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

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 $f(n,n^{\prime })={\left(n{n}^{\prime }\right)}^{\alpha }$ 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 $\alpha$ to yield physically sound solutions that satisfy the Lieb-Oxford lower bound for the exchange-correlation energy and exclude pinned states with the condition $n\left(\mathbf{k}\right)<1$ for all wave vectors $\mathbf{k}$. The results refine the constraints previously obtained from trial momentum distributions. We also compute the values for $\alpha$ 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 $\alpha \gtrsim 0.6$ and at $r_s\gtrsim 10$ for the density parameter, corresponding to relatively low densities.

Figure 1

Density-dependent bounds on the power functional for the 3DEG. The dashed lines refer to the results from the variational ansatz and the solid lines correspond to numerical results. The blue (squares) lines separate the regions with pinned states (below the curves) and without pinned states (above the curves). The red (circles) and orange (triangles) lines denote boundary of the regions where the LO bound is obeyed (below the curves) and the regions where it is violated (above the curves) for different values of the constant $C$.

Figure 2

Same as Fig. 1 but for the 2DEG. Here we only show the result for the tight LO-type bound in 2D ($C=1.96$) according to Ref. [21].

Figure 3

Exact $E_c$ (red, circles), ${\tilde{E}}_c$ (orange, squares), and $t_c$ (green, triangles) for the 3DEG. Within the shaded region the tight LO bound ($C=1.44$) is satisfied and the momentum distribution is strictly smaller than one. The solid black line denotes the violation of the LO bound when the kinetic contribution to the correlations energy is taken into account.

Figure 4

Same as Fig. 3 but for the 2DEG. In the shaded region the tight LO bound ($C=1.96$) is satisfied and the momentum distribution is not pinned, i.e., $n\left(k\right)<1$. The solid black line denotes the violation of the LO bound when the kinetic contribution to the correlations energy is taken into account.