which describe photodissociation.