How can I tell when a transition metal complex is low spin or high spin?
It depends on the values of splitting and pairing energies.....
According to ligand field theory, electron-electron repulsion causes normally degenerate orbitals to split. Ligand fields contact orbitals of the central atom; the orbitals that directly contact the ligand fields end up with higher energy than those orbitals that only have indirect contact with the ligand fields.
The reason is that when the orbital of the central atom has direct contact with the ligand field there is higher electron-electron repulsion because both the ligand field and the orbital contain electrons. As the repelling force between the ligand field and the orbital is high, some orbitals have higher energies than others. This splitting of energy varies according to molecular geometry because each geometry can hold different numbers of ligands and the orbitals have different shapes.
If the splitting energy is greater than the electron pairing energy then electrons fill all lower energy orbitals first. Then they pair up with electrons in these orbitals before moving to the higher energy orbitals. Electrons fall into the lowest energy state, and as the pairing energy is lower than the splitting energy, it is more energetically favourable for electrons to pair up and completely fill the low energy orbitals. Only then do the high energy orbitals start to be filled. This is a "low spin" complex.
If the pairing energy is greater than the splitting energy, the electrons occupy all the orbitals first and then pair up, irrespective of the energy of the orbitals. This is in accordance with Hund's rule that states that all orbitals must be occupied before any pairing of electrons occurs. This is a "high spin" complex.