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Generally people says that the state $1s(1)2s(1)$ $ ^1S$ is an excited state of Helium atom . Variation theorem guaranties that the expectation value of this state is greater than the ground state, that is $E[(1s(1)2s(1)]\geq e_0$ where $e_0$ is the ground state energy. For example since $1s(2)$ $ ^1S$ is the ground state we can show that the state $1s(1)2s(1)$ $ ^3S$ is an upper bound of the first excited state of Helium.

Now let $e_1$ be the first excited state of the Helium atom, is there any theorem that guaranties that we have $E[(1s(1)2s(1) ^1S ]\geq e_1$ ?

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  • $\begingroup$ Which state do you mean, the singlet of the triplet? $\endgroup$ – my2cts Nov 9 '20 at 13:08
  • $\begingroup$ I mean a Singlet $\endgroup$ – amilton moreira Nov 9 '20 at 13:08
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    $\begingroup$ The triplet 1s2s state is the lowest excited state. See physics.nist.gov/cgi-bin/ASD/… $\endgroup$ – my2cts Nov 9 '20 at 13:11
  • $\begingroup$ I know that triplet state is the lowest excited state. I am not asking if singlet sate is the lowest am asking if it is un upper bound of the excite states of helium $\endgroup$ – amilton moreira Nov 9 '20 at 13:14
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    $\begingroup$ "assuming that $1s(2)$ $ ^1S$ is the ground state " This is not an assumption but a hard fact. $\endgroup$ – my2cts Nov 9 '20 at 13:33
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There is no such theorem. What you have is experimental data and very accurate quantum chemical simulation (any atom with more than 1 electron for some reason is chemistry). There are many more excited states and the singlet certainly is not the highest of these. You could learn a lot by looking at the complete He I level set at https://physics.nist.gov/cgi-bin/ASD/energy1.pl?encodedlist=XXT2&de=0&spectrum=He+I&submit=Retrieve+Data&units=1&format=0&output=0&page_size=15&multiplet_ordered=0&conf_out=on&term_out=on&level_out=on&unc_out=1&j_out=on&lande_out=on&perc_out=on&biblio=on&temp=

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  • $\begingroup$ We can proof for example that the triplet is a upper bound of excited states assuming that singlet is a ground state $\endgroup$ – amilton moreira Nov 9 '20 at 13:16
  • $\begingroup$ @amiltonmoreira I get the impression that you limit yourself to the 1s2s configuration. $\endgroup$ – my2cts Nov 9 '20 at 13:31
  • $\begingroup$ I am not understanding what you mean $\endgroup$ – amilton moreira Nov 9 '20 at 13:32
  • $\begingroup$ @amiltonmoreira There are many excited states higher that the 1s2s triplet or singlet. Did you take a look at the NIST database? $\endgroup$ – my2cts Nov 9 '20 at 13:39
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    $\begingroup$ @amiltonmoreira Theory and experiment essentially give the same answers here. Besides, all physicists are interested in experimental data. $\endgroup$ – my2cts Nov 9 '20 at 13:56
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Yes, according to Wikipedia it is known as the Aufbau principle (or Madelung rule, as I've been taught in school).

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  • $\begingroup$ that is not a theorem $\endgroup$ – amilton moreira Nov 9 '20 at 12:58
  • $\begingroup$ Indeed it is not. By definition theorem is a general proposition not self-evident but proved by a chain of reasoning; a truth established by means of accepted truths. This principle, however is empirically measured, not theoretically postulated. $\endgroup$ – Krumuvecis Nov 9 '20 at 13:10

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