$$\tilde{\beta}(\theta)a=\ddot{f}(m)/\cos ^3\theta ,  \label{fstiff}$$ or, similar to results by Bartelt *et al*.,[@BEW] $$\frac{k_BT}{\tilde{\beta}(\theta )a}=-\frac{g^{\prime \prime }(\rho _0)}{k_BT}%
\cos ^3\theta .  \label{eq:gtostiffrho}$$ Thus, we only need $g^{\prime \prime }(\rho )$ to find the stiffness as a function of $m$ or $\theta $.

Of course, $\rho _0$ in $g^{\prime \prime }$ must be eliminated in favor of $%
m$ via Eq. (\[eq:saddlerho\]). The details for the general case are somewhat involved. Here, we simplify to the physically relevant case of $U=D$ and, defining $S\equiv H+U+D=H+2D$, just quote the results: $$\frac{g^{\prime \prime }\left( \rho _0\right) }{k_BT}=-m\left[ \ \frac{%
2\sinh \rho _0}{C(S,\rho_0)}+\coth \rho _0\right] \ +m^2
\label{eq:ddgfinal}$$ where $C(S,\rho_0) \equiv \cosh S -\cosh \rho_0$ and $\rho _0(m)$ is found by inverting $$\hspace{-.3in}m=\frac{\sinh \rho _0\sinh S}{C(S,\rho_0)\left[ \sinh S-C(S,\rho_0) \left( 1-e^{-2D}\right) \right] }.
\label{eq:rhotom}$$ Some details can be found in the Appendix. Since Eq. (\[eq:rhotom\]) is a quartic equation for $\cosh \rho _0$ or $e^{\rho _0}$, the explicit expression for $\rho _0(m)$ is rather opaque. However, at low-temperatures, a simpler formula emerges, as shown in the next section.

Low-T Solution: Simple Expression
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At low temperatures, we find that the appropriate root for $\rho _0$ diverges. Then we can write $\cosh \rho _0\approx \sinh \rho _0\approx
