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Appendix A: Derivation of Pauli function terms

Wavefunction plus abbreviation, where $ S = \braket{\phi_{1}\vert\phi_{2}}$

$\displaystyle \Psi$ $\displaystyle =$ $\displaystyle \frac{1}{\sqrt{2\pm2S^{2}}}(\phi_{1}(r_{1})\phi_{2}(r_{2})\pm\phi_{2}(r_{1})\phi_{1}(r_{2}))$  
  $\displaystyle =$ $\displaystyle \frac{1}{\sqrt{2\pm2S^{2}}} (12\pm21).$  

Verification of normalization:
$\displaystyle \braket{\Psi\vert\Psi}$ $\displaystyle =$ $\displaystyle \frac{1}{2\pm2 S^{2}} \braket{12\pm21\vert 12\pm21}$  
  $\displaystyle =$ $\displaystyle \frac{1}{2\pm2 S^{2}}\left(\braket{12\vert 12}\pm{12\vert 21}\right)$  
  $\displaystyle =$ $\displaystyle \frac{1}{2\pm2 S^{2}}\left(\braket{1\vert 1}\braket{2\vert 2}\pm\braket{1\vert 2}\braket{2\vert 1}\right)$  
  $\displaystyle =$ $\displaystyle 1.$  

Evaluate kinetic energy of these wavefunctions, using operators $ t_{1} = -\frac{1}{2} \nabla_{1}^{2}$ and $ t_{2} = -\frac{1}{2} \nabla_{2}^{2}$:
$\displaystyle \mathrm{KE}(12\pm21)$ $\displaystyle =$ $\displaystyle \braket{12\pm21\vert t_{1}+t_{2}\vert 12\pm21}$  
  $\displaystyle =$ $\displaystyle \frac{1}{2+2S^{2}} \cdot 2 \cdot \braket{12\pm21\vert t_{1}\vert 12\pm21}$  
  $\displaystyle =$ $\displaystyle \frac{1}{1+S^{2}}\left(\braket{12\vert t_{1}\vert 12}+\braket{21\vert t_{1}\vert 12}\pm 2 \braket{21\vert t_{1}\vert 21}\right)$  
  $\displaystyle =$ $\displaystyle \frac{1}{1+S^{2}}\left(\braket{1\vert t_{1}\vert 1}\braket{2\vert...
... 2}\braket{1\vert 1} \pm 2\braket{1\vert t_{1}\vert 2} \braket{2\vert 1}\right)$  
  $\displaystyle =$ $\displaystyle \frac{1}{1+S^{2}}\left(t_{11}+t_{22}\pm2 S t_{12}\right)$  

where
    $\displaystyle t_{11} = \braket{1\vert-\frac{1}{2} \nabla^{2}\vert 1} = \frac{3}{2}\,\frac{1}{s_{1}^{2}}$  
    $\displaystyle t_{22} = \braket{2\vert-\frac{1}{2} \nabla^{2}\vert 2} = \frac{3}{2}\,\frac{1}{s_{2}^{2}}$  
    $\displaystyle t_{12} = \braket{1\vert-\frac{1}{2} \nabla^{2}\vert 2} = S \left(\frac{3(s_{1}^{2}+s_{2}^{2})-2 x_{12}^{2}}{(s_{1}^{2}+s_{2}^{2})^{2}}\right)$  
    $\displaystyle S_{12} = \braket{1\vert 2} = \left(\frac{2}{s_{1}/s_{2}+s_{2}/s_{1}}\right)^{3/2} \exp(-x_{12}^{2}/(s_{1}^{2}+s_{2}^{2})).$  

Then we have
$\displaystyle E_{u}$ $\displaystyle =$ $\displaystyle \mathrm{KE}(12-21)-\mathrm{KE}(1) - \mathrm{KE}(2)$  
  $\displaystyle =$ $\displaystyle \frac{1}{1-S^{2}}(t_{11}+t_{22}-2 S t_{12})-\frac{1-S^{2}}{1-S^{2}}(t_{11}+t_{22})$  
  $\displaystyle =$ $\displaystyle \frac{S^{2}}{1-S^{2}} \left(t_{11} + t_{22} - \frac{2 t_{12}}{S}\right)$  

and
$\displaystyle E_{g}$ $\displaystyle =$ $\displaystyle \mathrm{KE}(12+21)-\mathrm{KE}(1) - \mathrm{KE}(2)$  
  $\displaystyle =$ $\displaystyle \frac{1}{1+S^{2}}(t_{11}+t_{22}-2 S t_{12})-\frac{1+S^{2}}{1+S^{2}}(t_{11}+t_{22})$  
  $\displaystyle =$ $\displaystyle -\frac{S^{2}}{1+S^{2}}\left(t_{11}+t_{22}-\frac{2 t_{12}}{S}\right).$  


next up previous contents
Next: Appendix B: Derivation of Up: Development of an electron Previous: Conclusion   Contents
Julius 2008-04-29