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

Abstract

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 2 GCF xn 4 ) total effort, where N GCF is the dimension of the GCF basis, n is the dimension of the orbital basis, and where the scale factor x depends on the number of electrons N and ranges from O(N 0 ) to O(N 2 ) 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, N GCF dimension, and orbital basis size. Timings are given for wave functions that correspond to configuration state function expansions over 10 73 in length, many orders of magnitude larger than can be considered using traditional electronic structure methods.

Multiconfiguration second‐order perturbation theory approach to strong electron correlation in chemistry and photochemistry

“…The future will tell whether these protocols will be capable to gain the same trust as CASSCF/CASPT2. We believe that both these models introduce limited gains, whereas an objective breakthrough would come from a formulation of CASSCF/CASPT2 based upon a nonlinear ansatz to the wave function in the spirit of the graphically contracted functions (GCFs) method of Shepard 119. Here in Uppsala, work has started in this direction, and we hope to report soon on what could be the next step in the history of CASSCF/CASPT2.…”

Second-order perturbation theory with a density matrix renormalization group self-consistent field reference function: Theory and application to the study of chromium dimer

“…Numerous attempts to overcome this limitation have been made, e.g., the restricted active space (RAS) method, 1, 2 general multi-configuration (GMC) method, 3 generalized a) Electronic mail: kura@ims.ac.jp. valence bond (GVB) method, 4 and perfect pairing and other valence-bond, geminal type theories, [5][6][7][8][9][10] and very recently, graphically contracted function, [11][12][13] complete-graph tensor network states 14,15 and correlator product states. 16 Recently, a novel multireference approach based on the density matrix renormalization group (DMRG) algorithm for quantum chemistry has been vigorously developed by several groups.…”

Efficient formulation of full configuration interaction quantum Monte Carlo in a spin eigenbasis via the graphical unitary group approach

“…Double excitation with a single overlapping index j but two different generators (1a) and (1b) can be treated in a similar way to single excitations, with the same weighting functions (51) and classification of remaining switches (49), but with a change of generator type at the overlap site, L ↔ R. Double excitations with an empty overlap range S 1 (32) (3c 0 , 3d 0 , 3e 0 and 3f 0 ) can be calculated as the product of two single excitations (34). However, e.g.…”

“…Based on the GUGA introduced by Shavitt 42,43 , Shepard et al 44,45 made extensive use of the graphical representation of spin eigenfunctions in form of Shavitt's distinct row table (DRT). In the multifacet graphically contracted method [46][47][48][49][50][51][52] the ground state wavefunction is formulated nonlinearly based on the DRT, conserving the total spin S.…”