A geometric approach to direct minimization

The approach presented, geometric direct minimization (GDM), is derived from purely geometrical arguments, and is designed to minimize a function of a set of orthonormal orbitals. The optimization steps consist of sequential unitary transformations of the orbitals, and convergence is accelerated using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) approach in the iterative subspace, together with a diagonal approximation to the Hessian for the remaining degrees of freedom. The approach is tested by implementing the solution of the self-consistent field (SCF) equations and comparing results with the standard direct inversion in the iterative subspace (DIIS) method. It is found that GDM is very robust and converges in every system studied, including several cases in which DIIS fails to find a solution. For main group compounds, GDM convergence is nearly as rapid as DIIS, whereas for transition metal-containing systems we find that GDM is significantly slower than DIIS. A hybrid procedure where DIIS is used for the first several iterations and GDM is used thereafter is found to provide a robust solution for transition metal-containing systems.