# Exact exchange-correlation kernels for optical spectra of model systems

## TL;DR

This paper calculates the exact exchange-correlation kernels for two model systems in time-dependent density functional theory, revealing their frequency and spatial dependence and emphasizing the importance of the exchange-correlation potential's accuracy.

## Contribution

It provides the first exact calculations of $f_{xc}$ for prototype systems, highlighting their simple form near the first excitation and complexity at higher energies.

## Key findings

- Exact $f_{xc}$ has weak frequency dependence near first excitation
- Spatial form of $f_{xc}$ is simple and non-local at low energies
- Higher excitations show complex spatial and frequency behavior

## Abstract

For two prototype systems, we calculate the exact exchange-correlation kernels $f_{\mathrm{xc}}(x,x',\omega)$ of time-dependent density functional theory. $f_{\mathrm{xc}}$, the key quantity for optical absorption spectra of electronic systems, is normally subject to uncontrolled approximation. We find that, up to the first excitation energy, the exact $f_{\mathrm{xc}}$ has weak frequency-dependence and a simple, though non-local, spatial form. For higher excitations, the spatial behavior and frequency-dependence become more complex. The accuracy of the underlying exchange-correlation potential is of crucial importance.