Globally convergent trust-region methods for self-consistent field electronic structure calculations

Abstract

As far as more complex systems are being accessible for quantum chemical calculations, the reliability of the algorithms used becomes increasingly important. Trust-region strategies comprise a large family of optimization algorithms that incorporates both robustness and applicability for a great variety of problems. The objective of this work is to provide a basic algorithm and an adequate theoretical framework for the application of globally convergent trust-region methods to electronic structure calculations. Closed shell restricted Hartree-Fock calculations are addressed as finite-dimensional nonlinear programming problems with weighted orthogonality constraints. A Levenberg-Marquardt-like modification of a trust-region algorithm for constrained optimization is developed for solving this problem. It is proved that this algorithm is globally convergent. The subproblems that ensure global convergence are easy-to-compute projections and are dependent only on the structure of the constraints, thus being extendable to other problems. Numerical experiments are presented, which confirm the theoretical predictions. The structure of the algorithm is such that accelerations can be easily associated without affecting the convergence properties.

On Augmented Lagrangian Methods with General Lower-Level Constraints

“…Minimization with orthogonality constraints [22,24,36,44]. Important problems on this class appear in many applications, such as the "ab initio" calculation of electronic structures.…”

Acceleration of self-consistent-field convergence in ab initio molecular dynamics and Monte Carlo simulations and geometry optimization

“…A major bottleneck is the self-consistent-field (SCF) procedure. So far, various efforts [4][5][6][7][8][9][10][11][12][13] have been carried out for accelerating the SCF convergence.…”

Inexact restoration method for minimization problems arising in electronic structure calculations

“…We studied the behavior of the Inexact Restoration algorithms in some typical electronic structure calculations arising in computational chemistry (from now on called "easy" problems) and in some designed problems known to display convergence instabilities [8,9] (called "hard" problems). Easy problems are standard organic molecules Carbon dioxide, Ethylene, Ethanol and Benzene, and some common biologically relevant molecules, as Alanine, Alanine dipeptide, Histidine and Tyrosine.…”

“…These easy examples were selected to illustrate the behavior of the algorithms in problems for which convergence is usually straightforward. On the other side, we also performed tests on some electronic structure calculations known to display multiple local minima and convergence problems: CrC, Cr 2 , Rh 2 and an arrangement of atoms of formula Li 9 F 9 [8,9].…”

“…However, the SCF-DIIS algorithm is very successful in many problems and its employment is widely extended in practical calculations. The Optimal Damping Algorithm (ODA) [7] and the trust-region methods given in [8,9] can be proved to be convergent, under different assumptions, to stationary points of (3,4). Other methods that incorporate trust-region concepts were given in [10,11].…”