Valence universal wave-operator

On the one hand, we can strive for a single cluster operator T, defining the valence universal wave operator U, U = exp(T), which will transform all the model space states ]< > ) into some linear combinations of the exact states jfl i), f = 1,2, , M, which in turn span the target space M, i.e.. 

It is interesting to note that ft in eq.(7.2.4) can be regarded as a valence universal wave-operator, i.e.,ft is also the wave-operator for the core-problem/94(a)/. This assertion follows from the simple observation that all T operators have destruction operators and hence they annihilate S. ... 

Clearly, with this choice, not only are the equations for the core cluster amplitudes decoupled from the rest, but also the equations for the T1n cluster— amplitudes have only T m>amplitudes with mvalence values. It should again be noted that O acting on a P for m this approach uses implicitly a valence-universal wave-operator. 

Mukherjee/69/, use of the sufficiency conditions (7.3.9) amounts in effect to assuming that ft is a valence-universal wave-operator. In fact Haque has explicitly demonstrated/123/ that the use of a valence-universal ft in the Fock-space Bloch equation leads automatically to eqn (7.3.9) with the ad-hoc sufficiency requirement. We give the sketch of a general proof here, since it shows that the extra information content of a Fock-space ft, as opposed to a Hilbert space, can be used to advantage for ensuring the connectivity of the cluster amplitudes of S/93/. For a valence-universal ft, the Fock-space Bloch equation (6.1.15) leads to... 

Mukherjee/91/ initially proved LCT for incomplete model spaces having n-hole n—particle determinants, showing also at the same time the validity of the core—valence separation. The corresponding open-shell perturbation theory of Brandow/20/ for such cases leads to unlinked terms and a breakdown of the core-valence separation, which used IN for O. Mukherjee emphasized that it is essential to have a valence-universal wave operator O within a Fock space formulation/91/ such that it also correlates the subduced valence sectors. Later on,... 

In the sprit of earlier discussions the valence universal wave operator O may be written as,... 

As our objective is to determine the amplitudes in Eq. 9, we would need to find the for the various sectors to completely define the valence universal wave operator (16, 22). The first step is to obtain which, alone, is... 

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. 

Among the fundamental reasons is the dilemma that the most straightforward formulation of an extensive theory leads inevitably to the appearance of the intruder problem and that it is hard to eliminate this problem without violating extensivity. In fact extensivity requires a Fock space formulation with a rnultiplicatively separable wave operator [12, 87]. This means that one formulates the wave operator and an effective Hamiltonian for the full valence space, for all possible particle numbers, i.e. that one uses a so-called valence universal theory. However then one can generally not avoid that external orbitals (i.e. which are not not in the valence space) get energies close to those of valence... 

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