Abstract
In this first part of the book we shall review some basics of quantum mechanics and the many-body theory for bound electronic systems that will form the foundations for the following treatment. This material can also be found in several standard text books.
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Notes
- 1.
This chapter is essentially a short summary of the second part of the book Atomic Many-Body Theory by Lindgren and Morrison [124], and the reader who is not well familiar with the subject is recommended to consult that book.
- 2.
Initially, we shall use the ‘hat’ symbol to indicate an operator, but later we shall use this symbol only when the operator character needs to be emphasized.
- 3.
Note that according to the quantum-mechanical picture the wave function has a single time also for a many-electron system. This question will be discussed further in later chapters.
- 4.
The symbol “\( = \):” indicates that this is a definition.
- 5.
In the case of an extended model space, we shall normally use the symbol \({\mathcal {E}}\) for the different energies of the model space.
- 6.
A closed diagram has the initial as well as the final state in the model space. Such a diagram can—in the case of complete model space—have no other free lines than valence lines. A diagram that is not closed is said to be open.
- 7.
The Rayleigh-Schrödinger and the linked-diagram expansions have the advantage compared to, for instance, the Brillouin-Wigner expansion, that they are size-extensive, which implies that the energy of a system increases linearly with the size of the system. This idea was actually behind the discovery of the linked-diagram theorem by Brueckner [40], who found that the so-called unlinked diagrams have a non-physical non-linear energy dependence and therefore must be eliminated in the complete expansion. The concept of size extensivity should not be confused with the term size consistency, introduced by People [197, 198], which implies that the wave function separates correctly when a molecule dissociates. The Rayleigh-Schrödinger or linked-diagram expansions are generally not size consistent. The coupled-cluster approach (to be discussed below), on the other hand, does have this property in addition to the property of size extensivity.
- 8.
The distinction between linked and connected diagrams should be noted. A linked diagram can be disconnected, if all parts are open, as defined in Sect. 2.4.
- 9.
It should be noted that this definition leads to different classification, depending on the effects studied. If only the energy or the effective Hamiltonan is studied, then a diagram would be reducible if it contains an intermediate model-space state. If, on the other hand, also the wave function or wave operator is studied, then the diagram would be reducible if there is an intermediate state free from photons.
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Lindgren, I. (2016). Time-Independent Formalism. In: Relativistic Many-Body Theory. Springer Series on Atomic, Optical, and Plasma Physics, vol 63. Springer, Cham. https://doi.org/10.1007/978-3-319-15386-5_2
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DOI: https://doi.org/10.1007/978-3-319-15386-5_2
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