Excitonic effects in time-dependent density-functional theory: An analytically solvable model

We investigate the description of excitonic effects within time-dependent density-functional theory (TDDFT). The exchange-correlation kernel fxc introduced in TDDFT allows a clear separation of quasiparticle and excitonic effects. Using a diagrammatic representation for fxc, we express its excitonic part f0.2em0exxcEx in terms of the effective vertex function Λ. The latter fulfills an integral equation that thereby establishes the exact correspondence between TDDFT and the standard many-body approach based on the Bethe-Salpeter equation (BSE). The diagrammatic structure of the kernel in the equation for Λ suggests the possibility of strong cancellation effects. Should the cancellation take place, already the first-order approximation to f0.2em0exxcEx is sufficient. A potential advantage of TDDFT over the many-body BSE method is thus dependent on the efficiency of the above-quoted cancellation. We explicitly verify this for an analytically solvable two-dimensional two-band model. The calculations confirm that the low-order f0.2em0exxcEx perfectly describes the bound exciton as well as the excitonic effects in the continuous spectrum in a wide range of the electron-hole coupling strength.