Goldstone expansion

T/ie Goldstone expansion is rederived by elementary time-independent methods, starting from Brillouin-Wigner (BW) perturbation theory. Interaction energy terms AE are expanded out of the BW energy denominators, and the series is then rearranged to obtained the linked-cluster result. ... 

In the Brueckner-Hartree-Fock (BHF) approximation, the Brueckner-Bethe-Goldstone (BBG) hole-line expansion is truncated at the two-hole-line level [5]. The short-range NN repulsion is treated by a resummation of the particle-particle ladder diagrams into a n effect vc i n tcract ion or G-matrix. Self-consistency is required at the level of the BHF single-particle spectrum eBHF(k),... 

It is also interesting to note that the explicit dependence on the quark chemical potential is communicated to the Goldstone excitations via the coefficients of the effective Lagrangian (see [31] for a review). For example is proportional to fj, in the high chemical potential limit and the low energy effective theory is a good expansion in the number of derivatives which allows to consistently incorporate in the theory the Wess-Zumino-Witten term [32] and its corrections. 

Draw all possible distinct Hugenholtz diagrams for a pertinent order of the perturbation expansion ) and generate from them Goldstone diagrams by (80). 

We may recall that the desirability of ensuring size-extensivity for a closed-shell state was one of the principal motivations behind the formulation of the MBPT for the closed-shells. The linked cluster theorem of Bruckner/25/, Goldstone/26/ and Hubbard/27/, proving that each term in the perturbation series for energy can be represented by a linked (connected) diagram directly reflects the size-extensivity of the theory. Hubbard/27/ and Coester/30/ even pointed out immediately after the inception of MBPT/25,26/, that the size-extensivity is intimately related to a cluster expansion structure of the associated wave-operator that is not just confined only to perturbative theory. The corresponding non-perturbative scheme for the closed-shells was first described by Coester and Kummel/30,31/ in nuclear physics and this was transcribed to quantum chemistry... 

The 1inked-cluster theorem for energy, from the above analysis, is a consequence of the connectivity of T, and the exponential structure for ft. Size-extensivity is thus seen as a consequence of cluster expansion of the wave function. Specfic realizations of the situation are provided by the Bruckner—Goldstone MBPT/25,26/, as indicated by Hubbard/27/, or in the non-perturbative CC theory as indicated by Coester/30,31/, Kummel/317, Cizek/32/, Paldus/33/, Bartlett/21(a)/ and others/30-38/. There are also the earlier approximate many-electron theories like CEPA/47/, Sinanoglu s Many Electron Theory/28/ or the Cl methods with cluster correction /467. 

The linked diagram expansion has, indeed, been derived by many authors and we shall, therefore, content ourselves with a brief outline of the Goldstone derivation here referring the interested reader elsewhere for full details.8-5... 

Step 4. When we happen to develop a perturbation expansion with respect to the Hartree-Fock reference state, the number of diagrams is further reduced. In this case, all diagrams produced by a contraction within an operator, which results in Goldstone terminology to a bubble ... 

A Computer-Algebraic Approach to the Derivation of Feynman-Goldstone Perturbation Expansions for Open-Shell Atoms and Molecules... 

The operators above in second quantization can be represented and manipulated also diagrammatically, that is by means of Feynman-Goldstone diagrams. Usually, these representations provide a simple and transparent approach to evaluate the various contributions from the perturbation expansions. There are different graphical representations known, which were developed by Goldstone [8],... 

Evaluation of the operator products as they occur either on the rhs of fhe Bloch Eq. (23) or in fhe definifion of fhe effective operators (15) and (49). The aim of this step is to bring all the creation and annihilation operators (in each term of the expansions) into the extended normal-order form (50). The resulf is a sequence of normal-ordered operator ferms (briefly referred to as Feynman-Goldstone diagrams). 

TABLE 10.1 Main commands of the GOLDSTONE program for deriving atomic and molecular perturbation expansions within an (one-electron) orbital picture ... 

Every term in this expansion (as indicated by the integer at the end of the line) represents a Feyman-Goldstone diagram and could be drawn also graphically. These are the (three) well-known diagrams of the second-order correlation energy for closed-shell systems as found at different places in the literature. If we assume a Hartree-Fock (one-particle) spectrum, instead,... 

These a posteriori corrections are based on a very simple idea which is suggested by the work of Brandow [10]. Brandow used the Brillouin-Wigner perturbation theory as a starting point for a derivation of the Goldstone linked diagram expansion by elementary time-independent methods . At a NATO Advanced Study Institute held in 1991, Wilson wrote [112] ... 

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