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Physical Review A

Semiclassical nonadiabatic dynamics with quantum trajectories

Vitaly A. Rassolov, Sophya V. Garashchuk

Published Mar 25, 2005• 46 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.71.032511

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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.
We present a time-dependent semiclassical method based on quantum trajectories. Quantum-mechanical effects are described via the quantum potential computed from the wave function density approximated as a linear combination of Gaussian fitting functions. The number of the fitting functions determines the accuracy of the approximate quantum potential (AQP). One Gaussian fit reproduces time-evolution of a Gaussian wave packet in a parabolic potential. The limit of the large number of fitting Gaussians and trajectories gives the full quantum-mechanical result. The method is systematically improvable from classical to fully quantum. The fitting procedure is implemented as a gradient minimization. We also compare AQP method to the widely used semiclassical propagator of Herman and Kluk by computing energy-resolved transmission probabilities for the Eckart barrier from the wave packet time-correlation functions. We find the results obtained with the Herman–Kluk propagator to be essentially equivalent to those of AQP method with a one-Gaussian density fit for several barrier widths.
The quantum trajectory method (QTM) is applied to nonadiabatic electronic transitions. Equations of motion in a Lagrangian framework are derived for the probability density, velocity, position, and action functions for a discretized wave packet moving along coupled potential energy surfaces. In solving these equations of motion, we obtain agreement with exact quantum results computed by solving the time-dependent Schrödinger equation on a space-fixed grid. On each of the coupled potential energy surfaces, the dynamics of the trajectories is fully quantum mechanical, i.e., there are no “surface–hopping transitions.” We present a detailed analysis of the QTM results including density changes, complex oscillations of the wave functions, phase space analysis, and a detailed discussion of the forces that contribute to the evolution the trajectories.
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.
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 coupled Schrödinger equations that describe nonadiabatic dynamics are recast using the Bohm formulation of quantum mechanics. The resulting coupled Bohm equations are solved numerically for two scattering models, giving results that are essentially identical to wave-packet solution of the original coupled Schrödinger equations. The classical limit of the set of coupled Bohm equations is then described, producing a mixed quantum-classical theory incorporating classical-like motion on each potential energy surface accompanied by quantum transitions between the quantum states. Numerical tests of the mixed quantum-classical method are in excellent agreement with the accurate full-quantum results for the model problems. The method is contrasted with the related surface-hopping approach. It is shown that computing the dynamics of a distribution of classical particles is more consistent and more accurate than computing the motion of independent point particles as with surface hopping.
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.