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The Journal of Chemical Physics

Quantum Dynamics of Anharmonic Oscillators. I. Simple Examples

Elliott B. Alterman, Charlotte T. Tahk, David J. Wilson

Published Jan 15, 1966• 14 citations
Publisher

Abstract

The dynamical behavior of two anharmonic oscillators, having a linear and a square-well potential, respectively, is studied. The expectation values of the coordinate and the mean spread of the oscillating wave packets are calculated as functions of time and presented graphically. Significant departures from the behavior of the corresponding classical systems are observed. These results cast some doubt on the use of classical-mechanical models in the theory of unimolecular rates.

DOI
10.1063/1.1726708

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Investigation of the dynamics of two coupled oscillators with mixed quantum-classical methods

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The dynamics of two coupled oscillators can become quite complex if anharmonic potential energy functions are employed. This type of system therefore represents a good model for an investigation of the performance of mixed quantum-classical methods. In this work, the motion of two coupled particles with a mass ratio of one to ten is studied with three different mixed quantum-classical methods in the presence of anharmonic potential terms for a comparison with exact quantum mechanical calculations. The mixed quantum-classical approaches include the multitrajectory Ehrenfest, the mixed quantum-classical Bohmian (MQCB), and the so-called coupled Schrödinger equations (CSE) formalisms. The analysis shows that while the description of a weakly anharmonic system by the Ehrenfest and MQCB schemes is accurate if proper sampling techniques are applied, both approximations break down rapidly if the anharmonic terms are increased. The performance of the simple CSE prescription, which corresponds to a reduction of the full two-dimensional wave function to two one-dimensional wave functions representing two quantum oscillators coupled via the potential energy in a classical fashion, decreases if the width of the initial wave packet is enlarged. The dependence of the CSE method on the diffuseness of the initial wave packet is therefore opposite to that of the MQCB method, which is more accurate for wide wave packets. Overall, the multitrajectory Ehrenfest ansatz is found to be most successful in reproducing the exact quantum results.

The Journal of Chemical Physics•May 11, 2006•6 citations
The dynamical behavior of two anharmonic oscillators, having a linear and a square-well potential, respectively, is studied. The expectation values of the coordinate and the mean spread of the oscillating wave packets are calculated as functions of time and presented graphically. Significant departures from the behavior of the corresponding classical systems are observed. These results cast some doubt on the use of classical-mechanical models in the theory of unimolecular rates.
Quantum calculations of the dynamics of a Morse oscillator with a Gaussian initial wave packet have been performed. As previously observed in other simple oscillators the classical and quantum expectation values for the coordinate exhibit significantly different behaviors, especially after the first few cycles. Some theories of chemical kinetics require the classical-mechanical calculation of lifetimes which may be quite long compared to vibrational periods. These theories deserve re-examination in view of the dynamical behavior observed in the present work.
The dynamics of an initially excital Gaussian wave packet (harmonic oscillator coherent state, HOCS) in a general one-dimensional anharmonic system is studied analytically with the aid of a second quantization formalism for times shorter than a classical oscillation period. The initial state population probability PQ(t), its classical counterpart PC(t), and the overlap probability PQC(t) between a time evolved wave packet and a classically moving Gaussian with stationary shape are calculated in order to analyze the spreading of the wave packet. It was found that, in general, PQ(t) and PC(t) agree very well if the potential energy function has no strongly changing low order derivatives. This agreement is held even if the wave packet spreads drastically within the time scale considered as can be demonstrated by analyzing PQC(t). The analytical expressions for PQ(t), PC(t), and PQC(t) are applied to a dissociative third-order oscillator and a strongly anharmonic fourth-order oscillator. While for a harmonic system, the initial decay time of PQ(t) and PC(t) (which are identical in this case) are only dependent on the energy expectation value 〈H〉 with respect to the initial HOCS, differences up to an order of magnitude occur for HOCS with the same energy in the model systems.
The origin of quantum stochasticity is studied via a simple model of decaying coupled anharmonic oscillators. It is shown that the decay rates of individual eigenstates, as well as mixed states, are never stochastic, even in regions where classical stochasticity exists. However, decay rates of (coherent) wave packets do show a transitional behavior, similar to the classical one, as the energy is raised. The transition is shown to arise from a competition between two rates, the dissociation rate, depending on the wave function amplitudes, and the dephasing rate, depending on the energy spectrum, and to occur when the dephasing becomes faster than the dissociation. It is suggested that only properties which depend simultaneously on the wave function-amplitude and the energy spectrum can display this transitional behavior. It is also shown that strong coupling is not a necessary condition for stochasticity.
The dynamics of two coupled oscillators can become quite complex if anharmonic potential energy functions are employed. This type of system therefore represents a good model for an investigation of the performance of mixed quantum-classical methods. In this work, the motion of two coupled particles with a mass ratio of one to ten is studied with three different mixed quantum-classical methods in the presence of anharmonic potential terms for a comparison with exact quantum mechanical calculations. The mixed quantum-classical approaches include the multitrajectory Ehrenfest, the mixed quantum-classical Bohmian (MQCB), and the so-called coupled Schrödinger equations (CSE) formalisms. The analysis shows that while the description of a weakly anharmonic system by the Ehrenfest and MQCB schemes is accurate if proper sampling techniques are applied, both approximations break down rapidly if the anharmonic terms are increased. The performance of the simple CSE prescription, which corresponds to a reduction of the full two-dimensional wave function to two one-dimensional wave functions representing two quantum oscillators coupled via the potential energy in a classical fashion, decreases if the width of the initial wave packet is enlarged. The dependence of the CSE method on the diffuseness of the initial wave packet is therefore opposite to that of the MQCB method, which is more accurate for wide wave packets. Overall, the multitrajectory Ehrenfest ansatz is found to be most successful in reproducing the exact quantum results.