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Variational Problems in Molecular Simulation

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Encyclopedia of Applied and Computational Mathematics

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Short Definition

At the basis of all computations in atomic and molecular physics and chemistry lies the fact that all their possible states are given as critical points of some functional. The ground states are minimizers. So, most computations in this area are based on the analytical study of the corresponding variational problems, and their numerical discretization.

General Presentation

When trying to understand the properties of a (nonrelativistic) molecule with M atomic nuclei and N electrons, the basic tool is the so-called Schrödinger Hamiltonian [29]

$$\displaystyle\begin{array}{rcl} H :& =& -\sum _{i=1}^{N}\frac{1} {2}\varDelta _{x_{i}} -\sum _{i=1}^{N}\sum _{ k=1}^{M} \frac{z_{k}} {\vert x_{i} -\bar{x}_{k}\vert } {}\\ & & +\sum _{1\leq i<j\leq N} \frac{1} {\vert x_{i} - x_{j}\vert }\,, {}\\ \end{array}$$

where for \(i = 1,\ldots ,M\), the i-th nucleus is supposed to be at \(\bar{x}_{i} \in \mathbb{R}^{3}\) and have charge z i . In writing the above Hamiltonian, we have...

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Authors and Affiliations

  1. CEREMADE, CNRS and Université Paris-Dauphine, Paris, France

    MariaJ. Esteban

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  1. MariaJ. Esteban
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Editors and Affiliations

  1. University of Texas at Austin, Austin, TX, USA

    Björn Engquist

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Esteban, M. (2015). Variational Problems in Molecular Simulation. In: Engquist, B. (eds) Encyclopedia of Applied and Computational Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70529-1_244

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