  This work presents a general framework for deriving exact and approximate
Newton self-consistent field (SCF) orbital optimization algorithms by
leveraging concepts borrowed from differential geometry. Within this framework,
we extend the augmented Roothaan--Hall (ARH) algorithm to unrestricted
electronic and nuclear-electronic calculations. We demonstrate that ARH yields
an excellent compromise between stability and computational cost for SCF
problems that are hard to converge with conventional first-order optimization
strategies. In the electronic case, we show that ARH overcomes the slow
convergence of orbitals in strongly-correlated molecules with the example of
several iron-sulfur clusters. For nuclear-electronic calculations, ARH
significantly enhances the convergence already for small molecules, as
demonstrated for a series of protonated water clusters.
