Multi-reference coupled cluster expansions

The CASSCF method itself is not very useful for anything else than systems with few electrons unless an effective method to treat dynamical correlation effects could be developed. The Multi-Reference Cl (MRCI) method was available but was limited due to the steep increase of the size of the Cl expansion as a function of the number of correlated electrons, the basis set, and the number of active orbitals in the reference function. The direct MRCI formulation by P. Siegbahn helped but the limits still prevented applications to larger systems with many valence electrons [20], The method is still used with some success due to recent technological developments [21], Another drawback with the MRCI approach is the lack of size-extensivity, even if methods are available that can approximately correct the energies. Multi-reference coupled-cluster methods are studied but have not yet reached a state where real applications are possible. 

A posteriori corrections can be developed for calculations performed by using the Brillouin-Wigner perturbation expansion. These a posteriori corrections can be obtained for the Brillouin-Wigner perturbation theory itself and, more importantly, for methods, such as limited configuration interaction or multi-reference coupled cluster theory, which can be formulated within the framework of a Brillouin-Wigner perturbation expansion. 

The formulation of a multi-reference bwcc theory can now proceed in two distinct ways. In the first option, we can formulate a multi-root version of the multi-reference BWCC theory which yields all roots of the d-dimensional 9 space simultaneously. This is the approach employed in most multi-reference coupled cluster formulations which are based on the Rayleigh-Schrodinger expansion. In the second option, we can use the state-specific wave operator (4.59) and formulate a state-specific (or single root) version of multi-reference bwcc theory [10]. 

Abstract The Brillouin-Wigner many-body problem in atomic and molecular physics and in quantum chemistry is described. The use of coupled cluster expansions, configuration interaction and perturbation series is considered both for the single-reference function case and for those cases requiring the use of a multi-reference formalism. 

However, although, starting from this point, many sophisticated methods for wave function expansion, for example, the coupled cluster approach, multi-configuration self-consistent-field method or multi-reference Cl methods, have been developed, the correlation problem faced many computational limitation, some of them almost insurmountable, due to the immense number of integrals to be evaluated. 

A brief introduction to coupled cluster theory has been given in Chapter 3, Section 3.3.2. Approximations based on cc expansions include, CCD, CCSD ccsdt, etc. and their multi-reference variants mr-ccd, mr-ccsd, mr-ccsdt, etc. The ccsD approximation is also known as the coupled-pair approximation (CPA) [75] or the coupled pair many electron theory (cpmet) [76,77]. 

---