## The Hartree-Fock Equations

Minimizing the energy with respect to changes in the Slater determinant
leads to the *Hartree-Fock equations*:

As before, *i* and *j* denote the quantum numbers necessary to
specify a single electron state. The sum over *j* runs over all occupied
states.
Although the expression looks complicated it is just an eigenvalue equation of
the form

where *h* is the simplified Hartree hamiltonian for non-interacting
electrons that we started with and the bracketed potential term is the
extra Hartree-Fock potential arising from including electron-electron
interactions with an antisymmetrized wave function. The Hartree-Fock potential
can be seen to have two main components. The first component *V*_{ee}
is the average repulsive Coulomb potential energy. It is also derived in
the full Hartree method. The second component *V*_{ex} is the
exchange interaction energy. It is a Hartree-Fock correction term providing a
lowering of the overall energy of interaction due to spin correlation keeping
like-spin electrons being apart.

The single-electron Hartree hamiltonian and the Coulomb component of the
Hartree-Fock potential are both local operators. They operate on the
*i*^{th} spin-orbital at its actual location **r**_{m}.
The exchange term, *V*_{ex}, is a __non-local__ operator. From
the above we can see that its effect on spin-orbital *i* is

For the *i*^{th} spin-orbital at **r**_{m}, it
operates at location **r**_{n}. The exchange term is the
manifestation of the Pauli exclusion principle. This can be seen more readily
if you consider expansion of the spin-orbital wave functions into spinor and
spatial components. The spinor components provide a kronicker delta between
the *i* and *j* wave functions. Thus, **only
alike** spin electrons are affected by this term. This creates what is
called the exchange hole.

In the general case, the Hartree-Fock equations given in Eq.(07) are
hard to solve. In practice a few other restrictions are thus imposed on
the spin-orbitals under consideration. The __First__ restriction is the
requirement that each spin-orbital can be separated into a spinor component and
a spatial (orbital) component. The __Second__ restriction is that we assume
the spin-orbitals are solutions to a spherically symmetric potential. This
latter restriction is called the *central field approximation* and it makes
possible a separation of the orbital component of the wave function into
a radial and an angular part.

For a closed shell system, where the total spin and angular momentum
is zero, the system is indeed spherically symmetric and that restriction
is automatically fulfilled. We can write the spin-orbital as

The *Y*_{l m}'s are the usual spherical harmonics and the
Hartree-Fock equations can now be simplified to the *Radial
Hartree-Fock equations*:

which can be solved to obtain the radial functions, *P*_{i}.

As can be seen from the above equations, the equation for *P*_{i}
depends on the radial functions for all the other *j*_{occupied}
electrons. Because of
this dependence, the Hartree-Fock equations are solved using an iterative
scheme. The starting point is an approximative description of the single
particle functions, *P(r)*. It could be hydrogen-like functions, but
usually some better approximation is used. With these starting functions the
Hartree-Fock potential is constructed and the eigenvalue equation, Eq.(11), is
solved. Then a new set of single-particle functions are obtained and a new
Hartree-Fock potential is constructed and again the eigenvalue equation is
solved. This is done over and over again until the radial functions as well as
the energy eigenvalues are stable.

We call this a *Self-Consistent Field* method.

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Created: November 12, 1998 --- Last Updated: November 28, 2001
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By Mark D. Pauli