## 3 Experimental Results and Discussion
### Engineering Problems
#### 3.7.3. Tersoff Potential Function Minimization Problem (TPF)  
Researchers have shown a lot of interest in the evaluation of inter atomic potentials for covalent systems, notably for silicon. One such potential is the Tersoff potential, which controls the interaction of silicon atoms with strong covalent bonding [54]. Two parameterizations of silicon have been provided by Tersoff and are known as Sc (B) and Sc (C). According to the Tersoff formulation, the binding energy is expressed as a sum over atomic sites in the form,  
\[E_{i}=\frac{1}{2}\sum\
olimits_{j\
eq i}f_{c}\big{(}r_{ij}\big{)}\big{(}V_{R} \big{(}r_{ij}\big{)}-B_{ij}V_{A}\big{(}r_{ij}\big{)}\big{)} \tag{43}\]  
where \(V_{R}\) is a repulsion term, \(V_{A}\) is an attraction term, \(f_{c}\big{(}r_{ij}\big{)}\) is a switching function, and \(B_{ij}\) is a many-body term that relies on the locations of atoms \(i\) and \(j\) as well as atom \(i\)'s neighbors. \[B_{ij}=\Big{(}1+\gamma^{n_{1}}\xi_{ij}^{n_{1}}\Big{)}^{-1/2n_{1}} \tag{44}\]  
\(n_{1}\) and \(\gamma\) are standard fitted parameters. For \(i\) and \(j\) the term \(\xi_{ij}^{n_{1}}\) is given as,  
\[\xi_{ij}=\sum\
olimits_{k\
eq i}f_{c}\big{(}r_{ij}\big{)}g\Big{(}\theta_{ijk} \Big{)}\exp\Big{(}\lambda_{3}^{3}\big{(}r_{ij}-r_{ik}\big{)}^{3}\Big{)} \tag{45}\]  
The bond angle between \(ij\) and \(ik\) is denoted as \(\theta_{ijk}\) which is given as,  
\[g\Big{(}\theta_{ijk}\Big{)}=1+c^{2}/d^{2}-c^{2}/\left(d^{2}+\Big{(}h-\cos \Big{(}\theta_{ijk}\Big{)}\Big{)}^{2}\right) \tag{46}\]  
\(\lambda_{3}\), \(c\), \(h\) and \(d\) are known as fitted terms. From Equation (41), \(V_{R}\) and \(V_{A}\) are expressed as,  
\[V_{R}\big{(}r_{ij}\big{)}=Ae^{-\lambda_{1}r_{ij}} \tag{47}\]  
\[V_{A}\big{(}r_{ij}\big{)}=Be^{-\lambda_{2}r_{ij}} \tag{48}\]  
Expressed as fitted parameters are \(A\), \(B\), \(\lambda_{1}\) and \(\lambda_{2}\). In Equation (43), the switching function \(f_{c}\big{(}r_{ij}\big{)}\) is expressed as,  
\[f_{c}\big{(}r_{ij}\big{)}= \begin{cases}1&if\ r_{ij}<R-D\\ \frac{1}{2}-\frac{1}{2}\sin\begin{bmatrix}\pi(r_{ij}-R)\\ 0\end{bmatrix}if\ R-D<r_{ij}<R+D\\ if\ r_{ij}\geq R+D\end{cases}  \tag{49}\]  
We must determine the energy of each atom in order to determine the minimal potential energy for atoms. Each atom, let us say atom \(i\) has a unique potential energy, \(E_{i}\) \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **GSLEO-AEFA** & **AEFA** & **PSO** & **DE** & **JAYA** & **SCA** & **CS** \\ \hline x(18) & 0.193 & \(-\)2.276 & 5.000 & 5.000 & 0.109 & 0.901 & \(-\)2.169 \\ x(19) & 0.601 & 0.835 & \(-\)1.502 & \(-\)4.099 & 0.002 & 0.314 & 0.090 \\ x(20) & \(-\)0.780 & 3.274 & \(-\)4.848 & 5.202 & \(-\)0.400 & \(-\)4.879 & \(-\)1.240 \\ x(21) & \(-\)0.879 & 1.261 & \(-\)0.337 & 5.188 & 1.740 & 0.073 & 1.516 \\ x(22) & 0.596 & \(-\)0.687 & \(-\)3.449 & \(-\)5.197 & 0.242 & \(-\)0.085 & 0.888 \\ x(23) & \(-\)1.042 & \(-\)0.199 & \(-\)1.141 & 4.858 & 0.795 & \(-\)0.164 & \(-\)1.337 \\ x(24) & 0.065 & 3.430 & 4.400 & 4.681 & \(-\)0.010 & \(-\)0.999 & 0.042 \\ x(25) & \(-\)0.130 & 0.348 & \(-\)0.642 & \(-\)4.883 & 1.285 & \(-\)1.179 & \(-\)1.877 \\ x(26) & \(-\)1.369 & 3.231 & \(-\)4.341 & 3.459 & 0.341 & 0.350 & \(-\)0.091 \\ x(27) & \(-\)0.494 & 2.142 & \(-\)0.456 & 4.257 & 0.277 & 0.044 & \(-\)1.566 \\ x(28) & \(-\)0.289 & 0.692 & \(-\)0.814 & \(-\)3.372 & \(-\)0.743 & \(-\)0.659 & 1.224 \\ x(29) & 0.147 & 4.094 & \(-\)3.390 & 5.390 & 0.003 & \(-\)0.579 & \(-\)0.373 \\ x(30) & 0.774 & 1.788 & \(-\)0.224 & 4.311 & 1.361 & 0.296 & 0.015 \\ f(x) & \(-\)26.027 & \(-\)7.072 & \(-\)15.126 & \(-\)10.615 & \(-\)14.233 & \(-\)11.513 & \(-\)11.646 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: _Cont._which is determined by Equation (43). Therefore, in order to compute Equation (43), which requires the computation of Equations (44)-(49) for each neighbor of that atom, to know the potential energy of a single atom.