Racah parameter
/ RAH-kah, B /
Two electrons crammed into the small d orbitals of a metal ion repel each other, and that repulsion costs energy. The Racah parameter is the number chemists use to measure how strong that electron-electron repulsion is. There are three of them in full — labeled A, B, and C — but the one you meet constantly is B, because it sets the energy gaps between the free-ion terms of the same spin and therefore controls much of a complex's spectrum.
Think of B as a yardstick for how tightly the d electrons are squeezed together. In a free gaseous ion, B has a characteristic value (a few hundred to about a thousand wavenumbers, depending on the metal and its charge). It is the natural unit in which Tanabe-Sugano diagrams are drawn — both axes are scaled by B — so when you fit a spectrum you extract two numbers at once: delta-o (the ligand-field splitting) and B (the repulsion). The energies of the spin-allowed bands depend on both, which is why you need ratios to disentangle them.
The reason B is so revealing is that it shrinks when the metal forms a complex, and the amount it shrinks measures covalency. In a complex the d electrons are no longer confined to the metal — they spread out a little onto the ligands because the bonding is partly covalent — so they repel each other less, and B drops below its free-ion value. This reduction is the nephelauxetic (cloud-expanding) effect, and the ratio B-in-complex over B-free is a direct gauge of how covalent the metal-ligand bond is. The honest caveat: this is exactly where the pure point-charge crystal-field picture fails — a reduced B is the spectroscopic confession that real bonds are not purely electrostatic, which is why ligand-field theory is needed.
Free Co3+ has a Racah B near 1100 wavenumbers, but in [Co(NH3)6]3+ the fitted B drops to roughly 650 — about 60 percent of the free-ion value. That nephelauxetic ratio tells you the cobalt-nitrogen bonds carry significant covalent character, sharing the d electrons out onto the ammonia ligands.
B drops from the free ion to the complex — covalency measured spectroscopically.
Crystal-field theory, treating ligands as bare point charges, predicts B should be unchanged in a complex; the fact that B always drops is direct experimental proof that the bonding is partly covalent — the gap the ligand-field model exists to fill.