Excitation energies from time-dependent density-functional theory beyond the adiabatic approximation

Time-dependent density-functional theory in the adiabatic approximation has been very successful for calculating excitation energies in molecular systems. This paper studies nonadiabatic effects for excitation energies, using the current–density functional of Vignale and Kohn [Phys. Rev. Lett. 77, 2037 (1996)]. We derive a general analytic expression for nonadiabatic corrections to excitation energies of finite systems and calculate singlet $s→s$ and $s→p$ excitations of closed-shell atoms. The approach works well for $s→s$ excitations, giving a small improvement over the adiabatic local-density approximation, but tends to overcorrect $s→p$ excitations. We find that the observed problems with the nonadiabatic correction have two main sources: (1) the currents associated with the $s→p$ excitations are highly nonuniform and, in particular, change direction between atomic shells, (2) the so-called exchange-correlation kernels of the homogeneous electron gas, $fxcL$ and $fxcT,$ are incompletely known, in particular in the high-density atomic core regions.