Direct Givens rotation method based on error back-propagation algorithm for self-consistent field solution

The self-consistent field (SCF) procedure is the standard technique for solving the Hartree-Fock and Kohn-Sham density functional theory calculations, while convergence is not theoretically guaranteed. Direct minimization methods, such as the augmented Lagrangian method (ALM) and second-order SCF (SOSCF), obtain the SCF solution by minimizing the Lagrangian with the gradient. In SOSCF, molecular orbitals are optimized by truncating the Taylor expansion of a unitary matrix represented in exponential form to ensure the orthonormality condition. This study proposes an alternative algorithm for direct-energy minimization to obtain an SCF solution using ALM Lagrangian by adopting sequential Givens rotations between occupied and virtual orbitals. The Givens rotation corresponds to unitary transformations that guarantee orthogonality and avoid variational collapse. Complex gradients for sequential Givens rotation were obtained by the error back-propagation method, which is based on the chain rule. Illustrative applications clarified the features of the present DGR methods by comparing with other SCF algorithms such as direct inversion in iterative subspace, SOSCF, and ALM.