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

Excitation Energies from Thermally-Assisted-Occupation Density Functional Theory: Theory and Computational Implementation