Approximate wavefunctions such as Hartree--Fock (HF) states need not
respect the symmetries of the molecular electronic Hamiltonian. In certain cases, the lowest-energy HF solution obtained by a variational scheme does not preserve the symmetries of the Hamiltonian. This broken symmetry HF solution captures some of the correlations associated with near-degeneracies in a symmetry-adapted construction. Broken symmetry HF solutions are, nevertheless, unphysical and cannot accurately represent the stationary states of a molecular system. By using projection operators, one can restore the physical character of the wavefunction while accessing the relevant correlations introduced by the broken symmetry mean field description.
In this work, we consider a single symmetry-projected Slater determinant as a working wavefunction ansatz. Originally proposed by Löwdin in 1955, the idea was mostly abandoned in the quantum chemistry community after decades of work. By borrowing techniques successful in the nuclear physics community, we use a rigorous, yet efficient mathematical apparatus to perform the projection before the variation of broken symmetry wavefunctions. The wavefunctions thus obtained have a multi-determinantal character and can account for significantly more correlations than a broken symmetry HF state in finite systems. The symmetry-projected HF approach is, nonetheless, not free of vices. The approach is neither size-consistent nor size-extensive.
In order to go beyond the symmetry projected HF wavefunction, we construct highly sophisticated multi-reference wavefunctions based on a small number of non-orthogonal Slater determinants. Chains of variational calculations are used to optimize wavefunctions suitable for an accurate description of ground and excited states, with well defined quantum numbers, which can account for both strong and weak correlation effects. Our results indicate that such expansions can produce fairly accurate results for small molecular systems. We hope our approach will eventually become yet another tool for the quantum chemist useful in situations where both strong and weak correlation effects are important.