Direct minimization on the complex Stiefel manifold in Kohn-Sham density functional theory for finite and extended systems

Direct minimization method on the complex Stiefel manifold in Kohn-Sham density functional theory is formulated to treat both finite and extended systems in a unified manner. This formulation is well-suited for scenarios where straightforward iterative diagonalization becomes challenging, especially when the Aufbau principle is not applicable. We present the theoretical foundation and numerical implementation of the Riemannian conjugate gradient (RCG) within a localized non-orthogonal basis set. Riemannian Broyden-Fletcher-Goldfarb-Shanno (RBFGS) method is tentatively implemented. Extensive testing compares the performance of the proposed methods and highlights that the quasi-Newton method is more efficient. However, for extended systems, the computational time required grows rapidly with respect to the number of $\mathbf{k}$-points.