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The Journal of Chemical Physics

Second-order quantized Hamilton dynamics coupled to classical heat bath

Eric M. Heatwole, Oleg V. Prezhdo

Published Jun 15, 2005• 19 citations
Publisher

Abstract

Starting with a quantum Langevin equation describing in the Heisenberg representation a quantum system coupled to a quantum bath, the Markov approximation and, further, the closure approximation are applied to derive a semiclassical Langevin equation for the second-order quantized Hamilton dynamics (QHD) coupled to a classical bath. The expectation values of the system operators are decomposed into products of the first and second moments of the position and momentum operators that incorporate zero-point energy and moderate tunneling effects. The random force and friction as well as the system-bath coupling are decomposed to the lowest classical level. The resulting Langevin equation describing QHD-2 coupled to classical bath is analyzed and applied to free particle, harmonic oscillator, and the Morse potential representing the OH stretch of the SPC-flexible water model.

DOI
10.1063/1.1931666

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Starting with a quantum Langevin equation describing in the Heisenberg representation a quantum system coupled to a quantum bath, the Markov approximation and, further, the closure approximation are applied to derive a semiclassical Langevin equation for the second-order quantized Hamilton dynamics (QHD) coupled to a classical bath. The expectation values of the system operators are decomposed into products of the first and second moments of the position and momentum operators that incorporate zero-point energy and moderate tunneling effects. The random force and friction as well as the system-bath coupling are decomposed to the lowest classical level. The resulting Langevin equation describing QHD-2 coupled to classical bath is analyzed and applied to free particle, harmonic oscillator, and the Morse potential representing the OH stretch of the SPC-flexible water model.
We derive a semi-classical Langevin equation that, in conjuction with a quantum fluctuation–dissipation theorem, can be used to run canonical dynamics and to determine thermal averages in the second-order quantized Hamilton dynamics (QHD-2). In contrast to our earlier work [J. Chem. Phys. 122 (2005) 234109], the bath is no longer treated classically, and the friction is represented at the second-order. As a result, the quantum fluctuation variables are kept close to their optimal values, similarly to the recently developed phase-space canonical averaging for QHD-2 [J. Chem. Phys. 126 (2007) 204108]. The theory is exact for the quantum harmonic oscillator and the free particle, and gives good agreement with the quantum answer for a variety of quartic potentials.
The quantization of Hamilton dynamics (QHD) [J. Chem. Phys. 113, 6557 (2000)] that efficiently generalizes classical mechanics to include quantum tunneling and zero-point energy effects is extended to a general position dependent potential. A Taylor series expansion of the potential is considered both around a fixed point and around the moving instantaneous value of the position variable. The equations-of-motion obtained for the moving frame are significantly simpler than for the fixed frame, while still satisfying the classical limit. The number of the QHD variables and the order of the Taylor expansion of the potential constitute two independent approximation parameters. Conservation of the total energy and the Heisenberg commutator relationship is established for the second-order QHD that includes linear and quadratic variables. The formal results are illustrated by examples, including the harmonic oscillator, tunneling in a doublewell potential, and energy exchange between coupled Morse oscillators representing the stretching vibrations of the SPC-F water molecule commonly used in molecular dynamics simulations. QHD provides an improvement over classical mechanics in all cases. The QHD data agree with the exact data in the large ℏ quantum limit and in the ℏ=0 classical limit, with deviations observed for the intermediate ℏ values, depending on the system and initial conditions. QHD provides a quantitative short time approximation. The second-order QHD becomes inappropriate when quantum wave packets split. Higher-order QHD approximations are required in such cases. The second-order QHD is particularly suitable for the description of zero-point energy effects and properly treats moderate tunneling events.
Quantized Hamilton dynamics (QHD) is a simple and elegant extension of classical Hamilton dynamics that accurately includes zero-point energy, tunneling, dephasing, and other quantum effects. Formulated as a hierarchy of approximations to exact quantum dynamics in the Heisenberg formulation, QHD has been used to study evolution of observables subject to a single initial condition. In present, we develop a practical solution for generating canonical ensembles in the second-order QHD for position and momentum operators, which can be mapped onto classical phase space in doubled dimensionality and which in certain limits is equivalent to thawed Gaussian. We define a thermal distribution in the space of the QHD-2 variables and show that the standard beta=1/kT relationship becomes beta'=2/kT in the high temperature limit due to an overcounting of states in the extended phase space, and a more complicated function at low temperatures. The QHD thermal distribution is used to compute total energy, kinetic energy, heat capacity, and other canonical averages for a series of quartic potentials, showing good agreement with the quantum results.
A conceptually simple approximation to quantum mechanics, quantized Hamilton dynamics (QHD) includes zero-point energy, tunneling, dephasing, and other important quantum effects in a classical-like description. The hierarchy of coupled differential equations describing the time evolution of observables in QHD can be mapped in the second order onto a classical system with double the dimensionality of the original system. While QHD excels at dynamics with a single initial condition, the correct method for generating thermal initial conditions in QHD remains an open question. Using the coherent state representation of thermodynamics of the harmonic oscillator (HO) [Schnack, Europhys. Lett. 45, 647 (1999)], we develop canonical averaging for the second order QHD [Prezhdo, J. Chem. Phys. 117, 2995 (2002)]. The methodology is exact for the free particle and HO, and shows good agreement with quantum results for a variety of quartic potentials.
The quantized Hamilton dynamics (QHD) method, which was introduced and developed in J. Chem. Phys. 113, 6557 (2000) to the second order, is extended to the third and fourth orders. The QHD formalism represents an extension of classical mechanics and allows for the derivation of a hierarchy of equations of motion which converge with the quantum-mechanical limit. Here, the second, third, and fourth order QHD approximations are applied to two model problems: the decay of a particle in a metastable cubic potential and the intermode energy exchange observed in the Henon–Heiles system. The QHD results exhibit good convergence with the quantum data with increasing order yet preserve the computational efficiency of classical calculations. The second order QHD approximation already does an excellent job in maintaining the zero-point energy in the Henon–Heiles system and describing moderate tunneling events in the metastable potential. Extensions to higher orders substantially improve the QHD results for deep tunneling and are capable of describing the finer details of energy exchange.
Using Terwiel's cumulants the Markovian approximation to arrive to the Quantum Master Equation, for a system interacting with a thermal bath, is revisited.The second order weak coupling approximation is analyzed, then a Kossakowski-Lindblad form for the generator is written in terms of the position and momentum operators.A weak coupling approximation for the stochastic non-Markovian wave function is worked out.A free particle model interacting with a thermal quantum bath is studied in the context of Schr ödinger-Langevin picture.A phenomenological point of view is introduced in order to overcome certain difficulties in the time evolution -in the second order approximation -for the free particle Hamiltonian.