Nonadiabatic Systems Research Articles

High-fidelity first principles nonadiabaticity: diabatization, analytic representation of global diabatic potential energy matrices, and quantum dynamics.

Nonadiabatic dynamics, which goes beyond the Born-Oppenheimer approximation, has increasingly been shown to play an important role in chemical processes, particularly those involving electronically excited states. Understanding multistate dynamics requires rigorous quantum characterization of both electronic and nuclear motion. However, such first principles treatments of multi-dimensional systems have so far been rather limited due to the lack of accurate coupled potential energy surfaces and difficulties associated with quantum dynamics. In this Perspective, we review recent advances in developing high-fidelity analytical diabatic potential energy matrices for quantum dynamical investigations of polyatomic uni- and bi-molecular nonadiabatic processes, by machine learning of high-level ab initio data. Special attention is paid to methods of diabatization, high fidelity construction of multi-state coupled potential energy surfaces and property surfaces, as well as quantum mechanical characterization of nonadiabatic nuclear dynamics. To illustrate the tremendous progress made by these new developments, several examples are discussed, in which direct comparison with quantum state resolved measurements led to either confirmation of the observation or sometimes reinterpretation of the experimental data. The insights gained in these prototypical systems greatly advance our understanding of nonadiabatic dynamics in chemical systems.

A partially linearized spin-mapping approach for nonadiabatic dynamics. II. Analysis and comparison with related approaches.

In a previous paper [J. R. Mannouch and J. O. Richardson, J. Chem. Phys. 153, 194109 (2020)], we derived a new partially linearized mapping-based classical-trajectory technique called the spin partially linearized density matrix (spin-PLDM) approach. This method describes the dynamics associated with the forward and backward electronic path integrals using a Stratonovich-Weyl approach within the spin-mapping space. While this is the first example of a partially linearized spin-mapping method, fully linearized spin-mapping is already known to be capable of reproducing dynamical observables for a range of nonadiabatic model systems reasonably accurately. Here, we present a thorough comparison of the terms in the underlying expressions for the real-time quantum correlation functions for spin-PLDM and fully linearized spin mapping in order to ascertain the relative accuracy of the two methods. In particular, we show that spin-PLDM contains an additional term within the definition of its real-time correlation function, which diminishes many of the known errors that are ubiquitous for fully linearized approaches. One advantage of partially linearized methods over their fully linearized counterparts is that the results can be systematically improved by re-sampling the mapping variables at intermediate times. We derive such a scheme for spin-PLDM and show that for systems for which the approximation of classical nuclei is valid, numerically exact results can be obtained using only a few "jumps." Additionally, we implement focused initial conditions for the spin-PLDM method, which reduces the number of classical trajectories that are needed in order to reach convergence of dynamical quantities, with seemingly little difference to the accuracy of the result.

A partially linearized spin-mapping approach for nonadiabatic dynamics. I. Derivation of the theory

We present a new partially linearized mapping-based approach for approximating real-time quantum correlation functions in condensed-phase nonadiabatic systems, called the spin partially linearized density matrix (spin-PLDM) approach. Within a classical trajectory picture, partially linearized methods treat the electronic dynamics along forward and backward paths separately by explicitly evolving two sets of mapping variables. Unlike previously derived partially linearized methods based on the Meyer-Miller-Stock-Thoss mapping, spin-PLDM uses the Stratonovich-Weyl transform to describe the electronic dynamics for each path within the spin-mapping space; this automatically restricts the Cartesian mapping variables to lie on a hypersphere and means that the classical equations of motion can no longer propagate the mapping variables out of the physical subspace. The presence of a rigorously derived zero-point energy parameter also distinguishes spin-PLDM from other partially linearized approaches. These new features appear to give the method superior accuracy for computing dynamical observables of interest when compared with other methods within the same class. The superior accuracy of spin-PLDM is demonstrated in this paper through application of the method to a wide range of spin-boson models as well as to the Fenna-Matthews-Olsen complex.

Benchmarking Quasiclassical Mapping Hamiltonian Methods for Simulating Electronically Nonadiabatic Molecular Dynamics.

Quasi-classical mapping Hamiltonian methods have recently emerged as a promising approach for simulating electronically nonadiabatic molecular dynamics. The classical-like dynamics of the overall system within these methods makes them computationally feasible, and they can be derived based on well-defined semiclassical approximations. However, the existence of a variety of different quasi-classical mapping Hamiltonian methods necessitates a systematic comparison of their respective advantages and limitations. Such a benchmark comparison is presented in this paper. The approaches compared include the Ehrenfest method, the symmetrical quasi-classical (SQC) method, and five variations of the linearized semiclassical (LSC) method, three of which employ a modified identity operator. The comparison is based on a number of popular nonadiabatic model systems; the spin-boson model, a Frenkel biexciton model, and Tully's scattering models 1 and 2. The relative accuracy of the different methods is tested by comparing with quantum-mechanically exact results for the dynamics of the electronic populations and coherences. We find that LSC with the modified identity operator typically performs better than the Ehrenfest and standard LSC approaches. In comparison to SQC, these modified methods appear to be slightly more accurate for condensed phase problems, but for scattering models there is little distinction between them.

Microcanonical rates from ring-polymer molecular dynamics: Direct-shooting, stationary-phase, and maximum-entropy approaches.

We address the calculation of microcanonical reaction rates for processes involving significant nuclear quantum effects using ring-polymer molecular dynamics (RPMD), both with and without electronically non-adiabatic transitions. After illustrating the shortcomings of the naive free-particle direct-shooting method, in which the temperature of the internal ring-polymer modes is set to the translational energy scale, we investigate alternative strategies based on the expression for the microcanonical rate in terms of the inverse Laplace transform of the thermal reaction rate. It is shown that simple application of the stationary-phase approximation (SPA) dramatically improves the performance of the microcanonical rates using RPMD, particularly in the low-energy region where tunneling dominates. Using the SPA as a Bayesian prior, numerically exact RPMD microcanonical rates are then obtained using maximum entropy inversion of the thermal reaction rates for both electronically adiabatic and non-adiabatic model systems. Finally, the direct-shooting method is revisited using the SPA-determined temperature for the internal ring-polymer modes, leading to a simple, direct-simulation method with improved accuracy in the tunneling regime. This work suggests a general strategy for the extraction of microcanonical dynamical quantities from RPMD (or other approximate thermal) simulations.

Multistate ring polymer instantons and nonadiabatic reaction rates.

We present two multistate ring polymer instanton (RPI) formulations, both obtained from an exact path integral representation of the quantum canonical partition function for multistate systems. The two RPIs differ in their treatment of the electronic degrees of freedom; while the Mean-Field (MF)-RPI averages over the electronic state contributions, the Mapping Variable (MV)-RPI employs explicit continuous Cartesian variables to represent the electronic states. We compute both RPIs for a series of model two-state systems coupled to a single nuclear mode with electronic coupling values chosen to describe dynamics in both adiabatic and nonadiabatic regimes. We show that the MF-RPIs for symmetric systems are in good agreement with the previous literature, and we show that our numerical techniques are robust for systems with non-zero driving force. The nuclear MF-RPI and the nuclear MV-RPI are similar, but the MV-RPI uniquely reports on the changes in the electronic state populations along the instanton path. In both cases, we analytically demonstrate the existence of a zero-mode, and we numerically find that these solutions are true instantons with a single unstable mode as expected for a first order saddle point. Finally, we use the MF-RPI to accurately calculate rate constants for adiabatic and nonadiabatic model systems with the coupling strength varying over three orders of magnitude.

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. E. Runeson and J. O. Richardson, 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 to 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 light-harvesting 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.

Analysis of transient-absorption pump-probe signals of nonadiabatic dissipative systems: "Ideal" and "real" spectra.

We introduce and analyze the concept of the "ideal" time and frequency resolved transient-absorption pump-probe (PP) signal. The ideal signal provides the most direct link between the "real" (measurable) PP signal and the material system dynamics. The simulation of PP signals involves two steps. (i) The ideal signal, which exhibits perfect time and frequency resolution, is calculated. For this purpose, the probe pulse is replaced by an auxiliary continuous-wave pulse. (ii) The real signal is obtained by the convolution of the ideal signal with the appropriate time- and frequency-gate function, which depends on the envelope of the actual probe pulse. This concept has been used to simulate integral and dispersed PP signals for a model system exhibiting nonadiabatic and dissipative dynamics. The ideal signal is computed with the two-pulse equation-of-motion phase-matching approach which has been extended to take excited-state absorption into account. We demonstrate how the ideal signal, an object exhibiting the features of moving wave packets as well as stationary spectra, is related to real signals detected with short (good temporal resolution) or long (good frequency resolution) probe pulses.

Three-dimensional nonadiabatic electron- optical gyrotron systems with inhomogeneous current density

The results of a numerical study of the effect of threedimensional perturbations of the space charge density distribution in a nonadiabatic electron-optic systems of gyrotrons are presented. Data of direct numerical simulation using the 3-D program CST Studio Suite are described.

Coupled-Trajectory Mixed Quantum-Classical Algorithm: A Deconstruction

We analyze a mixed quantum-classical algorithm recently derived from the exact factorization equations [Min, Agostini, Gross, PRL 115, 073001 (2015)] to show the role of the different terms in the algorithm in bringing about decoherence and wavepacket branching. The algorithm has the structure of Ehrenfest equations plus a "coupled-trajectory" term for both the electronic and nuclear equations, and we analyze the relative roles played by the different nonadiabatic terms in these equations, including how they are computed in practice. In particular, we show that while the coupled-trajectory term in the electronic equation is essential in yielding accurate dynamics, that in the nuclear equation has a much smaller effect. A decoherence time is extracted from the electronic equations and compared with that of augmented fewest-switches surface-hopping. We revisit a series of nonadiabatic Tully model systems to illustrate our analysis.

Symmetric quasi-classical dynamics with quasi-diabatic propagation scheme.

We apply a recently developed quasi-diabatic (QD) scheme to the symmetric quasi-classical (SQC) approach for accurate quantum dynamics propagation. By using the adiabatic states as the QD states during a short-time quantum dynamics propagation, the QD scheme allows for directly interfacing the diabatic SQC method with commonly used adiabatic electronic structure calculations, thus alleviating any non-trivial theoretical efforts to reformulate SQC in the adiabatic representation. Furthermore, the QD scheme ensures a stable propagation of the dynamics and allows using a much larger time step compared to directly propagating SQC dynamics in the adiabatic representation. This is due to the fact that the QD scheme does not explicitly require non-adiabatic couplings that could exhibit highly peaked values during non-adiabatic dynamics propagation. We perform the QD-SQC calculations with a wide range of model non-adiabatic systems to demonstrate the accuracy of the proposed scheme. This study opens up the possibility for combining accurate diabatic quantum dynamics methods such as SQC with any adiabatic electronic structure calculations for non-adiabatic on-the-fly propagations.

A path integral methodology for obtaining thermodynamic properties of nonadiabatic systems using Gaussian mixture distributions.

We introduce a new path integral Monte Carlo method for investigating nonadiabatic systems in thermal equilibrium and demonstrate an approach to reducing stochastic error. We derive a general path integral expression for the partition function in a product basis of continuous nuclear and discrete electronic degrees of freedom without the use of any mapping schemes. We separate our Hamiltonian into a harmonic portion and a coupling portion; the partition function can then be calculated as the product of a Monte Carlo estimator (of the coupling contribution to the partition function) and a normalization factor (that is evaluated analytically). A Gaussian mixture model is used to evaluate the Monte Carlo estimator in a computationally efficient manner. Using two model systems, we demonstrate our approach to reduce the stochastic error associated with the Monte Carlo estimator. We show that the selection of the harmonic oscillators comprising the sampling distribution directly affects the efficiency of the method. Our results demonstrate that our path integral Monte Carlo method's deviation from exact Trotter calculations is dominated by the choice of the sampling distribution. By improving the sampling distribution, we can drastically reduce the stochastic error leading to lower computational cost.

Coupled forward-backward trajectory approach for nonequilibrium electron-ion dynamics

We introduce a simple ansatz for the wavefunction of a many-body system based on coupled forward and backward-propagating semiclassical trajectories. This method is primarily aimed at, but not limited to, treating nonequilibrium dynamics in electron-phonon systems. The time-evolution of the system is obtained from the Euler-Lagrange variational principle, and we show that this ansatz yields Ehrenfest mean field theory in the limit that the forward and backward trajectories are orthogonal, and in the limit that they coalesce. We investigate accuracy and performance of this method by simulating electronic relaxation in the spin-boson model and the Holstein model. Although this method involves only pairs of semiclassical trajectories, it shows a substantial improvement over mean field theory, capturing quantum coherence of nuclear dynamics as well as electron-nuclear correlations. This improvement is particularly evident in nonadiabatic systems, where the accuracy of this coupled trajectory method extends well beyond the perturbative electron-phonon coupling regime. This approach thus provides an attractive route forward to the ab-initio description of relaxation processes, such as thermalization, in condensed phase systems.

Quasi-Diabatic Representation for Nonadiabatic Dynamics Propagation.

We develop a nonadiabatic dynamics propagation scheme that allows interfacing diabatic quantum dynamics methods with commonly used adiabatic electronic structure calculations. This scheme uses adiabatic states as the quasi-diabatic (QD) states during a short-time quantum dynamics propagation. At every dynamical propagation step, these QD states are updated based on a new set of adiabatic basis. Using the partial linearized density matrix (PLDM) path-integral method as one specific example for diabatic dynamics approaches, we demonstrate the accuracy of the QD scheme with a wide range of model nonadiabatic systems as well as the on-the-fly propagations with density functional tight-binding (DFTB) calculations. This study opens the possibility to combine accurate diabatic quantum dynamics methods with adiabatic electronic structure calculations for nonadiabatic dynamics propagations.

Nonadiabatic semiclassical dynamics in the mixed quantum-classical initial value representation.

We extend the Mixed Quantum-Classical Initial Value Representation (MQC-IVR), a semiclassical method for computing real-time correlation functions, to electronically nonadiabatic systems using the Meyer-Miller-Stock-Thoss (MMST) Hamiltonian in order to treat electronic and nuclear degrees of freedom (dofs) within a consistent dynamic framework. We introduce an efficient symplectic integration scheme, the MInt algorithm, for numerical time evolution of the phase space variables and monodromy matrix under the non-separable MMST Hamiltonian. We then calculate the probability of transmission through a curve crossing in model two-level systems and show that MQC-IVR reproduces quantum-limit semiclassical results in good agreement with exact quantum methods in one limit, and in the other limit yields results that are in keeping with classical limit semiclassical methods like linearized IVR. Finally, exploiting the ability of the MQC-IVR to quantize different dofs to different extents, we present a detailed study of the extents to which quantizing the nuclear and electronic dofs improves numerical convergence properties without significant loss of accuracy.

Path integral molecular dynamics with surface hopping for thermal equilibrium sampling of nonadiabatic systems.

In this work, a novel ring polymer representation for a multi-level quantum system is proposed for thermal average calculations. The proposed representation keeps the discreteness of the electronic states: besides position and momentum, each bead in the ring polymer is also characterized by a surface index indicating the electronic energy surface. A path integral molecular dynamics with surface hopping (PIMD-SH) dynamics is also developed to sample the equilibrium distribution of the ring polymer configurational space. The PIMD-SH sampling method is validated theoretically and by numerical examples.

The Symmetrical Quasi-Classical Model for Electronically Non-Adiabatic Processes Applied to Energy Transfer Dynamics in Site-Exciton Models of Light-Harvesting Complexes.

In a recent series of papers, it has been illustrated that a symmetrical quasi-classical (SQC) windowing model applied to the Meyer-Miller (MM) classical vibronic Hamiltonian provides an excellent description of a variety of electronically non-adiabatic benchmark model systems for which exact quantum results are available for comparison. In this paper, the SQC/MM approach is used to treat energy transfer dynamics in site-exciton models of light-harvesting complexes, and in particular, the well-known 7-state Fenna-Mathews-Olson (FMO) complex. Again, numerically "exact" results are available for comparison, here via the hierarchical equation of motion (HEOM) approach of Ishizaki and Fleming, and it is seen that the simple SQC/MM approach provides very reasonable agreement with the previous HEOM results. It is noted, however, that unlike most (if not all) simple approaches for treating these systems, because the SQC/MM approach presents a fully atomistic simulation based on classical trajectory simulation, it places no restrictions on the characteristics of the thermal baths coupled to each two-level site, e.g., bath spectral densities (SD) of any analytic functional form may be employed as well as discrete SD determined experimentally or from MD simulation (nor is there any restriction that the baths be harmonic), opening up the possibility of simulating more realistic variations on the basic site-exciton framework for describing the non-adiabatic dynamics of photosynthetic pigment complexes.

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.

How large are nonadiabatic effects in atomic and diatomic systems?

With recent developments in simulating nonadiabatic systems to high accuracy, it has become possible to determine how much energy is attributed to nuclear quantum effects beyond zero-point energy. In this work, we calculate the non-relativistic ground-state energies of atomic and molecular systems without the Born-Oppenheimer approximation. For this purpose, we utilize the fixed-node diffusion Monte Carlo method, in which the nodes depend on both the electronic and ionic positions. We report ground-state energies for all systems studied, ionization energies for the first-row atoms and atomization energies for the first-row hydrides. We find the ionization energies of the atoms to be nearly independent of the Born-Oppenheimer approximation, within the accuracy of our results. The atomization energies of molecular systems, however, show small effects of the nonadiabatic coupling between electrons and nuclei.

Semiclassical quantization of nonadiabatic systems with hopping periodic orbits.

We present a semiclassical quantization condition, i.e., quantum-classical correspondence, for steady states of nonadiabatic systems consisting of fast and slow degrees of freedom (DOFs) by extending Gutzwiller's trace formula to a nonadiabatic form. The quantum-classical correspondence indicates that a set of primitive hopping periodic orbits, which are invariant under time evolution in the phase space of the slow DOF, should be quantized. The semiclassical quantization is then applied to a simple nonadiabatic model and accurately reproduces exact quantum energy levels. In addition to the semiclassical quantization condition, we also discuss chaotic dynamics involved in the classical limit of nonadiabatic dynamics.