The quantum trajectory dynamics in imaginary time with the momentum-dependent quantum potential [ J. Chem. Phys. 2010, 132, 014112], which can be used for calculations of low-lying energy eigenstates or for Boltzmann evolution, is extended to many dimensions. Evolution of a nodeless initial wave function represented in exponential form is described in terms of a trajectory ensemble. In the Lagrangian frame of reference the trajectories evolve according to the classical equations of motion under the influence of the inverted classical potential and the quantum potential defined by the gradient of the trajectory momenta. The initial wave function is sampled by the trajectories randomly, which makes the approach scalable to high dimensions. Another practical feature is approximate computation of the quantum potential and force from the global least squares fit of the trajectory momenta in a small basis. The importance sampling and the Eulerian frame-of-reference formulation, compensating for divergent dynamics in bound potentials, are introduced to reduce the number of trajectories in multidimensional applications. The zero-point energies are computed for model and chemical (H 2 O and SO 2 ) systems.
The quantum trajectory dynamics in imaginary time with the momentum-dependent quantum potential [ J. Chem. Phys. 2010, 132, 014112], which can be used for calculations of low-lying energy eigenstates or for Boltzmann evolution, is extended to many dimensions. Evolution of a nodeless initial wave function represented in exponential form is described in terms of a trajectory ensemble. In the Lagrangian frame of reference the trajectories evolve according to the classical equations of motion under the influence of the inverted classical potential and the quantum potential defined by the gradient of the trajectory momenta. The initial wave function is sampled by the trajectories randomly, which makes the approach scalable to high dimensions. Another practical feature is approximate computation of the quantum potential and force from the global least squares fit of the trajectory momenta in a small basis. The importance sampling and the Eulerian frame-of-reference formulation, compensating for divergent dynamics in bound potentials, are introduced to reduce the number of trajectories in multidimensional applications. The zero-point energies are computed for model and chemical (H 2 O and SO 2 ) systems.
The Boltzmann evolution of a wavefunction can be recast as imaginary-time dynamics of the quantum trajectory ensemble. The quantum effects arise from the momentum-dependent quantum potential--computed approximately to be practical in high-dimensional systems--influencing the trajectories in addition to the external classical potential [S. Garashchuk, J. Chem. Phys. 132, 014112 (2010)]. For a nodeless wavefunction represented as ψ(x, t) = exp(-S(x, t)/ħ) with the trajectory momenta defined by ∇S(x, t), analysis of the Lagrangian and Eulerian evolution shows that for bound potentials the former is more accurate while the latter is more practical because the Lagrangian quantum trajectories diverge with time. Introduction of stationary and time-dependent components into the wavefunction representation generates new Lagrangian-type dynamics where the trajectory spreading is controlled improving efficiency of the trajectory description. As an illustration, different types of dynamics are used to compute zero-point energy of a strongly anharmonic well and low-lying eigenstates of a high-dimensional coupled harmonic system.
The quantum trajectory dynamics is extended to the wave function evolution in imaginary time. For a nodeless wave function a simple exponential form leads to the classical-like equations of motion of trajectories, representing the wave function, in the presence of the momentum-dependent quantum potential in addition to the external potential. For a Gaussian wave function this quantum potential is a time-dependent constant, generating zero quantum force yet contributing to the total energy. For anharmonic potentials the momentum-dependent quantum potential is cheaply estimated from the global Least-squares Fit to the trajectory momenta in the Taylor basis. Wave functions with nodes are described in the mixed coordinate space/trajectory representation at little additional computational cost. The nodeless wave function, represented by the trajectory ensemble, decays to the ground state. The mixed representation wave functions, with lower energy contributions projected out at each time step, decay to the excited energy states. The approach, illustrated by computing energy levels for anharmonic oscillators and energy level splitting for the double-well potential, can be used for the Boltzmann operator evolution.
In this chapter we formulate quantum trajectory dynamics in imaginary time which seamlessly connects to Bohmian dynamics in real time. Approximate determination of the quantum potential and mixed representation of wavefunctions with nodes make this trajectory approach stable and scalable to high dimensions, while including the T&F Cat # K11224, Chapter 23, Page 362, 15-9-2010 leading quantum effects into wavefunction evolution. Reaction probabilities, energy eigenstates and reaction rate constants can be obtained by propagating an ensemble of quantum trajectories in real and/or imaginary time.
The quantum nature of nuclei often affects molecular structure and properties, which are associated with the motion of protons and other light nuclei at low temperature. However, incorporation of the nuclear quantum effects into theoretical studies of large molecular systems is an outstanding challenge in theoretical chemistry. In this paper, the de Broglie-Bohm formulation of the time-dependent Schrödinger equation is used as a framework for the development of approximate quantum corrections to the dynamics of a trajectory ensemble, representing a time-dependent wave function. Specifically, the trajectory-centered local Least Square Fit (L-LSF) to the trajectory attributes is introduced, formally analyzed and illustrated on model potentials, typically used to benchmark approximate and semiclassical dynamics methods. The L-LSF method improves the dynamics accuracy, compared to the globally defined approximations, developed previously, and is shown to converge to the exact (generally impractical for large systems) limit of quantum dynamics.
Ensembles of quantum trajectories are evolved to study time-dependent reaction dynamics in multidimensional systems with up to 25 vibrational modes. The equations of motion are formulated in curvilinear reaction path coordinates and all coupling terms are retained, including those involving curvature of the reaction path. The model potential is a Gaussian barrier along the translational coordinate coupled to M vibrational modes. Spatial derivatives needed to propagate the trajectories are evaluated by least squares fitting in a contracted basis set. Stable propagation of the trajectory ensembles was carried out until complete bifurcation into reflected and reactive subensembles. The reaction probabilities were evaluated by Monte Carlo integration of the multidimensional smooth transmitted densities. Computational results, including trajectory plots and time-dependent reaction probabilities, are presented for M = 1, 5, and 25 vibrational modes.
In a previous publication [J. Chem. Phys. 118, 9911 (2003)], the derivative propagation method (DPM) was introduced as a novel numerical scheme for solving the quantum hydrodynamic equations of motion (QHEM) and computing the time evolution of quantum mechanical wave packets. These equations are a set of coupled, nonlinear partial differential equations governing the time evolution of the real-valued functions C and S in the complex action, S=C(r,t) + iS(r,t)/Planck's over 2pi, where Psi(r,t)=exp(S). Past numerical solutions to the QHEM were obtained via ensemble trajectory propagation, where the required first- and second-order spatial derivatives were evaluated using fitting techniques such as moving least squares. In the DPM, however, equations of motion are developed for the derivatives themselves, and a truncated set of these are integrated along quantum trajectories concurrently with the original QHEM equations for C and S. Using the DPM quantum effects can be included at various orders of approximation; no spatial fitting is involved; there is no basis set expansion; and single, uncoupled quantum trajectories can be propagated (in parallel) rather than in correlated ensembles. In this study, the DPM is extended from previous one-dimensional (1D) results to calculate transmission probabilities for 2D and 3D wave packet evolution on coupled Eckart barrier/harmonic oscillator surfaces. In the 2D problem, the DPM results are compared to standard numerical integration of the time-dependent Schrodinger equation. Also in this study, the practicality of implementing the DPM for systems with many more degrees of freedom is discussed.