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I'm starting to read about two-electron atoms from physics of atoms and molecules by Bransden.

Analyzing Helium spectrum I noticed that the energy of $He^+(1s)+e^-$ is greater than the energy of He, where the energy of $He^+(1s)+e^-$ is calculated using the $E_n$'s formula of hydrogenic atom with Z=2 (-2.0 a.u.). Obviously in order to obtain $He^+(1s)+e^-$ I have to excite He, so it is natural that the energy of $He^+(1s)+e^-$ is bigger that energy of He. But how is it possible that an atom with Z=2 when it has 1 electron this is less bound and when it has 2 electrons these are more bound? It appears to me that the one-electron should be more bound because the nuclear charge is Z=2 and there aren't repulsive effects.

Sorry for bad English and thanks you all.

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  • $\begingroup$ <I have to excite He, so it is natural that the energy of He+(1s)+e− is bigger that energy of He.> ..why you say that as for excitation you must have put in work/energy and it will get less bounded. $\endgroup$ – drvrm Aug 20 '18 at 16:22
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Helium has two electrons. So you have to be careful which ionisation energy you are talking about. If it's the one to remove the first, the second, or all electrons.

The first ionisation energy is $ 24.59 \, eV $, experimentally. This makes it go from $He$ to $He^+ + e^-$.

The second ionsation energy is $54.42 \, eV$, which can be calculated theoretically because the Helium ion is a hydrogenic atom. This makes it go from $He^+$ to $He^{++} + 2e^-$.

So you see that it is easier to remove the first electron, when you start from a 2 electron situation. This is because the second electron is more loosely bound to the nucleus, since the nuclear charge $Z$ is shielded by the first electron and becomes an effective $Z_{eff} < 2$.

When you solve the Helium atom, you treat the electron-electron repulsion as a pertubration.
You start with the energy of the second electron to be $-54.42\, eV$ (from the hydrogenic atom formula) and you add the postive repulsive energy that it gets from $e-e$ interaction. This gives you an energy of $-24.39 \, eV$, which means you have to provide $24.39 \, eV$ in order to free (ionise) the first electron.

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