  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
$f(n,n')=(n n')^\alpha$ 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
$n({\mathbf k})<1$ for all wave vectors ${\mathbf k}$. The results refine the
constraints previously obtained from trial momentum distributions. We also
compute the values for $\alpha$ 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.
