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

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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 fₓcᵉx in terms of the effective vertex function Lambda. The latter fulfills an integral equation which thereby establishes the exact correspondence between TDDFT and the standard many-body approach based on 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 fₓcᵉx 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 fₓcᵉx 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.