Augmented Roothaan-Hall Hessian Applied to Spin-Restricted Open-Shell Density Functional Theory

We generalize the augmented Roothaan–Hall (ARH) Hessian formalism to the self-consistent field (SCF) optimization of spin-restricted open-shell (RO) wave functions, encompassing high-spin, low-spin, and two-determinant electronic states. A detailed ARH formulation is presented. We demonstrate that ARH is a highly efficient optimization algorithm for rapidly identifying accurate SCF solutions, primarily owing to its systematic construction of an effective Hessian, particularly in the case of Euclidean quadratic energy functions. The ARH is built upon a universal energy formulation, including grid-based integration, for spin-restricted closed-shell, spin-unrestricted, and RO density functional theory (DFT), thereby unifying and simplifying their numerical implementation. The performance of the present method is evaluated using two benchmarking studies. First, for a series of iron–sulfur clusters exhibiting different spin states, which represent notoriously challenging SCF problems, the ARH algorithm demonstrates superior convergence efficiency relative to L-BFGS and truncated Newton methods, requiring much fewer RO-SCF iterations to achieve convergence. Second, the ARH approach tends to avoid convergence to higher-energy stationary points in two-determinant RO-SCF calculations for singlet excited states of selected photoactive compounds. Finally, an application of the ARH-based RO-SCF is illustrated by an investigation of the mechanistic origin of the spin-crossover phenomenon in a Ni(II)-porphyrin complex utilized as a contrast agent.