Quantum-Classical Path Integral Methodology

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Nancy Makri of the University of Illinois is supported by an award from the Chemical Theory, Models and Computational Methods program in the Division of Chemistry to develop new, rigorous computational methods to study electron and proton transfer processes. Many dynamical processes in large chemical or biological systems can be described successfully and very efficiently through classical trajectory simulations. However, chemical and biological reactions frequently involve the transfer of electrons or protons. These light particles often need to be studied using quantum mechanics. Electron and many proton transfer reactions exhibit pronounced quantum effects and classical trajectories are not adequate for modeling such processes. Numerical solution of the quantum mechanical equations of motion, however is extremely computationally demanding. The most profitable approach for systems of any size is the combination of quantum and classical trajectory simulation tools. It is not always easy to combine classical and quantum mechanics without introducing severe approximations which may result in inaccurate results. This is because classical mechanics is localized in space, i.e., at each point in time, each atom has a definite position and momentum. Quantum mechanics is usually represented in terms of waves, which are delocalized. Makri addresses this apparent incompatibility via a novel quantum-classical path integral (QCPI) approach, which uses a local description of the quantum particles, eliminating the need for approximations and allowing simulations with unprecedented accuracy. Makri and her research group teach a hands-on lecture course entitled "Music, light and the atom" which she developed. It is aimed at high school students, presenting the basic ideas of quantum mechanics through analogies to music. The QCPI formulation treats the interaction between quantum and classical degrees of freedom in full atomistic detail through a dynamical phase along each quantum-classical path. However, the number of quantum paths grows exponentially with propagation time, and each quantum path specifies a different classical trajectory. By exploiting phase relations that make distinct contributions to decoherence, Makri has shown that the QCPI expression can be evaluated with only modest computational effort. The proposed work introduces new ideas based on quantum interference, zero-point energy and spontaneous phonon emission that can be used to further accelerate QCPI calculations by orders of magnitude, leading to a highly accurate methodology applicable to complex chemical processes with effort comparable to that employed in typical molecular dynamics simulations.