Brueckner-Goldstone MBPT

In Chapter 2, the author describes state- and property-specific quantum chemistry including comments on certain aspects of the separation of electron correlation into its dynamical and nondynamical parts. The account contains also a historic appraisal of multiconfigurational schemes comparing Brueckner-Goldstone many-body perturbation theory, MBPT,... 

Hence, Brueckner-Goldstone many-body perturbation theory (MBPT) makes it possible to write RSPT in the form. 

Brueckner-Goldstone (BG) MBPT [91,92] was used for the determination of atomic correlation energies and polarizabilities and soon applied... 

In any case, the general formulation of the MBPT—primarily due to Brueckner [31], Goldstone [32], Hugenholtz [33], and Hubbard [34]— has shed much lighten the structure of fully correlated, exact A-fermion wave functions, and was essential to the development of perturbative-type methods, including CC theory. For this reason, the next section is devoted to this topic. 

The conjecture that the unlinked terms exactly cancel out in every order of the RS MBPT, as implied by Brueckner s observation [31] that this is the case up to and including the fourth order, was soon proved by Goldstone [32]. At almost the same time, independent derivations were also provided by Hugenholtz [33] and Hubbard [34]. 

At that time, the best-known method for electron correlation in molecules was undoubtedly configuration interaction (Cl). This tool had developed in the hands of Slater and Condon, with early applications by Boys, Parr, Matsen, and their coworkers around 1950 (see Ref. [2] for an excellent review). Somewhat less known to the quantum chemistry community was the parallel development in the mid 1950s of the correlation problem in physics that originated with Brueckner [3] and Goldstone [4], termed many-body perturbation theory (MBPT) because it was applicable to many-electron systems. This feature, that we now call size-extensivity [5], was not shared by Cl, but was a necessity for the physics applications to nuclear matter and the electron gas. Important questions at this time included the correlation treatment of the high- and low-density electron... 

Many-body methods, based on the linked-cluster expansion (LCE), were first developed by Brueckner [1] and Goldstone [2] in the 1950s for nuclear physics problems. Perturbation-theory applications to atomic and molecular systems (in a numerical, one-center frame) were pioneered by Kelly [3] in the early 1960s. Basis sets were later introduced, first in second-order [4] and then in third-order [5]. The 1970s saw a proliferation of molecular applications with basis sets, under the names of many-body perturbation theory (MBPT) [6] or the Moller-Plesset method [7]. Nowadays, many-body methods offer some of the most powerful tools in the quantum chemistry arsenal, in particular the coupled-cluster (CC) method, and are available in many widely used quantum chemistry program packages. 

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