\left( 
                 \left| h_i  - h_{j_1 }\right|-2\right) 
             \right] 
         }  \right.\\[3ex]
   & & 
     + \sum\limits_{j_2}
           {\left[
              2 J_2 \delta 
                \left(
                  \left| h_i  - h_{j_2 }\right|
                \right) + \left( 2 J_2  + K_2\right)  \delta 
                \left( 
                  \left| h_i  - h_{j_2 } \right| - 3 
                \right) 
           \right] }  \\
   & &
     + 
     \left. 
       J_2 \sum\limits_{j_3} 
         {\left[ 
               \delta 
               \left( \left| h_i  - h_{j_3 } \right| - 2 \right) 
               + \delta 
               \left( 
                  \left| h_i  - h_{j_3}\right| - 4 
               \right) 
         \right]} 
     \right\}+ N J_0, 
\end{array}$$

where

$$\begin{aligned}
J_0 &=&  -\frac{1}{2} \epsilon^{BB}_{1}+ \epsilon_{1}, \\
J_1 &=& -\frac{1}{2} ( \epsilon^{BB}_{2} + \epsilon^{BB}_{3}+ 
2\epsilon^{BB}_{4} )+ \epsilon_{2} + \epsilon_{3}+ \epsilon_{4}, \\
K_1 &=& -\frac{1}{2} ( \epsilon^{BB}_{1} + 2\epsilon^{BB}_{3}+ 
\epsilon^{BB}_{4} )+ \epsilon_{1} + \epsilon_{3}+ \epsilon_{4}, \\
J_2 &=& -\frac{1}{2} \epsilon^{BB}_{4}+ \epsilon_{4}, \\
K_2 &=& -\frac{1}{2} \epsilon^{BB}_{3} + \epsilon_{3}- 
\epsilon_{4}, \\
\epsilon_{i}&=& \epsilon^{AB}_{i}- \epsilon^{AA}_{i},\end{aligned}$$

and sums over $j_1$, $j_2$, and $j_3$ denotes summing over first, second, and third neighbors of a column at site $i$. In what follows, we will treat $J_1$, $J_2$, $K_1$, and $K_2$ as model parameters. It is worth noting that this BCCSOS Hamiltonian of overlayer-induced faceting at constant coverage can also be used to study a clean bcc(111) surface by setting $\epsilon^{AB}_{i}=\epsilon^{AA}_{i}=\epsilon^{BB}_{i}$.

 Energies of (111), (211), and (110) faces at 