On the convergence of SCF algorithms for the Hartree-Fock equations

Abstract

Abstract. The present work is a mathematical analysis of two algorithms, namely the Roothaan and the level-shifting algorithms, commonly used in practice to solve the Hartree-Fock equations. The level-shifting algorithm is proved to be well-posed and to converge provided the shift parameter is large enough. On the contrary, cases when the Roothaan algorithm is not well defined or fails in converging are exhibited. These mathematical results are confronted to numerical experiments performed by chemists.Mathematics Subject Classification.

Accelerating self-consistent field convergence with the augmented Roothaan–Hall energy function

“…Unfortunately, SCF convergence without any accelerating technique is always problematic in many cases. 10 To achieve and accelerate the SCF convergence, a variety of methods has been developed such as simple density mixing between the previous and current density matrices, level shifting, 11 fractional electron occupations, 12 the optimal damping algorithm ͑ODA͒, [13][14][15] the direct inversion iterative subspace ͑DIIS͒ approach, 16,17 energy-DIIS ͑EDIIS͒, 18,19 density subspace minimization ͑DSM͒, 20,21 and the ground-state-directed optimization scheme. 22,23 Since the DIIS-based algorithms are particularly robust and efficient in most molecular systems, we focus on the DIIS procedure for SCF convergence in this work.…”

s∗-compressibility of the discrete Hartree-Fock equation

“…The convergence of this algorithm for a sufficiently large parameter b, to "self-consistent-field" solutions of the Hartree-Fock equations (1.4) has been proven by Cancès and Le Bris [9]. In a previous paper [16], we have studied the asymptotic behaviour of solutions within such an iteration scheme and of the final "selfconsistent-field" solutions of the Hartree-Fock equation.…”

Tensors-structured numerical methods in scientific computing: Survey on recent advances

“…e − x 2 diag(e −x 2 ) ∆ −1 2 1, ε = 10 −6 , 10 −7 , 10 −8 2 9 5.0 9.4 7.8 3.8 3.6/3.6/3.6 2 10 5.1 9.4 7.7 3.9 3.6/3.6/3.6 2 11 5.2 9.3 7.5 3.9 3.7/3.7/3.7 Table 3.2: QTT 2 -ranks of functional N × N -arrays on large grids, N = 2 p .…”

“…The Hartree-Fock (HF) equation for determination of the ground state of a molecular system consisting of M nuclei and N electrons is given by the following nonlinear eigenvalue problem in L 2 (R 3 ), 9) with F Φ being the non-linear Fock operator…”

“…In this case, instead of truncated preconditioned inverse iteration, one can employ the Struncated Krylov subspace-type iteration (the so-called DIIS iteration [9,82]) applied to the Galerkin system in the problem adapted basis, (see [58]), leading to the numerical cost O(N log N ). Figure 4.2 (left) shows convergence of the SCF iteration for all electron case of H 2 O.…”