We present a new approach for calculating quantum time correlation functions for systems whose dynamics exhibits relevant nonadiabatic effects. The method involves partial linearization of the full quantum path-integral expression for the time correlation function written in the nonadiabatic mapping Hamiltonian formalism. Our analysis gives an algorithm which is both numerically efficient and accurate as we demonstrate in test calculations on the spin-boson model where we find results in good agreement with exact calculations. The accuracy of our new approach is comparable to that of calculations performed using other approximate methods over a relatively broad range of model parameters. However, our method converges relatively quickly when compared with most alternative schemes. These findings are very encouraging in view of the application of the new method for studying realistic nonadiabatic model problems in the condensed phase.

We revisit the statistical mechanics of charge fluctuations in capacitors. In constant-potential classical molecular simulations, the atomic charge of electrode atoms are treated as additional degrees of freedom which evolve in time so as to satisfy the constraint of fixed electrostatic potential for each configuration of the electrolyte. The present work clarifies the role of the overall electroneutrality constraint, as well as the link between the averages computed within the Born-Oppenheimer approximation and that of the full constant-potential ensemble. This allows us in particular to derive a complete fluctuation-dissipation relation for the differential capacitance, that includes a contribution from the charge fluctuations (around the charges satisfying the constant-potential and electroneutrality constraints) also present in the absence of an electrolyte. We provide a simple expression for this contribution from the elements of the inverse of the matrix defining the quadratic form of the fluctuating charges in the energy. We then illustrate numerically the validity of our results, and recover the expected result for an empty capacitor with structureless electrodes at large inter-electrode distances. By considering a variety of liquids between graphite electrodes, we confirm that this contribution to the total differential capacitance is small compared to that induced by the thermal fluctuations of the electrolyte.

This paper presents a new approach to propagating the density matrix based on a time stepping procedure arising from a Trotter factorization and combining the forward and backward incremental propagators. The sums over intermediate states of the discrete quantum subsystem are implemented by a Monte Carlo surface hopping-like procedure, while the integrals over the continuous variables are performed using a linearization in the difference between the forward and backward paths of these variables leading to classical-like equations of motion with forces determined by the quantum subsystem states. The approach is tested on several models and numerical convergence is explored.

The quantum-classical Liouville equation provides a description of the dynamics of a quantum subsystem coupled to a classical environment. Representing this equation in the mapping basis leads to a continuous description of discrete quantum states of the subsystem and may provide an alternate route to the construction of simulation schemes. In the mapping basis the quantum-classical Liouville equation consists of a Poisson bracket contribution and a more complex term. By transforming the evolution equation, term-by-term, back to the subsystem basis, the complex term (excess coupling term) is identified as being due to a fraction of the back reaction of the quantum subsystem on its environment. A simple approximation to quantum-classical Liouville dynamics in the mapping basis is obtained by retaining only the Poisson bracket contribution. This approximate mapping form of the quantum-classical Liouville equation can be simulated easily by Newtonian trajectories. We provide an analysis of the effects of neglecting the presence of the excess coupling term on the expectation values of various types of observables. Calculations are carried out on nonadiabatic population and quantum coherence dynamics for curve crossing models. For these observables, the effects of the excess coupling term enter indirectly in the computation and good estimates are obtained with the simplified propagation.

A new semiclassical approach to implementing the mapping Hamiltonian formulation of nonadiabatic dynamics is presented. The approach involves using initial distributions of mapping oscillator variables that focus the sampling in such a way as to recover individual trajectory motion over the occupied state potential surface. The usual semiclassical implementation of the mapping Hamiltonian approach only recovers this feature after ensemble averaging. We test the approach on several model problems and show that it converges with very few trajectories compared to the usual approach.

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We show that quantum time correlation functions including electronically nonadiabatic effects can be computed by using an approach in which their path integral expression is linearized in the difference between forward and backward nuclear paths while the electronic component of the amplitude, represented in the mapping formulation, can be computed exactly, leading to classical-like equations of motion for all degrees of freedom. The efficiency of this approach is demonstrated in some simple model applications. In statistical mechanics, time correlation functions are central quantities bridging the microscopic dynamics and fluctuations of a given system with macroscopic, phenomenological quantities, such as transport coefficient, relaxation times, and rates (1, 2). Although relatively standard numerical methods provide a viable tool for their evaluation in classical systems (3, 4), full quantum mechanical calculation of time correlations functions is currently out of the realm of affordable computation. Consequently, many approximate techniques have been developed to tackle this problem (5-9). In this article, we present a mixed quantum-classical approach, belonging to the family of so-called linearization methods (10)(11)(12)(13)(14)(15), that addresses the evaluation of time correlation functions of nuclear or electronic operators evolving in the presence of nonadiabatic effects. TheoryWe begin by rewriting the function in a basis set defined as the tensor product of nuclear positions and diabatic electronic statesHere, the Hamiltonian Ĥ contains the nuclear kinetic energy and an electronic part, ĥ el , with matrix elements h ,Ј (R), is the density matrix of the system, and we have chosen the operator B to be diagonal in the nuclear space. A convenient representation to account for the effects of the electronic transitions on the nuclear degrees of freedom is offered by the mapping Hamiltonian method (16-23). In this context, the n diabatic states are substituted by n harmonic oscillators with occupation number limited to 0 or 1, i.e., ͉␣͘ 3 ͉m ␣ ͘ ϭ ͉0 1 , . . . , 1 ␣ , . . . , 0 n ͘, and the electronic Hamiltonian becomeswhere q and p are the positions and momenta of the oscillators. While leaving the nuclear motion unaltered, the mapping simplifies the electronic problem considerably and has been applied to study nonadiabatic dynamics in many semiclassical calculations (16-23). Its advantages become apparent once a hybrid momentum-coordinate representation is introduced for the propagators in the correlation function, for examplewhereͬ.[4]The transition amplitude between mapping states m ␣ and m ␤ is determined by a quadratic Hamiltonian that depends parametrically on the nuclear path. Thus, this amplitude can be evaluated exactly, for example, by using a semiclassical expression. A particularly convenient semiclassical choice, both computationally and from a theoretical viewpoint, is the Herman-Kluk representation (24), which provides us with the following expression for the amplitude ͗m ␤ ͉e Ϫ͑i/ -h͒ĥm͑RN...

We demonstrate the accuracy and efficiency of a recently introduced approach to account for nuclear quantum effects (NQEs) in molecular simulations: the adaptive quantum thermal bath (adQTB). In this method, zero-point energy is introduced through a generalized Langevin thermostat designed to precisely enforce the quantum fluctuation−dissipation theorem. We propose a refined adQTB algorithm with improved accuracy and report adQTB simulations of liquid water. Through extensive comparison with reference path integral calculations, we demonstrate that it provides excellent accuracy for a broad range of structural and thermodynamic observables as well as infrared vibrational spectra. The adQTB has a computational cost comparable to that of classical molecular dynamics, enabling simulations of up to millions of degrees of freedom.