Brillouin-Wigner configuration interaction theory

In the method of configuration interaction, the total atomic or molecular wave function, f, is written as a linear combination of some known AT-electron determinantal functions, i. 

The algebraic approximation results in the domain of the operator being restricted to a finite-dimensional subspace S of Hilbert space. For an AT-electron system, the algebraic approximation may be implemented by defining a suitable orthonormal basis set of M N) one-electron spin orbitals (most often solutions of the matrix Hartree-Fock equations) and then constructing all unique IV-electron determinants. The number of unique determinants that can be formed is given by 

In practice, difficulties arise in setting up and solving secular equations of high order, except in the case of small basis sets. For most adequate basis sets, only a small subset of the determinantal functions can be used in the expansion. It is then usual to write the expansion (4.127) in the form 

Multi-Reference configuration interaction is robust and thus, for example, Meissner et al. wrote [155]  

The multi-reference configuration interaction (mrci) method with single and doubles (mr-cisd) is one of the few quantum chemical methods which are used in routine calculations for systems requiring a multi-reference description. The main reason for that is its formal and computational simplicity and resistance to the intruder-state problem which frequently occurs in other multi-reference-type calculations. 

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Single-reference Brillouin-Wigner configuration interaction theory... 

Single state coupled cluster expansions and multireference coupled cluster expansions based on the generalized Brillouin-Wigner pertm bation theory have been described elsewhere [19]. The generalized Brillouin-Wigner perturbation theory can also be applied to the configuration interaction problem. 

Keywords Many-body theory Brillouin-Wigner theory State-specific multireference correlation problem Many-body perturbation theory Coupled cluster theory Configuration interaction Collaborative virtual enviroments... 

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 and limited configuration interaction Let us write the exact Schrodinger equation as... 

Multi-reference Brillouin-Wigner perturbation theory for limited configuration interaction... 

If j) is a determinant related to one of the reference determinants by a double replacement, then k) involves, at most, quadruple replacements with respect to 1 ) in eq. (4.193). Repeated application of the Lippmann-Schwinger-file equation [160] leads to higher order replacements. If we restrict the degree of replacement admitted in (4.193) then we realize a limited multi-reference configuration interaction method. It is this realization of the multi-reference limited configuration interaction method that we use to obtain an a posteriori correction based on Brillouin-Wigner perturbation theory. 

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