How big a Hilbert space do you need?

How big a Hilbert space do I need to describe the electronic properties of a molecule?

Specifically, suppose in the molecule there are N valence electrons.

One must decide then how many spatial orbitals are required and how many Slater determinants? This issue is brought out in this review paper. Benzene is an illuminating case. McWeeny shows that a "brute force" approach based on molecular orbital theory requires hundreds and sometimes thousands of Slater determinants to obtain results of comparable accuracy to that obtained with a valence bond description.The latter uses just six localised orbitals and two Slater determinants (corresponding to the two Kekule structures).

Why does this matter? First, the priority of chemical insight favours the valence bond description over the "black box" approach embodied in the molecular orbital theory approach. The issues are described nicely in this Nature paper, which emphasises that the delocalised molecular orbitals are physically misleading. Second, computational efficiency may be more likely to be obtained with the simpler description.

A key question is whether one can codify these issues in a systematic way (perhaps with ideas from quantum information theory) to develop quantitative criteria to decide what is the minimal number of orbitals and Slater determinants.

Thanks to Anthony Jacko for providing the cartoon.