The Hartree-Fock and LCAO approximations, when applied to the electronic Schroedinger equation, lead to what are called the Roothaan-Hall equations.

• In this expression, e are the orbital energies, and S is a measure of the extent to which basis functions interact, or overlap. S is called the overlap matrix.

• F is the Fock matrix; it is analogous to the Hamiltonian in the Schroedinger equation. It is a sum of three terms (in atomic units):
  
  • The first term, H^core, accounts for the kinetic and potential energies of individual electrons:
    

  • F^Coulomb represents the electrostatic interactions between electrons:
    

  • The final term, F^exchange, has no classical meaning; it is related to electron correlation, the tendency of electrons of the same spin to repel each other more than electrons of opposite spins.
    

• P in the above expressions is the density matrix, the elements of which are the squares of the molecular orbital coefficients summed over all occupied orbitals.

• [The LCAO page has some further explanation of this idea.] It is given by:
  

the evaluation of which is the most time-consuming step of an MO calculation.