Time-dependent density functional theory based on a noncollinear formulation of the exchange-correlation potential

In this study we have introduced a formulation of time-dependent density functional theory (TDDFT) based on a noncollinear exchange-correlation potential. This formulation is a generalization of conventional TDDFT. The form of this formulation is exactly the same as that of the conventional TDDFT for the excitation energies of transitions that do not involve spin flips. In addition, this noncollinear TDDFT formulation allows for spin-flip transitions. This feature makes it possible to resolve more fully excited state spin multiplets, while for closed-shell systems, the spin-flip transitions will result in singlet-triplet excitations and this excitation energy calculated from this formulation of TDDFT is exactly the same as that from ordinary TDDFT. This formulation is applied to the dissociation of $H2$ in its $Σg+1$ ground state and $Σu+1$ and $Σu-3$ excited states with $Σu-3$ $(Ms=+1)$ as the reference state and the multiplets splitting of some atoms.