Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Condensed matter

Exchange-correlation kernel in time-dependent density functional theory

F. Aryasetiawan, O. Gunnarsson

Published Oct 31, 2002• 32 citations
Publisher

Abstract

In time-dependent density-functional theory, an important quantity is the exchange-correlation kernel whose knowledge allows for the calculations of the excitation spectrum of electronic systems. In most applications, the exchange-correlation kernel is approximated by the adiabatic local-density approximation, ignoring the frequency dependence and nonlocality. To gain insight into the nature of the exact exchange-correlation kernel, we have considered a two-dimensional one-band Hubbard model. The calculated exact exchange-correlation kernel reveals a number of striking features. It has a weak energy dependence up to the main excitation energy. On the other hand, the exact kernel shows a strong energy dependence in energy regions, where there are many-body excitations that are not contained in the local-density approximation, even though these excitations have small weight. It is found that a static approximation can well reproduce the main excitation peaks, but the satellite structures arising from many-body interaction are unlikely to be accounted for with a static approximation.

DOI
10.1103/physrevb.66.165119

Related papers

Reconstructing the adiabatic exchange-correlation kernel of time-dependent density-functional theory

M. Thiele, Stephan Kümmel

The interacting and the Kohn-Sham static density-density response functions for different one-dimensional two-electron singlet systems are reconstructed numerically. From their inverse we obtain the exact static exchange-correlation kernel. This quantity represents the adiabatically exact approximation of the frequency-dependent exchange-correlation kernel that is crucial for time-dependent linear density-response theory. We investigate its performance for nonlocal perturbations and analyze its sum rule properties. We also compute the adiabatically exact transition energies that follow from the static kernel within linear-response theory.

Physical Review A•Jul 24, 2009•10 citations

Exact exchange-correlation kernels for optical spectra of model systems

Mike Entwistle, Rex W. Godby

For two prototype systems, we calculate the exact exchange-correlation kernels ${f}_{\mathrm{xc}}(x,{x}^{\ensuremath{'}},\ensuremath{\omega})$ 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 nonlocal, 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.

arXiv:1812.05512•Physical review. B./Physical review. B•Apr 3, 2019•11 citations

Time-dependent density-functional theory for strongly interacting electrons

Luis Cort, Daniel Karlsson, Giovanna Lani +1 more

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 previously [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 nonlocality 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.

arXiv:1703.00724•Physical Review A•Apr 11, 2017•10 citations

Insights from exact exchange-correlation kernels

Nick Woods, Mike Entwistle, Rex W. Godby

The exact exchange-correlation (xc) kernel ${f}_{\text{xc}}(x,{x}^{\ensuremath{'}},\ensuremath{\omega})$ of linear response time-dependent density functional theory is computed over a wide range of frequencies for three canonical one-dimensional finite systems. Methods used to ensure the numerical robustness of ${f}_{\text{xc}}$ are set out. The frequency dependence of ${f}_{\text{xc}}$ is found to be largely due to its analytic structure, i.e., its singularities at certain frequencies, which are required in order to capture particular transitions, including those of double excitation character. However, within the frequency range of the first few interacting excitations, ${f}_{\text{xc}}$ is approximately $\ensuremath{\omega}$ independent, meaning the exact adiabatic approximation ${f}_{\text{xc}}(\ensuremath{\omega}=0)$ remedies the failings of the local density approximation and random phase approximation for these lowest transitions. The key differences between the exact ${f}_{\text{xc}}$ and its common approximations are analyzed, and cannot be eliminated by exploiting the limited gauge freedom in ${f}_{\text{xc}}$. The optical spectrum benefits from using as accurate as possible an ${f}_{\text{xc}}$ and ground-state xc potential, while maintaining exact compatibility between the two is of less importance.

arXiv:2101.12207•Physical review. B./Physical review. B•Mar 29, 2021•9 citations

Exchange and Correlation Kernels at the Resonance Frequency: Implications for Excitation Energies in Density-Functional Theory

Xavier Gonze, Matthias Scheffler

Specific matrix elements of exchange and correlation kernels in time-dependent density-functional theory are computed. The knowledge of these matrix elements not only constrains approximate time-dependent functionals. It also allows one to link different practical approaches to excited states, based either on density-functional theory or on many-body perturbation theory, despite the approximations that have been performed to derive them.

arXiv:cond-mat/9905119•Physical Review Letters•May 31, 1999•63 citations

Time-Dependent Density-Functional Theory and Strongly Correlated Systems: Insight from Numerical Studies

Claudio Verdozzi

We illustrate the scope of time-dependent density-functional theory for strongly correlated (lattice) models out of equilibrium. Using the exact many-body time evolution, we reverse engineer the exact exchange correlation (xc) potential v_(xc) for small Hubbard chains exposed to time-dependent fields. We introduce an adiabatic local density approximation to v_(xc) for the 1D Hubbard model and compare it to exact results, to gain insight about approximate xc potentials. Finally, we provide some remarks on the v-representability for the 1D Hubbard model.

Physical Review Letters•Oct 13, 2008•128 citations
In time-dependent density-functional theory, an important quantity is the exchange-correlation kernel whose knowledge allows for the calculations of the excitation spectrum of electronic systems. In most applications, the exchange-correlation kernel is approximated by the adiabatic local-density approximation, ignoring the frequency dependence and nonlocality. To gain insight into the nature of the exact exchange-correlation kernel, we have considered a two-dimensional one-band Hubbard model. The calculated exact exchange-correlation kernel reveals a number of striking features. It has a weak energy dependence up to the main excitation energy. On the other hand, the exact kernel shows a strong energy dependence in energy regions, where there are many-body excitations that are not contained in the local-density approximation, even though these excitations have small weight. It is found that a static approximation can well reproduce the main excitation peaks, but the satellite structures arising from many-body interaction are unlikely to be accounted for with a static approximation.
The interacting and the Kohn-Sham static density-density response functions for different one-dimensional two-electron singlet systems are reconstructed numerically. From their inverse we obtain the exact static exchange-correlation kernel. This quantity represents the adiabatically exact approximation of the frequency-dependent exchange-correlation kernel that is crucial for time-dependent linear density-response theory. We investigate its performance for nonlocal perturbations and analyze its sum rule properties. We also compute the adiabatically exact transition energies that follow from the static kernel within linear-response theory.
For two prototype systems, we calculate the exact exchange-correlation kernels ${f}_{\mathrm{xc}}(x,{x}^{\ensuremath{'}},\ensuremath{\omega})$ of time-dependent density functional theory. fxc{f}_{\mathrm{xc}}fxc​, 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 fxc{f}_{\mathrm{xc}}fxc​ has weak frequency dependence and a simple, though nonlocal, 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.
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 previously [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 nonlocality 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.
The exact exchange-correlation (xc) kernel ${f}_{\text{xc}}(x,{x}^{\ensuremath{'}},\ensuremath{\omega})$ of linear response time-dependent density functional theory is computed over a wide range of frequencies for three canonical one-dimensional finite systems. Methods used to ensure the numerical robustness of fxc{f}_{\text{xc}}fxc​ are set out. The frequency dependence of fxc{f}_{\text{xc}}fxc​ is found to be largely due to its analytic structure, i.e., its singularities at certain frequencies, which are required in order to capture particular transitions, including those of double excitation character. However, within the frequency range of the first few interacting excitations, fxc{f}_{\text{xc}}fxc​ is approximately $\ensuremath{\omega}$ independent, meaning the exact adiabatic approximation ${f}_{\text{xc}}(\ensuremath{\omega}=0)$ remedies the failings of the local density approximation and random phase approximation for these lowest transitions. The key differences between the exact fxc{f}_{\text{xc}}fxc​ and its common approximations are analyzed, and cannot be eliminated by exploiting the limited gauge freedom in fxc{f}_{\text{xc}}fxc​. The optical spectrum benefits from using as accurate as possible an fxc{f}_{\text{xc}}fxc​ and ground-state xc potential, while maintaining exact compatibility between the two is of less importance.
We illustrate the scope of time-dependent density-functional theory for strongly correlated (lattice) models out of equilibrium. Using the exact many-body time evolution, we reverse engineer the exact exchange correlation (xc) potential v_(xc) for small Hubbard chains exposed to time-dependent fields. We introduce an adiabatic local density approximation to v_(xc) for the 1D Hubbard model and compare it to exact results, to gain insight about approximate xc potentials. Finally, we provide some remarks on the v-representability for the 1D Hubbard model.