Evaluation of the Spin−Orbit Interaction within the Graphically Contracted Function Method

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The graphically contracted function (GCF) method is extended to include an effective one-electron spin-orbit (SO) operator in the Hamiltonian matrix construction. Our initial implementation is based on a multiheaded Shavitt graph approach that allows for the efficient simultaneous computation of entire blocks of Hamiltonian matrix elements. Two algorithms are implemented. The SO-GCF method expands the spin-orbit wave function in the basis of GCFs and results in a Hamiltonian matrix of dimension N(dim)=N(alpha)((S(max) + 1)(2) - S(min)(2)). N(alpha) is the number of sets of nonlinear arc factor parameters, and S(min) and S(max) are respectively the minimum and maximum values of an allowed spin range in the wave function expansion. The SO-SCGCF (SO spin contracted GCF) method expands the wave function in a basis of spin contracted functions and results in a Hamiltonian matrix of dimension N(dim) = N(alpha). For a given N(alpha) and spin range, the number of parameters defining the wave function is the same in the two methods after accounting for normalization. The full Hamiltonian matrix construction with both approaches scales formally as O(N(alpha)(2)omegan(4)) for n molecular orbitals. The omega factor depends on the complexity of the Shavitt graph and includes factors such as the number of electrons, N, and the number of interacting spin states. Timings are given for Hamiltonian matrix construction for both algorithms for a range of wave functions with up to N = n = 128 and that correspond to an underlying linear full-CI CSF expansion dimension of over 10(75) CSFs, many orders of magnitude larger than can be considered using traditional CSF-based spin-orbit CI approaches. For Hamiltonian matrix construction, the SO-SCGCF method is slightly faster than the SO-GCF method for a given N(alpha) and spin range. The SO-GCF method may be more suitable for describing multiple states, whereas the SO-SCGCF method may be more suitable for describing single states.