with the predicted repulsive $\epsilon_2$.) It is also not obvious [*a priori*]{} whether multi-atom interactions also contribute significantly. (For homoepitaxy, the only semiempirical result is that they are insignificant for Ag on Ag(001);[@VA] however, it is likely that semiempirical calculations will underestimate multi-atom interactions.)

To address these questions, we are currently carrying out calculations[@STK] using the VASP package.[@VASP] Preliminary results for Cu(001) suggest that $\epsilon_2$ is indeed attractive, and that $\epsilon_2/\epsilon_1$ is about 1/8; however, there are indications of a [*repulsive*]{} right-triangle trio interaction $\epsilon_{RT}$ with sizable magnitude (perhaps comparable to $|\epsilon_2|$, consistent with [*a priori*]{} expectations[@TLE-Unertl; @TLE-Maine]), which would diminish rather than enhance the effect of $\epsilon_2$.

In summary, NNN interactions may well account for a significant fraction, perhaps even a majority, of the discrepancy between NN Ising model calculations and experimental measurements of the orientation dependence of the reduced stiffness;[@dieluweit02] the effect is even somewhat greater than estimated by the Twente group[@ZP1; @ZP2]. However, inclusion of $\epsilon_2$ is not the whole answer, nor, seemingly, is consideration of $\epsilon_{RT}$. One possible missing ingredient is other multi-site interactions, most notably the linear trio $\epsilon_{LT}$ consisting of 3 colinear atoms (a pair of NN legs and an apex angle of $180^{\circ}$). 