Quantum Time Correlation Functions Research Articles

A model of pre-reaction complexes

One of the main objectives of computational chemistry studies is the interpretation of chemical mechanisms whose data were obtained from quantum mechanical software packages. In these mechanisms, starting from the reactant, the reactant and the product states are connected through intermediates and transition states, in an alternating order. Generally, the reactivity of a given pathway is determined by the overall barrier height. Sometimes, a comparison becomes necessary for a given step with another possible step that can be realized within the same mechanism, and in this case as well the step with the lowest-lying transition state is preferred. However, there are some exceptions in the order of these states. There are states known as pre-reaction complexes that reside between a reactant, or an intermediate, and a transition state. Excluding some first principle quantum dynamics studies, the effect of a pre-reaction complex is generally not discussed in most of the computational chemistry studies, and the overall barrier heights become the main considerations, as usual.In this work, a mathematical model that allows obtaining the rate constant of a pathway without a pre-reaction complex, and that of a pathway with a pre-reaction complex was described. The model was built using time-dependent quantum perturbation theory and quantum time-correlation functions. This work can improve our understanding of the nature of chemical reactions that include pre-reaction complexes by enabling us to differentiate the seemingly identical reaction pathways.

Comparison of Matsubara dynamics with exact quantum dynamics for an oscillator coupled to a dissipative bath.

We report the first numerical calculations in which converged Matsubara dynamics is compared directly with exact quantum dynamics with no artificial damping of the time-correlation functions (TCFs). The system treated is a Morse oscillator coupled to a harmonic bath. We show that, when the system-bath coupling is sufficiently strong, the Matsubara calculations can be converged by explicitly including up to M = 200 Matsubara modes, with the remaining modes included as a harmonic "tail" correction. The resulting Matsubara TCFs are in near-perfect agreement with the exact quantum TCFs, for non-linear as well as linear operators, at a temperature at which the TCFs are dominated by quantum thermal fluctuations. These results provide compelling evidence that incoherent classical dynamics can arise in the condensed phase at temperatures at which the statistics are dominated by quantum (Boltzmann) effects, as a result of smoothing of imaginary-time Feynman paths. The techniques developed here may also lead to efficient methods for benchmarking system-bath dynamics in the overdamped regime.

Imaginary-time open-chain path-integral approach for two-state time correlation functions and applications in charge transfer.

Quantum time correlation functions (TCFs) involving two states are important for describing nonadiabatic dynamical processes such as charge transfer (CT). Based on a previous single-state method, we propose an imaginary-time open-chain path-integral (OCPI) approach for evaluating the two-state symmetrized TCFs. Expressing the forward and backward propagation on different electronic potential energy surfaces as a complex-time path integral, we then transform the path variables to average and difference variables such that the integration over the difference variables up to the second order can be performed analytically. The resulting expression for the symmetrized TCF is equivalent to sampling the open-chain configurations in an effective potential that corresponds to the average surface. Using importance sampling over the extended OCPI space via open path-integral molecular dynamics, we tested the resulting path-integral approximation by calculating the Fermi's golden rule CT rate constant within a widely used spin-boson model. Comparing with the real-time linearized semiclassical method and analytical result, we show that the imaginary-time OCPI provides an accurate two-state symmetrized TCF and rate constant in the typical turnover region. It is shown that the first bead of the open chain corresponds to physical zero-time and that the endpoint bead corresponds to final time t; oscillations of the end-to-end distance perfectly match the nuclear mode frequency. The two-state OCPI scheme is seen to capture the tested model's electronic quantum coherence and nuclear quantum effects accurately.

Application of the imaginary time hierarchical equations of motion method to calculate real time correlation functions.

We investigate the application of the imaginary time hierarchical equations of motion method to calculate real time quantum correlation functions. By starting from the path integral expression for the correlated system-bath equilibrium state, we first derive a new set of equations that decouple the imaginary time propagation and the calculation of auxiliary density operators. The new equations, thus, greatly simplify the calculation of the equilibrium correlated initial state that is subsequently used in the real time propagation to obtain the quantum correlation functions. It is also shown that a periodic decomposition of the bath imaginary time correlation function is no longer necessary in the new equations such that different decomposition schemes can be explored. The applicability of the new method is demonstrated in several numerical examples, including the spin-Boson model, the Holstein model, and the double-well model for proton transfer reaction.

Time-dependent self-consistent harmonic approximation: Anharmonic nuclear quantum dynamics and time correlation functions

Most material properties of great physical interest are directly related to nuclear dynamics, e.g. the ionic thermal conductivity, Raman/IR vibrational spectra, inelastic X-ray, and Neutron scattering. A theory able to compute from first principles these properties, accounting for the anharmonicity and quantum fluctuations in the nuclear energy landscape that can be implemented in systems with hundreds of atoms is missing. Here, we derive an approximate theory for the quantum time evolution of lattice vibrations at finite temperature. This theory introduces the time dynamics in the Self-Consistent Harmonic Approximation (SCHA) and shares with the static case the same computational cost. It is nonempirical, as pure states evolve according to the Dirac least action principle and the dynamics of the thermal ensemble conserves both energy and entropy. The static SCHA is recovered as a stationary solution of the dynamical equations. We apply perturbation theory around the static SCHA solution and derive an algorithm to compute efficiently quantum dynamical response functions. Thanks to this new algorithm, we have access to the response function of any general external time-dependent perturbation, enabling the simulation of phonon spectra without following any perturbative expansion of the nuclear potential or empirical methods. We benchmark the algorithm on the IR and Raman spectroscopy of high-pressure hydrogen phase III, with a simulation cell of 96 atoms. Our work also explores the nonlinear regime of the dynamical nuclear motion, providing a paradigm to simulate the interaction with intense or multiple probes, as in pump-probe spectroscopy, or chemical reactions involving light atoms, as the proton transfer in biomolecules

Ehrenfest and classical path dynamics with decoherence and detailed balance.

We present a semiclassical approach for nonadiabatic molecular dynamics based on the Ehrenfest method with corrections for decoherence and detailed balance. Decoherence is described via a coherence penalty functional that drives dynamics away from regions in Hilbert space characterized by large values of coherences. Detailed balance is incorporated by modification of the off-diagonal matrix elements with a quantum correction factor used in semiclassical approximations to quantum time-correlation functions. Both decoherence and detailed balance corrections introduce nonlinear terms to the Schrödinger equation. At the same time, the simplicity of fully deterministic dynamics and a single trajectory for each initial condition is preserved. In contrast, surface hopping is stochastic and requires averaging over multiple realization of the stochastic process for each initial condition. The Ehrenfest-decoherence-detailed-balance (Ehrenfest-DDB) method is adapted to the classical path approximation and ab initio time-dependent density functional theory and applied to an experimentally studied nanoscale system consisting of a fluorophore molecule and an scanning tunneling microscopy tip and undergoing current-induced charge injection, cooling, and recombination. Ehrenfest-DDB produces time scales that are similar to those obtained with decoherence induced surface hopping, which is a popular nonadiabatic molecular dynamics technique applied to condensed matter. At long times, Ehrenfest-DDB dynamics slows down considerably because the detailed balance correction makes off-diagonal elements go to zero on approach to Boltzmann equilibrium. The Ehrenfest-DDB technique provides efficient means to study quantum dynamics in large systems.

Evaluation of the quantum time-correlation functions employing the Hamilton–Jacobi dynamics framework

The quantum Hamilton–Jacobi equation (QHJE) is formally equivalent to the time-dependent Schrodinger equation, and the solutions to the QHJE can be easily interpreted in terms of trajectories providing a link between classical and quantum mechanics. The trajectory-based approaches to quantum molecular dynamics are, generally, appealing because they circumvent exponential scaling associated with exact quantum methods with the system size, and because, unlike classical molecular dynamics, such methods incorporate dominant quantum effects due to delocalization of wavefunctions describing the nuclei. We explore the utility of the QHJE framework for calculations of the time-correlation functions (TCFs) involving quantum evolution defined by the Boltzmann density operator and by the Hamiltonian time-evolution operator. The implementation is based on solutions to the imaginary-time counterpart to the QHJE, which yield approximations to the ground state wavefunction. The resulting nodeless wavefunction is used to generate a basis in coordinate space, which is efficient for evaluation of the low-lying excited states and of the quantum TCFs, including the Kubo-transformed TCFs, at low temperature. The QHJE/basis approach is illustrated on several model systems in and out of thermal equilibrium, i.e., the $${\rm H}_2$$ dimer and bound anharmonic potentials. If a system exhibits large amplitude motion, e.g., in case of the nonequilibrium dynamics, then the real-time trajectory propagation provides an alternative to the basis representation, as demonstrated on a model describing the inversion mode of the ammonia molecule and ion.

An open-chain imaginary-time path-integral sampling approach to the calculation of approximate symmetrized quantum time correlation functions.

We introduce a scheme for approximating quantum time correlation functions numerically within the Feynman path integral formulation. Starting with the symmetrized version of the correlation function expressed as a discretized path integral, we introduce a change of integration variables often used in the derivation of trajectory-based semiclassical methods. In particular, we transform to sum and difference variables between forward and backward complex-time propagation paths. Once the transformation is performed, the potential energy is expanded in powers of the difference variables, which allows us to perform the integrals over these variables analytically. The manner in which this procedure is carried out results in an open-chain path integral (in the remaining sum variables) with a modified potential that is evaluated using imaginary-time path-integral sampling rather than requiring the generation of a large ensemble of trajectories. Consequently, any number of path integral sampling schemes can be employed to compute the remaining path integral, including Monte Carlo, path-integral molecular dynamics, or enhanced path-integral molecular dynamics. We believe that this approach constitutes a different perspective in semiclassical-type approximations to quantum time correlation functions. Importantly, we argue that our approximation can be systematically improved within a cumulant expansion formalism. We test this approximation on a set of one-dimensional problems that are commonly used to benchmark approximate quantum dynamical schemes. We show that the method is at least as accurate as the popular ring-polymer molecular dynamics technique and linearized semiclassical initial value representation for correlation functions of linear operators in most of these examples and improves the accuracy of correlation functions of nonlinear operators.

Quasi-classical approaches to vibronic spectra revisited.

The framework to approach quasi-classical dynamics in the electronic ground state is well established and is based on the Kubo-transformed time correlation function (TCF), being the most classical-like quantum TCF. Here we discuss whether the choice of the Kubo-transformed TCF as a starting point for simulating vibronic spectra is as unambiguous as it is for vibrational ones. Employing imaginary-time path integral techniques in combination with the interaction representation allowed us to formulate a method for simulating vibronic spectra in the adiabatic regime that takes nuclear quantum effects and dynamics on multiple potential energy surfaces into account. Further, a generalized quantum TCF is proposed that contains many well-established TCFs, including the Kubo one, as particular cases. Importantly, it also provides a framework to construct new quantum TCFs. Applying the developed methodology to the generalized TCF leads to a plethora of simulation protocols, which are based on the well-known TCFs as well as on new ones. Their performance is investigated on 1D anharmonic model systems at finite temperatures. It is shown that the protocols based on the new TCFs may lead to superior results with respect to those based on the common ones. The strategies to find the optimal approach are discussed.

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.

Thermal quantum time-correlation functions from classical-like dynamics

ABSTRACTThermal quantum time-correlation functions are of fundamental importance in quantum dynamics, allowing experimentally measurable properties such as reaction rates, diffusion constants and vibrational spectra to be computed from first principles. Since the exact quantum solution scales exponentially with system size, there has been considerable effort in formulating reliable linear-scaling methods involving exact quantum statistics and approximate quantum dynamics modelled with classical-like trajectories. Here, we review recent progress in the field with the development of methods including centroid molecular dynamics , ring polymer molecular dynamics (RPMD) and thermostatted RPMD (TRPMD). We show how these methods have recently been obtained from ‘Matsubara dynamics’, a form of semiclassical dynamics which conserves the quantum Boltzmann distribution. We also apply the Matsubara formalism to reaction rate theory, rederiving t → 0+ quantum transition-state theory (QTST) and showing that Matsubara-TST, like RPMD-TST, is equivalent to QTST. We end by surveying areas for future progress.

Non-equilibrium dynamics from RPMD and CMD.

We investigate the calculation of approximate non-equilibrium quantum time correlation functions (TCFs) using two popular path-integral-based molecular dynamics methods, ring-polymer molecular dynamics (RPMD) and centroid molecular dynamics (CMD). It is shown that for the cases of a sudden vertical excitation and an initial momentum impulse, both RPMD and CMD yield non-equilibrium TCFs for linear operators that are exact for high temperatures, in the t = 0 limit, and for harmonic potentials; the subset of these conditions that are preserved for non-equilibrium TCFs of non-linear operators is also discussed. Furthermore, it is shown that for these non-equilibrium initial conditions, both methods retain the connection to Matsubara dynamics that has previously been established for equilibrium initial conditions. Comparison of non-equilibrium TCFs from RPMD and CMD to Matsubara dynamics at short times reveals the orders in time to which the methods agree. Specifically, for the position-autocorrelation function associated with sudden vertical excitation, RPMD and CMD agree with Matsubara dynamics up to O(t4) and O(t1), respectively; for the position-autocorrelation function associated with an initial momentum impulse, RPMD and CMD agree with Matsubara dynamics up to O(t5) and O(t2), respectively. Numerical tests using model potentials for a wide range of non-equilibrium initial conditions show that RPMD and CMD yield non-equilibrium TCFs with an accuracy that is comparable to that for equilibrium TCFs. RPMD is also used to investigate excited-state proton transfer in a system-bath model, and it is compared to numerically exact calculations performed using a recently developed version of the Liouville space hierarchical equation of motion approach; again, similar accuracy is observed for non-equilibrium and equilibrium initial conditions.

Asymptotic neutron scattering laws for anomalously diffusing quantum particles.

The paper deals with a model-free approach to the analysis of quasielastic neutron scattering intensities from anomalously diffusing quantum particles. All quantities are inferred from the asymptotic form of their time-dependent mean square displacements which grow ∝t(α), with 0 ≤ α < 2. Confined diffusion (α = 0) is here explicitly included. We discuss in particular the intermediate scattering function for long times and the Fourier spectrum of the velocity autocorrelation function for small frequencies. Quantum effects enter in both cases through the general symmetry properties of quantum time correlation functions. It is shown that the fractional diffusion constant can be expressed by a Green-Kubo type relation involving the real part of the velocity autocorrelation function. The theory is exact in the diffusive regime and at moderate momentum transfers.

Gas phase infrared spectra from quasi-classical Kubo time correlation functions

We generalise the recently developed phase integration method (PIM) to obtain a computable approximation of the Kubo expression for quantum time correlation functions. Our scheme combines exact sampling of the quantum thermal density with classical dynamics to provide a quasi-classical approximation for the correlation function. The method will be specialised to the evaluation of the momentum autocorrelation function, with the goal to compute infrared spectra of simple molecules in the gas phase. Application to two simple but interesting benchmark systems shows that the approach is accurate and stable over a broad range of temperatures.

A new class of ensemble conserving algorithms for approximate quantum dynamics: Theoretical formulation and model problems.

We develop two classes of quasi-classical dynamics that are shown to conserve the initial quantum ensemble when used in combination with the Feynman-Kleinert approximation of the density operator. These dynamics are used to improve the Feynman-Kleinert implementation of the classical Wigner approximation for the evaluation of quantum time correlation functions known as Feynman-Kleinert linearized path-integral. As shown, both classes of dynamics are able to recover the exact classical and high temperature limits of the quantum time correlation function, while a subset is able to recover the exact harmonic limit. A comparison of the approximate quantum time correlation functions obtained from both classes of dynamics is made with the exact results for the challenging model problems of the quartic and double-well potentials. It is found that these dynamics provide a great improvement over the classical Wigner approximation, in which purely classical dynamics are used. In a special case, our first method becomes identical to centroid molecular dynamics.

Boltzmann-conserving classical dynamics in quantum time-correlation functions: "Matsubara dynamics".

We show that a single change in the derivation of the linearized semiclassical-initial value representation (LSC-IVR or "classical Wigner approximation") results in a classical dynamics which conserves the quantum Boltzmann distribution. We rederive the (standard) LSC-IVR approach by writing the (exact) quantum time-correlation function in terms of the normal modes of a free ring-polymer (i.e., a discrete imaginary-time Feynman path), taking the limit that the number of polymer beads N → ∞, such that the lowest normal-mode frequencies take their "Matsubara" values. The change we propose is to truncate the quantum Liouvillian, not explicitly in powers of ħ(2) at ħ(0) (which gives back the standard LSC-IVR approximation), but in the normal-mode derivatives corresponding to the lowest Matsubara frequencies. The resulting "Matsubara" dynamics is inherently classical (since all terms O(ħ(2)) disappear from the Matsubara Liouvillian in the limit N → ∞) and conserves the quantum Boltzmann distribution because the Matsubara Hamiltonian is symmetric with respect to imaginary-time translation. Numerical tests show that the Matsubara approximation to the quantum time-correlation function converges with respect to the number of modes and gives better agreement than LSC-IVR with the exact quantum result. Matsubara dynamics is too computationally expensive to be applied to complex systems, but its further approximation may lead to practical methods.

Can the ring polymer molecular dynamics method be interpreted as real time quantum dynamics?

The ring polymer molecular dynamics (RPMD) method has gained popularity in recent years as a simple approximation for calculating real time quantum correlation functions in condensed media. However, the extent to which RPMD captures real dynamical quantum effects and why it fails under certain situations have not been clearly understood. Addressing this issue has been difficult in the absence of a genuine justification for the RPMD algorithm starting from the quantum Liouville equation. To this end, a new and exact path integral formalism for the calculation of real time quantum correlation functions is presented in this work, which can serve as a rigorous foundation for the analysis of the RPMD method as well as providing an alternative derivation of the well established centroid molecular dynamics method. The new formalism utilizes the cyclic symmetry of the imaginary time path integral in the most general sense and enables the expression of Kubo-transformed quantum time correlation functions as that of physical observables pre-averaged over the imaginary time path. Upon filtering with a centroid constraint function, the formulation results in the centroid dynamics formalism. Upon filtering with the position representation of the imaginary time path integral, we obtain an exact quantum dynamics formalism involving the same variables as the RPMD method. The analysis of the RPMD approximation based on this approach clarifies that an explicit quantum dynamical justification does not exist for the use of the ring polymer harmonic potential term (imaginary time kinetic energy) as implemented in the RPMD method. It is analyzed why this can cause substantial errors in nonlinear correlation functions of harmonic oscillators. Such errors can be significant for general correlation functions of anharmonic systems. We also demonstrate that the short time accuracy of the exact path integral limit of RPMD is of lower order than those for finite discretization of path. The present quantum dynamics formulation also serves as the basis for developing new quantum dynamical methods that utilize the cyclic nature of the imaginary time path integral.

Correlation Functions in Open Quantum-Classical Systems

Quantum time correlation functions are often the principal objects of interest in experimental investigations of the dynamics of quantum systems. For instance, transport properties, such as diffusion and reaction rate coefficients, can be obtained by integrating these functions. The evaluation of such correlation functions entails sampling from quantum equilibrium density operators and quantum time evolution of operators. For condensed phase and complex systems, where quantum dynamics is difficult to carry out, approximations must often be made to compute these functions. We present a general scheme for the computation of correlation functions, which preserves the full quantum equilibrium structure of the system and approximates the time evolution with quantum-classical Liouville dynamics. Several aspects of the scheme are discussed, including a practical and general approach to sample the quantum equilibrium density, the properties of the quantum-classical Liouville equation in the context of correlation function computations, simulation schemes for the approximate dynamics and their interpretation and connections to other approximate quantum dynamical methods.

Gas phase infrared spectra via the phase integration quasi-classical method

We apply the recently developed phase integration method (PIM) (Monteferrante et al. Mol Phys. 2011;109:3015–3027) to the calculation of infrared spectra of gas phase molecules. The PIM combines a generalised Monte Carlo sampling of the exact thermal density of the system with classical molecular dynamics to obtain approximate time quantum correlation functions. To describe the molecules, we adopt very simple analytical potentials that have, however, proved interesting, and surprisingly challenging, benchmarks for approximate quantum dynamical schemes. We show that, in contrast with two other commonly applied methods, our spectra do not exhibit spurious features or unphysical shifts depending on the temperature. Identifying the positions of the peaks requires only a few tens of trajectories, while an accurate evaluation of the relative intensities of the peaks is computationally more demanding.

Communication: Predictive partial linearized path integral simulation of condensed phase electron transfer dynamics

A partial linearized path integral approach is used to calculate the condensed phase electron transfer (ET) rate by directly evaluating the flux-flux/flux-side quantum time correlation functions. We demonstrate for a simple ET model that this approach can reliably capture the transition between non-adiabatic and adiabatic regimes as the electronic coupling is varied, while other commonly used semi-classical methods are less accurate over the broad range of electronic couplings considered. Further, we show that the approach reliably recovers the Marcus turnover as a function of thermodynamic driving force, giving highly accurate rates over four orders of magnitude from the normal to the inverted regimes. We also demonstrate that the approach yields accurate rate estimates over five orders of magnitude of inverse temperature. Finally, the approach outlined here accurately captures the electronic coherence in the flux-flux correlation function that is responsible for the decreased rate in the inverted regime.