Table 2 Interatomic potentials near the Pareto frontiers in Fig. 2

SC⁷⁹

\(E_i = \mathop {\sum }\limits_j \frac{{644.52}}{{r^9}}f(r) - \left( {\mathop {\sum }\limits_j \frac{{527.62}}{{r^6}}f(r)} \right)^{0.5}\)

GP1

\(E_i = \mathop {\sum}\limits_j {(r^{10.21 - 5.47r} - 0.21^r)f(r)} + 0.97\left( {\mathop {\sum}\limits_j {0.33^rf(r)} } \right)^{ - 1}\)

GP2

\(E_i = 7.33\mathop {\sum}\limits_j {r^{3.98 - 3.94r}f(r)} + \left( {27.32 - \mathop {\sum}\limits_j {\left( {11.13 + 0.03r^{11.74 - 2.93r}} \right)f(r)} } \right)\left( {\mathop {\sum}\limits_j {f(r)} } \right)^{ - 1}\)

EAM2⁵⁴

\(\begin{array}{l}E_i = \mathop {\sum}\limits_j {E_1\left( {e^{ - 2\alpha (r - r_0)} - 2e^{ - \alpha (r - r_0)}} \right)f(r)} + F\left( {\mathop {\sum}\limits_j {r^6(e^{ - \beta r} + 2^9e^{ - 2\beta r})f(r)} } \right)\\ \mathop {\sum}\limits_i {F\left( {\bar \rho _i} \right) = E\left( L \right) - } \frac{1}{2}\mathop {\sum}\limits_i {\mathop {\sum}\limits_j {E_1\left( {e^{ - 2\alpha (r - r_0)} - 2e^{ - \alpha (r - r_0)}} \right)f(r)} } \\ E\left( L \right) = - E_{{\mathrm{sub}}}(1 + a^ \ast )e^{ - a^ \ast }\\ a^ \ast = (a/a_0 - 1)/(E_{{\mathrm{sub}}}/9B\Omega )^{1/2}\end{array}\)

ABCHM⁵⁸

\(\begin{array}{l}E_i = \mathop {\sum }\limits_j \varphi (r)f(r) + 1.57 \cdot 10^{ - 5}\left( {\mathop {\sum}\limits_j {\psi (r)f(r)} } \right)^2 - \left( {\mathop {\sum}\limits_j {\psi (r)f(r)} } \right)^{0.5}\\ \varphi (r) = \left\{ {\begin{array}{*{20}{c}} {e^{0.82 + 16.01r - 15.73r^2 + 3.80r^3},} & {1 < r < 1.9} \\ { + \;0.62(4.43 - r)^3,} & {1.9 < r < 4.43} \\ { - 3.02(4.17 - r)^3,} & {1.9 < r < 4.17} \\ { + 2.84(4.04 - r)^3,} & {1.9 < r < 4.04} \\ { - 0.41(3.62 - r)^3,} & {1.9 < r < 3.62} \\ { + 0.65(3.13 - r)^3,} & {1.9 < r < 3.13} \\ { + 0.81(2.56 - r)^3,} & {1.9 < r < 2.56} \end{array}} \right.\\ \psi (r) = \left\{ {\begin{array}{*{20}{c}} {0.21(4.43 - r)^3,} & {1.9 < r < 4.43} \\ { + 0.36(3.62 - r)^3} & {1.9 < r < 3.62} \end{array}} \right.\end{array}\)

CuNi⁵⁵

\(\begin{array}{l}E_i = \frac{1}{2}\mathop {\sum}\limits_j {\left( {D_M\left[ {1 - e^{ - \alpha _M\left( {r - R_M} \right)}} \right]^2 - D_M} \right)} f(r) + F\left( {\bar \rho _i} \right)\\ \bar \rho _i = \mathop {\sum}\limits_j {\tanh \left( {20r^2} \right)\left\{ {r^6\left( {e^{ - \beta r} + 2^9e^{ - 2\beta r}} \right) + \frac{{\sigma ^{(1)}}}{{\mu ^{(1)}}}e^{ - \frac{1}{2}\left[ {\mu ^{(1)}\left( {r - R_B} \right)} \right]^2} - 0.1\sigma ^{(1)}e^{ - \frac{1}{2}\left[ {\mu ^{(1)}\left( {r - \left( {R_B + 0.5} \right)} \right)} \right]^2}} \right\}f(r)} \\ \mathop {\sum}\limits_i {F\left( {\bar \rho _i} \right) = E\left( L \right) - } \frac{1}{2}\mathop {\sum}\limits_i {\mathop {\sum}\limits_j {\left( {D_M\left[ {1 - e^{ - \alpha _M\left( {r - R_M} \right)}} \right]^2 - D_M} \right)} f(r)} \\ E\left( L \right) = - E_{{\mathrm{sub}}}(1 + a^ \ast )e^{ - a^ \ast }\\ a^ \ast = (a/a_0 - 1)/(E_{{\mathrm{sub}}}/9B\Omega )^{1/2}\end{array}\)

EAM1⁵⁴

\(\begin{array}{l}E_i = \mathop {\sum}\limits_j {\left( {\left[ \begin{array}{l}E_1\left( {e^{ - 2\alpha _1(r - r_0^{(1)})} - 2e^{ - \alpha _1(r - r_0^{(1)})}} \right) + \\ E_2\left( {e^{ - 2\alpha _2(r - r_0^{(2)})} - 2e^{ - \alpha _2(r - r_0^{(2)})}} \right) + \delta \end{array} \right]f(r) - \mathop {\sum}\limits_{n = 1}^3 {\left( {H\left( {r_s^{(n)} - r} \right)S_n(r_s^{(n)} - r)^4} \right)} } \right)} + F\left( {\bar \rho _i} \right)\\ if(\bar \rho _i < 1):F\left( {\bar \rho _i} \right) = F^{(0)} + 0.5F^{(2)}\left( {\bar \rho _i - 1} \right)^2 + \mathop {\sum}\limits_{n = 1}^4 {\left( {q_n\left( {\bar \rho _i - 1} \right)^{n + 2}} \right)} \\ else:F\left( {\bar \rho _i} \right) = \frac{{F^{(0)} + 0.5F^{(2)}\left( {\bar \rho _i - 1} \right)^2 + q_1\left( {\bar \rho _i - 1} \right)^3 + Q_1\left( {\bar \rho _i - 1} \right)^4}}{{1 + Q_2(\bar \rho _i - 1)^3}}\\ where:\bar \rho _i = \mathop {\sum}\limits_j {\left( {\left[ {ae^{ - \beta _1(r - r_0^{(3)})^2} + e^{ - \beta _2(r - r_0^{(4)})}} \right]f(r)} \right)} \end{array}\)