The Hartree-Fock Equations

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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 Vee is the average repulsive Coulomb potential energy. It is also derived in the full Hartree method. The second component Vex 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 ith spin-orbital at its actual location rm. The exchange term, Vex, is a non-local operator. From the above we can see that its effect on spin-orbital i is

For the ith spin-orbital at rm, it operates at location rn. 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 Yl 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, Pi.

As can be seen from the above equations, the equation for Pi depends on the radial functions for all the other joccupied 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
By Mark D. Pauli