  The Holstein Hamiltonian describes itinerant electrons whose site density
couples to local phonon degrees of freedom. In the single site limit, at
half-filling, the electron-phonon coupling results in a double well structure
for the lattice displacement, favoring empty or doubly occupied sites. In two
dimensions, and on a bipartite lattice in $d \geq 2$, an intersite hopping
causes these doubly occupied and empty sites to alternate in a charge density
wave (CDW) pattern when the temperature is lowered. Because a discrete symmetry
is broken, this occurs in a conventional second-order transition at finite
$T_{\rm cdw}$. In this paper, we investigate the effect of changing the phonon
potential energy to one with an intrinsic double well structure even in the
absence of an electron-phonon coupling. While this aids in the initial process
of pair formation, the implications for subsequent CDW order are non-trivial.
One expects that, when the electron-phonon coupling is too strong, the double
wells become deep and the polaron mass large, an effect which reduces $T_{\rm
cdw}$. We show here the existence of regions of parameter space where the
double well potential, while aiding local pair formation, does so in a way
which also substantially enhances long range CDW order.
