Perturbation expansion, single-reference

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. 

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... 

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]. 

For open-shell systems, therefore, the model space is no longer simple and sometimes not even known before the computations really start. For closed-shell systems, in contrast, the model space can be formed by a single many-electron state, which is taken as the reference state and which is sufficient in order to classify the single-electron states into particle (unoccupied) and hole (occupied or core) states, respectively. In open-shell systems, instead, the valence shells are neither empty nor completely filled. To facilitate the handling of such open shells, we shall provide (and discuss) a orbital notation in Subsection 3.4 which is appropriate for the derivation of perturbation expansions. 

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]. 

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