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Perturbation theory - repulsion. Variation Method: Compare with Experiment

The 2-electron hamiltonian consists of two single-electron parts for each electron and their interaction - repulsion.
The repulsion term needs to be evaluted as matrix element between the states of independent electrons.

The MULTIPOLE EXPANSION in terms of spherical functions (and Legendre polynomials of the angle - see below)

The six-dimensional integrals are transformed to a SUM over terms with two independent integrals over
each of the pair of angles (theta, phi, collectively Omega or r-hat)
and a two r-variable integral non-separable due to the r-larger -- r-smaller terms)

For ground state (1s)(1s) on "both sides" the matrix element reduces to one term only
      small_0010.png

       small_0010.png 
In the last part the general matrix elements for (n, l)(n',l')  -  (n'' l'') (n''' l''') is illustrated - some aspects with the "triangle relation"

Here we worked on the sheets of the presentation - first mentioning the r-larger r-smaller origin, The Legendre polynomial expansion
The notation 
      small_0011.png

       small_0011.png
Legendre expansion is in terms of cosine of the angle between the two position vectors
http://en.wikipedia.org/wiki/Legendre_polynomials#Orthogonality      and  also in the next section
http://en.wikipedia.org/wiki/Legendre_polynomials#Applications_of_Legendre_polynomials_in_physics

      small_0012.png

       small_0012.png

FOR THE GROUND STATE
The angular part evaluates to 1, it remains to evaluate the radial integral. The radial functions are in general
polynomials times an exponential
      small_0013.png

       small_0013.png

      small_0014.png

       small_0014.png
After evaluation by quite a few elementary terms the final result is obtained - a rather simple expression.

A n important feature - this term depends on Z -linear proportionality. Thus the repulsion scales with Z
(note that all the single particle terms scale with Z2 ) 
      small_0015.png

       small_0015.png       
Here we discussed the result - with reference to the table below (there compared with the variational method)
            small_0020.png

       small_0020.png
Here in the last part we sketched how the different terms influence the resulting ground state energy
Experimental value is LOWER than the evaluated - "with perturbation theory approximate wavefunction"  - see variational theorem
Variational theorem: The ground state energy is smaller or equal to any approximated function
EXPECTATION VALUE (i.e. that matrix element ... see the application )
      small_0030.png

       small_0030.png
VARIATIONAL METHOD - look for a wavefunction which gives lowest value
of the expectation value

And here we sketch how to look - effective Z - which we denote z

      small_0040.png

       small_0040.png
In the lower part of the above - we sketch how we define something to vary - effective Z - which we denote z
We evaluate the T and V for each electron and the repulsion - simply by recognizing how they depend on the
Z-value of the wavefunction. The potential term scales with Z 2  - but the term contains explicitely only one Z.
Thus the dependence will be zZ. The kinetic energy term scales with Z2  - but contains no explicit Z. Thus it must
depend on z2 . The repulsion also has no Z-dependence explicitely, but was evaluated to 5/8 Z, i.e. it must become
... see the next picture ... 5/8 z
With this replacements we are ready to find the "best value" of effective z  - varying the z and finding the minimum.
I.e. we find the z for which the expression gives a minimum - zero derivative
(using the variation theorem )
      small_0050.png

       small_0050.png     
(this table was also us ed to discuss the perturbation result )
  small_0051.png

       small_0051.png
The  variational method - best possible effective z - but INDEPENDENT ELECTRONS
The discrepancy - can be solved by going beyond the independent electrons.
This is often called ELECTRON CORRELATION - the motion is not independent - it is correlated
One possible way to include correlation - beyond a product function.
This can be accomplished by a SUM OF PRODUCT FUNCTIONS
(we shall explore this for many-electron systems - configuration mixing)

Alternatively - assume a function of "correlated variables", as e.g. the distance between the electrons
Hylleraas wavefunction (there are no simple references to be found on Wikipedia or similar sites.
But Wolfram (Mathematica) has a nice demonstration of the variational method and the Hylleraas
approach - http://demonstrations.wolfram.com/VariationalCalculationsOnTheHeliumIsoelectronicSeries/
Unfortunately, their "player" must be installed to experience the variation. The text is interesting
even without the player)

      small_0060.png

       small_0060.png
The "correlation variables" used by Hylleraas are in the above inset.

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