Wigner phase space method: Analysis for semiclassical applications
15 Aug 1976-Journal of Chemical Physics (American Institute of Physics)-Vol. 65, Iss: 4, pp 1289-1298
TL;DR: In this article, the suitability of the Wigner method as a tool for semiclassical dynamics is investigated, and qualitatively introduced the difficulties encountered in some applications, and derive quantitative means of surmounting these difficulties.
Abstract: We investigate the suitability of the Wigner method as a tool for semiclassical dynamics. In spite of appearances, the dynamical time evolution of Wigner phase space densities is found not to reduce to classical dynamics in most circumstances, even as h→0. In certain applications involving highly ’coherent’ density matrices, this precludes direct h‐expansion treatment of quantum corrections. However, by selective resummation of terms in the Wigner–Moyal series for the quantum phase space propagation it is possible to arrive at a revised or renormalized classicallike dynamics which solves the difficulties of the direct approach. In this paper, we review the Wigner method, qualitatively introduce the difficulties encountered in certain semiclassical applications, and derive quantitative means of surmounting these difficulties. Possible practical applications are discussed.
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TL;DR: The ab initio multiple spawning (AIMS) method is a time-dependent formulation of quantum chemistry, whereby the nuclear dynamics and electronic structure problems are solved simultaneously as mentioned in this paper. But it does not consider the nonadiabatic effects which are crucial in modeling dynamics on multiple electronic states.
Abstract: The ab initio multiple spawning (AIMS) method is a time-dependent formulation of quantum chemistry, whereby the nuclear dynamics and electronic structure problems are solved simultaneously. Quantum mechanical effects in the nuclear dynamics are included, especially the nonadiabatic effects which are crucial in modeling dynamics on multiple electronic states. The AIMS method makes it possible to describe photochemistry from first principles molecular dynamics, with no empirical parameters. We describe the method and present the application to two molecules of interest in organic photochemistryethylene and cyclobutene. We show that the photodynamics of ethylene involves both covalent and ionic electronic excited states and the return to the ground state proceeds through a pyramidalized geometry. For the photoinduced ring opening of cyclobutene, we show that the disrotatory motion predicted by the Woodward−Hoffmann rules is established within the first 50 fs after optical excitation.
724 citations
TL;DR: The semiclassical (SC) initial value representation (IVR) as mentioned in this paper provides a potentially practical way for adding quantum mechanical effects to classical molecular dynamics (MD) simulations of the dynamics of complex molecular systems (i.e., those with many degrees of freedom).
Abstract: The semiclassical (SC) initial value representation (IVR) provides a potentially practical way for adding quantum mechanical effects to classical molecular dynamics (MD) simulations of the dynamics of complex molecular systems (i.e., those with many degrees of freedom). It does this by replacing the nonlinear boundary value problem of semiclassical theory by an average over the initial conditions of classical trajectories. This paper reviews the background and rebirth of interest in such approaches and surveys a variety of their recent applications. Special focus is on the ability to treat the dynamics of complex systems, and in this regard, the forward−backward (FB) version of the approach is especially promising. Several examples of the FB-IVR applied to model problems of many degrees of freedom show it to be capable of describing quantum effects quite well and also how these effects are quenched when some of the degrees of freedom are averaged over (“decoherence”).
708 citations
TL;DR: In this article, a review of the quantum phase-space distribution functions with emphasis on both the fundamental characteristics and practical applications of the distribution functions is given, with particular attention to the Wigner distribution function and the Husimi distribution function.
698 citations
TL;DR: In this article, a two-photon or multiphoton process that is resonant with an excited electronic state can be used to assist chemistry on the ground state potential energy surface by controlling the delay between a pair of ultrashort (femtosecond) laser pulses.
Abstract: We present a novel approach to the control of selectivity of reaction products The central idea is that in a two‐photon or multiphoton process that is resonant with an excited electronic state, the resonant excited state potential energy surface can be used to assist chemistry on the ground state potential energy surface By controlling the delay between a pair of ultrashort (femtosecond) laser pulses, it is possible to control the propagation time on the excited state potential energy surface Different propagation times, in turn, can be used to generate different chemical products There are many cases for which selectivity of product formation should be possible using this scheme We illustrate the methodology with numerical application to a variety of model two degree of freedom systems with two inequivalent exit channels Branching ratios obtained using a swarm of classical trajectories are in good qualitative agreement with full quantum mechanical calculations
604 citations
TL;DR: In this paper, an unconventional time dependent formula for total photodissociation cross sections is presented, which is directly applicable to polyatomic photofragmentation, and as a spinoff is derived the polyatomic generalization of the (diatomic) reflection method.
Abstract: An unconventional time dependent formula for total photodissociation cross sections shows the importance of short time dynamics in direct photofragmentation. This is exploited to provide a systematic expansion in powers of h/ for the cross section. The lowest order term is a classical cross section which is shown to be an improvement upon the venerable reflection approximation. Terms to higher order in h/ lead to even greater improvements in accuracy as shown by simple numerical examples. Our formulas are directly applicable to polyatomic photofragmentation, and as a spinoff we derive the polyatomic generalization of the (diatomic) reflection method.
571 citations
References
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Book•
01 Jan 1965
TL;DR: Au sommaire as discussed by the authors developed the concepts of quantum mechanics with special examples and developed the perturbation method in quantum mechanics and the variational method for probability problems in quantum physics.
Abstract: Au sommaire : 1.The fundamental concepts of quantum mechanics ; 2.The quantum-mechanical law of motion ; 3.Developing the concepts with special examples ; 4.The schrodinger description of quantum mechanics ; 5.Measurements and operators ; 6.The perturbation method in quantum mechanics ; 7.Transition elements ; 8.Harmonic oscillators ; 9.Quantum electrodynamics ; 10.Statistical mechanics ; 11.The variational method ; 12.Other problems in probability.
8,141 citations
TL;DR: In this article, the Boltzmann formula for the probability of a configuration is given in classical theory by means of a probability function, and the result discussed is developed for the correction term.
Abstract: The probability of a configuration is given in classical theory by the Boltzmann formula $\mathrm{exp}[\ensuremath{-}\frac{V}{\mathrm{hT}}]$ where $V$ is the potential energy of this configuration. For high temperatures this of course also holds in quantum theory. For lower temperatures, however, a correction term has to be introduced, which can be developed into a power series of $h$. The formula is developed for this correction by means of a probability function and the result discussed.
6,791 citations
TL;DR: In this article, the Boltzmann formula for lower temperatures has been developed for a correction term, which can be developed into a power series of h. The formula is developed for this correction by means of a probability function and the result discussed.
Abstract: The probability of a configuration is given in classical theory by the Boltzmann formula exp [— V/hT] where V is the potential energy of this configuration. For high temperatures this of course also holds in quantum theory. For lower temperatures, however, a correction term has to be introduced, which can be developed into a power series of h. The formula is developed for this correction by means of a probability function and the result discussed.
5,865 citations
Book•
01 Jan 1966
3,150 citations