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Erschienen in:
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2010 | OriginalPaper | Buchkapitel

3. The Many-Body Problem in Atoms and Molecules

verfasst von : Ivan Hubač, Stephen Wilson

Erschienen in: Brillouin-Wigner Methods for Many-Body Systems

Verlag: Springer Netherlands

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Abstract

The many-body problem in atomic and molecular physics and in quantum chemistry is described. The second-quantization formulation is introduced together with the diagrammatic techniques which form an essential ingredient of many-body methodology. The use of coupled cluster expansions, mixing of configurations (or configuration interaction) and perturbation theory series for the description of electron correlation effects is considered.

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Vorheriges Kapitel Brillouin-Wigner Perturbation Theory
Nächstes Kapitel Brillouin-Wigner Methods for Many-Body Systems
Fußnoten
1
For an interesting account of the history of quantum electrodynamics, see Schweber’s QED and the Men Who Made It: Dyson, Feynman, Schwinger and Tomonaga [21].
 
2
The modern formalism of relativistic quantum mechanics, quantum field theory, employs Feynman’s picture in which the negative energy states moving backwards in time, corresponds to positive energy states of antiparticles moving forward in time. The advantage of the Feynman picture is that it is applicable to both fermions and bosons.
 
3
Čížek has published a short historical review [60] of coupled cluster theory.
 
4
Ursell had published a paper [62] in the Proceedings of the Cambridge Philosophical Society entitled The evaluation of Gibbs phase-integral for imperfect gases in 1927 whilst Mayer published a paper [63] in 1937 entitled The Statistical Mechanics of Condensing Systems. I in the Journal of Chemical Physics.
 
5
A recent review of the group function model has been given by McWeeny [73].
 
6
In the preface to the second edition of his Methods of Molecular Quantum Mechanics in 1989, McWeeny comments on the methods of theoretical physics which have been introduced into quantum chemistry. These methods are “still unfamiliar in chemistry” and, “because of their complexity and sophistication, their use has been widely resisted”, but “their power and generality is now beyond doubt”.
 
7
Note that the same expressions can be obtained by using the Baker–Hausdorf–Campbell theorem [83, 84]. We can write
$$\mathcal{H}{e}^{T}\vert {\Phi }_{ 0}\rangle = {\mathcal{E}}_{0}{e}^{T}\vert {\Phi }_{ 0}\rangle$$
or
$$\langle {\Phi }_{0}\vert {e}^{-T}\mathcal{H}{e}^{T}\vert {\Phi }_{ 0}\rangle = {\mathcal{E}}_{0}.$$
Now, using Baker–Hausdorf–Campbell theorem, we have
$$\{{e}^{-T}\mathcal{H}{e}^{T}\} = \{{\mathcal{H}}_{ N}{e}^{T}\}_{ lc},$$
$$\langle {\Phi }_{0}\vert \{{\mathcal{H}}_{N}{e}^{T}\}_{ lc}\vert {\Phi }_{0}\rangle = {\mathcal{E}}_{0}$$
and for the amplitudes
$$\langle {\Phi }^{\text{ exc}}\vert \{{\mathcal{H}}_{N}{e}^{T}\}\vert {\Phi }_{ 0}\rangle = 0,$$
which is equivalent to eq. (3.252).
 
8
These details can be found in a number of review articles. See, for example, the reviews by Hubač and Čársky [85], by Paldus and Čížek [29], by Bartlett and his coworkers (see [86] and references therein), and by Urban et al. [87].
 
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Metadaten
Titel
The Many-Body Problem in Atoms and Molecules
verfasst von
Ivan Hubač
Stephen Wilson
Copyright-Jahr
2010
Verlag
Springer Netherlands
Buch
Brillouin-Wigner Methods for Many-Body Systems

Print ISBN: 978-90-481-3372-7

Electronic ISBN: 978-90-481-3373-4

Copyright-Jahr: 2010

DOI
https://doi.org/10.1007/978-90-481-3373-4_3