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Perturbation Theory and Molecular Dynamics

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

In the framework of Quantum Mechanics the dynamics of a molecule is governed by the (time‐dependent) Schrödinger equation , involvingnuclei and electrons coupled through electromagnetic interactions. While the equation is mathematically well-posed, yielding the existence ofa unique solution, the complexity of the problem makes the exact solution unattainable. Even for small molecules, the large number of degrees offreedom prevents from direct numerical simulation, making an approximation scheme necessary.

Indeed, one may exploit the smallness of the electron/nucleus mass ratio to introduce a convenient computational scheme leading to approximatesolutions of the original time‐dependent problem. In this article we review the standard approximation scheme (dynamical Born–Oppenheimer approximation ) together with its ramifications and some recent generalizations, focusing on mathematically rigorousresults.

The success of this approximation scheme is rooted in...

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Abbreviations

Adiabatic decoupling :

In a complex system (either classical or quantum), the dynamical decoupling between the slow and the fast degrees of freedom.

Adiabatic perturbation theory :

A mathematical algorithm which exploits the adiabatic decoupling of degrees of freedom in order to provide an approximated (but yet accurate) description of the slow part of the dynamics. In the framework of QMD, it is used to approximately describe the dynamics of nuclei, the perturbative parameter ε being related to the small electron/nucleus mass ratio.

Electronic structure problem:

The problem consisting in computing, at fixed positions of thenuclei, the energies (eigenvalues) and eigenstates corresponding to the electrons. An approximate solution is usually obtained numerically.

Molecular dynamics :

The dynamics of the nuclei in a molecule. While a first insight in the problem can be obtained by using classical mechanics (Classical Molecular Dynamics), a complete picture requires quantum mechanics (Quantum Molecular Dynamics) Perturbation Theory in Quantum Mechanics. This contribution focuses on the latter viewpoint.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “La Sapienza”, Roma, Italy

    Gianluca Panati

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  1. Gianluca Panati
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Panati, G. (2009). Perturbation Theory and Molecular Dynamics. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_399

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