In this paper, we introduce an algebraic approach to electronic structure calculations. Our approach constructs a Jordan algebra based on the second-quantized electronic Hamiltonian. From the structure factor of this algebra, we show that we can calculate the energy of the ground electronic state of the Hamiltonian operator. We apply our method to several generalized Hubbard models and show that we can usually obtain a significant fraction of the correlation energy for low-to-moderate values of the electronic repulsion parameter while still retaining the O(L(3)) scaling of the Hartree-Fock algorithm. This surprising result, along with several other observations, suggests that our algebraic approach represents a new paradigm for electronic structure calculations which opens up many new directions for research.