We present here an overview of the electronic structure of atoms. We begin with the mean-field approximation. This scheme is sometimes also called the Hartree approximation, and is the most basic starting point when discussing many-electron systems. In this approach, the atomic states are distinguished from one another by their electronic configuration. An electronic configuration is, in general, degenerate with a number of other configurations. However, when we take into account the corrections due to the deviation of the Coulomb interaction away from the mean field, the energy levels are split into a number of distinct levels, and each of these split energy levels is called a multiplet. In order to demonstrate this point, we introduce the Slater determinant. After this, we discuss the Coulomb integral and the exchange integral. In particular, because of the Pauli principle, the exchange integral exists only between electrons with the same spin orientations. This allows us to explain Hund's rule, that is, the multiplet that has the largest value of composite spin has the lowest energy.
Mean-field approximation and electronic configurations
The usual starting point for discussing the electronic structure of atoms is the mean-field approximation.
The motion of an electron is affected by attractive Coulomb interaction due to the positive charge Ze of the nucleus and repulsive Coulomb interaction due to the other electrons. The latter is time dependent owing to the motion of the other electrons, but we may, as an approximation, replace these electrons by an appropriate charge distribution and consider the Coulomb force due to it.