Finite-Temperature Spin-Adapted ROKS and TDDFT

Abstract

Finite-temperature conditions constitute a central regime of interest in quantum chemistry. Nonetheless, a consistent incorporation of finite-temperature effects into density functional theory (DFT) for both ground and excited states, while rigorously preserving the spin-symmetry associated with the $\hat{S}^2$ operator, has not yet been achieved. In this work, we develop a finite-temperature extension of restricted open-shell Kohn-Sham (ROKS) theory and spin-adapted time-dependent density functional theory (TDDFT), providing unified frameworks for the description of ground and excited states, respectively. For finite-temperature ROKS, we construct a canonical ensemble by using integer high-spin ROKS components, where each component $I$ has the same spin number $|SS \rangle$ and its weight $w_I$ follows the Boltzmann distribution. For finite-temperature spin-adapted TDDFT, every integer component $I$ supplies an ordinary zero-temperature spin-adapted TDDFT matrix pair $(\Mmat_I,\Nmat_I)$. These matrices are arranged in one unified spatial-orbital order and then averaged. In the zero-temperature single-component limit, the theory reduces to ordinary high-spin ROKS and the corresponding spin-adapted TDDFT. As a numerical application, we apply the theory to a diradicaloid and use the calculated excitations to interpret the thermally activated absorption observed in variable-temperature UV/Vis spectroscopy.