## Hartree-Fock equations - The Formulae

#### Oktober - November 1995 version

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

(6)

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

(7)

where h is

(8)

The last term, u HF (i) is called the Hartree-Fock potential. From the above we can see that its effect on $\psi_a(i)$ is

(9)

Due to the presence of the last term, the exchange potential , the potential is non-local. The exchange term is a consequence of the Pauli principle and the requirement of antisymmetric wave functions. In the general case the Hartree-Fock equations given in Eq.(eq:hf), are hard to solve. In practice a few other restrictions are thus imposed on the spin-orbitals, psi . The first restriction is the requirement that the spin-orbital can be separated into one spin and one orbital part. The second restriction is that we assume that the psi's are solutions to a spherical symmetric potential. This later restriction is called the central field approximation and it make possible a separation of the orbital part of the wave function into one radial and one angular part. For a closed shell system, where the total spin and angular momentum is zero, the system is indeed spherical symmetric and that restriction is automatically fulfilled. We can now write the spin-orbital as a product of the radial, the angular and the spin part

(10)

The Y l m 's are spherical harmonics and the Hartree-Fock equations can now be simplified to the radial Hartree-Fock equations

(11)

which can be solved to obtain the radial functions, P . As can be seen from the above equations, the equation for Pa depends on the radial functions for all the other electrons. Because of that one uses 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.(7), 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,Eq.(7), 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.

The Herman-Skillman program which we use, is not of Hartree-Fock, but of Hartree type. That means that the exchange potential term (the non-local one), is not included.

Instead, it is simulated by a statistical exchange term. One consequenceis that there is only one average potential for all orbitals considered. About the Herman-Skillman