Restrained Optimization of Broken-Symmetry Determinants

The goal of this work is to provide an overview on how Lagrange multipliers can be used in self-consistent-field algorithms for convergence control. In particular, a restrained optimization scheme is proposed with the purpose to guide any set of initial guess molecular orbitals to those of a Slater determinant which (1) has certain expectation values (i.e., molecular properties) within a predefined range and (2) corresponds to a true (local) energy minimum, if such a minimum exists within the property range chosen. The most important practical application of such an algorithm is the calculation of minimum-energy broken-symmetry determinants, which are often used in Kohn-Sham density functional calculations to model antiferromagnetically coupled states. This is achieved by (a) combining a constrained optimization with a subsequent optimization where the constraints are released, i.e., the Lagrange multipliers are set to zero. Alternatively, (b) a modified scheme may be employed, where the constraints are released as soon as the energy starts to rise in the SCF algorithm. Both alternatives are combined with both intermediate and final automatic control of whether the properties of interest are within the desired range. A parameter study using scheme (a) is carried out for a set of small test systems. Applications to a synthetically available trinuclear copper complex and to Crabtree and Brudvig's water-oxidizing complex demonstrate the value and limitations of restrained optimization. (C) 2009 Wiley Periodicals, Inc. Int J Quantum Chem 109: 2430-2446, 2009