Rigorous Excited-State Entropy in Finite-Temperature Time-Dependent Density Functional Theory

We present a formal derivation of the excited-state difference density matrix at finite electronic temperature within linear-response Time-Dependent Density Functional Theory (TDDFT). Unlike previous models that use a diagonal approximation for excited-state occupation numbers, our approach incorporates the full density matrix and orbital relaxations via the Z-vector formalism. The method, implemented in the Time-Dependent Density Functional Tight-Binding (TD-DFTB) framework, is applied to the torsional rotation of ethylene and the charge transfer in p-nitroaniline. We show that defining the excited-state entropy through the full density matrix leads to an improved description along the potential energy surface. This framework offers a rigorous starting point for the treatment of excited states in warm dense matter and other systems where fractional occupations are significant.