Intrinsic convergence in closed-shell SCF calculations. A general criterion

Abstract

It is shown that the intrinsic convergence of the classical SCF algorithm for closed shells is governed by the eigenvalues of a supermatrix Q whose elements involve orbital excitation energies and electron repulsion integrals over occupied and virtual orbitals. In general, the SCF process is intrinsically convergent if all eigenvalues of Q have absolute values less than unity, and intrinsically divergent if one ore more is greater than unity. It is also shown how the symmetry preserving features of the SCF algorithm enable one to avoid certain divergences provided one starts a calculation with symmetry adapted orbitals. In cases of intrinsic convergence it is proved that the ratio of successive energy increments produced by the SCF algorithm is equal to the square of the largest eigenvalue of Q. The implications of the theory are illustrated by calculations on linear H8, on the cyclic planar ion S4N3+, and on model Q matrices filled with random numbers over the interval (−0.5,0.5).

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

“…The behavior of the Roothaan algorithm is notably investigated in [3,9] and in [24,25]. In [3,9] convergence difficulties are demonstrated for elementary two-dimensional models; in [24,25], a stability condition of the Roothaan algorithm in the neighbourhood of a minimum of the Hartree-Fock energy is given for the so-called closed-shell systems.…”

“…The behavior of the Roothaan algorithm is notably investigated in [3,9] and in [24,25]. In [3,9] convergence difficulties are demonstrated for elementary two-dimensional models; in [24,25], a stability condition of the Roothaan algorithm in the neighbourhood of a minimum of the Hartree-Fock energy is given for the so-called closed-shell systems. More sophisticated SCF algorithms for solving the Hartree-Fock equations have also been proposed to improve the convergence using various techniques like for instance relaxation (or damping [6,26]) or level-shifting [21].…”

Convergence properties of Hartree–Fock SCF molecular calculations

“…3 the theorem is proven and in Sec. 4 we discuss in this context a current technique to achieve or accelerate SCF convergence. The close connection between SCF convergence and H F stability properties is again analyzed, showing advantages and limitations of forcedconvergence techniques.…”

“…Different techniques for achieving or speeding up SCF convergence in molecular ab initio calculations have also been discussed [3]. In closed-shell RHF calculations, convergence of the SCF iterative scheme has been analyzed and a general criterion to assess convergence was found [4], requiring the eigenvalues of a certain matrix to have absolute values lower than unity. This general convergence criterion may be related to the HF stability problem [5] both of RHF [6] and UHF [7] approaches.…”

From atoms to crystals: a mathematical journey

“…[137,193,194,195]. The first attempt to understand mathematically the algorithms of computational chemistry dates back to [3], but a complete understanding of most of the algorithms commonly used appeared only recently [53,50].…”