Bohmian quantum-classical approach

The following three sections describe the Bohmian quantum-classical approach [22,23] that uniquely solves the quantum back-reaction branching problem, the stochastic mean-field approximation [20] (SMF) that both resolves the back-reaction problem and incorporates the quantum decoherence and Franck-Condon overlap effects into NA-MD, and the quantized mean-field method [21] (QMF) that takes into account ZPE. The Bohmian and QMF approaches are illustrated by a model designed to capture some features of the O2 dissociation on a Pt surface. The concluding section summarizes the features of the methods and discusses further avenues for development and consideration. 

Burghardt, I., Parlant, G. On the dynamics of coupled Bohmian and phase-space variables a new hybrid quantum-classical approach. J. Chem. Phys. 120... 

Numerical illustration of the Bohmian quantum-classical and quantized mean-field approaches... 

The Bohmian quantum-classical and QMF approaches [21,22] have been applied to a model intended as a simplified representation of gaseous oxygen interacting with a platinum surface, Ref. [13,91]. The model consists of a light particle q with mass m colliding with a heavier particle Q with mass M. The heavy particle is bound to an immobile surface. The total Hamiltonian for the system is given by... 

The Bohmian quantum-classical and QMF approaches have been tested with a model application designed to simulate the interaction of an oxygen molecule with a platinum surface [13,21,22,91]. With trajectory branching the Bohmian quantum-classical method recovers the correct asymptotic behavior of the scattering probability of the quantum subsystem. The QMF approach shows improvement in both the short and long time scattering probabilities. The improvement is achieved due to the proper treatment of ZPE. 

To recapitulate, the Bohmian quantum-classical, stochastic mean-field and quantized mean-field approaches described above are capable of reproducing quantum solvent effects that are crucial in simulation of NA chemical processes. The approaches are computationally simple and are particularly suitable for studies of large chemical systems. 

The question then arises if a convenient mixed quantum-classical description exists, which allows to treat as quantum objects only the (small number of) degrees of freedom whose dynamics cannot be described by classical equations of motion. Apart in the limit of adiabatic dynamics, the question is open and a coherent derivation of a consistent mixed quantum-classical dynamics is still lacking. All the methods proposed so far to derive a quantum-classical dynamics, such as the linearized path integral approach [2,6,7], the coupled Bohmian phase space variables dynamics [3,4,9] or the quantum-classical Li-ouville representation [11,17—19], are based on approximations and typically fail to satisfy some properties that are expected to hold for a consistent mechanics [5,19]. 

Recently [8-11] an alternative treatment to mix quantum mechanics with classical mechanics, based on Bohmian quantum trajectories was proposed. Briefly, the quantum subsystem is described by a time-dependent Schrodinger equation that depends parametrically on classical variables. This is similar to other approaches discussed above. The difference comes from the way the classical trajectories are calculated. In our approach, which was called mixed quantum-classical Bohmian (MQCB) trajectories, the wave packet is used to define de Broglie-Bohm quantum trajectories [12] which in turn are used to calculate the force acting on the classical variables. 

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