e^{\rho _0}/2$. Of course, $H\propto 1/T$ so that $\cosh S\approx e^S/2$. With these approximations, Eq. (\[eq:rhotom\]) becomes quadratic in $%
e^{\rho _0}$: $$\begin{aligned}
  \label{eq:lowT}
  m = \frac{ e^{\rho_0 + S}  }{(e^S-e^{\rho_0})[e^S-(e^S-e^{\rho_0})(1-e^{-2 D})]}\end{aligned}$$ Likewise, the expression for $g^{\prime \prime}(\rho_0)$, Eq. (\[eq:ddgfinal\]), becomes $$\begin{aligned}
  \label{eq:lowTddg}
\frac{g^{\prime \prime }\left( \rho_0 \right)}{k_B T}
&=&-m \left[\ \frac{2 e^{\rho_0}}{(e^S - e^{\rho_0})} + 1 \right]\ + m^2.\end{aligned}$$ Solving for $e^{\rho_0}$ in Eq. (\[eq:lowT\]) and inserting the solution into Eq. (\[eq:lowTddg\]) gives $$\begin{aligned}
   \label{eq:ddglowt}
\frac{g^{\prime \prime }\left( \rho_0\right)}{k_B T}
&=&-m  \sqrt{(1 - m)^2 + 4 m e^{-2D}}.\end{aligned}$$ so that, from Eq. (\[eq:gtostiffrho\]), and recalling $D \! =\! -\epsilon_2/2k_BT$, we arrive at our [*main result*]{}, a simple, algebraic expression for $\tilde{\beta}$ as a function of $m$: $$\begin{aligned}
  \label{eq:nnnstiffness}
  \frac{k_B T}{\tilde{\beta}a} = \frac{m \sqrt{(1-m)^2 + 4me^{\epsilon_2/k_BT}}}
  {( 1 + m^2 )^{3/2}  }.\end{aligned}$$ We examine Eq. (\[eq:nnnstiffness\]) in several different limiting cases. When $\epsilon_2=0$, this reduces to $$\begin{aligned}
 \label{eq:stiffd=0}
  \frac{k_B T}{\tilde{\beta}a} = \frac{m + m^2  }{\left( 1 + m^2 \right)^{3/2}  }.\end{aligned}$$ as found in a previous study involving only NN interactions.[@dieluweit02] Interestingly, at $\theta = 45^\circ$, Eq. (\[eq:nnnstiffness\]) shows a simple dependence on $\epsilon_2$, namely, $$\begin{aligned}
  \label{eq:m=1}
     \frac{k_B T}{\tilde{\beta}a}=\frac{e^{\epsilon_2/2k_BT}}{\sqrt{2}}.\end{aligned}$$ Of course, this reduces to the venerable Ising result of $1/\sqrt{2}$ in 