The quantum trajectory framework incorporates quantum effects on dynamics through the quantum potential acting on a trajectory ensemble in addition to the classical potential. A global quadratic approximation to the quantum potential makes the method practical in many dimensions and captures dominant quantum effects in semiclassical systems. In this paper the approach is further developed to describe the "double well" dynamicsa prototype of the proton transfer reactionswhich exhibits the "hard" quantum effect of tunneling. Accurate description is achieved by combining the approximate quantum trajectory dynamics with the population amplitudes in the reactant and product wells. The quantum trajectory dynamics is defined by the asymptotic classical potentials. The population amplitudes represented in a small basis describe transfer between the wells. The method is exact if the reactant/product potentials are quadratic and the basis size is sufficiently large. In the semiclassical regime the trajectory dynamics is approximate and the basis size can be as small as two functions. The approach is fully compatible with the trajectory description of multidimensional systems capturing quantum tunneling along the reactive coordinate and zero-point energy flow among all degrees of freedom.
The quantum trajectory framework incorporates quantum effects on dynamics through the quantum potential acting on a trajectory ensemble in addition to the classical potential. A global quadratic approximation to the quantum potential makes the method practical in many dimensions and captures dominant quantum effects in semiclassical systems. In this paper the approach is further developed to describe the "double well" dynamicsa prototype of the proton transfer reactionswhich exhibits the "hard" quantum effect of tunneling. Accurate description is achieved by combining the approximate quantum trajectory dynamics with the population amplitudes in the reactant and product wells. The quantum trajectory dynamics is defined by the asymptotic classical potentials. The population amplitudes represented in a small basis describe transfer between the wells. The method is exact if the reactant/product potentials are quadratic and the basis size is sufficiently large. In the semiclassical regime the trajectory dynamics is approximate and the basis size can be as small as two functions. The approach is fully compatible with the trajectory description of multidimensional systems capturing quantum tunneling along the reactive coordinate and zero-point energy flow among all degrees of freedom.
A method of molecular dynamics with quantum corrections, practical for studies of large molecular systems, is reviewed. The approach is based on the Bohmian formulation of the time-dependent Schrödinger equation in which a wavefunction is represented by an ensemble of interdependent trajectories. The quantum effects come from the quantum potential acting on trajectories on par with the usual classical potential. The quantum potential is determined from the evolving nuclear wavefunction, i.e. from the quantum trajectory (QT) ensemble itself. For practical and conceptual reasons the quantum potential and corresponding quantum nuclear effect are computed only for the selected light nuclei. For studies of reactive chemical processes, the classical potential is computed on-the-fly using the density functional tight binding method of electronic structure. A massively parallel implementation, based on the message passing interface allows for efficient simulations of ensembles of thousands of trajectories describing systems of up to 200 atoms. As a biochemical application, the approximate QT approach is used to model the tunnelling-dominated proton transfer in soybean-lipoxygenase-1. A materials science application is represented by a study of the nuclear quantum effect on adsorption of hydrogen and deuterium on a C37H15 molecule, which is a model ‘flake’ of graphene.
Dynamics based on quantum trajectories with approximate quantum potential is generalized to nonadiabatic systems and its semiclassical properties are discussed. The formulation uses the mixed polar-coordinate space representation of a wave function. The polar part describes the overall time evolution of the wave-function components semiclassically using the single-surface approximate quantum potential. The coordinate part represents a complex``population'' amplitude, which in case of localized coupling can be solved for quantum mechanically in an efficient manner. In the high-energy regime this is accomplished by using a small basis determined by the coupling between surfaces. An illustration is given for a typical curve-crossing problem. The energy-resolved probabilities obtained from the time evolution of two wave packets for a wide range of energies are in excellent agreement with exact results for energies above the threshold of the diabatic reaction, including the case of total nonadiabatic transition.
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 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.
The quantum trajectory approach is generalized to arbitrary coordinate systems, including curvilinear coordinates. This allows one to perform an approximate quantum trajectory propagation, which scales favorably with system size, in the same framework as standard quantum wave packet dynamics. The trajectory formulation is implemented in Jacobi coordinates for a nonrotating triatomic molecule. Wave packet reaction probabilities are computed for the O(3P) + H2 --> OH + H reaction using the approximate quantum potential. The latter is defined by the nonclassical component of the momentum operator expanded in terms of linear and exponential functions. Unlike earlier implementations with linear functions, the introduction of the exponential function provides an accurate description of asymptotic dynamics for this system and gives good agreement of approximate reaction probabilities with accurate quantum calculations.
Dynamics of quantum trajectories provides an efficient framework for description of various quantum effects in large systems, but it is unstable near the wave function density nodes where the quantum potential becomes singular. A mixed coordinate space/polar representation of the wave function is used to circumvent this problem. The resulting modified trajectory dynamics associated with the polar representation is nonsingular and smooth. The interference structure and the nodes of the wave function density are described, in principle, exactly in the coordinate representation. The approximate version of this approach is consistent with the semiclassical linearized quantum force method [S. Garashchuk and V. A. Rassolov, J. Chem. Phys. 120, 1181 (2004)]. This approach is exact for general wave functions with the density nodes in a locally quadratic potential.