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

Note that is not simply a sum of terms for electron 1 and electron 2, . Therefore, it is not possible to write the wave function as a simple product of the form , nor is it even possible to use the special form we introduced for identical particles

because these simple products are not correct solutions to the Schrödinger equation.

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

For this imaginary helium atom, the wave function can be expressed as an antisymmetric product, and because we have included spin, in the ground state, both electrons can be in the 1s() spatial orbital without having the wave function vanish. Recall that, when , the 1s orbital is

Multiplying this by a spin wave function, the wave function for one of the electrons is

Given this form, the antisymmetrized two-electron wave function becomes

which, by virtue of the spin wave functions, does not vanish.

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

in Rydbergs. This comes from the fact that the energy of one electron interacting with a nucleus of charge is in Rydbergs. So, the ground-state energy would be -8 Ry. In real helium, the electron-electron Coulomb repulsion increases the ground state energy above this value (the experimentally measured ground-state energy is -5.8 Ry). The tendency is for the electrons to arrange themselves such that the Coulomb repulsion is as small as possible.