Electronic excitations lie at the origin of most of the commonly measured spectra. However, the first-principles computation of excited states requires a larger effort than ground-state calculations, which can be very efficiently carried out within density-functional theory. On the other hand, two theoretical and computational tools have come to prominence for the description of electronic excitations. One of them, many-body perturbation theory, is based on a set of Green's-function equations, starting with a one-electron propagator and considering the electron-hole Green's function for the response. Key ingredients are the electron's self-energy Σ and the electron-hole interaction. A good approximation for Σ is obtained with Hedin's $\mathrm{GW}$ approach, using density-functional theory as a zero-order solution. First-principles $\mathrm{GW}$ calculations for real systems have been successfully carried out since the 1980s. Similarly, the electron-hole interaction is well described by the Bethe-Salpeter equation, via a functional derivative of Σ. An alternative approach to calculating electronic excitations is the time-dependent density-functional theory (TDDFT), which offers the important practical advantage of a dependence on density rather than on multivariable Green's functions. This approach leads to a screening equation similar to the Bethe-Salpeter one, but with a two-point, rather than a four-point, interaction kernel. At present, the simple adiabatic local-density approximation has given promising results for finite systems, but has significant deficiencies in the description of absorption spectra in solids, leading to wrong excitation energies, the absence of bound excitonic states, and appreciable distortions of the spectral line shapes. The search for improved TDDFT potentials and kernels is hence a subject of increasing interest. It can be addressed within the framework of many-body perturbation theory: in fact, both the Green's functions and the TDDFT approaches profit from mutual insight. This review compares the theoretical and practical aspects of the two approaches and their specific numerical implementations, and presents an overview of accomplishments and work in progress.

An ab initio approach to the calculation of excitonic effects in the optical absorption spectra of semiconductors and insulators is formulated. It starts from a quasiparticle band structure calculation and is based on the relevant Bethe-Salpeter equation. An application to bulk silicon shows a substantial improvement with respect to previous calculations in the description of the experimental spectrum, for both peak positions and line shape.

Starting from the many-body Bethe-Salpeter equation we derive an exchange-correlation kernel $f_{\mathrm{xc}}$ that reproduces excitonic effects in bulk materials within time-dependent density functional theory. The resulting $f_{\mathrm{xc}}$ accounts for both self-energy corrections and the electron-hole interaction. It is static, nonlocal, and has a long-range Coulomb tail. Taking the example of bulk silicon, we show that the $-\alpha {/q}^2$ divergency is crucial and can, in the case of continuum excitons, even be sufficient for reproducing the excitonic effects and yielding excellent agreement between the calculated and the experimental absorption spectrum.

We report the first ab initio quasiparticle calculation in a real cluster ${\mathrm{Na}}_4$ within Hedin's $\mathrm{GW}$ approximation for the valence electron self-energy. Our approach avoids the summations over empty states, and also eliminates the problem of residual interactions between the periodic images. Self-energy corrections open the local density approximation gap by more than 2 eV; finite-size effects on screening are shown to play an important role. The absorption spectrum calculated by including excitonic effects using our ab initio screened interaction gives a good account of the experimental photodepletion data.

We discuss the effects of a static long-range contribution $-\alpha {/q}^2$ to the exchange-correlation kernel $f_{\mathrm{xc}}\left(\mathbf{q}\right)$ of time-dependent density functional theory. We show that the optical absorption spectrum of solids exhibiting a strong continuum excitonic effect is considerably improved with respect to calculations where the adiabatic local-density approximation is used. We discuss the limitations of this simple approach, and in particular that the same improvement cannot be found for the whole spectral range including the valence plasmons and bound excitons. On the other hand, we also show that within the range of validity of the method, the parameter $\alpha$ depends linearly on the inverse of the dielectric constant, and we demonstrate that this fact can be used to predict continuum excitonic effects in semiconductors. Results are shown for the real and imaginary part of the dielectric function of Si, GaAs, AlAs, diamond, MgO, SiC and Ge, and for the loss function of Si.