For CIS and TD-DFT calculations, the particle and hole densities defined in Section 2.2 coincide precisely with the attachment density and the detachment density, respectively, quantities that were originally defined in a manner that is not be limited to single-excitation theories. This definition sheds additional light on the nature of ∆P elec and ∆P hole in TD-DFT, and is the topic of this section First, let us diagonalize a difference density matrix ∆P and express the result in the form