electron-pairing energy

Electrons hate to share a room. Two electrons forced into the same orbital, even with opposite spins, sit close together and repel each other, costing energy. There is also a subtler bonus that vanishes when you pair them: parallel-spin electrons in different orbitals enjoy a quantum-mechanical stabilization (the exchange energy) that pairing throws away. The total price you pay to make two electrons occupy one orbital instead of staying apart is the electron-pairing energy, written P.

P has two parts, both opposing pairing. The first is plain Coulomb repulsion: cram two like charges into one small region and they push back. The second is the loss of exchange energy, the extra stability that a set of unpaired, parallel-spin electrons enjoys (this is the deep reason behind Hund's rule, which says electrons fill empty orbitals singly first). Because pairing costs P, an electron will only pair up if doing so is cheaper than the alternative. In a transition-metal complex the alternative is to climb the crystal field gap delta into a higher orbital. So the contest is delta versus P, electron by electron.

This single quantity is the gatekeeper of the high-spin/low-spin choice. If the splitting delta is bigger than P, paying P to pair within the lower set is the bargain, and you get low-spin. If delta is smaller than P, climbing into the upper set is cheaper, electrons stay unpaired, and you get high-spin. P is roughly fixed for a given ion (it grows for smaller, more compact ions where electrons are squeezed together), so in practice it is the ligand-controlled delta that does the deciding — which is exactly why the spectrochemical series matters.