Double excitation effect in non-adiabatic time-dependent density functional theory with an analytic construction of the exchange–correlation kernel in the common energy denominator approximation

Time-dependent density functional (response) theory (TDDF(R)T) is applied almost exclusively in its adiabatic approximation (ATDDFT), which is restricted to predominantly single electronic excitations and neglects additional roots of the TDDFT eigenvalue problem stemming from the interaction between single and double excitations. We incorporate the effect of the latter interaction into a non-adiabatic frequency-dependent and spatially non-local Hartree-exchange–correlation (Hxc) kernel fCEDAHxc (r1, r2, ω), the explicit analytical expression of which is derived for interacting single and double excitations well separated from the other excitations, within the common energy denominator approximation (CEDA) for the Kohn–Sham (KS) and interacting density response functions, χs and χ, respectively. The kernel fCEDAHxc (r1, r2, ω) obtained from the direct analytical inverse of χCEDAs and χCEDA is a sum of the delta-function and non-local orbital-dependent spatial terms with frequency-dependent factors, with which fCEDAHxc acquires a modulated quadratic dependence on ω. The effective incorporation in fCEDAHxc of the complete manifold of excited states (through the delta function term) represents an extension of the kernel reported by Maitra, Zhang, Cave, and Burke [J. Chem. Phys., 2004, 120, 5932]. In the TDDFT eigenvalue equations considered in the diagonal approximation, fCEDAHxc generates two excitation energies ωq and ωq+1, which both correspond to the same single KS excitation ωsq, thus producing the effect of the single–double excitation interaction.