Hartree-Fock Theory
The Hartree-Fock Limit
In order to solve the Hartree-Fock equations it was necessary to expand the spatial molecular orbitals in terms of a set of known basis functions. The quality of the predictions, thus, depends on the ability of the basis set to span the space needed to describe the molecule. Indeed, the quality of the predictions is found, in practice, to improve with increasing basis set size up to a point. Eventually, adding functions to the basis set does not lower the expectation value of the energy nor improve the variational wave function; the expectation value of the energy and the wave function are said to have achieved their Hartree-Fock limits. In the literature it is common to reserve the terms ``Hartree-Fock energy'' and ``Hartree-Fock wave function'' for these limits and denote the less accurate values by the ``SCF'' designator. The Hartree-Fock limit and its discrepancy from experiment is due to the inadequacies of Slater determinants, as previously covered in section 1.7, and approximations in the Hartreee-Fock method. In order to understand the latter, examine the form of the Fock operator in equation 1.56. The Coulomb and exchange operators nested inside the Fock operator depend on the other occupied orbitals and describe the motion of the ith electron in the average electric field generated by the N-1 other electrons. Remember, it was this dependence which made it necessary to perform the SCF iterations. Because the Hartree-Fock method treats each electron in the average field of the others it is known as an independent particle model. In reality, the motions of the electrons are not independent and the Hartree-Fock model cannot account for this even if the basis set is infinite.