\rangle$, and becomes zero when $\alpha _{c} =0$. Hence $d ^{(2)}$ represents the covalent correlation, and we shall include both $d ^{(1)}$ and $d ^{(2)}$ for the blue dash curve in Fig. 2 (a). By including the correlation contributions, we have $$\begin{aligned}
E_{b} =  tr (\rho ^{(1)} + \rho ^{(2)}) H_{sb} + \int _{ {\bf r} ^{ \prime },{\bf r} ^{ \prime \prime } \in \Omega} d ^{3} r ^{ \prime } d ^{3} r ^{\prime \prime} U ( | {\bf r} ^{\prime} - {\bf r} ^{\prime \prime} | ) \langle {\bf r} ^{\prime} | \rho ^{(1)} | {\bf r} ^{\prime} \rangle \langle {\bf r} ^{\prime \prime} | \rho ^{(2)} | {\bf r} ^{\prime \prime} \rangle +\end{aligned}$$ $$tr H_{sb} d ^{(1)} + \int _{ {\bf r} ^{\prime},{\bf r} ^{\prime \prime} \in \Omega} d ^{3} r ^{\prime} d ^{3} r ^{\prime \prime} U ( | {\bf r} ^{\prime} - {\bf r} ^{\prime \prime} | ) \langle {\bf r} ^{\prime} | \rho ^{(1)} | {\bf r} ^{\prime} \rangle \langle {\bf r} ^{\prime \prime} | d ^{(1)} | {\bf r} ^{\prime \prime} \rangle + \text{ \ \ \ \ \ }$$ $$\int _{ {\bf r} ^{ \prime },{\bf r} ^{\prime \prime} \in \Omega} 