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 f(xc) introduced in TDDFT allows a clear separation of quasiparticle and excitonic effects. Using a diagrammatic representation for f(xc), we express its excitonic part f (Ex)(xc) in terms of the effective vertex function Lambda. 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 Lambda suggests the possibility of strong cancellation effects. Should the cancellation take place, already the first-order approximation to f (Ex)(xc) 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 f (Ex)(xc) 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.