  This paper investigates the relation between the density-scaling exponent
$\gamma$ and the virial potential-energy correlation coefficient $R$ at several
thermodynamic state points in three dimensions for the generalized $(2n,n)$
Lennard-Jones (LJ) system for $n=4, 9, 12, 18$, as well as for the standard
$n=6$ LJ system in two, three, and four dimensions. The state points studied
include many low-density states at which the virial potential-energy
correlations are not strong. For these state points we find the roughly linear
relation $\gamma\cong 3nR/d$ in $d$ dimensions. This result is discussed in
light of the approximate "extended inverse power law" description of
generalized LJ potentials [N. P. Bailey et al., J. Chem. Phys. 129, 184508
(2008)]. In the plot of $\gamma$ versus $R$ there is in all cases a transition
around $R\approx 0.9$, above which $\gamma$ starts to decrease as $R$
approaches unity. This is consistent with the fact that $\gamma\rightarrow
2n/d$ for $R\rightarrow 1$, a limit that is approached at high densities and/or
temperatures at which the repulsive $r^{-2n}$ term dominates the physics.
