Convergence and computational efficiency enhancements in the iterative solution of the G‐particle‐hole hypervirial equation

Abstract

The G-particle-hole hypervirial (GHV) equation has been recently reported (Valdemoro et al.). This equation is the newest member of the family of equations which can be obtained by applying a matrix-contracting mapping (Valdemoro, An R Soc Esp Fís 1983, 79, 106; Valdemoro, Phys Rev A 1985, 31, 2114 Valdemoro, in Density Matrices and Density Functionals, Reidel: Dordrecht, 1987; p 275.) to the matrix representation in the N-electron space of the Schrödinger, Liouville and hypervirial equations. The procedure that we have applied in order to solve the GHV equation exploits the stationary property of the hypervirials (Hirschfelder, J Chem Phys 1960, 33, 1462Hirschfelder and Epstein, Phys Rev 1961, 123, 1495) and follows the general lines of Mazziotti's variational approach for solving the antiALCOBA ET AL. 184101). In this article, we report how the method's convergence has been significantly enhanced and how its computational scaling has been considerably reduced (in both floating-point operations and storage). The results for a variety of atomic and molecular calculations confirming these methodological improvements are reported here.

Symmetry-adapted formulation of the combined G-particle-hole hypervirial equation and Hermitian operator method

“…The approximation algorithm which is now being used is a recently published modification of Nakatsuji-Yasuda's one [12,27]. Proceeding in this way, the solution of the GHV equation may be obtained by iteratively solving a set of differential equations to minimize the 2-order error matrix resulting from the deviation from exact fulfilment of the equation [11]. As a result, an approximated G-particle-hole matrix corresponding to the eigenstate being considered is obtained [11].…”

“…The accuracy of the results obtained with the GHV method when studying the ground state of molecular systems at equilibrium geometry was excellent when compared with the equivalent Full Configuration Interaction (FCI) quantities [9,[11][12][13]. However, the study of the excited states is still a partially open question [14,15].…”

Optimized Solution Procedure of the G-Particle−Hole Hypervirial Equation for Multiplets: Application to Doublet and Triplet States

“…53,14,61 The accuracy of the results obtained with this method when studying singlet ground-and excited-states with weak to moderate multiconfigurational character of a set of atoms and molecules was excellent compared to the equivalent full configuration interaction (FCI) quantities. 53,14,61 The purpose of the current work is to investigate the behavior of the GHV methodology in the study of highspin doublet and triplet states occurring in a variety of systems. It must be noted that an alternative approach for treating these systems has been reported within the framework of the ACSE.…”

“…14 The integration of the differential equation is carried out until either the least-squares error of the GHV equation, or the least-squares error of its contraction into the 1-electron space, the first-order contracted Schr€ odinger equation, 30,36 ceases to decrease. 53,14 The computational efficiency of the GHV method has recently been significantly enhanced through the use of sum factorization and matrix-matrix multiplication at computational costs of K 6 in floating point operations and K 4 in storage, where K is the number of orbitals forming the basis set. 14 For the sake of comparison, MP2, CCSD, and CCSD(T) methods scale in floating point operations as K 5 , K 6 , and K 7 , respectively.…”

“…However, in the search for a method of solving the second-order contracted Schr€ odinger equation (2-CSE), 2-5 a set of approximated construction algorithms for the RDMs were recently devised. 13,14,[15][16][17][18][19][20][21][22][23][24][25][26] These algorithms jointly with the conditions imposed by the system Hamiltonian and those N-representability conditions which are known, lead to a direct determination of the 2-RDM without a previous computation of the wave function. [2][3][4][5] The 2-CSE was initially derived in 1976 in first quantization by Cho, 27 Cohen and Frishberg, 28 and Nakatsuji 29 and deduced later on in second quantization by Valdemoro 30 through the contraction of the matrix representation of the Schr€ odinger equation into the two-electron space.…”