Complementary active space

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 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. 

---