Here, represents the mean value of the kinetic and potential energy of a single electron (in the spin orbital ) in the bare nuclear field:

.

J_ij is the Coulomb integral:

Physically, the Coulomb interaction is the integral over the energy required to bring charge increment for electron 1 from infinity to position, x, in the field of the charge increment of electron 2 (The charge increment is the product of the probability density of the relevant wave function times dx). In other words, the Coulomb integral corresponds to the coulomb interaction of one electron with the averaged electric field generated by the remaining (n-1) electrons.

In case of a closed shell system, the integration over the spin and spatial variables can be written as a product and can be carried out separately. The spin integration determines independently, whether the two electrons possess a or b spin, always equal to1 ( different to the Exchange integral!).*

K_ij is the exchange integral

has no classical analogue. It occurs only between electrons in different orbitals with parallel spins. It is a quantum-mechanical manifestation of the Coulomb term and comes from exchanging the labeling of the indistinguishable electrons. Physically, that results from the fact that electrons with parallel spin like to keep as far away from each other as possible. Thus, it has no effect on electrons with antiparallel spin (See Chapter 3).

The following selection rule can therefore be defined:

To summarize, the total energy contains one-electron () and two-electron (J_ij, K_ij) terms.

Now the variational energy must be minimized. With the given term for the total electronic energy, , only one Slater determinant, constructed from N orthonormal spin orbitals, must be found which minimizes the expectation value of the energy:

.

is a functional of the wave function. The set of {} which minimizes the energy (inf means lower limit in the above equation) must be determined. This is achieved by varying the orbital functions, , until the minimum is reached (functional variation). Forming the required orthonormal set of {} can be done by the method of Lagrange undetermined multipliers.

last changes: 01.04.2008, AS
questions & comments to: axel.schulz@uni-rostock.de