Linear response time-dependent density functional theory without unoccupied states: The Kohn-Sham-Sternheimer scheme revisited

Abstract

The Sternheimer approach to time-dependent density functional theory in the linear response regime is attractive because of its computational efficiency. The latter results from avoiding the explicit calculation of unoccupied orbitals and from the basic structure of the Sternheimer equations, which naturally lend themselves to far-reaching parallelization. In this article, we take a fresh look at the frequency-dependent Sternheimer equations. We first give a complete, self-contained derivation of the equations that complements previous derivations. We then discuss several aspects of an efficient numerical realization. As a worked example, we compute the photoabsorption spectra of small hydrogenated silicon clusters and confirm that for these the quality of the Kohn-Sham eigenvalues is more important than the effects of the exchange-correlation kernel. Finally, we demonstrate how triplet excitations can readily be computed from the Sternheimer approach.

Frequency-dependent Sternheimer linear-response formalism for strongly coupled light-matter systems

“…An alternative approach which rather computes the response equations instead of the response function is the frequency-dependent Sternheimer equation [32]. Formulated within the framework of TDDFT, this method computes the linear density response due to an external weak perturbation [37,38,47] as well as non-linear responses [37,48]. This approach has several advantages the main one being that it relies only on the occupied Kohn-Sham orbitals thereby restricting the computation complexity for very large systems.…”

“…Within the context of TDDFT there exist different formalism for computing optical excitation spectra [48]. The frequency-dependent Sternheimer method formulated within TDDFT is a perturbative approach on the Kohn-Sham orbitals that computes the density response without relying explicitly on unoccupied Kohn-Sham orbitals [37,38]. Based on this advantage, an extension of this approach to the setting of QEDFT to treat complex atomic and molecular systems coupled to arbitrary large but finite number of photon modes is an important alternative method to existing QEDFT methods [25,29].…”

“…These two quantities depends on the first-order perturbed Kohn-Sham orbitals ϕ (±) k (r, ω) and photon responses δq (±) α (ω). It is important to note that the first-order response of the Kohn-Sham orbitals must satisfy the orthogonality condition with the groundstate Kohn-Sham orbitals [37,38]:…”

“…Details about the numerical treatment of Eqs. ( 21) and ( 22) have been discussed in the TDDFT framework of the frequency-dependent Sternheimer method [37,38]. Therefore, we only summarize features in the numerical application of these equations.…”

“…Recent applications of the Sternheimer equation have also been used to compute the frequencydependent electronic response [34][35][36][37]. The Sternheimer equation has been formulated within the framework of time-dependent density-functional theory (TDDFT) that allows to study dynamic response of much larger complex systems as it includes only occupied orbitals [37,38]. One of a few advantages the Sternheimer approach has over real-time TDDFT is that it is formulated in the frequency space and the responses at different frequencies can be computed independently of each other allowing for the use of parallelization schemes that speed up the computation.…”