HELIUM - PART 3

2010.09.21 previous lecture note 2010.09.28 next lecture note

HELIUM -PART 3

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The spectrum of helium. For the first column - the singlet and triplet parts are separated.
To the right is the case when one electron is free.
The original picture is in the EXAM SLIDES
Doubly excited states - the electrons would be bound, but because of their repulsion
the total energy is high so that one electron can be ejected.
Thus we get autoionizing states

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Part of the table - just for helium. This is about the role of the 'perturbation term' ⁵/₈ Z E₀

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The whole table, with only perturbation part

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Evaluation of the repulsion - the multipole expansion

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Spherical harmonics integral - for ground state we only need one term - all three zeros

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b3-repulsion.png

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Done - perturbation theory for the ground state. How is it for the excited states, when antisymmetric functions
must be used

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This is originally from the many-electron atoms EXAM SLIDES
excited states, when antisymmetric functions must be used

d-15-excited-state-evaluate.png

VARIATIONAL METHOD FOR GROUND STATE OF HELIUM

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LATER WE SHALL USE A SLIGHTLY more free theory - without assumed normalization.
That is important to get "variational principle for Schrödinger equation" - this all later

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Details of the variational calculation

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d-6-variational.png

d-7-variational.png
The final results are summarized in the table which is in the EXAM SLIDES.
It can also be obtained from the spreadsheet (we have both xls (microsoft) - and ods (the world- OpenOffice) formats (advice: use ODS, OpenOffice)
You can also see the HTML version for quick information (without the calculations).

SUMMARY OF HELIUM

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The case of disappearing excited state.
A circle can not have first excited state - it has an infinite excitation energy

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As the rotator gets 2 identical ends, three, four and then general N corners (balls),
the region of variable phi shrinks from 2pi to pi (N=2), 2pi/3 (N=3) etc.
Consequently, the first excited state gets HIGHER and HIGHER

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This is an illustration of what the "identical" and "symmetry" can mean: it means eliminating the regions of variable(s)
which are not physical (i.e. not a part of the consistent model)

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MANY ELECTRON ATOMS
2010.09.21 previous lecture note 2010.09.28 next lecture note