2010.10.14 previous lecture note                                                  2010.10.21 next lecture note

Hartree-Fock - Derivation + DFT + Configuration Interaction

Configuration - how are the electrons distributed over the states.
For the ground state - the natural configuration is the N lowest energy states
But there can be a different configuration - e.g. one electron is in a higher
state than it should be
Why is this relevant - see below
Configuration metaphore - chesboard (starting configuration - the king and queen
and the bishops .... the towers ... not much exchange symmetry there ... )
1-intro.png
We also revisited the Lagrange Multiplier  -  the paroboloid - minimum on a LINE
(the limiting surface is an inclied plane (Schiefe Ebene in German I hope - skråplanet in Norwegian)  )
There it is easy to see how the minimum slides along the inclied plane until it hits the limiting line g(x,y)=0 ....

We looked back at last lecture - the single-particle operators and the pairs
11-Counting_Lithium.png
Deriving hartree fock by the variational prescription revisited
See the Schrödinger equation from variational "principle" ... last time
2-derive-H.F.-.png
the Schrödinger equation from variational "principle" ... last time ... is repeated here in the bottom
One more visit to counting the terms
3-derive-H.F.-sin-particle.png
Taking the pairs - and doing the analogue of the Schrödinger equation from variational "principle"
- in practice it meant slashing away the Dirac   < bra |      from all the    < a | O | b   >

First we get back the Hartree terms
4-derive-H.F.-two-particle.png

But the last exchange term is quite a different kind of animal ...

When we look at it we discover that it behaves as a NON-LOCAL OPERATOR
5-derive-H.F.-exchange.png
Above is the explicit form of the non-local exchange interaction

6--H.F.-exchange.png
It is in this case a demonstration of the "exclusion" principle in space
7-nonlocal-exchange.png
... often talked about as a repulsion "hole"
   
8-nonlocal-DFT.png

Link to Schrodinger Inc: http://www.schrodinger.com/  in particular: Jaguar: http://www.schrodinger.com/products/14/7
(this last link might change next year, but JAGUAR is our desired product)

Here we look at an important question:
we obtain the Hartree-fock Lagrange multipliers = SIngle electron selfconsistent energies
Is the total electron energy equal to the sum - the answer is NO!
It is related to
Koopmans' theorem http://en.wikipedia.org/wiki/Koopmans%27_theorem

81-sum-orbital-energies.png
You must subtract the pair interaction, because it is counted twice
                                                                                                  
Expansion in a basis - we have some notes in the PHYS208  notes: web.ift.uib.no/AMOS/PHYS208/2010.10.18/index.html#expand
Now we come back to the    Configuration Interaction

8-config-interaction.png

     
We apply the expansion twice
9-config-interaction.png
And here three times - and so on
So expansion over configurations follows naturally from this
a-config-interaction.png
AND THE CONFIGURATION INTERACTION thus means removing the INDEPENDENT PARTICLES feature

Thus   CONFIGURATION INTERACTION   means ELECTRON CORRELATIONS

Link to diagonalization in PHYS208 notes: http://web.ift.uib.no/AMOS/PHYS208/2010.10.18/index.html#diag
2010.10.14 previous lecture note                                                  2010.10.21 next lecture note