# Time-dependent density-functional theory for strongly interacting electrons

## TL;DR

This paper analytically investigates the exact exchange-correlation kernel in time-dependent density functional theory for strongly interacting electrons, revealing the dominant instantaneous terms and the conditions under which the adiabatic approximation fails.

## Contribution

It provides an analytical expansion of the exchange-correlation kernel in the strong interaction limit and compares it with the strictly correlated electron approach, highlighting the approximation's validity.

## Key findings

- Leading terms in the kernel act instantaneously in the strong interaction limit.
- Memory effects appear only in higher-order terms of the expansion.
- The adiabatic strictly correlated electron approximation matches the exact leading terms.

## Abstract

We consider an analytically solvable model of two interacting electrons that allows for the calculation of the exact exchange-correlation kernel of time-dependent density functional theory. This kernel, as well as the corresponding density response function, is studied in the limit of large repulsive interactions between the electrons and we give analytical results for these quantities as an asymptotic expansion in powers of the square root of the interaction strength. We find that in the strong interaction limit the three leading terms in the expansion of the kernel act instantaneously while memory terms only appear in the next orders. We further derive an alternative expansion for the kernel in the strong interaction limit on the basis of the theory developed in [Phys. Chem. Chem. Phys. 18, 21092 (2016)] using the formalism of strictly correlated electrons in the adiabatic approximation. We find that the first two leading terms in this series, corresponding to the strictly correlated limit and its zero-point vibration correction, coincide with the two leading terms of the exact expansion. We finally analyze the spatial non-locality of these terms and show when the adiabatic approximation breaks down. The ability to reproduce the exact kernel in the strong interaction limit indicates that the adiabatic strictly correlated electron formalism is useful for studying the density response and excitation properties of other systems with strong electronic interactions.