# Converging self-consistent field equations in quantum chemistry – recent achievements and remaining challenges

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m2an0596.dvi ESAIM: M2AN ESAIM: Mathematical Modelling and Numerical Analysis Vol. 41, No 2, 2007, pp. 281–296 www.edpsciences.org/m2an DOI: 10.1051/m2an:2007022 CONVERGING SELF-CONSISTENT FIELD EQUATIONS IN QUANTUM CHEMISTRY – RECENT ACHIEVEMENTS AND REMAINING CHALLENGES ∗ Konstantin N. Kudin1 and Gustavo E. Scuseria2 Abstract. This paper reviews popular acceleration techniques to converge the non-linear self-consistent field equations appearing in quantum chemistry calculations with localized basis sets. The different methodologies, as well as their advantages and limitations are discussed within the same framework. Several illustrative examples of calculations are presented. This paper attempts to describe recent achievements and remaining challenges in this field. Mathematics Subject Classification. 35P30, 65B99, 65K10, 81-08. Received: October 10, 2005. 1. Introduction In the course of several decades of dedicated effort, quantum chemistry techniques have become highly accurate and quite accessible to non-experts. Many users of modern quantum chemistry programs usually lack expertise in the numerical approaches which the methods are based on, and would very much prefer to use these computational programs in a fully “black box” manner, while still expecting qualitatively good predictions. The Hartree-Fock (HF) method [15] or Density-Functional theory (DFT) [17,21] are among the least expensive yet still universally applicable and reliable approaches. These methods utilize orbitals or the density matrix as the basic variables. One of the most practical ways to solve the non-linear equations appearing in the HF or DFT method is to iterate some initial density (orbitals) until convergence, which means that a stationary point has been found and the input and output densities match [30]. This procedure is called the self-consistent field (SCF) iteration. Since in most cases the stationary point reached by the SCF iteration is actually a minimum, we will use these two terms inte

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