Notes on time-dependent properties and linear response.

Carsten A. Ullrich

This chapter discusses the linear-response framework of time-dependent density-functional theory (TDDFT). It begins with a thorough and self-contained exposition of general linear-response theory for many-body systems. The time-dependent as well as the frequency-dependent response are discussed. Several exact representations and properties of the density-density response function are reviewed, such as the Lehmann representation and fluctuation-dissipation theorem, and the Kramers-Kronig relations. It is shown how several important spectroscopic observables are obtained from linear-response theory. Then, the TDDFT formulation of linear-response theory is derived. The key quantity in this formalism is the so-called exchange-correlation kernel. The remainder of the chapter demonstrates how this linear-response formalism can be used to calculate excitation energies in principle exactly with TDDFT. The Casida equation is derived, as well as several approximate schemes. A comparison is made with time-dependent Hartree-Fock theory.

Makoto Ohya

Ofir E. Alon, Alexej I. Streltsov, Lorenz S. Cederbaum

A unified view on linear response of interacting systems utilizing multiconfigurational time-dependent Hartree methods is presented. The cases of one-particle and two-particle response operators for identical particles and up to all-system response operators for distinguishable degrees-of-freedom are considered. The working equations for systems of identical bosons and identical fermions, as well for systems of distinguishable particles are explicitly derived. These linear-response theories - applicable for discrete excitation spectra - provide numerically exact excitation energies and system's properties, when numerical convergence is achieved in the calculations.

Kota Hanasaki, Sandra Luber

We implemented ab initio Hubbard parameter calculation schemes in the k-point sampling real-time TDDFT (RT-TDDFT) program in CP2K. We propose a new linear-response-based calculation scheme for energy-dependent Hubbard parameters. Our scheme extends the minimum-tracking linear-response method proposed in [Moynihan et al., arXiv preprint arXiv:1704.08076(2017); E. B. Linscott et al., Phys. Rev. B 98, 235157 (2018)] to realize the calculation of energy-dependent Hubbard parameters that reflect the exchange-correlation (xc) effects included in the xc-functional. We discuss the properties of the minimum-tracking linear-response method in comparison to another promising scheme, ACBN0 [Agapito et al., Phys. Rev. X, 5, 011006 (2015)]. We show that, while neither clearly outperforms the other in the accuracy of static property calculations, each has a distinct dynamical application depending on its theoretical formulation.

Tamás S. Bíró, Antal Jakovác