The hydrogen atom is the only atom for which exact solutions of the Schrödinger equation exist. For any atom that contains two or more electrons, no solution has yet been discovered (so no solution for the helium atom exists!) and we need to introduce approximation schemes.
Let us consider the helium atom. The nucleus has a charge of , and if we place the nucleus at the origin, there will be an electron at a position with spin and an electron at position and spin . As usual, we consider the nucleus to be fixed. The classical energy is then
This means that the wave function depends on the full set of 6 coordinates or if spherical coordinates are used, and 2 spin coordinates , , and that this dependence is not simple! In fact, as the number of electrons increases, the number of variables on which depends increases as well. For an atom with electrons, the wave function depends on coordinates! Thus, it is clear that the wave function for a many-electron atom is a very unwieldy object!
As a side bar, we note that the 1998 Nobel prize in chemistry was awarded to Walter Kohn for the development of an extremely elegant theory of electronic structure known as density functional theory. In this theory, it is shown that the wave function , which depends on coordinates, can be replaced by a much simpler object called the electron density denoted . This object depends on only three variables for a system of any number of electrons. In density functional theory, it is shown that any physical quantity can be computed from this electron density .
Consider now an imaginary form of helium in which the two electrons do not interact. For this simplfied case, the energy is simply
Given this wave function, the energy would just be the sum of
the energies of two electrons interacting with a nucleus of charge .
We would need two quantum numbers and for this, and from our
study of hydrogen-like atoms, the energy would be