An efficient procedure to compute Hamiltonian matrix elements and reduced one- and two-particle density matrices for electronic wave functions using a new graphical-based nonlinear expansion form is presented. This method is based on spin eigenfunctions using the graphical unitary group approach ( GUGA), and the wave function is expanded in a basis of product functions ( each of which is equivalent to some linear combination of all of the configuration state functions), allowing application to closed-and open-shell systems and to ground and excited electronic states. In general, the effort required to construct an individual Hamiltonian matrix element between two product basis functions HMN) < M vertical bar H vertical bar N > scales as O (beta n(4)) for a wave function expanded in n molecular orbitals. The prefactor, itself scales between N-0 and N-2, for N electrons, depending on the complexity of the underlying Shavitt graph. Timings with our initial implementation of this method are very promising. Wave function expansions that are orders of magnitude larger than can be treated with traditional CI methods require only modest effort with our new method.