Asymptotic Behavior of the Exchange‐Correlation Energy Density and the Kohn‐Sham Potential in Density Functional Theory: Exact Results and Strategy for Approximations

2020 ◽  
Vol 60 (8-9) ◽  
pp. 805-822 ◽  
Author(s):  
Eli Kraisler
Keyword(s):  
Density Functional Theory ◽  
Asymptotic Behavior ◽  
Energy Density ◽  
Density Functional ◽  
Correlation Energy ◽  
Exact Results ◽  
Functional Theory ◽  
Exchange Correlation

scholarly journals Asymptotic Behavior of the Exchange-Correlation Energy Density and the Kohn-Sham Potential in Density Functional Theory: Exact Results and Strategy for Approximations

2019 ◽  
Author(s):  
Eli Kraisler
Keyword(s):  
Density Functional Theory ◽  
Asymptotic Behavior ◽  
Energy Density ◽  
Density Functional ◽  
Correlation Energy ◽  
Electron Systems ◽  
Functional Theory ◽  
Exchange Correlation ◽  
Exact Exchange ◽  
Exact Factorization

The present work is a review of two analytical properties of the exact exchange-correlation (xc) functional in density-functional theory. These properties are the asymptotic behavior of the xc energy density per particle and the asymptotic behavior of the Kohn-Sham potential, in finite many-electron systems. The derivation of the asymptotic forms for both quantities is reviewed, employing the concepts of the adiabatic connection and of the xc hole with relation to the first quantity and the electron exact factorization approach for the second one. Furthermore, it is shown that the correct asymptotic behavior of one of the aforementioned quantities does not guarantee a correct behavior of the other. In this process, a new quantity, the xc hole response function, is defined and its exact exchange part is analytically derived. The extent to which existing xc approximations satisfy the named exact properties is reviewed and the relationship between correct asymptotics and freedom from one-electron self-interaction in DFT is discussed. Finally, a strategy for development of advanced approximations for exchange and correlation with a correct asymptotic behavior is suggested.<br>

scholarly journals Asymptotic Behavior of the Exchange-Correlation Energy Density and the Kohn-Sham Potential in Density Functional Theory: Exact Results and Strategy for Approximations

2019 ◽  
Author(s):  
Eli Kraisler
Keyword(s):  
Density Functional Theory ◽  
Asymptotic Behavior ◽  
Energy Density ◽  
Density Functional ◽  
Correlation Energy ◽  
Electron Systems ◽  
Functional Theory ◽  
Exchange Correlation ◽  
Exact Exchange ◽  
Exact Factorization

The present work is a review of two analytical properties of the exact exchange-correlation (xc) functional in density-functional theory. These properties are the asymptotic behavior of the xc energy density per particle and the asymptotic behavior of the Kohn-Sham potential, in finite many-electron systems. The derivation of the asymptotic forms for both quantities is reviewed, employing the concepts of the adiabatic connection and of the xc hole with relation to the first quantity and the electron exact factorization approach for the second one. Furthermore, it is shown that the correct asymptotic behavior of one of the aforementioned quantities does not guarantee a correct behavior of the other. In this process, a new quantity, the xc hole response function, is defined and its exact exchange part is analytically derived. The extent to which existing xc approximations satisfy the named exact properties is reviewed and the relationship between correct asymptotics and freedom from one-electron self-interaction in DFT is discussed. Finally, a strategy for development of advanced approximations for exchange and correlation with a correct asymptotic behavior is suggested.<br>

Exchange-Correlation Energy Densities and Response Potentials: Connection Between Two Definitions and Analytical Model for the Strong-Coupling Limit of a Stretched Bond

2019 ◽  
Author(s):  
S. Giarrusso ◽  
Paola Gori-Giorgi
Keyword(s):  
Density Functional Theory ◽  
Strong Coupling ◽  
Density Functional ◽  
Correlation Energy ◽  
Order Term ◽  
Strong Coupling Limit ◽  
Local Form ◽  
Coupling Constant ◽  
Functional Theory ◽  
Exchange Correlation

We analyze in depth two widely used definitions (from the theory of conditional probablity amplitudes and from the adiabatic connection formalism) of the exchange-correlation energy density and of the response potential of Kohn-Sham density functional theory. We introduce a local form of the coupling-constant-dependent Hohenberg-Kohn functional, showing that the difference between the two definitions is due to a corresponding local first-order term in the coupling constant, which disappears globally (when integrated over all space), but not locally. We also design an analytic representation for the response potential in the strong-coupling limit of density functional theory for a model single stretched bond.<br>

scholarly journals Exchange-Correlation Energy Densities and Response Potentials: Connection Between Two Definitions and Analytical Model for the Strong-Coupling Limit of a Stretched Bond

2019 ◽  
Author(s):  
S. Giarrusso ◽  
Paola Gori-Giorgi
Keyword(s):  
Density Functional Theory ◽  
Strong Coupling ◽  
Density Functional ◽  
Correlation Energy ◽  
Order Term ◽  
Strong Coupling Limit ◽  
Local Form ◽  
Coupling Constant ◽  
Functional Theory ◽  
Exchange Correlation

We analyze in depth two widely used definitions (from the theory of conditional probablity amplitudes and from the adiabatic connection formalism) of the exchange-correlation energy density and of the response potential of Kohn-Sham density functional theory. We introduce a local form of the coupling-constant-dependent Hohenberg-Kohn functional, showing that the difference between the two definitions is due to a corresponding local first-order term in the coupling constant, which disappears globally (when integrated over all space), but not locally. We also design an analytic representation for the response potential in the strong-coupling limit of density functional theory for a model single stretched bond.<br>

scholarly journals Approximate bounds and temperature dependence of adiabatic connection integrands for the uniform electron gas

2021 ◽  
Author(s):  
Brittany P. Harding ◽  
Zachary Mauri ◽  
Aurora Pribram-Jones
Keyword(s):  
Density Functional Theory ◽  
Density Functional ◽  
Correlation Energy ◽  
Electron Gas ◽  
High Energy ◽  
Zero Temperature ◽  
Temperature Dependent ◽  
Functional Theory ◽  
Exchange Correlation ◽  
Uniform Electron Gas

Thermal density functional theory is commonly used in simulations of warm dense matter, a highly energetic phase characterized by substantial thermal effects and by correlated electrons demanding quantum mechanical treatment. The numerous approximations for the exchange-correlation energy component in zero-temperature density functional theory, though often used in these high-energy-density simulations with Fermi-weighted electronic densities, are known to miss temperature-dependent effects in the electronic structure of these systems. In this work, the temperature-dependent adiabatic connection is demonstrated and analyzed using a well-known parameterization of the uniform electron gas free energy. Useful tools based on this formalism for analyzing and constraining approximations of the exchange-correlation at zero temperature are leveraged for the finite-temperature case. Inspired by the Lieb-Oxford inequality, which provides a lower bound for the ground-state exchange-correlation energy, bounds for the exchange-correlation at finite-temperatures are approximated for various degrees of electronic correlation.

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