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Again, the HartreeFock method seeks to approximately solve the
electronic Schrödinger equation, and it assumes that the
wavefunction can be approximated by a single Slater determinant
made up of one spin orbital per electron. Since the energy
expression is symmetric, the variational theorem holds, and
so we know that the Slater determinant with the lowest energy
is as close as we can get to the true wavefunction for the assumed
functional form of a single Slater determinant. The HartreeFock
method determines the set of spin orbitals which minimize the
energy and give us this ``best single determinant.''
So, we need to minimize the HartreeFock energy expression with respect
to changes in the orbitals
.
We have also been assuming that the orbitals are orthonormal,
and we want to ensure that our variational procedure leaves them
orthonormal. We can accomplish this by Lagrange's method of
undetermined multipliers, where we employ a functional
defined as

(18) 
where are the undetermined Lagrange multipliers and
is the overlap between spin orbitals and , i.e.,

(19) 
Setting the first variation
, and working
through some algebra, we eventually arrive at the HartreeFock equations
defining the orbitals:

(20) 
where is the energy eigenvalue associated with orbital
.
The HartreeFock equations can be solved numerically (exact HartreeFock),
or they can be solved in the space spanned by a set of basis functions
(HartreeFockRoothan equations). In either case, note that the solutions
depend on the orbitals. Hence, we need to guess some initial orbitals
and then refine our guesses iteratively. For this reason, HartreeFock
is called a selfconsistentfield (SCF) approach.
The first term above in square brackets,

(21) 
gives the Coulomb interaction of an electron in spin orbital with
the average charge distribution of the other electrons. Here we see
in what sense HartreeFock is a ``mean field'' theory. This is called
the Coulomb term, and it is convenient to define a Coulomb operator
as

(22) 
which gives the average local potential at point due
to the charge distribution from the electron in orbital .
The other term in brackets in eq. (20) is harder to explain and
does not have a simple classical analog. It arises from the antisymmetry
requirement of the wavefunction. It looks much like the Coulomb term,
except that it switches or exchanges spin orbitals and .
Hence, it is called the exchange term:

(23) 
We can define an exchange operator in terms of its action on an
arbitrary spin orbital :

(24) 
In terms of these Coulomb and exchange operators, the HartreeFock equations
become considerably more compact.

(25) 
Perhaps now it is more clear that the HartreeFock equations are eigenvalue
equations. If we realize that

(26) 
then it becomes clear that we can remove the restrictions in
the summations, and we can introduce a new operator, the Fock
operator, as

(27) 
And now the HartreeFock equations are just

(28) 
Introducing a basis set transforms the HartreeFock equations into the
Roothaan equations. Denoting the atomic orbital basis functions
as , we have the expansion

(29) 
for each spin orbital . This leads to

(30) 
Left multiplying by
and integrating
yields a matrix equation

(31) 
This can be simplified by introducing the matrix element notation
Now the HartreeFockRoothaan equations can be written in matrix form
as

(34) 
or even more simply as matrices

(35) 
where is a diagonal matrix of the orbital
energies . This is like an eigenvalue equation except for
the overlap matrix . One performs a transformation of
basis to go to an orthogonal basis to make vanish. Then
it's just a matter of solving an eigenvalue equation (or, equivalently,
diagonalizing !). Well, not quite. Since
depends on it's own solution (through the orbitals), the process must be
done iteratively.
This is why the solution of the HartreeFockRoothaan equations are
often called the selfconsistentfield procedure.
Next: About this document ...
Up: Introduction to HartreeFock
Previous: Energy Expression
David Sherrill
20020530