The Quantum-Classical Path Integral (QCPI)

A rigorous formulation of quantum-classical dynamics with molecular dynamics scaling

Many chemical and biological processes involve a change of electronic state, which requires a quantum mechanical treatment.  The relevant electronic states are coupled to each other and also to the nuclear degrees of coordinates of the nuclei involved in the process.  The interaction of electronic and nuclear degrees of freedom, in particular the required reorganization of the atoms, is crucial in determining the kinetics of such processes.

Quantum mechanics is a nonlocal theory which requires computational effort that scales exponentially with the number of coupled degrees of freedom.  A classical treatment of the nuclear degrees of freedom, governed by Newtonian forces on each electronic state, is efficient and often sufficiently accurate.  However, accounting for the change of electronic states in a classical trajectory treatment presents a major challenge because classical trajectories are local in space, while quantum mechanical wavefunctions are delocalized.  As a result, attempts to combine quantum and classical tools tend to rely on various assumptions and uncontrolled approximations.

Recent work in our group addressed these challenges by developing a quantum-classical methodology that is based on Feynman’s path integral formulation. The quantum-classical path integral (QCPI) methodology treats the coupled electronic states by full quantum mechanics, while the dynamical effects of the nuclear degrees of freedom are captured through classical trajectories. Because quantum mechanical paths are local in space, the interaction between quantum and classical degrees of freedom can be treated accurately, in full detail, and without the need for any assumptions.  The QCPI formulation allows a faithful treatment of quantum interference among electronic states and its decoherence through coupling to the nuclear degrees of freedom, leading to correct branching ratios and product distributions.

The major numerical challenge of path integral methods is the exponential proliferation of quantum paths with the number of time steps, the number of which quickly reaches astronomical values.  The problem is even worse in the quantum-classical implementation of the path integral: each path of the quantum system generally gives rise to a distinct classical trajectory, thus the required number of trajectories also grows exponentially with propagation time.  This exponential proliferation of trajectories is the quantum-classical manifestation of time nonlocality, familiar from influence functional approaches.  Dealing with this extremely challenging problem seemed possible only through a thorough understanding of the physics of environment-induced decoherence.

Our analysis showed that the effects of a condensed phase environment on the dynamics of a quantum system have two contributions.  The first, which corresponds to a classical decoherence mechanism, dominates completely at high temperature/low-frequency solvents and/or when the system-environment interaction is weak.  Within the QCPI framework, the memory associated with classical decoherence is removable.  A second, nonlocal in time, quantum decoherence process, is operative at low temperatures, although the contribution of the classical decoherence mechanism continues to play the most prominent role there.  The classical decoherence is analogous to the treatment of light absorption via an oscillating dipole, while quantum decoherence is primarily associated with spontaneous emission, whose description requires quantization of the radiation field.  The QCPI methodology takes advantage of the memory-free nature of solvent trajectories that evolve on a single electronic state to account for all classical decoherence effects on the dynamics of the quantum system using a single trajectory per initial condition.

However, inclusion of the residual quantum decoherence is necessary for preserving the detailed balance property and obtaining correct state populations. Accounting for quantum decoherence processes requires the inclusion of phase factors accumulated along all classical trajectories that evolve subject to forces from the quantum paths.  Thus, inclusion of this important contribution would entail evaluation of the full sum over all paths of the electronic system.  Fortunately, the quantum-classical memory is eventually quenched, implying that the QCPI is amenable to iterative decompositions.

To further accelerate convergence of the iterative QCPI methodology, we have developed a dynamically consistent state hopping (DCSH) procedure, which captures some of the quantum memory effects through a judicious choice of the solvent trajectory retained past the memory interval. This scheme significantly reduces the length of quantum memory that needs to be treated explicitly, leading to significant reduction in the number of classical trajectories required for convergence.

The QCPI code has been adapted to work with two widely used molecular dynamics packages, NAMD and LAMMPS.  We have applied QCPI to simulate charge transfer in solution without any ad hoc assumptions in the dynamics.

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