Figure courtesy of Joel Ong

\(\huge\uparrow\)
\(E\)

\(\left|\sigma_{ss}\right> = {1 \over \sqrt 2}{\color{blue}{\left|s_1\right>}} + {1 \over \sqrt 2}{\color{orange}{\left|s_2\right>}}\) \(\left|\sigma_{ss}\right> = u_{11}{\color{blue}{\left|s_1\right>}} + u_{12}{\color{orange}{\left|s_2\right>}}\)

\(\left|\sigma_{ss}^*\right> = {1 \over \sqrt 2}{\color{blue}{\left|s_1\right>}} - {1 \over \sqrt 2}{\color{orange}{\left|s_2\right>}}\) \(\left|\sigma_{ss}^*\right> = u_{21}{\color{blue}{\left|s_1\right>}} + u_{22}{\color{orange}{\left|s_2\right>}}\)

\(\Delta E = \sqrt{\left({\color{orange}{E_2}} - {\color{blue}{E_1}}\right)^2 + \left(\int {\color{blue}{\left<s_1\right|}}\hat{H}{\color{orange}{\left|s_2\right>}}\ \mathrm{d}^3 x\right)^2} \) \[\Delta E = \int {\color{blue}{\left<s_1\right|}}\hat{H}{\color{orange}{\left|s_2\right>}}\ \mathrm{d}^3 x \]