  The time-dependent density functional theory (TDDFT) has been broadly used to
investigate the excited-state properties of various molecular systems. However,
the current TDDFT heavily relies on outcomes from the corresponding
ground-state density functional theory (DFT) calculations which may be prone to
errors due to the lack of proper treatment in the non-dynamical correlation
effects. Recently, thermally-assisted-occupation density functional theory
(TAO-DFT) [J.-D. Chai, \textit{J. Chem. Phys.} \textbf{136}, 154104 (2012)], a
DFT with fractional orbital occupations, was proposed, explicitly incorporating
the non-dynamical correlation effects in the ground-state calculations with low
computational complexity. In this work, we develop time-dependent (TD) TAO-DFT,
which is a time-dependent, linear-response theory for excited states within the
framework of TAO-DFT. With tests on the excited states of H$_{2}$, the first
triplet excited state ($1^3\Sigma_u^+$) was described well, with non-imaginary
excitation energies. TDTAO-DFT also yields zero singlet-triplet gap in the
dissociation limit, for the ground singlet ($1^1\Sigma_g^+$) and the first
triplet state ($1^3\Sigma_u^+$). In addition, as compared to traditional TDDFT,
the overall excited-state potential energy surfaces obtained from TDTAO-DFT are
generally improved and better agree with results from the equation-of-motion
coupled-cluster singles and doubles (EOM-CCSD).
