In a model calculation their energy was comparable to $\epsilon_{RT}$,[@TLE-Unertl; @TLE-Maine] albeit with half as many occurrences per atom in the monolayer phase. The corrections due to $\epsilon_{LT}$ would be more complicated than simple shifts in the effective values of $\epsilon_1$ and $\epsilon_2$. Since direct interactions are probably important, there is no way to escape doing a first-principles computation; we continue to use the VASP package to extend our preliminary calculations.[@STK] A more daunting (at least for lattice-gas afficionadoes) possibility is that long-range intrastep elastic effects may be important. Ciobanu and Shenoy have made noteworthy progress in understanding how this interaction contributes to the orientation dependence of noble-metal steps.[@shenoy]

Calculational Details
=====================

Partition Function
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To carry out the sum in Eq. (\[eq:partfunction\]), we consider the Fourier transform of $Z(Y)$: $$\begin{aligned}
W\left( \mu \right) &\equiv &\int_{-\infty }^\infty dY~e^{i\mu Y}Z\left(
Y\right)  \nonumber  \label{eq:fourier} \\
&=&\sum_{\left\{ \Delta \right\} }\exp \sum_{j=1}^L\left( i\mu \Delta
_j-K\left( \Delta _j\right) \right)  \nonumber \\
&=&\left[ \sum_{\Delta =-\infty }^\infty \exp \left( i\mu \Delta -K\left(
\Delta \right) \right) \right] ^L,\end{aligned}$$ where $K\left( \Delta \right) \equiv \left( V+H\left| \Delta \right|
+U\left| \Delta -1\right| +D\left| \Delta +1\right| \right) $ is the energy in Eq. (\[eq:interfaceEnergy\]), associated with adjacent columns with height difference 