An efficient recursive algorithm to compute wave function optimization gradients for the graphically contracted function method

An efficient recursive algorithm is presented to compute orbital‐level Hamiltonian matrices for wave functions expanded in a basis of graphically contracted functions (GCF). Each GCF depends on a nonlinear set of parameters called arc factors. The orbital‐level Hamiltonian matrices characterize the dependence of the energy on the wave function changes associated with a subset of these nonlinear parameters corresponding to an individual molecular orbital. From these Hamiltonian matrices, gradients with respect to the nonlinear arc factor parameters may be computed and other arc factor optimization algorithms may be used. The recursive algorithm allows the orbital‐level Hamiltonian matrices to be computed with O(N 2GCFωn4) total effort, where NGCF is the dimension of the GCF basis, n is the dimension of the orbital basis, and where 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. This effort is about two to five times that required to compute an energy expectation value for a given set of arc factors; thus the energy and gradient have the same scaling behavior with increasing molecule size, NGCF dimension, and orbital basis size. Timings are given for wave functions that correspond to configuration state function expansions over 1073 in length, many orders of magnitude larger than can be considered using traditional electronic structure methods. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010