Effect of Trio Interactions
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In addition to the NNN interaction, trio (3-atom, non-pairwise) interactions may well influence the stiffness. The strongest such interaction is most likely that associated with 3 atoms forming a right isosceles triangle, whose sides are at NN distance and hypotenuse at NNN separation. In a lattice gas model, there is a new term with $\epsilon_{RT}$ times the occupation numbers of the 3 sites.[@LG] Note that this trio interaction energy $\epsilon_{RT}$ is in addition to the contribution $2\epsilon_1 \! +\! \epsilon_2$ of the constituent pair interactions. If we count broken trios and weight each by $R$, we find an additional contribution to Eq. (\[eq:interfaceEnergy\]) of $R$ times $$\begin{aligned}
  \label{eq:trio}
4|\Delta_i| +2\delta_{\Delta_i,0} +2 = 2|\Delta_i| + |\Delta_i \! +\! 1| + |\Delta_i \! -\! 1| +2,\end{aligned}$$ where we have converted the Kronecker delta at $i\! =\! 0$ to make better contact with Eq. (\[eq:interfaceEnergy\]). Thus, without further calculation we can include the effect of this trio by replacing $H$ by $H\! +\! 2R$, $U$ by $U\! +\! R$, $D$ by $D\! +\! R$, and (trivially) $V$ by $V\! +\! 2$.

By arguments used at the end of Section IIA, we recognize $R\! =\! -\scriptstyle{\frac{1}{2}}\textstyle{\epsilon_{RT}}$. Consequently, the effective 