Perturbation expansion, multi-reference

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... 

A further restriction on the use of many-body perturbation techniques arises from the (quasi-) degenerate energy structure, which occurs for most open-shell atoms and molecules. In these systems, a single reference state fails to provide a good approximation for the physical states of interest. A better choice, instead, is the use of a multi-configurational reference state or model space, respectively. Such a choice, when combined with configuration interactions calculations, enables one to incorporate important correlation effects (within the model space) to all orders. The extension and application of perturbation expansions towards open-shell systems is of interest for both, the traditional order-by-order MBPT [1] as well as in the case of the CCA [17]. 

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. 

Brillouin-Wigner perturbation theory was, however, used as a step in the development of an acceptable many-body perturbation theory most notably by Brandow [67] in his pioneering work on multi-reference formalisms for the many-body problem. In a review entitled Linked-Cluster Expansions for the Nuclear Many-Body Problem and published in 1967, B.H. Brandow writes ... 

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. 

Equations (2.222) and (2.229) are the first two terms in the general Rayleigh-Schrodinger perturbation expansion for the multi-reference 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. 

A brief introduction to the perturbation theory as a many-body method has been given in Chapter 3, Section 3.2.3. Approximations based on pt expansions include mbpt2, mbpt3, mbpt4, etc. and their multi-reference variants mr-mbpt2, MR-MBPT3, MR-MBPT4, etC. ... 

In the discussion given in the previous subsection, we have specified the wave operator f by means of the multi-reference BriUouin-Wigner perturbation expansion defined by eqs. (4.59) and (4.60) ... 

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. 

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