The electronic structure of small sodium clusters is investigated using a basis set motivated by path integral Monte Carlo calculations. The goal of this study is to develop a small basis set that can be used to study large sodium clusters. Past and present path integral calculations suggest that the correlated electronic wave function is characterized by electron density located between several atoms, rather than on the atoms. This is in agreement with previous quantum chemical and density functional calculations, which find the total electron density has maxima in the same locations. A basis set used in previous quantum chemical calculations is modified by moving valence orbitals from atoms to locations between atoms to create a non-atom-based basis set. This results in a smaller basis set, as there are fewer non-atom-based sites than atoms; this has the advantage of allowing the study of larger clusters. The vertical and adiabatic ionization potentials are evaluated using both the old and the new basis sets and are compared with experimental results. In addition, the electronic absorption spectrum is calculated and compared with both theoretical and experimental values. It is found that the non-atom-based basis set gives agreement with experimental values as well or better than the relatively larger atom-based basis set, indicating that the new basis set provides the opportunity to study larger clusters.