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Physical Review A

Adiabatic connection in density-functional theory: Two electrons on the surface of a sphere

Michael Seidl

Published Jun 13, 2007• 40 citations
Publisher

Abstract

Two interacting electrons that are confined to the surface of a sphere have a uniform ground-state (surface) density. The Schr\"odinger equation of this helium-type two-electron system is solved here accurately for different values $\ensuremath{\alpha}∊\mathbb{R}$ of a constant that is multiplied to the electron-electron repulsion ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}}_{\mathit{ee}}$. The correlation structure in the resulting wave functions is analyzed for different values of $\ensuremath{\alpha}$. The asymptotic limits $\ensuremath{\alpha}\ensuremath{\rightarrow}0$ and $\ensuremath{\alpha}\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$ are treated analytically. Using these results, the ISI (interaction-strength interpolation) model for the density-functional ${E}_{\mathit{xc}}[\ensuremath{\rho}]$ of the exchange-correlation energy in the real system with $\ensuremath{\alpha}=1$ is tested against the exact adiabatic connection in density-functional theory.

DOI
10.1103/physreva.75.062506

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Two interacting electrons that are confined to the surface of a sphere have a uniform ground-state (surface) density. The Schr"odinger equation of this helium-type two-electron system is solved here accurately for different values \ensuremathα∊R\ensuremath{\alpha}∊\mathbb{R}\ensuremathα∊R of a constant that is multiplied to the electron-electron repulsion V\ifmmode^\elseˆ\fiee{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{V}}_{\mathit{ee}}V\ifmmode^\elseˆ\fi​ee​. The correlation structure in the resulting wave functions is analyzed for different values of \ensuremathα\ensuremath{\alpha}\ensuremathα. The asymptotic limits \ensuremathα\ensuremath→0\ensuremath{\alpha}\ensuremath{\rightarrow}0\ensuremathα\ensuremath→0 and \ensuremathα\ensuremath→\ifmmode±\else\textpm\fi\ensuremath∞\ensuremath{\alpha}\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}\ensuremathα\ensuremath→\ifmmode±\else\textpm\fi\ensuremath∞ are treated analytically. Using these results, the ISI (interaction-strength interpolation) model for the density-functional Exc[\ensuremathρ]{E}_{\mathit{xc}}[\ensuremath{\rho}]Exc​[\ensuremathρ] of the exchange-correlation energy in the real system with \ensuremathα=1\ensuremath{\alpha}=1\ensuremathα=1 is tested against the exact adiabatic connection in density-functional theory.
The adiabatic connection formula of ground-state density functional theory relates the correlation energy to a coupling-constant integral over a purely potential contribution, and is widely used to understand and improve approximations. The corresponding formula for thermal density functional theory is cast as an integral over temperatures instead, ranging upward from the system's physical temperature. We also show how to relate different correlation components to each other, either in terms of temperature or coupling-constant integrations. We illustrate our results on the uniform electron gas.
We present a new method for realizing the adiabatic connection approach in density functional theory, which is based on combining accurate variational quantum Monte Carlo calculations with a constrained optimization of the ground state many-body wavefunction for different values of the Coulomb coupling constant. We use the method to study an electron gas in the presence of a cosine-wave potential. For this system we present results for the exchange-correlation hole and exchange-correlation energy density, and compare our findings with those from the local density approximation and generalized gradient approximation. 1. Introduction 2. The Adiabatic connection 3. Quantum Monte Carlo realization 3.1 Variational Monte Carlo 3.2 Fixed-density variance minimization 4. Cosine-wave jellium 4.1 Many-body wavefunction 5. Results and discussion 1 Introduction Density functional theory (DFT) (1, 2) is the main computational tool for the treatment of many-body effects in solid state electroni...
Electrons can have a given smooth density distribution $\ensuremath{\rho}(r),$ even if their Coulomb interaction is scaled to infinity. This strong-interaction limit of density-functional theory provides essential information on the correlation energy of real electron systems. The simple concept of strictly correlated electrons (SCE) is analyzed here as a model for that limit. SCE is solved exactly for any one-dimensional (1D) N-electron density and, in particular, for any 3D spherical two-electron system, such as the helium atom. Both the SCE interaction energy and the SCE external potential, which are obtained here as density functionals, obey all the relations known for the corresponding quantities in the unknown true strong-interaction limit. At large but finite interaction, the electrons are still strongly correlated, performing zero-point oscillations about the SCE limit.
We consider the high-density-limit correlation energy E(c) in D > or = 2 dimensions for the (1)S ground states of three two-electron systems: helium (in which the electrons move in a Coulombic field), spherium (in which they move on the surface of a sphere), and hookium (in which they move in a quadratic potential). We find that the E(c) values are strikingly similar, depending strongly on D but only weakly on the external potential. We conjecture that for large D, the limiting correlation energy E(c) approximately -delta(2)/8 in any confining external potential, where delta = 1/(D-1).
We study the properties of the Hooke's law correlation energy (Ec{E}_{\mathrm{c}}Ec​), defined as the correlation energy when two electrons interact via a harmonic potential in a DDD-dimensional space. More precisely, we investigate the 1S{}^{1}S1S ground-state properties of two model systems: the Moshinsky atom (in which the electrons move in a quadratic potential) and the spherium model (in which they move on the surface of a sphere). A comparison with their Coulombic counterparts is made that highlights the main differences of the Ec{E}_{\mathrm{c}}Ec​ in both the weakly and strongly correlated limits. Moreover, we show that the Schr\"odinger equation of the spherium model is exactly solvable for two values of the dimension (D=1and3D=1 \text{and} 3D=1and3) and that the exact wave function is based on Mathieu functions.
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.