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We introduce a new implementation of time-dependent density-functional theory which allows the emph{entire} spectrum of a molecule or extended system to be computed with a numerical effort comparable to that of a emph{single} standard ground-state calculation. This method is particularly well suited for large systems and/or large basis sets, such as plane waves or real-space grids. By using a super-operator formulation of linearized time-dependent density-functional theory, we first represent the dynamical polarizability of an interacting-electron system as an off-diagonal matrix element of the resolvent of the Liouvillian super-operator. One-electron operators and density matrices are treated using a representation borrowed from time-independent density-functional perturbation theory, which permits to avoid the calculation of unoccupied Kohn-Sham orbitals. The resolvent of the Liouvillian is evaluated through a newly developed algorithm based on the non-symmetric Lanczos method. Each step of the Lanczos recursion essentially requires twice as many operations as a single step of the iterative diagonalization of the unperturbed Kohn-Sham Hamiltonian. Suitable extrapolation of the Lanczos coefficients allows for a dramatic reduction of the number of Lanczos steps necessary to obtain well converged spectra, bringing such number down to hundreds (or a few thousands, at worst) in typical plane-wave pseudopotential applications. The resulting numerical workload is only a few times larger than that needed by a ground-state Kohn-Sham calculation for a same system. Our method is demonstrated with the calculation of the spectra of benzene, C$_{60}$ fullerene, and of chlorofyll a.
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Linear-response time-dependent density-functional theory (TDDFT) can describe excitonic features in the optical spectra of insulators and semiconductors, using exchange-correlation (xc) kernels behaving as $-1/k^{2}$ to leading order. We show how excitons can be modeled in real-time TDDFT, using an xc vector potential constructed from approximate, long-range corrected xc kernels. We demonstrate for various materials that this real-time approach is consistent with frequency-dependent linear response, gives access to femtosecond exciton dynamics following short-pulse excitations, and can be extended with some caution into the nonlinear regime.
Time-dependent density-functional theory (TDDFT) is a computationally efficient first-principles approach for calculating optical spectra in insulators and semiconductors, including excitonic effects. We show how exciton wave functions can be obtained from TDDFT via the Kohn-Sham transition density matrix, both in the frequency-dependent linear-response regime and in real-time propagation. The method is illustrated using one-dimensional model solids. In particular, we show that our approach provides insight into the formation and dissociation of excitons in real time. This opens the door to time-resolved studies of exciton dynamics in materials by means of real-time TDDFT.
Excitons are electron-hole pairs appearing below the band gap in insulators and semiconductors. They are vital to photovoltaics, but are hard to obtain with time-dependent density-functional theory (TDDFT), since most standard exchange-correlation (xc) functionals lack the proper long-range behavior. Furthermore, optical spectra of bulk solids calculated with TDDFT often lack the required resolution to distinguish discrete, weakly bound excitons from the continuum. We adapt the Casida equation formalism for molecular excitations to periodic solids, which allows us to obtain exciton binding energies directly. We calculate exciton binding energies for both small- and large-gap semiconductors and insulators, study the recently proposed bootstrap xc kernel [S. Sharma et al., Phys. Rev. Lett. 107, 186401 (2011)], and extend the formalism to triplet excitons.
I show that the so-called causality paradox of time-dependent density functional theory arises from an incorrect formulation of the variational principle for the time evolution of the density. The correct formulation not only resolves the paradox in real time, but also leads to a new expression for the causal exchange-correlation kernel in terms of Berry curvature. Furthermore, I show that all the results that were previously derived from symmetries of the action functional remain valid in the present formulation. Finally, I develop a model functional theory which explicitly demonstrates the workings of the new formulation.
The accurate description of the optical spectra of insulators and semiconductors remains an important challenge for time-dependent density-functional theory (TDDFT). Evidence has been given in the literature that TDDFT can produce bound as well as continuum excitons for specific systems, but there are still many unresolved basic questions concerning the role of dynamical exchange and correlation (xc). In particular, the role of the long spatial range and the frequency dependence of the xc kernel $f_{rm xc}$ for excitonic binding are still not very well explored. We present a minimal model for excitons in TDDFT, consisting of two bands from a one-dimensional Kronig-Penney model and simple approximate xc kernels, which allows us to address these questions in a transparent manner. Depending on the system, it is found that adiabatic xc kernels can produce a single bound exciton, and sometimes two bound excitons, where the long spatial range of $f_{rm xc}$ is not a necessary condition. It is shown how the Wannier model, featuring an effective electron-hole interaction, emerges from TDDFT. The collective, many-body nature of excitons is explicitly demonstrated.
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