Wave-function Cluster Expansion

There are at least three types of cluster expansions, perhaps the most conventional simply being based on an ordinary MO-based SCF solution, on a full space entailing both covalent and ionic structures. Though the wave-function has delocalized orbitals, the expansion is profitably made in a localized framework, at least if treating one of the VB models or one of the Hubbard/PPP models near the VB limit -and really such is the point of the so-called Gutzwiller Ansatz [52], The problem of matrix element evaluation for extended systems turns out to be somewhat challenging with many different ideas for their treatment [53], and a neat systematic approach is via Cizek s [54] coupled-cluster technique, which now has been quite successfully used making use [55] of the localized representation for the excitations. 

Also a cluster expansion based on the set of Kekule structures is possible [63], and indeed (in the nomenclature used here) evidently yields the first suggested many-body resonating VB solution scheme (earlier many-body approaches seeking solutions to VB models without much attention to the chemically appealing local spin pairing). 

This scheme in its lowest order with spin pairing constrained to nearest neighbors (i.e., Kekule structures) has now been rather widely studied [64], and extensions beyond nearest-neighbor spin pairing evidently can be made to a limited extent for modest improvement [65] or substantially further for quite high accuracy [66], 

Further there are yet some other seemingly exotic wavefunction AnsStze which might be classified as cluster expansions [67], These involve phrases such as flux phase , spiral phase , and commensurate flux phase . 

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Coupled cluster calculations are similar to conhguration interaction calculations in that the wave function is a linear combination of many determinants. However, the means for choosing the determinants in a coupled cluster calculation is more complex than the choice of determinants in a Cl. Like Cl, there are various orders of the CC expansion, called CCSD, CCSDT, and so on. A calculation denoted CCSD(T) is one in which the triple excitations are included perturbatively rather than exactly. 

In addition to the encouraging numerical results, the canonical transformation theory has a number of appealing formal features. It is based on a unitary exponential and is therefore a Hermitian theory it is size-consistent and it has a cost comparable to that of single-reference coupled-cluster theory. Cumulants are used in two places in the theory to close the commutator expansion of the unitary exponential, and to decouple the complexity of the multireference wave-function from the treatment of dynamic correlation. 

Cl expansion of the CCSD/EOMCCSD wave function which can be easily determined using the CCSD/EOMCCSD cluster and excitation amplitudes fo(At), C(r)) md r (At)- As in the case of... 

In this section we shall discuss an approach which is neither variational nor perturba-tional. This approach also has its origin in nuclear physics and was introduced to quantum chemistry by Sinanoglu47, It is based on a cluster expansion of the wave function. A systematic method for the calculation of cluster expansion components of the exact wave function was developed by C ek48 The characteristic feature of this approach is the expansion of the wave function as a linear combination of Slater determinants. Formally, this expansion is similar to the ordinary Cl expansion. The cluster expansion, however, gives us not only the physical insight of the correlation energy but it also shows the connections between the variational approaches (Cl) and the perturbational approaches (e.g. MB-RSPT). 

Let us compare our cluster expansion with the well known Cl expansion of the wave function... 

In Section V we presented a method which is neither perturbational nor variational but which possesses certain features of both. Here, we shall make a few remarks on another promising method of this kind. Instead of the cluster expansion of the wave function this method is based on the idea of direct combination of the perturbation expansion with the variational method. 

There is an intimate connection between the cluster-expansion of a wave-function and the property of size-extensivity. To describe this aspect in the simplest manner,it is pertinent to recall first the closed-shell ground state. The ways to encompass the open-shell states can then be indicated as appropriate extensions and generalizations of the closed-shell cluster expansion strategy. 

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. 

Recently Geerstsen and Oddershede /106/ utilized the CC ground state function for calculating EE using polarization propagator. This approach may also be viewed as a semi-cluster expansion strategy. However a clear connection with the wave function approach is difficult to establish in a propagator theory (unless it is consistent /63,84/) and we shall not elaborate further on this theory. 

As explained in Sec.2, the full cluster—expansion theories in Hilbert space are designed to compute wave-functions for the open-shell states that are explicitly size-extensive with respect to the total number of electrons N. The underlying cluster structure of all these developments is what was envisaged by Silverstone and Sinanoglu/48/s... 

The cluster expansion methods are based on an excitation operator, which transforms an approximate wave function into the exact one according to the exponential ansatz... 

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