The evaluation of spin-density matrices within the graphically contracted function method

An efficient algorithm is presented to compute spin-density matrices from wave functions expanded in a basis of graphically contracted functions (GCF). The GCFs are based on the graphical unitary group approach (GUGA), which is a "spin-free" formulation of the electronic wave function. The spin-density matrix elements are computed from one-particle and two-particle charge-density matrix elements. The recursive algorithm allows the spin-density matrix to be computed with O(N² _GCFωn²) total effort where N_GCF is the dimension of the GCF basis and n is the dimension of the orbital basis. The scale factor ω depends on the number of electrons N and ranges from O(N⁰) to O(N²) depending on the complexity of the underlying Shavitt graph. Because the "spin-free" GCF formulation eliminates the need to expand the wave function in a spin-dependent Slater determinant basis, it is possible to treat wave functions with large numbers of electrons and orbitals. Timings are given for wave functions that correspond to determinantal expansions over 10 ²⁰⁰ in length. The implementation is applicable to arbitrary spin states and to both ground and excited electronic states.