Size extensive formulations

SIZE-EXTENSIVE FORMULATIONS WITH INCOMPLETE MODEL SPACE (IMS)... 

We define an operator as closed , if its action on any model function G P produces only internal excitations within the IMS. An operator is quasi-open , if there exists at least one model function which gets excited to the complementary model space R by its action. Obviously, both closed and quasi-open operators are all labeled by only active orbitals. An operator is open , if it involves at least one hole or particle excitation, leading to excitations to the g-space by acting on any P-space function. It was shown by Mukheijee [28] that a size-extensive formulation within the effective Hamiltonians is possible for an IMS, if the cluster operators are chosen as all possible quasi-open and open excitations, and demand that the effective Hamiltonian is a closed operator. Mukhopadhyay et al. [61] developed an analogous Hilbert-space approach using the same idea. We note that the definition of the quasi-open and closed operators depends only on the IMS chosen by us, and not on any individual model function. 

An important insight in the development of size-extensive formulations in a IMS was the realization that the intermediate normalization convention for the wave operator, viz. PflP = P, should be abandoned in favor of a more appropriate normalization [28,61]. For the IMS, in general, products of quasi-open operators may lead to internal excitations, or may even be closed, so that if we choose il = 2 exp(7 )l< X< l, with = Top-f Tq-op, then powers of Tq. op coming from the exponential might lead from 4>p to internal excitations to some other model function or it may contain closed operators. We would have to bear this in mind while developing our formalism, and would not force POP = P in our developments. 

So far, we have considered only wave functions that are variationally determined. For technical reasons, many popular methods do not employ the variation principle for the construction of the wave function. As discussed in Section 13.1.4, the exponential ansatz does not lend itself easily to variational treatments and the wave functions are instead generated by a different principle. Nevertheless, we shall see that the exponential ansatz may still be given a size-extensive formulation. 

The main reason why existing MR CC methods as well as related MR MBPT cannot be considered as standard or routine methods is the fact that both theories suffer from the Intruder state problem or generally from the convergence problems. As is well known, both MR MBPT/CC theories are built on the concept of the effective Hamiltonian that acts in a relatively small model or reference space and provides us with energies of several states at the same time by diagonalization of the effective Hamiltonian. In order to warrant size-extensivity, both theories employ the complete model space formulations. Although conceptually simpler, the use of the complete model space makes the calculations rather... 

Very similar in spirit to CEPA, but formulated as a functional to be made stationary, is the coupled-pair functional (CPF) approach of Ahlrichs and co-workers [28]. CPF can be viewed as modifying the CISD energy functional to obtain size-extensivity for the special case of noninteracting two-electron systems. One disadvantage of some of the CEPA methods is that, unlike CISD or CCSD, the results are not invariant to a unitary transformation that mixes occupied orbitals with one another. CPF... 

The CASSCF method itself is not very useful for anything else than systems with few electrons unless an effective method to treat dynamical correlation effects could be developed. The Multi-Reference Cl (MRCI) method was available but was limited due to the steep increase of the size of the Cl expansion as a function of the number of correlated electrons, the basis set, and the number of active orbitals in the reference function. The direct MRCI formulation by P. Siegbahn helped but the limits still prevented applications to larger systems with many valence electrons [20], The method is still used with some success due to recent technological developments [21], Another drawback with the MRCI approach is the lack of size-extensivity, even if methods are available that can approximately correct the energies. Multi-reference coupled-cluster methods are studied but have not yet reached a state where real applications are possible. 

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 next set of open-shell cluster expansion theories to appear on the scene emphasized the size-extensivity feature (al), and all of them were designed to compute energy differences with a fixed number of valence electrons. Several related theories may be described here - (i) the level-shift function approach in a time-dependent CC framework by Monkhorst/56/ and later generalizations by Dalgaard and Monkhorst/57/, also by Takahasi and Paldus/105/, (ii) the CC-based linear response theory by Mukherjee and Mukherjee/58/, and generalized later by Ghosh et a 1/59.60.107/,(iii)the closely related formulations by Nakatsuji/50,52/ and Emrich/62/ and (iv) variational theories by Paldus e t a I / 54/ and Saute et. al /55/ and by Nakatsuji/50/. 

The next two formulations advocated maintaining size-extensivity with respect to the total electron number N, i.e., they subscribed to the... 

There are mainly two inter-related reasons why on would like to formulate a size-extensive theory involving incomplete model spaces (i) to bypass intruders, (ii) for calculating the low-lying EE s of a closed-shell ground state, one feels that a set of hole-particle determinants would suffice as a choice for a reasonable model space, which involves valence holes as well as valence particles. This is an IMS. 

Incomplete model spaces generate disconnected diagrams not only for the h used by Hose and Kaldor, but for other perturbative and CC formulations [20, 45, 46]. The effects of such diagrams on the size extensivity of the calculated energies depend both on the particular h and on the kind of model space used [21, 47-52]. Most of the recent work on... 

Coupled Cluster based size-extensive intermediate hamiltonian formalisms were developed by our group [33-35] by way of transcribing a size-extensive CC formulation in an incomplete model space in the framework of intermediate hamiltonians. In this method, there are cluster operators correlating the main model space. There are no cluster operators for the intermediate space. This formulation thus is conceptually closer to the perturbative version of Kirtman... 

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