As we have seen, the electronic Hamiltonian contains two terms that act on one electron at a time, the kinetic energy and the electron-nucleus attraction, and a term that describes the pairwise repulsion of electrons. The latter depends on the coordinates of two electrons at the same time, and has turned out to be a practical computational bottleneck, which can be passed only for very small systems :
To avoid this problem the independent particle approximation is introduced : the interaction of each electron with all the others is treated in an average way. Suppose :
Then the Schrödinger equation which initially depended on the coordinates x (representing spatial and spin coordinates) of all electrons can be reduced to a set of equations :
The wavefunctions are called one-electron spin-orbitals.
The obvious problem is that for each electron the potential due to all other electrons has to be known, but initially none of these is known. In practice trial orbitals are used which are iteratively modified until a self-consistent solution (a "Self-Consistent Field") is obtained, which can be expressed as a solution to the Hartree-Fock equations :
It is important to realize that convergence of the SCF procedure is by no means
guaranteed. Many techniques have been developed over the years to speed up
convergence, and to solve even difficult cases. In practice, difficulties often
occur with systems with an unusual structure, where the electrons "do not know
where to go".
The eigenvalues are interpreted as orbital energies. The orbital energies have an attractively simple physical interpretation : they give the amount of energy necessary to take the electron out of the molecular orbital, which corresponds to the negative of the experimentally observable ionization potential (Koopmans' Theorem):
In addition to being a solution of the electronic Schrödinger equation the wavefunction must be normalized and satisfy the Pauli principle. The normalization condition is connected with the interpretation of the wavefunction as a distribution function which when integrated over entire space should give a value of one :
in "bra-ket" notation :
The Pauli principle states that the wavefunction must change sign when two independent electronic coordinates are interchanged :
For a two-electron system the spin-orbitals and (in which sigma is either alpha or beta spin state) can be combined as follows :
According to the definition of a determinant this antisymmetrized product is equal to :
This type of wavefunction is known as a Slater determinant, commonly abbreviated as :
An important property of the SCF method is that its solutions satisfy the Variation Principle, which states that the expectation value of the energy evaluated with an inexact wavefunction is always higher than the exact energy :
As a consequence the lowest energy is associated with the best approximate
wavefunction and energy minimization is equivalent with wavefunction
The energies of Slater determinants from a Hartree-Fock calculation are readily expressed in one- and two-electron integrals. For the ground state it is :
Here we have used the following abbreviations :
The two-electron integral (ii|jj) which describes the repulsion between two electrons each localized in one orbital is called a Coulomb integral, (ij|ij) for which a classical picture cannot be drawn so easily is called the Exchange integral.
In many cases it is advantageous to apply the restriction that electrons with
opposite spin pairwise occupy the same spatial orbital. This leads to the
Restricted Hartree Fock method (RHF), as opposed to the Unrestricted version
(UHF). An important advantage of the RHF method is that the magnetic moments
associated with the electron spin cancel exactly for the pair of electrons in
the same spatial orbital, so that the SCF wavefunction is an eigenfunction of
the spin operators and . Note that the UHF wavefunction is more flexible
than the RHF wavefunction, thus can approximate the exact solution better and
give a lower energy.
In practice RHF is mostly used for closed shell systems, UHF for open
shell species. RHF models for open shell systems and more advanced models can
used when necessary.
The total energy for a closed shell ground state RHF model can be written as :
The orbital energy in this case is :