Diagrams linked

Nakatsuji and Yasuda [56, 57] focused on the term appearing in A, which according to the perturbative expansion could be interpreted as a linked diagram of two A elements. In analogy to the Dyson equation, they proposed to estimate the A with a procedure whose main step, expressed in a spin-orbital basis, may be written... 

I. Lindgren, The Rayleigh-Schrodinger perturbation and the linked-diagram theorem for a multi configurational model space, J. Phys. B At. Mol. Opt. Phys. 7 (1974) 2441. 

The coupled cluster (CC) method is actually related to both the perturbation (Section 5.4.2) and the Cl approaches (Section 5.4.3). Like perturbation theory, CC theory is connected to the linked cluster theorem (linked diagram theorem) [101], which proves that MP calculations are size-consistent (see below). Like standard Cl it expresses the correlated wavefunction as a sum of the HF ground state determinant and determinants representing the promotion of electrons from this into virtual MOs. As with the Mpller-Plesset equations, the derivation of the CC equations is complicated. The basic idea is to express the correlated wave-function Tasa sum of determinants by allowing a series of operators 7), 73,... to act on the HF wavefunction ... 

The inspection of the K dependence shows that the unlinked diagrams such as 96 scale as K1 while the other (linked) diagrams as K°. Equation (2-95) is shown to behave in the same fashion, i.e.,... 

Table VII Fourth-order linked-diagram triple-excitation and quadruple excitation components of the electron correlation energy in a number of atoms and small molecules...

It is clear that the new generation of computers - supercomputers when used in conjunction with theoretical developments such as the linked diagram theory or the unitary group theory, is going to have a significant impact on the accuracy attainable in quantum chemistry and on the size of problem which may be treated. 

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... 

The relationship between the coupled-electron pair approximation (c.e.p.a.) and the many-body perturbation theory has been discussed in detail by Ahl-richs.149 All of the methods denoted by c.e.p.a. (x) (x=0, 1, 2, 3) may be related to the summation of certain classes of diagrams in the many-body perturbation theory to infinite order. For example, c.e.p.a. (0), which is Cizek s linear approximation or Hurley s c.p.a. (0) ansatz150 is equivalent to the summation of all double-excitation linked diagrams in the perturbation series. This is also denoted d.e.m.b.p.t. (double excitation many-body perturbation theory) by some workers.151 168... 

Hence, a statement of the linked diagram theorem is that... 

We express contributions to ft(n) diagrammatic ally, but they can be automatically identified through Eq. (23) with contributions to the (n). These are shown for the first- and second-order MBPT wave function in Fig. 1. In the wave operator expansion, we encounter disconnected but linked diagrams, i.e., diagrams that are composed of two or more separate parts like... 

The subscripted sum Y L is used to indicate that the only terms which are included are the so-called linked diagrams [245], i.e. those Feynman graphs which contain no external lines or any part of the diagram completely disconnected from the rest. 

The failing of MBPT is that it is basically an order-by-order perturbation approach. For difficult correlation problems it is frequently necessary to go to high orders. This will be the case particularly when the single determinant reference function offers a poor approximation for the state of interest, as illustrated by the foregoing examples at 2.0 R. A practical solution to this problem is coupled-cluster (CC) theory. In fact, CC theory simplifies the whole concept of extensive methods and the linked-diagram theorem into one very simple statement the exponential wavefunction ansatz. 

Similarly, the exact wave function I P) = I Pq) is given by aU non-vacuum linked diagrams, namely. 

However, a far more pervasive way to remove the perturbative nature of MBPT is offered by coupled-cluster theory, and this takes us to its origin. The usual expression for the linked diagram theorem in the time-dependent development is as a logarithm. Inverting the procedure, Hubbard [16] seems to have been the first to recognize that the linked diagram wavefunction above can be most conveniently written as... 

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