The orbital approximation

Schrödinger's equation cannot be solved precisely for any atom with more than one electron. Numerical solutions by using the computers can be performed to a high degree of accuracy, and these depict that the equation does work, at least for fairly light atoms in which relativistic effects are negligible. For many purposes it is an adequate estimation to represent the wave function of each electron by an atomic orbital identical to the solutions for the hydrogen atom. The limitation of orbital approximation is that electron repulsion is included only approximately and the way where electrons move to avoid each other, termed as electron correlation is neglected.

I = -?1

A state of an atom is represented by an electron configuration displaying which orbitals are occupied by electrons. The hydrogen's ground state is written (1s)1 with one electron in the 1s orbital; two excited states are (2s)1 and (2p)1. the ground state is (1s)2; (1s)1(2s)1 and (1s)1(2p)1 are excited states, for helium with two electrons. The energy needed to excite or remove one electron is suitably represented by an orbital energy, usually written with the Greek letter ε. Similar convention is employed as in hydrogen, with zero being taken like the ionization limit, the electron's energy is removed from the atom. So energies of bound orbitals are negative.

The ionization energy needed to eliminate an electron from an orbital with energy ε1 is then which is generally known as Koopmans' theorem, even though it is better called Koopmans' approximation, because it depends on the limitations of the orbital approximation.