The Hartree-Fock approximation

1. The Hartree-Fock approximation Assume that the wavefunction can be approximated by a Slater-determinant. Vary the orbitals and use the variation principle to minimize the energy.

2. The Hartree-Fock approximation (continued) closed-shell system: energy: molecular orbitals (MO) Fock-operator MO-energy Fock-equation:

3. The Hartree-Fock approximation (continued) energy: kinetic energy and nuclear attraction Coulomb-interaction exchange interaction

4. The Hartree-Fock approximation (continued) electron-electron repulsion MO-energy

5. The Hartree-Fock approximation (continued) MO-energy Which MO’s should we choose to construct the Slater-determinant? Hund’s first rule: The Slater-determinant has the lowest energy if it’s constructed from the MOs with the lowest MO-energy.

6. The Hartree-Fock approximation (continued) ionization energy electron affinity occupied orbitals empty or virtual orbital MO-energy Koopmans-theorem:

7. The Hartree-Fock approximation (continued) The Fock-operator’s effect depends on the solution of the Fock-equation. An iterative procedure is needed to solve the Fock-equation, hence the name Self-Consisten Field (SCF) for the Hartree- Fock approximation.

8. The Hartree-Fock approximation (continued) atomic orbitals parameters to be varied Roothaan- or secular equation: Linear combination of atomic orbitals (LCAO) approximation:

9. The Hartree-Fock approximation (continued) • The Hartree-Fock procedure • choose basis functions {ci} • calculate the overlap matrix S • calculate the contribution of the one-electron operator to F • calculate two-electron integrals • make an initial guess for the MOs • calculate the total energy • calculate the Coulomb-integrals • calculate the exchange-integrals • compute the Fock-matrix F • solve the Roothaan-equation • compare the old an new MO’s • if old an new MOs are different then repeat this block