Size-extensivity Ansatz

It is of some note that many of the models may be (and often were) obtained by-passing the derivational approach here. Basically each model may be viewed as represented by the first terms in a graph-theoretic cluster expansion [80]. Once the space on which the model to be represented is specified, the interactions in the orthogonal-basis cases are just the simplest additive few-site operators possible. For the nonorthogonal bases the overlaps are just the simplest multiplicative operators possible, while the associated Hamiltonian operators are the simplest associated derivative operators. These ideas lead [80] to proper size-consistency and size extensivity. Similar sorts of ideas apply in developing wavefunction Ansatze or ground-state energy expansions for the various models. 

Presently, the widely used post-Hartree-Fock approaches to the correlation problem in molecular electronic structure calculations are basically of two kinds, namely, those of variational and those of perturbative nature. The former are typified by various configuration interaction (Cl) or shell-model methods, and employ the linear Ansatz for the wave function in the spirit of Ritz variation principle (c/, e.g. Ref. [21]). However, since the dimension of the Cl problem rapidly increases with increasing size of the system and size of the atomic orbital (AO) basis set employed (see, e.g. the so-called Paldus-Weyl dimension formula [22,23]), one has to rely in actual applications on truncated Cl expansions (referred to as a limited Cl), despite the fact that these expansions are slowly convergent, even when based on the optimal natural orbitals (NOs). Unfortunately, such limited Cl expansions (usually truncated at the doubly excited level relative to the IPM reference, resulting in the CISD method) are unable to properly describe the so-called dynamic correlation, which requires that higher than doubly excited configurations be taken into account. Moreover, the energies obtained with the limited Cl method are not size-extensive. 

Another important development was the reahzation of the importance of the size-consistency and size-extensivity in the studies of associative or dissociative chemical processes by Primas [67], as well as his clear delineation of the relationship between the configuration interaction and the exponential coupled-cluster Ansatze. 

Moreover, once the cluster Ansatz is introduced (for an option of directly solving Bloch equations without invoking the cluster Ansatz, see Ref. [200]), it is essential that the so-called complete model space (CMS), spanned by configurations involving all possible occupancies of valence or active (spin) orbitals, be used, lest the desirable property of size-extensivity be violated. This requirement, however, leads not only to highly dimensional (and thus computationally demanding) model spaces, but, most importantly, to the occurrence of the so-called intruder states. 

Abstract We present in this paper a comprehensive study of the various aspects of size extensivity of a set of unitary group adapted multi-reference coupled cluster (UGA-MRCC) theories recently developed by us. All these theories utilize a Jez-iorski-Monkhorst (JM) inspired spin-free cluster Ansatz of the forml P) = = exp(r ), where is... 

The coupled-cluster electronic state is uniquely defined by the set of the cluster amplitudes and these amplitudes are used to obtain the coupled-cluster energy from Eq. (33). Due to the fact that the Ansatz of the coupled-cluster wave function has the exponential parametrization [Eq. (28)] the energy is size-extensive. This is an obvious advantage of the coupled-cluster formalism compared to some other techniques (e.g. configuration interaction). For a general discussion of coupled-cluster theory and the coupled-cluster equations see Refs. [5, 36]. 

Let us now turn to the coupled pair methods. The conventional discussion of these approaches starts from the well known exp(T) ansatz for the wave operator. Since this ansatz is reviewed elsewhere, we will choose an alternative route, which is in fact more straightforward. From a practical (or technical) point of view, pair techniques aim at a size-extensive treatment at the computational expense of a Cl calculation with single and double excitations (CI(SD)), which requires including effects of higher excitations in an approximate manner. This goal has been approached in basically three different ways ... 

The use of the exponential ansatz in formulating a quantum mechanical many-body theory - a theory which is extensive and scales linearly with the number of electrons studied - was first realized in nuclear physics by Coester and Kiimmel [64-66], The origins of the cluster approach to many-fermion systems, goes back to the early 1950s, when the first attempts were made to understand the correlation effect in an electron gas [67,68] and in nuclear matter [69], For both of these systems, it was absolutely essential that the method employed scaled linearly with the number of particles, i.e. that it is size extensive . 

The arguments for size-extensivity in CCPT follow closely those for linked coupled-cluster theory in Section 13.3. Thus, whereas the order-by-order size-extensivity of the energy follows ftom the exponential ansatz for the wave function, the termwise size-extensivity follows from the use of a similarity-transformed Hamiltonian, assuming that both the zero-order Fock operator and the first-order fluctuation operator separate for noninteracting systems. 

In contrast to Cl, the CC approaches, even at the SR level, very efficiently account for the dynamic correlation thanks to the exponential CC Ansatz for the wave operator. The general form of the CC wave function also automatically guarantees the size-extensivity of the computed energies [as do, in fact, the individual linked diagrams of the many-body perturbation theory (MBPT)]. Unfortunately, this size-extensive property is of a little use when the nondynamic correlation is not properly accounted for. Indeed, the CCSD PECs often display an artificial "hump in the region of intermediate internuclear separations, as well as grossly erroneous asymptotic behavior in the completely dissociated limit [cf., e.g. the CCSD PECs for N2 in Refs. (5,9)]. 

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 product state thus represents not only a solution for the supersystem (10.2.27) but also a size-extensive solution (10.2.28). Since this argument may be extended to open-shell systems, we conclude that the Hartree-Fock model is size-extensive. In Section 4.3, we demonstrated that exponentially generated wave functions are in general size-extensive. The size-extensivity of the Hartree-Fock model is therefore a direct consequence of the exponential ansatz (10.1.8). 

Chapter 13 discusses coupled-cluster theory. Important concepts such as connected and disconnected clusters, the exponential ansatz, and size-extensivity are discussed the Unked and unlinked equations of coupled-clustCT theory are compared and the optimization of the wave function is described. Brueckner theory and orbital-optimized coupled-cluster theory are also discussed, as are the coupled-cluster variational Lagrangian and the equation-of-motion coupled-cluster model. A large section is devoted to the coupled-cluster singles-and-doubles (CCSD) model, whose working equations are derived in detail. A discussion of a spin-restricted open-shell formalism concludes the chapter. 

This is without doubt a significant improvement since the application of the Born-Handy ansatz as a full replacement of the COM separation is not restricted by the size of the system under investigation. However, its applicability is unfortunately limited to adiabatic systems only. The Born-Handy formulation yields only the adiabatic corrections to the B-0 results. Beyond this approach we enter the enigmatic region, which is usually denoted as the break-down of B-O approximation. Therefore our main goal is here to find an extension of the Born-Handy formula , which is valid both in the adiabatic limit as well as beyond. 

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