Let's assume a wave function of the Slater determinant form and find an expression for the expectation
value of the energy. We've written a Slater determinant as a ket vector in shorthand notation,
allowing us to make use of Dirac notation for such things as overlap. In this context, recall that
where the basis vectors 
is expanded in are every possible value of x with contraction
coefficients identified as the value of 
at x. Thus placing an operator (such as
) inside the bracket, we get the expectation value of the observable associated with that
operator. Since is the energy operator,
is the differential of all the spin and space coordinates of all the electrons.
With much foresight, we continue to simplify the problem by writing as a sum of one- and
two-electron operators
This will allow us to more precisely develop the electronic energy by it's components.
First, examine the core hamiltonian 
.
The nature of this is best evidenced by example, so we turn to the familiar helium atom,
. Look at one term in the above sum, for the sake
of illustration take 
.
Here 
is defined as 
. In the first two terms of equation 28, the integrations over
electron two's coordinates can be carried out irrespective of electron one's, and give the two terms of
equation 29. The last two terms integrate to zero due to the orthogonality of 
and
.
Repeating this for 
we get exactly the same thing, and we see
Profound, isn't it? Seems that every occupied spin orbital 
yields a term of the form
to the one electron energy.
Now look at 
.
Continuing to work in the helium atom example (realize that this could be two electron
system) pick (i,j) = (1,2) and look at that one term.
Unfortunately, the 
operator prevents seperation of the integrations over the
electronic coordinates of electron 1 and electron 2. It cannot be assured that the last two terms are
zero. In general, they are not. However, since the 
and 
are dummy variables,
the first and second terms of equation 33 are equal, as are the last two. Thus for the two
electron operator ,
where
The construction 
is called an antisymmetrized two electron integral in
physicists notation.
By working in the spin orbital basis, much trouble is avoided. In fact, by extending the results shown
previously to the general case, we can now write down the HF energy for a given set of occupied spin
orbitals.
Now move on and consider working in the spatial orbital basis, where
This is more natural, since our intuition is usually based on having a region of space which describes
the location (more or less) of electrons, one of alpha spin and one of beta spin.
Some of quantum chemistry is formulated entirely in terms of spin orbitals, for various reasons.
For our purposes, we will work entirely in the spatial orbital basis. This will cause things to get
somewhat murky soon, but in the long run it will be simpler.
At any rate, in the two electron system we adore so much, we can identify the two occupied spin
orbitals and as the spin up and spin down halves of the single lowest lying spatial
orbital, a 1s in helium or the 
bonding orbital in H
for example. These can be more
precisely defined as
This changes the way we write slater determinants. Using an overbar to denote 
spin occupation
of a spatial orbital, 
,
Rethinking the one electron integrals for this case,
The notation 
is used to denote an integral over only spatial coordinates,
what remains after the spin integrations have been carried out, giving a factor of 1 or 0.
That was a neat closed shell system. How about something like
?
The coefficient 
here is related to the occupation of spatial orbital i, and will be more
precisely defined later.
The two electron integrals are a tad more involved, but we go about it in essentially the same manner.
Yes, I know. Very confusing. But it's all just notation, and can be understood. In physicist's
notation (equivalent to Dirac notation), 
refers to the two
electron integral where 
and 
are functions of electron 1, while 
and 
are functions of electron 2. Chemist's notation (with the square brackets []) places the functions of
electron 1 on the left and the functions of electron 2 on the right. When the two functions of a
single electron
are not of the same spin, the whole integral goes to zero, otherwise the spin integrates out to 1.
Hence the spin-free notation 
.
What occurs when the two electrons are of parallel spin, requiring distinct spatial orbitals and a
wavefunction something like 
? The same general form is present, and a
related antisymmetrized two electron integral is evaluated. In this case,
.
Another bit of notation, which should be apparent. 
and 
.
is termed a coulomb integral and has the physically reassuring interpretation of somehow
accounting for electronic repulsion between electrons in molecular orbital i and molecular orbital
j. 
, the exchange integral, has no classical analog and no true physical
interpretation. Many have
tried to come up with something, and a typical attempt says that it ``correlates the motions of electron
i and j when they have parallel spins, lowering the energy since those electrons avoid each other
better.'' Whatever.
Best to just move on to a general energy equation in the spatial MO basis. In summary:
for each electron in orbital i.
for each pair of electrons, and
for each pair of parallel spin.
