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Now that we have a form for the wavefunction and a simplified notation
for the Hamiltonian, we have a good starting point to tackle the
problem. Still, how do we obtain the molecular orbitals?
We state that the HartreeFock wavefunction will have the form of a Slater
determinant, and that the energy will be given by the usual
quantum mechanical expression (assuming the wavefunction is normalized):

(14) 
For symmetric energy expressions, we can employ the variational
theorem, which states that the energy is always an upper bound to the
true energy. Hence, we can obtain better approximate wavefunctions
by varying their parameters until we minimize the energy
within the given functional space. Hence, the correct molecular orbitals
are those which minimize the electronic energy !
The molecular orbitals can be obtained numerically using integration
over a grid, or (much more commonly) as a linear combination of a set
of given basis functions (socalled ``atomic orbital'' basis functions,
usually atomcentered Gaussian type functions).
Now, using some tricks we don't have time to get into, we can
rewrite the HartreeFock energy in terms of integrals
of the one and twoelectron operators:

(15) 
where the one electron integral is

(16) 
and a twoelectron integral (Chemists' notation) is

(17) 
There exist efficient computer algorithms for computing such one
and twoelectron integrals.
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David Sherrill
20020530