Configuration interaction first-order interacting space

In principle, one can extract from G(ti)) the complete series of the primary (one-hole, Ih) and excited (shake-up) states of the cation. In practice, one usually restricts the portion of shake-up space to be spanned to the 2h-lp (two-hole, one-particle) states defined by a single-electron transition, neglecting therefore excitations of higher rank (3h-2p, 4h-3p. ..) in the ionized system. In the so-called ADC[3] scheme (22), elertronic correlation effects in the reference ground state are included through third-order. In this scheme, multistate 2h-lp/2h-lp configuration interactions are also accounted for to first-order, whereas the couplings of the Ih and 2h-lp excitation manifolds are of second-order in electronic correlation. 

Equation [1] is an internally contracted configuration space, doubly excited with respect to the CAS reference function 0) = G4SSCF) one or two of the four indices p,q,r,s must be outside the active space. The functions of Eq. [1] are linear combinations of CFs and span the entire configuration space that interacts with the reference function. Labeling the compound index pqrs as (i or v, we can write the first-order equation as... 

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 ... 

Finally, in certain problems involving mainly excited states, it is preferable to first partition the total function space into two or more (rarely) zero order separately optimized spaces, each representing one or more moieties of physical significance, which are fhen allowed to mix. This implies the application of nonorthonormal configuration interaction (NONCI). 

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... 

In addition to MP2, MP3, and MP4 calculations, CCSD(T), CASSCF, FOCI (First-Order Configuration Interaction), and sometimes SOCI (Second-Order Configuration Interaction) approaches have been used to ensure the convergence of the results. The complete definitions of the variational spaces used are given in [61,62,63]. Electronically-excited states have been obtained by means of the MC/P method, recently developed in our group [64,65] it couples a variational treatment to deal with the nondynamic correlation effects and a perturbation treatment to account for the dynamic correlation effects as well as the non-dynamic effects not treated at the variational level becanse of their limited contributions to the phenomena investigated. All electronic transitions reported here are vertical transition energies. 

The MCSCF provides a good first-order description covering the static electron correlation due to degeneracy problems. Dynamic electron correlation should be addressed with the MCSCF wave function as a reference. The multireference configuration interaction, or MRCI, generates excited determinants from all (or selected) determinants included in the MCSCF. The complete active space perturbation theory, second order (CASPT2) is a more economical approach. Both methods can be applied to compute excited states. 

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