Space excitation level truncation

In 1988, Olsen and co-workers46 presented the restricted active space (RAS) Cl, which specifies the Cl space in an a priori manner reminiscent of the second-order Cl (SOCI) and its derivatives. Olsen partitions the orbitals into three subspaces, labeled RAS I, RAS II, and RAS III. Typically, RAS I contains occupied and possibly very important virtual orbitals, RAS II contains the most important virtuals, and RAS III contains the less important virtuals. The Cl space includes all determinants with a minimum of p electrons in RAS I and a maximum of q electrons in RAS III. There is no restriction on RAS II, which is akin to the complete active space. Using this simple procedure, it is possible to formulate any Cl space truncated according to excitation level (e.g., CISD, CISDT, etc.) as well as excitation class selected MR-CI spaces, such as SOCI and CISD[TQ], The RAS Cl method is discussed more fully in section 4.8. 

Any Cl space truncated according to excitation level may be formulated within the RAS Cl framework the occupied orbitals are placed in RAS I, and the unoccupied orbitals are placed in RAS III, and the RAS II subspace is absent. The maximum number of electrons in RAS III is set equal to the maximum excitation level, and the minimum number of electrons in RAS I is simply the total number of electrons N minus the maximum excitation level. A full Cl can be obtained by applying trivial restrictions, such as a minimum of zero electrons in RAS I and a maximum of N electrons in RAS III. 

Any Cl truncated below the full excitation level necessarily suffers from size-consistency problems. The severity of this shortcoming depends very much on the case it is certainly a problem of major concern for large-scale singlereference CI(SD) calculations which are now feasible (e.g. correlating 28 electrons in using a basis of 1 SO contracted Gauss type orbitals ) it is a minor problem for MR-CI(SD) treatments based on a sufficiently large reference space or CI(SDTQ) calculations as far as they are feasible at present. - ... 

The Cl approach to the many-electron problem suffers from two main disadvantages. First, since the Cl model is not manifestly separable, it does not provide size-extensive energies. In Section 4.3.2, we found that size-extensivity follows only when the variational space of the compound system is equal to the direct product of the fragment spaces. This requirement cannot be satisfied for Cl wave functions truncated at a fixed excitation level relative to the reference configuration. For example, if two fragments are each described at the CISD level, then a size-extensive treatment of the compound system requires the wave function to be augmented with certain triples and quadruples - namely, those that represent products of single and double excitations in the two subsystems. Such wave functions are not defined within the hierarchy of truncated Cl wave functions based on excitation levels. 

A simple way to implement n-particle space truncation is to use the uncorrelated wave function (which as noted above is a very substantial fraction of the exact wave function) to classify terms in the n-particle space. If we consider the Hartree-Fock determinant, for example, we can construct all CSFs in the full n-particle space by successively exciting one, two,.., electrons from the occupied Hartree-Fock MOs to unoccupied MOs. For cases in which a multiconfigurational zeroth-order wave function is required, the same formal classification can be applied. Since only singly and doubly excited CSFs can interact with the zeroth-order wave func tion via the Hamiltonian in Eq. (1), it is natural to truncate the n-particle expansion at this level, at least as a first approximation. We thus obtain single and double excitations from Hartree-Fock (denoted SDCI) or its multiconfigurational reference analog, multireference Cl (MRCI). 

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