PHYS261 - Atomic Physics and Physical Optics

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Helium - 28.08.2014

1. Evaluation of repulsion; Perturbation Theory; Multipole expansion
2. Pauli principle and Spin - Parahelium, Orthohelium

1. Evaluation of repulsion; Perturbation Theory; Multipole expansion

We start by mentioning the Fine structure constant - supplements atomic units;
It does express the "weakness" of electro(magnetic) interactions

Then we start looking at the numerical values from the last lecture - starting the perturbation theory
At the bottom of the frame - the spreadsheet of the numbers (this is expanded below)
xcf_a010.png

xcf_a010.png

Independent electrons with neglected repulsion give 4*13.6*2 = 108.8 eV for the total binding of the
two electrons.

Further we shall look how the repulsion can be evaluated

c000-product-function-perturbation.png

Spreadsheet, includes the above reported repulsion contribution 5/8*27.2*Z eV
The variational method will be discussed in the next lectures

A000-spreadsheet-energies.png

A000-spreadsheet-energies.png

Perturbation theory

from astronomy, applied mathematics - http://en.wikipedia.org/wiki/Perturbation_theory- see also
http://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)
in particular (as sketched below)
                         http://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)#First_order_corrections
(Note that the word "perturbation" is somewhat special; outside of mathematical sciences
"perturbed"    means "thrown into a state of agitated confusion"     or "worried or upset by something"

xcf_a020.png

The energy correction (1. order)
is the same number as the EXPECTATION VALUE of the perturbation term
In 2-electron system (He and related IONS) this expectation value is a 6-dimensional integral

Here we sketch the evaluation, below we go shortly through the details
xcf_a030.png

xcf_a030.png

How to evaluate the EXPECTATION VALUE of the perturbation term - the electron-electron repulsion
In 2-electron system (He and related IONS) this expectation value is a 6-dimensional integral

g010_Repulsion_term.png

g010_Repulsion_term.png

General method - valid for any pair of "orbitals" (the electronic states are often called orbitals)
Multipole expansion, spherical harmonics.
An integral over any spherical harmonics (except of 00 ) over the sphere surface (4 pi) is allways ZERO
This follows from orthonormalization, for example

g020_repulsion_6dim_to_double_radial.png

g020_repulsion_6dim_to_double_radial.png

So this reduces to a double radial integration, and the radial 1s functions have a simple form.
The integrals of powers with exponentials are rather easily evaluated

g030-radial-integration-steps.png

g030-radial-integration-steps.png

Here we reproduce overview of the steps - dividing the integral into 2 terms according to what r_< and r_>
mean in different integration regions

g040-double-integral-two-parts.png

g040-double-integral-two-parts.png

The result has already been used above.
Note that the same approach can be used for ANY PAIR OF HYDROGEN-like ORBITALS

g050-expansion_evaluated_to_17_Z_eV.png