Optimization of Grand-Canonical Self-Consistent-Field on Manifolds

Self-consistent-field (SCF) in the grand-canonical (GC) ensemble faces convergence difficulties with significant fractional occupation at low temperatures. By recognizing the orbital coefficient matrix and the fractional occupation vectors as two independent variables, this work provides a new viewpoint upon GC-SCF as an optimization problem on a product manifold of a flag manifold and a Euclidean space. During the optimization process, the manifold is automatically adjusted to the orbitals, which are divided into three partitions based on the occupation numbers, so that the occupation numbers are optimized properly, avoiding notorious gradient explosion and vanishing. Important concepts in manifold optimization and their specific expressions in GC-SCF are given. Three optimization algorithms -- Newton's method, augmented-Roothaan-Hall method and limited-memory Broyden-Fletcher-Goldfarb-Shanno method -- are implemented. The efficiency of our method is demonstrated via numerical benchmarks on three molecules at the room temperature and certain electrochemical potentials. Additionally, it is shown that Fermi-Dirac distribution, beside the Hartree-Fock-Roothaan equation, is a necessary condition for self-consistency in GC.