Here, the first term represents the one-photon wavefunctions that arise due to the quantum fluctuations of the photon while the second term is the correction due to the dipole self-energy contribution (after variational SCF procedure is performed). 147 For the sake of brevity, we refer the reader to Ref. 168 and Ref. 171 for additional details and discussion on the scQED-DFT formulation. 3.2.5 QED Time-dependent Density Functional Theory Time-dependent DFT (TD-DFT) in the linear response framework 173 has been used ubiquitously over the last couple decades to describe electronic excitations in all manner of chemical systems, due to its computational simplicity and feasibility for large systems from 100s to 1000s of atoms in size. For the cavity QED community, it is natural to extend this powerful method to describe the coupled electron-photon system. Recent work on developing a new scheme to include the additional photonic DOFs are underway with promising results in a variety of molecular systems. The Casida-like generalized eigenvalue equation in the random phase approximation (RPA) can be recast in an increased dimensional space to include the excitation and de-excitation transition densities for both the electron and photon subsystems including all terms in ĤPF (Eq. 55), resulting in the following 93 generalized Casida equation