We've derived a complete many-electron Hamiltonian operator. Of course, the Schrödinger
equation involving it is intractable, so let's consider a simpler problem, involving the
one-electron hamiltonian
which involves no electron-electron interaction. This soluable in the B-O
approximation (recall the hydrogen atom by letting M=1). Call the solutions to the one-electron
Schrödinger equation 
. These will be molecular spin
orbitals when we get around to
it, but for now let it suffice to know they satisify the eigenequation
with the interpretation that electron i occupies spin orbital 
with energy 
.
If we ignore electron-electron interaction in 
, we construct a simpler system
with Hamiltonian
It will have eigenfunctions which are simple products of occupied spin orbitals, and thus an energy
which is a sum of individual orbital energies, as
This kind of wavefunction is called a Hartree Product, and it is not physically realistic. In the
first place, it is an independent-electron model, and we know electrons repel each other.
Secondly, it does not satisfy the antisymmetry principle due to Pauli which states that the sign of
the wavefunction must be inverted under the operation of switching the coordinates of any two
electrons, or
Part of the proof of equation 13 acknowledges this is not so for a Hartree Product.
To remedy this, first consider a two-electron system, such as helium. Two equivalent Hartree
Product wavefunctions for this system are
Obviously, neither of these is appropriate. However, using the old ``by inspection...'' trick, we
notice that
does. The mathematical form of this wavefunction can be generated by a determinant of 
's,
The familiar Pauli exclusion principle follows directly from this example. When we attempt to doubly
occupy a spin orbital 
by putting electron 1 and electron 2 in it, what happens?
Equation 17 can be generalized to give the N electron Slater determinant
A shorthand notation for a Slater determinant has been introduced, where all the diagonal elements in
the determinant are written in order as a ``ket'' vector. Equation 19 can thus be written as
where the normalization constant is absorbed into the notation.
Now we have written down a wave function appropriate for use in the case where
. In HF theory, we make some simplifications so many-electron atoms and
molecules be treated this way. By tacitly assuming that each electron moves in a
percieved electric field generated by the stationary nuclei and the average spatial distribution of all
the other electrons, it essentially becomes an independant-electron problem. The HF Self Consistent
Field procedure (SCF) will be bent on constructing each 
to give the lowest energy.