Algebraic approximation in many-body perturbation theory

Many-body perturbation theory is developed within the algebraic approximation, i.e., parametrization of state functions by expansion in a finite basis set. This considerably extends the applicability of the theory to molecules, allowing all two-, three-, and four-body contributions to the energy to be evaluated through third order. Within this context, a comparison is presented between perturbation calculations and previously reported configuration-interaction calculations which employed the same basis sets. The 2/1 Pad\'e approximants to the energy are constructed and upper bounds to the energy expectation values are determined. Two zeroth-order Hamiltonians are used, and the convergence of the resulting perturbation series is compared. Both of these perturbation expansions yield Pad\'e approximants to the energy which are within 0.9% of the corresponding configuration-interaction results. The variation-perturbation upper bounds for the energy are all within 3.2% of the corresponding configuration-interaction bounds. Considering this agreement and the tractability of the diagrammatic perturbative scheme, it appears that the perturbation expansion is highly competitive.