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arXiv:1602.03624 · Physical Review A

Validity of power functionals for a homogeneous electron gas in reduced-density-matrix-functional theory

A. Putaja, F. G. Eich, Tim Baldsiefen, E. Räsänen

Published Mar 10, 2016• 3 citations
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Abstract

Physically valid and numerically efficient approximations for the exchange and correlation energy are critical for reduced-density-matrix-functional theory to become a widely used method in electronic structure calculations. Here we examine the physical limits of power functionals of the form $f(n,{n}^{\ensuremath{'}})={(n{n}^{\ensuremath{'}})}^{\ensuremath{\alpha}}$ for the scaling function in the exchange-correlation energy. To this end we obtain numerically the minimizing momentum distributions for the three- and two-dimensional homogeneous electron gas, respectively. In particular, we examine the limiting values for the power $\ensuremath{\alpha}$ to yield physically sound solutions that satisfy the Lieb-Oxford lower bound for the exchange-correlation energy and exclude pinned states with the condition $n(\mathbf{k})<1$ for all wave vectors $\mathbf{k}$. The results refine the constraints previously obtained from trial momentum distributions. We also compute the values for $\ensuremath{\alpha}$ that yield the exact correlation energy and its kinetic part for both the three- and two-dimensional electron gas. In both systems, narrow regimes of validity and accuracy are found at $\ensuremath{\alpha}\ensuremath{\gtrsim}0.6$ and at ${r}_{s}\ensuremath{\gtrsim}10$ for the density parameter, corresponding to relatively low densities.

DOI
10.1103/physreva.93.032503

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Physically valid and numerically efficient approximations for the exchange and correlation energy are critical for reduced-density-matrix-functional theory to become a widely used method in electronic structure calculations. Here we examine the physical limits of power functionals of the form f(n,n\ensuremath′)=(nn\ensuremath′)\ensuremathαf(n,{n}^{\ensuremath{'}})={(n{n}^{\ensuremath{'}})}^{\ensuremath{\alpha}}f(n,n\ensuremath′)=(nn\ensuremath′)\ensuremathα for the scaling function in the exchange-correlation energy. To this end we obtain numerically the minimizing momentum distributions for the three- and two-dimensional homogeneous electron gas, respectively. In particular, we examine the limiting values for the power \ensuremathα\ensuremath{\alpha}\ensuremathα to yield physically sound solutions that satisfy the Lieb-Oxford lower bound for the exchange-correlation energy and exclude pinned states with the condition n(k)<1n(\mathbf{k})<1n(k)<1 for all wave vectors k\mathbf{k}k. The results refine the constraints previously obtained from trial momentum distributions. We also compute the values for \ensuremathα\ensuremath{\alpha}\ensuremathα that yield the exact correlation energy and its kinetic part for both the three- and two-dimensional electron gas. In both systems, narrow regimes of validity and accuracy are found at \ensuremathα\ensuremath≳0.6\ensuremath{\alpha}\ensuremath{\gtrsim}0.6\ensuremathα\ensuremath≳0.6 and at rs\ensuremath≳10{r}_{s}\ensuremath{\gtrsim}10rs​\ensuremath≳10 for the density parameter, corresponding to relatively low densities.
Reduced density-matrix functional theory (RDMFT) has become an appealing alternative to density-functional theory to describe electronic properties of strongly correlated systems. Here we derive exact conditions for the suitability of RDMFT to describe the two-dimensional homogeneous electron gas, which is the base system for semiconductor quantum dots and quantum Hall devices, for example. Following the method of Cioslowski and Pernal [J. Chem. Phys. 111, 3396 (1999)] we focus on the properties of power functionals of the form $f(n,{n}^{\ensuremath{'}})={(n{n}^{\ensuremath{'}})}^{\ensuremath{\alpha}}$ for the scaling function in the exchange-correlation energy. We show that in order to have stable and analytic solutions, and for fff to satisfy the homogeneous scaling constraint, the power is restricted to $1/4\ensuremath{\leqslant}\ensuremath{\alpha}\ensuremath{\leqslant}3/4$. Applying a reasonable ansatz for the momentum distribution and the lower bound for the exchange-correlation energy tightens the physical regime further to $0.64\ensuremath{\lesssim}\ensuremath{\alpha}\ensuremath{\leqslant}0.75$.
The subject of this study is the exchange-correlation-energy functional of reduced density-matrix functional theory. Approximations of this functional are tested by applying them to the homogeneous electron gas. We find that two approximations recently proposed by Gritsenko et al., [J. Chem. Phys. 122, 204102 (2005)] yield considerably better correlation energies and momentum distributions than previously known functionals. We introduce modifications to these functionals, which, by construction, reproduce the exact correlation energy of the homogeneous electron gas.
Scaling arguments show that the exchange-correlation energy density of a spin-unpolarized homogeneous electron gas equals $A{\ensuremath{\omega}}^{4∕3}$, where $\ensuremath{\omega}$ is given by a simple functional of the momentum distribution n(k)n(k)n(k) and AAA is a constant. This functional possesses the property of yielding the true n(k)n(k)n(k) (including the discontinuity at the Fermi momentum, which is not reproduced by any of the presently known density matrix functionals) upon minimization of the energy density eee constrained by the electron density $\ensuremath{\rho}$. At the limit of $\ensuremath{\rho}\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\omega}$ coincides with $\ensuremath{\rho}$ and the dependence of $\ensuremath{\omega}$ on n(k)n(k)n(k), which can be readily recovered from the actual values of eee and n(k)n(k)n(k), is given by an explicit analytical formula. Application of this formalism to general Coulombic systems yields a local version of the density matrix theory, in which the exchange-correlation energy is given by a spatial integral of an expression that depends on the amplitudes of natural orbitals at a given point and the respective occupation numbers.
We propose a method that employs functionals of the one-electron reduced density matrix (density matrix) to capture long-range effects of electron correlation. The complementary short-range regime is treated with density functionals. In an effort to find approximations for the long-range density-matrix functional, a modified power functional is applied to the homogeneous electron gas with Coulomb interactions replaced by their corresponding long-range counterparts. For the power $\ensuremath{\beta}=1/2$ and the range-separation parameter $\ensuremath{\omega}=1/{r}_{s}$, the functional reproduces the correlation and the kinetic correlation energies with a remarkable accuracy for intermediate and large values of rs{r}_{s}rs​. Analysis of the Euler equation corresponding to this functional reveals correct rs{r}_{s}rs​ expansion of the correlation energy in the limit of large rs{r}_{s}rs​. The first expansion coefficient is in very good agreement with that obtained from the modified Wigner-Seitz model.
Accurate treatment of the electronic correlation in inhomogeneous electronic systems, combined with the ability to capture the correlation energy of the homogeneous electron gas, allows to reach high predictive power in the application of density-functional theory. For two-dimensional systems we can achieve this goal by generalizing our previous approximation [Phys. Rev. B 79, 085316 (2009)] to a parameter-free form, which reproduces the correlation energy of the homogeneous gas while preserving the ability to deal with inhomogeneous systems. The resulting functional is shown to be very accurate for finite systems with an arbitrary number of electrons with respect to numerically exact reference data.
We derive an intrinsically temperature-dependent approximation to the correlation grand potential for many-electron systems in thermodynamical equilibrium in the context of finite-temperature reduced-density-matrix-functional theory (FT-RDMFT). We demonstrate its accuracy by calculating the magnetic phase diagram of the homogeneous electron gas. We compare it to known limits from highly accurate quantum Monte Carlo calculations as well as to phase diagrams obtained within existing exchange-correlation approximations from density functional theory and zero-temperature RDMFT.