Model space complete

K. F. Freed, Tests and applications of complete model space quasidegenerate many-body perturbation theory for molecules, in Many-Body Methods in Quantum Chemistry (U. Kaldor, ed.), Springer, Berlin, 1989, p. 1. 

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... 

Turning now back to the single-root MR BWCC approach, we derive the basic equations for the effective Hamiltonian and cluster amplitudes in the spin-orbital form without the use of the BCH formula. We limit ourselves to a complete model space which implies that amplitudes corresponding to internal excitations (i.e. excitations within the model space) are equal to zero. In our derivation we shall work with the Hamiltonian in the normal-ordered-product form, i.e. 

Here a complete model space (P-space), defined on eigenfunctions of Ho, representing all possible distributions of electrons between open shells, is utilized. In our case the model space consists of the ls22s22p4 and ls22p6 configurations for the even parity states and of the ls22s2p5 configuration for odd parity states. The wave-operator may be written as... 

The essential requirement for arriving at the proof of the above linked valence expansion/3/ was the assumption that the model space is complete. With a complete model space, the model space functions separate into proper fragments as well, and Brandow s theory is thus additionally size-consistent. 

It should be emphasized here that the convenience of working in a complete model space for arriving at the connectedness of was first shown by... 

Mukherjee et. alZ68-69/ formulated an explicitly connected CC theory for complete model space by invoking a valence universal cluster operator Q of the form... 

Kutzelnigg et. al795/ have discussed in detail the Fock—space classification of operators for quasi— complete model space. They also introduced a new type od IMS, called the isolated incomplete model space(IIMS). In IIMS, products of q—open operators are all q-open, never closed. As a result O. = 1., just as in a CMS. THe resulting CC equations for IIMS have thus exacly the same structure as in the CMS. 

For complete model spaces, since IN is valid, we have... 

The above formulation is quite general and applies equally well to quasi-complete model spaces having m holes and n partic 1es.When there are several p-h valence ranks in the parent model space, the situation is fairly complicated. The subduced model spaces in this case may belong to the parent model space itself. The valence-universality of ft in such a situation implies that ft is the wave—operator for all the subduced model spaces, in addition to those which have same number of electrons as in the parent model space. It appears that a more convenient route to solve this problem is to redefine the core in such a way that holes for the problem become particles and treat it as an IMS involving valence particles only. 

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. 

Let us consider a complete model space spanned by M reference configurations, so that the model function for the ground state is expressed as... 

Another way to reduce the computational demand would be to work with an incomplete model (or active) space. Just like in the case of effective Hamiltonian-based theories, the choice of a complete model space (CMS) in a state-specific formalism... 

In our formalism, we choose in every all open and quasi-open operators. For an arbitrary IMS, a given quasi-open operator, acting on a given model function, may lead to excitation to some specific model function, but there would be at least one model function which, when acted upon by this quasi-open operator, would lead to excitations out of the IMS. A closed operator, by contrast, cannot lead to excitations out of the IMS by its action on any function in the IMS. Clearly, any pair of model functions and (f> can be reached with respect to each other by either a quasi-open or a closed operator, but not both. This follows from the definition of these operators. For an arbitrary IMS, it is possible to remain within the IMS if a quasi-open operator acts on a specific model function. On another model function, it may lead to excitation out of the IMS. The QCMS (Quasi-Complete Model Space) is a special class of IMS, where we group orbitals into various subsets, labeled A, B, etc. and form a model space spanned by model functions... 

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