UMRSF-TDDFT: Unrestricted Mixed-Reference Spin-Flip-TDDFT

An unrestricted version of Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory (UMRSF-TDDFT) was developed based on unrestricted Kohn-Sham orbitals (UKS) with a new molecular orbital (MO) reordering scheme. Additionally, a simple yet accurate method for estimating \( \langle S^2 \rangle \) expectation values was devised. UMRSF-TDDFT was benchmarked against cases where DFT, TDDFT, and SF-TDDFT traditionally fail to provide accurate descriptions. In an application to the ground and excited states of a Be atom, UMRSF-TDDFT successfully recovers the degenerate states, with its energies slightly reduced compared to its RO counterpart, due to the additional variational flexibility of UKS. A clear difference between UMRSF and U-SF-TDDFT is evident in the bond breaking of the hydrogen fluoride system, as the latter misses an important configuration. In the case of the Jahn-Teller distortion of trimethylenemethane (TMM), the relative singlet energy compared to the triplet is lower by 0.1 eV and 0.2 eV for UMRSF and U-SF-TDDFT, respectively, than that of MRSF-TDDFT. The reduction in UMRSF energy is attributed to spatial orbital relaxations, whereas the reduction in U-SF-TDDFT energy results from spin contamination. Overall, the additional orbital relaxations afforded by unrestricted Kohn-Sham (UKS) orbitals in UMRSF-TDDFT lead to lower total system energies compared to their restricted open-shell counterparts. This enhancement adds a practical and accurate quantum chemical theory to the existing RO variant for addressing challenging systems where traditional quantum theories suffer.