We can take a look at the excited states of carbon to get an appreciation of Hund's rules. The following chart shows the states of a carbon atom. For most states, a basis of is assumed and the state of the sixth electron is given. Some states have other excited electrons and are indicated by a superscript. Different states are not shown since the splitting is small. Electric dipole transitions are shown changing by one unit.
The ground state has and as we predicted. Other states labeled are the ones that Hund's first two rules determined to be of higher energy. They are both spin singlets so its the symmetry of the space wavefunction that is making the difference here.
Jim Branson 2013-04-22