Atomic physics Part PHYS261

LECTURE NOTE 2013.09.17 to index 2013.09.17 - previous lecture note 2013.10.08 - next lecture note

Many Electron atoms - part 2

SCF-potentials and Total energy of N-electrons in a Slater Determinant

Explaining the periodic table - SCF-potentials

In this we are trying to explain why the potentials which are not purely Coulomb - or Kepler - lead to different
energies for the same n but different orbital momentum quantum number l through the "centrifugal potential" addition
(l (l+1)) / r²

We modelled the SCF potential by its limiting behaviour - close to the nucleus - Coulomb for charge number Z
far away - Coulomb for charge number 1 - the last electron does not see the whole charge, that is screened away
by the (Z-1) others.

So why are the energies different? The pictures show it.
a010.png

a010.png
Periodic table is thus explained by the centrifugal barrier.
Because of that the s-states of n+1 - shell come below the d-states of the n - shell

At the end we mention what else can be studied BEYOND the independent electrons and the SCF
TERMS - splitting of configurations, rest-interactions after the SCF is determined. This and configuration mixing
will be discuessed later

Total energy of N-electrons in a Slater Determinant

Now we turn to a more formal work with the SCF. Instead of the qualitative picture of the
AVERAGE POTENTIAL above, we start with Schrödinger equation for N electrons

First N=2, already mainly done. Consider only the antisymmetric spatial function
(for (n,l)(n',l') configuration ) 2-terms in each wavefunction, 3 terms in the hamiltonian,
thus 12 terms. We shall see that many will be zero
and then many will be identical.

We shall arrive at two statements:
1. for single-particle terms -
   The sum over coordinates turns into a sum over orbitals - with no additional N-factors (or n as used in the note)
   The n!-terms in the normalizationcancel with the number of n! various terms - which are identical
2. for pair interaction:
    The sum over pairs of coordinates turns into a sum over the pairs of orbitals - but with a doubling
    due to the DIRECT and EXCHANGE terms

We start with the normalization - and go straight to the single particle terms
    a020.png

a020.png

Here we state why the terms become zero. The single particle operators can have terms of type <a|O|b> wit a different from b,
but then there will be <b|a> which gives the zero
a030.png

       a030.png
Thus this explains the first "theorem" here at least for 2 electrons, but as well for N-electrons:
1. for single-particle terms -
   The sum over coordinates turns into a sum over orbitals - with no additional N-factors (or n as used in the note)
   The n!-terms in the normalizationcancel with the number of n! various terms - which are identical

Now we go over to the PAIR term in case of n=2, helium
a040.png

a040.png
In the helium case we see the origin of the direct and exchange term. But the sum over pairs - it is ONLY one pair.
But we see why the normalization cancels with the number of IDENTICAL REPETITIONS - in this case TWO.

And we start on the Lithium N=3 case - continued below

Here is the helium revisited and summarized.
b01.png

b01.png

b02.png

       b03.png
Lithium case - 3! is 6 ( n! )(n!) is 36. But as shown only 12 of these will be nonzero - and six of them will be direct
           6 of them will be direct.
And then it will be repeated for all coordinate pairs - and the number will simply fit
so that this demonstrates the second "theorem"
2. for pair interaction:
    The sum over pairs of coordinates turns into a sum over the pairs of orbitals - but with a doubling
    due to the DIRECT and EXCHANGE terms

b04.png

... from the text
b05.png

b05.png

Thus we have really shown that
1. for single-particle terms -
   The sum over coordinates turns into a sum over orbitals - with no additional N-factors (or n as used in the note)
   The n!-terms in the normalizationcancel with the number of n! various terms - which are identical
2. for pair interaction:
    The sum over pairs of coordinates turns into a sum over the pairs of orbitals - but with a doubling
    due to the DIRECT and EXCHANGE terms