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9.8: Electron-electron interaction - ground state by perturbation theory

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    The hydrogen wavefunctions are only a choice of basis set: the hydrogenic potential ignores the electron-electron repulsion. A simple approach is to treat this as a perturbation and to use degenerate perturbation theory.

    The perturbing potential is just \(V = e^2/4\pi \epsilon_0 r_{12}\) where \(r_{12} = |{\bf r_1 − r_2}|\). The unperturbed spatial ground state is just a product of the hydrogenic ones with Z=2 for helium:

    \[u_{100}(r_1)u_{100}(r_2) = \frac{Z^3}{\pi a^3_0} e^{−Zr_1/a_0} e^{−Zr_2/a_0} \nonumber\]

    so by perturbation theory, the energy shift due to this potential is given by:

    \[\langle u_{100}(r_1)u_{100}(r_2)|e^2 /4\pi \epsilon_0 r_{12}|u_{100}(r_1)u_{100}(r_2) \rangle \nonumber\]

    The electron-electron repulsion is over 30% of the unperturbed energy \((4Z\mu e^4/\hbar^2 )\), so perturbation theory may seem inappropriate. Strictly, it isn’t even the right integral, as it neglects correlation. But in fact the value of this integral is \(5Z\mu e^4/8 \hbar^2\) within 5% of the actual energy.

    Note also that the radial wavefunctions are different for 2s and 2p, so the electron-electron interation splits the degeneracy between 1s2s and 1s2p configurations.

    This page titled 9.8: Electron-electron interaction - ground state by perturbation theory is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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