The Linked Diagram Theorem

The contributions to the first term which have a non-linear dependence on N 

When the single determinant many-electron functions are constructed from canonical Hartree-Fock orbitals, the excited functions, and , are doubly excited with respect to the reference function . The second term in the third order energy expression cancels diagonal components for which p = v in the first term. The principal term in the fourth order energy expression has the form 

Both and can involve double replacements with respect to the single determinantal reference function constructed from canonical Hartree-Fock orbitals. The intermediate state can involve single, double, triple and quadruple replacements with respect to the reference function. The renormalization terms in (44) involve only double replacement. Brueckner91 showed that diagonal terms in the principal term which have a non linear dependence of N are completely cancelled by the renormalization terms. This cancellation is incomplete if the level of replacement employed in generating the intermediate states is restricted. 

In Goldstone s treatment, the model state at / = —oo is allowed to evolve to the fully interacting state at t = 0. The time-dependent Schrodinger equation 

The time evolution operator, Ux(t, /0) itself satisfies the time-dependent Schro-dinger equation 

---

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. 

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

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. 

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... 

Coupled electron pair and cluster expansions. - The linked diagram theorem of many-body perturbation theory and the connected cluster structure of the exact wave function was first established by Hubbard211 in 1958 and exploited in the context of the nuclear correlation problem by Coester212 and by Coester and Kummel.213 Cizek214-216 described the first systematic application to molecular systems and Paldus et al.217 described the first ab initio application. The analysis of the coupled cluster equations in terms of the many-body perturbation theory for closed-shell molecular systems is well understood and has been described by a number of authors.9-11,67,69,218-221 In 1992, Paldus221 summarized the situtation for open-shell systems one must nonetheless admit... 

One important property of Eq. (23), which manifests itself very easily in the diagrammatic evaluation of the perturbation expansions, is the linked-diagram theorem (LOT). According to this theorem, only a selected class of terms survives in the perturbation expansion of the wave operator, while all unlinked diagrams cancel in each order of perturbation theory in addition, this theorem ensures the size consistency of MBPT and coupled-cluster theory. Based on LDT, Eq. (23) simplify to... 

It must be based either directly or indirectly on the linked diagram theorem of many-body perturbation theory so as to ensure that the calculated energies and other expectation values scale linearly with particle number... 

We know that the Rayleigh-Schrodinger perturbation theory series leads directly to the many-body perturbation theory by employing the linked diagram theorem. This theory uses factors of the form Eq—Ek) as denominators. Furthermore, this theory is fully extensive it scales linearly with electron number. The second term... 

Size-extensivity of MP Methods and the Linked Diagram Theorem... 

It is useful to start with the linked diagram theorem and to use only linked diagrams in the perturbation expansion. Clearly, in this way the time-consuming handling of unphysical terms, which will cancel anyway for a size-extensive PT method, can be avoided. This is the strategy of MBPT, which can be considered as that PT which is size-extensive for any number of electrons since it is based on the linked diagram theorem. MP theory is an MBPT with the MP perturbation operator and, therefore, it is reasonable to speak of MBPT(2), MBPT(3), etc., instead of MP2, MP3, etc. 

---