Single-reference function perturbation expansions

We are now in a position to obtain perturbation expansions by expanding the inverse operator in the effective Hamiltonian, the wave operator and the reaction operator. We begin, as we did in our discussion of the partitioning technique, by considering the case of a single-reference function and then turn our attention to the multi-reference function case. 

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. 

By following procedures similar to those employed in the case of a single-reference function, the exact wave function f o, for a = 1,2,. .., d, in the Brillouin-Wigner perturbation theory can be written as the expansion... 

Just as single reference Cl can be extended to MRCI, it is also possible to use perturbation methods with a multi-detenninant reference wave function. Formulating MR-MBPT methods, however, is not straightforward. The main problem here is similar to that of ROMP methods, the choice of the unperturbed Hamilton operator. Several different choices are possible, which will give different answers when the tlieory is carried out only to low order. Nevertheless, there are now several different implementations of MP2 type expansions based on a CASSCF reference, denoted CASMP2 or CASPT2. Experience of their performance is still somewhat limited. 

The second step of the calculation involves the treatment of dynamic correlation effects, which can be approached by many-body perturbation theory (62) or configuration interaction (63). Multireference coupled-cluster techniques have been developed (64—66) but they are computationally far more demanding and still not established as standard methods. At this point, we will only focus on configuration interaction approaches. What is done in these approaches is to regard the entire zeroth-order wavefunc-tion Tj) or its constituent parts double excitations relative to these reference functions. This produces a set of excited CSFs ( Q) that are used as expansion space for the configuration interaction (Cl) procedure. The resulting wavefunction may be written as... 

When the reference wave function contains substantial multi-reference character, a perturbation expansion based on a single determinant will display poor convergence. If the reference wave function suffers from symmetry breaking (Section 3.8.3), the... 

Cluster expansion representation of a wave-function built from a single determinant reference function [1] has been eminently successful in treating electron correlation effects with high accuracy for closed shell atoms and molecules. The cluster expansion approach provides size-extensive energies and is thus the method of choice for large systems. The two principal modes of cluster expansion developments in Quantum Chemistry have been the use of single reference many-body perturbation theory (SR-MBPT) [2] and the non-perturbative single reference Coupled Cluster (SRCC) theory [3,4]. While the former is computationally economical for the first few orders of the perturbation expansion... 

In this section, we present Brillouin-Wigner perturbation theory in both its single reference and its multireference form. This will serve both to emphasize the similarity of single reference and multireference formulations of Brillouin-Wigner perturbation theory and to establish notation for later sections. In section 3.1, we define the basic concepts of any perturbation theory. The definition of single and multireference spaces is considered in section 3.2 and the model wave function is described in section 3.3. The Brillouin-Wigner expansion is developed in section 3.4. 

In the previous section, we have given the Brillouin-Wigner perturbation expansion for the exact wave function for state a developed with respect to some single reference or multireference model function In this section, we define the Brillouin-Wigner wave operator and the corresponding Bloch-like equation [64]. 

M0ller-Plesset second-order perturbation theory [78,162] is the most widely used approach to the electron correlation problem in contemporary ab initio molecular electronic structure studies [163-168], For systems which are well described by a single determinantal reference function, this theory - based on the use of Rayleigh-Schrodinger perturbation theory to describe electron correlation corrections to the Hartree-Fock independent electron model - affords an approach which combines accuracy with computational efficiency. The method, which is often designated mp2 , is based on the lowest order of the many-body perturbation theory expansion to take account of correlation effects. 

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