Hartree-Fock method

The Hartree-Fock method is an independent particle model which uses variational methods to find the best individual electron spin-orbitals.

Let us consider a many-electron atom with atomic number . The task is to find an equation describing the effective potential felt by a single electron.

Several effects have to be taken into account:

  • the nucleus attraction:
  • the interaction with the other electrons
  • the necessity for the system wavefunction to be a Slater determinant (antisymmetric)

Given the set of the electrons' wavefunctions , which constitute a starting ground state (a Slater determinant), we can define a charge distribution

We can now introduce the Hartree potential as
This potential represents the coulomb electron-electron repulsion. However it does not take into account spin effects and consider also the interaction of the electron with itself. These effects are taken into account by Fock-potential, which hasn't a classical interpretation
The sum is the Hartree-Fock potential

The Hartree-Fock hamiltonian for a single electron is then:[1]

The equation we'd like to solve is then

It's important to note that both and , which enter in the definition of and depend on the solutions of the hamiltonian equation and are not given "a priori". This is the reason why Hartree Fock equation is not, in fact, a Schrödinger equation for until we fix in some way and

Hartree-Fock method consist in a self-consistent procedure:

  • Chose an initial charge density, which means a starting ground state and compute the corresponding
  • solve the hartree equation with that and , find the energy corresponding to it and find the new wavefunctions
  • compute
  • put and in the hartree-fock potential and solve again the equation. If and are better approximations for the ground state according to the variational principle.
  • repeat the procedure until you reach a satisfactory agreement between the previous solutions and the latest.

Central field approximation

We can introduce further simplifications if we build as a central potential, since we are thus allowed to separate variables in the Hartree-Fock equation. Solutions are then of the form

This decomposition is a feature shared by central potentials. However is different from the radial function of hydrogenlike solutions. Furthermore, since the dependence of Coulomb potential has been broken, energy eigenvalues depend not only on but also on : Hartree-Fock model thus provides an explanation for the l-ordering observed in atomic spectra. Indeed, because of the centrifugal term which grows with , lower implies lower energy (other things being equal). Electronic configuration of atoms is expressed specifying the number of electrons in each eigenstate: for example the expression means that 2 electrons are in the state , 2 electrons in the state and 4 electrons in . In Hartree-Fock approximation there is no need for specifying and values since the energy does not depend on them.

  1. same hamiltonian without is the Hartree hamiltonian, less correct but much easier to solve