## The Hartree-Fock method

#### Oktober - November 1995 version

As mentioned in the introduction, the Hartree-Fock method is based on the independent particle approximation. It means that each electron is assumed to move in an average potential created by the nucleus and the other electrons.

The approximative Hamiltonian can then be written as a sum of one-particle Hamiltonians

(1)

where N is the number of electrons in our system. The simplest form of an eigenfunction to Happrox. will be a product function

(2)

where a , b etc stand for the quantum numbers necessary to specify a single electron state. Any permutation of the single-particle functions, which we can call spin-orbitals, will also lead to an eigenfunction of Happrox. . However, in order to satisfy the Pauli exclusion principle the total wave function, Psi, must be antisymmetric with respect to the interchange of any two of the electrons. We can form a wave function with this property by antisymmetrizing the product function above. We write this antisymmetrized wave function as

(3)

and it is called a Slater determinant. As a simple example consider an atom with two electrons, than the antisymmetrized wave function will be

(4)

If we want the total wave function, Psi to consist of one Slater determinant, how do we chose the ``best'' one? The expectation value of the total energy for a state represented by the Slater determinant is given by the expectation value of the total Hamiltonian

(5)

The ``best'' determinant for the ground state should be the one which minimize the expectation value of H.

We thus minimize the energy with respect to changes in the Slater determinant Psi. This leads to the Hartree-Fock equations