Valence universal

By adopting the no-pair approximation, a natural and straightforward extension of the nonrelativistic open-shell CC theory emerges. The multireference valence-universal Fock-space coupled-cluster approach is employed [25], which defines and calculates an effective Hamiltonian in a low-dimensional model (or P) space, with eigenvalues approximating some desirable eigenvalues of the physical Hamiltonian. The effective Hamiltonian has the form [26]... 

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.. 

The former approach is referred to as the valence universal (VU) or Pock space MR CC method [51-54] and the latter one as the state universal (SU) or Hilbert space method [55]. In spite of a great number of papers devoted to both the VU and SU approaches, very few actual applications have been carried out since their inception more than two decades ago. Certainly, no general-purpose codes have been developed. This is not so much due to the increased complexity of the MR formalism relative to the SR one, as it is due to a number of genuine obstacles that have yet to be overcome. 

C. A. Russell, The History of Valency (University Press, Leicester, 1971). 

Jeziorski B, Paldus J (1990) Valence universal exponential ansatz and the cluster structure of multireference configuration interaction wave function. J Chem Phys 90 2714-2731... 

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... 

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 what follows, it will be useful to classify the parent n-valence IMS belonging to the space P, the complementary"active space, whose union with P forms the CMS, as R and the true virtual space containing inactive labels as Q. The cluster operators inducing the transition P+R are all labelled by valence lines, and will henceforth be denoted as quasi-open. The operators making transitions P- Q will be called open. The operators connecting model spaces only will be called closed. In general, for a valence-universal O, it so happens that products of quasi-open... 

We show below, following Mukherjee/91-94/, how a connected H can he obtained using a Valence-universal O which uses a normalization convention different from intermediate normalization. We start from the Fock-space Bloch Equation ... 

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

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. 

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 major difference between the Fock-space CC methods and the EOMCC theory lies in the fact that in VUCC approaches different sectors of the Fock space are considered simultaneously (cf. the valence universality condition that VUCC theories must obey [8,12,13]), whereas the EOMCC theory treats different sectors of the Fock space as separate prob-... 

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