UMRSF-TDDFT: Unrestricted\nMixed-Reference Spin-Flip-TDDFT

An unrestricted version of Mixed-Reference Spin-Flip\nTime-Dependent\nDensity Functional Theory (UMRSF-TDDFT) was developed based on unrestricted\nKohn–Sham orbitals (UKS) with a new molecular orbital (MO)\nreordering scheme. Additionally, a simple yet accurate method for\nestimating ⟨<i>S</i><sup>2</sup>⟩ expectation\nvalues was devised. UMRSF-TDDFT was benchmarked against cases where\nDFT, TDDFT, and SF-TDDFT traditionally fail to provide accurate descriptions.\nIn an application to the ground and excited states of a Be atom, UMRSF-TDDFT\nsuccessfully recovers the degenerate states, with its energies slightly\nreduced compared to its RO counterpart, due to the additional variational\nflexibility of UKS. A clear difference between UMRSF and U-SF-TDDFT\nis evident in the bond breaking of the hydrogen fluoride system, as\nthe latter misses an important configuration. In the case of the Jahn–Teller\ndistortion of trimethylenemethane (TMM), the relative singlet energy\ncompared to the triplet is lower by 0.1 and 0.2 eV for UMRSF and U-SF-TDDFT,\nrespectively, than that of MRSF-TDDFT. The reduction in UMRSF energy\nis attributed to spatial orbital relaxations, whereas the reduction\nin U-SF-TDDFT energy results from spin contamination. Overall, the\nadditional orbital relaxations afforded by unrestricted Kohn–Sham\n(UKS) orbitals in UMRSF-TDDFT lead to lower total system energies\ncompared to their restricted open-shell counterparts. This enhancement\nadds a practical and accurate quantum chemical theory to the existing\nRO variant for addressing challenging systems where traditional quantum\ntheories suffer.