|
1.
<math>
\Psi_{\textrm{MO}}=\vert n\bar{n}p\bar{p}\vert
</math>
;
<math>
\Psi_{\textrm{VB}}=\vert \left( n-\lambda p\right)\left( \bar{n}-\lambda\bar{p}\right)\left( n+\lambda p\right)\left(\bar{n}+\lambda\bar{p}\right)\vert
</math>
2.
<math>
\begin{matrix}
\Psi_{\textrm{VB}}
&=&\vert \left( n\bar{n} -\lambda p\bar{n} -\lambda n\bar{p} +\lambda^2 p\bar{p} \right) \left( n\bar{n} +\lambda p\bar{n} +\lambda n\bar{p} +\lambda^2 p\bar{p} \right) \vert
\end{matrix}
</math>
<math>
\begin{matrix}
\Psi_{\textrm{VB}}
&=&\vert n\bar{n}n\bar{n} \vert + \lambda\vert n\bar{n}p\bar{n} \vert + \lambda\vert n\bar{n}n\bar{p} \vert + \lambda^2 \vert n\bar{n}p\bar{p} \vert
- \lambda \vert p\bar{n}n\bar{n} \vert -\lambda^2 \vert p\bar{n}p\bar{n} \vert -\lambda^2 \vert p\bar{n}n\bar{p} \vert -\lambda^3 \vert p\bar{n}p\bar{p} \vert
\end{matrix}
</math>
<math>
\begin{matrix}
- \lambda \vert n\bar{p}n\bar{n} \vert -\lambda^2 \vert n\bar{p}p\bar{n} \vert -\lambda^2 \vert n\bar{p}n\bar{p} \vert -\lambda^3 \vert n\bar{p}p\bar{p} \vert
+ \lambda^2 \vert p\bar{p}n\bar{n} \vert +\lambda^3 \vert p\bar{p}p\bar{n} \vert +\lambda^3 \vert p\bar{p}n\bar{p} \vert +\lambda^4 \vert p\bar{p}p\bar{p} \vert
\end{matrix}
</math>
After eliminating all determinants having two orbitals with the same spin, there remains :
<math>
\begin{matrix}
\Psi_{\textrm{VB}}
&=& \lambda^2 \vert n\bar{n}p\bar{p} \vert -\lambda^2 \vert p\bar{n}n\bar{p} \vert -\lambda^2 \vert n\bar{p}p\bar{n} \vert + \lambda^2 \vert p\bar{p}n\bar{n} \vert
\end{matrix}
</math>
After permuting the columns and changing signs accordingly, there remains : <math>
\Psi_{\textrm{VB}}=4\lambda^2\vert n\bar{n}p\bar{p} \vert=\Psi_{\textrm{MO}} </math> (if one includes normalization factors).
3.
<math>\Phi_1=\vert \left( n-\lambda p \right)\left( \bar{n}-\lambda\bar{p} \right)\left( n+\lambda p \right) \vert </math>, <math>\Phi_2=\vert \left( n+\lambda p \right)\left( \bar{n}+\lambda\bar{p} \right)\left( n-\lambda p \right) \vert </math>
Permuting the first and third orbitals in <math>\Phi_2</math> and changing the sign accordingly, we get <math>-\Phi_2</math> that has maximum orbital correspondence with <math>\Phi_1</math> :
<math>-\Phi_2</math> =<math>\vert \left( n-\lambda p \right) \left( \bar{n}+\lambda\bar{p} \right) \left( n+\lambda p \right) \vert </math>.
4.
<math>\Phi_1</math> and <math>-\Phi_2</math> differ by only one orbital, <math>\left( \bar{n}-\lambda \bar{p} \right)</math> in <math>\Phi_1</math> which becomes <math>\left( \bar{n}+\lambda \bar{p} \right)</math> in <math>-\Phi_2</math>. Therefore the matrix element <math> \langle \Phi_1 \vert \hat{H} \vert -\Phi_2 \rangle </math> is a simple <math>\beta</math> integral, necessarily negative. Hence, the lowest ionized state is <math> \frac{1}{\sqrt{2}}\left(\Phi_1-\Phi_2\right)</math> while the higher ionized state is <math> \frac{1}{\sqrt{2}}\left(\Phi_1+\Phi_2\right)</math>.
5.
<math>\Phi_1=\vert n\bar{n}n\vert -\lambda\vert p\bar{n}n\vert -\lambda\vert n\bar{p}n\vert +\lambda^2\vert p\bar{p}n\vert+\lambda\vert n\bar{n}p\vert-\lambda^2\vert p\bar{n}p\vert-\lambda^2\vert n\bar{p}p\vert+\lambda^3\vert \bar{p}p\vert</math>
<math>\Phi_1=+2\lambda^2\vert p\bar{p}n \vert +2\lambda\vert n\bar{n}p \vert</math>
In the same way, one shows that <math>\Phi_2=-2\lambda^2\vert p\bar{p}n \vert +2\lambda\vert n\bar{n}p \vert</math>. It follows that :
<math>\left(\Phi_1-\Phi_2\right)\propto \vert n\bar{n}p \vert </math> (lowest ionized state in MO theory)
<math>\left(\Phi_1+\Phi_2\right)\propto \vert p\bar{p}n \vert </math> (higher ionized state in MO theory).
It is concluded that 1) VB theory yields two ionization potentials for H<math>{}_2</math>O, in agreement with experiment, and 2) that these ionization potentials are exactly the same as the ones found in elementary MO theory.
| 
