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arXiv:1208.4707

Reduced density matrix functional theory at finite temperature. III. Application to the electron gas: Correlation effects

Tim Baldsiefen, Eberhard K. U. Gross

Published Aug 23, 2012• 4 citations
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Abstract

Based on our derivation of finite temperature reduced density matrix functional theory and the discussion of the performance of its first-order functional this work presents several different correlation-energy functionals and applies them to the homogeneous electron gas. The zero temperature limits of the correlation-energy and the momentum distributions are investigated and the magnetic phase diagrams in collinear spin configuration are discussed.

DOI
10.48550/arxiv.1208.4707

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Based on our derivation of finite temperature reduced density matrix functional theory and the discussion of the performance of its first-order functional this work presents several different correlation-energy functionals and applies them to the homogeneous electron gas. The zero temperature limits of the correlation-energy and the momentum distributions are investigated and the magnetic phase diagrams in collinear spin configuration are discussed.
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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.
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.
The formalism developed in the first paper of the series [arXiv:0901.1060] is applied to two thermodynamic systems: (i) of three global observables (the energy, the total electron number and the spin number), (ii) of one global observable (the internal electron energy) and two local (position-dependent) observables (the total electron density and the spin density). The two-component potential of the many-electron system of interest is constructed of a scalar external potential and a collinear magnetic field (coupled only with the spin operator). Various equilibrium characteristics of two systems are defined and investigated. Conditions for the equivalence between two systems (the same equilibrium density matrix demanded) are derived and thoroughly discussed. The applicability of the Hohenberg-Kohn theorem is extended to the thermodynamic spin-density functional theory. Obtained results provide a rigorous mathematical foundation for future derivation of the zero-temperature limit of this theory and determination of its chemical reactivity indices.
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(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 $\ensuremath{\alpha}$ 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.