First order interacting space

The first-order interacting space restriction is generally used to reduce the number of double substitutions included in single- or multi-reference CISD... 

According to simple first-order perturbation theory, the configurations considered in an MR-CI treatment should span the first-order interacting space relative to the reference wavefunction To. This space comprises all configurations which have a non-vanishing matrix element... 

An alternative method, named internally contracted Cl, was suggested by Meyer and was applied by Werner and Reinsch in the MCSCF self-consistent electron-pair (SCEP) approach. Here only one reference state is used, the entire MCSCF wavefunction. The Cl expansion is then in principle independent of the number of configurations used to build the MCSCF wavefunction. In practice, however, the complexity of the calculation also strongly depends on the size of the MCSCF expansion. A general configuration-interaction scheme which uses, for example, a CASSCF reference state, therefore still awaits development. Such a Cl wavefunction could preferably be used on the first-order interacting space, which for a CASSCF wavefunction can be obtained from single and double substitutions of the form ... 

For the OSS states, the set of excitation operators ( / that generate the full singly and doubly (SD) excited manifold were given in Ref. (13), Eqs. 59-70. The subset of excitation operators generating the first order interacting space and thus the first order wave function J,... 

We can again use the same technique as in the UGA CC method to derive explicit form of these matrix elements, and we list them in Table 2 for the relevant first order interacting space configurations j) = G/ o When evaluating the third order (EN3) energy E 3 Eqs. 32 and 33, we can employ the same expressions for off-diagonal Vjj s as in the MP case (cf. Eq. 23), namely... 

The classification of CSFs into single, double,... excitations is straightforward and unambiguous for a closed-shell Hartree-Fock reference function. In open-shell or multireference cases, there are more possibilities for defining these excited CSFs, some of which interact with the reference space in the lowest order of perturbation theory, and some of which do not. It is very common to exclude the latter excitations. This is termed restricting the wave function to the first-order interacting space. ... 

Due to the fact that at most single and double excitations can directly interact through the BO Hamiltonian with the zero-order wavefunction 0), it is sufficient to terminate the expansion at this level known as first-order interacting space ... 

The second equality of P only arises if the entire Mr space is diagonalized, which is not necessary though. Since the linearly independent singly and doubly excited CSFs (determinants) g relative to < gMr span exactly the first-order interacting space [46, 47], it appears natural to choose the Ap first-order functions f f as the external functions for a minimal dynamic correction to 1... 

---