A model that is wrong on purpose
By now you can read any complex on sight: brackets mark the coordination sphere, you count donor atoms for the coordination number, and you balance charges to pin down the metal's oxidation state. Werner handed you the *shape* — six ligands around iron means an octahedron. But the puzzle that opened this rung is still unanswered. Why is one cobalt complex pink and another blue? Why is [Fe(H2O)6]2+ magnetic while [Fe(CN)6]4- is not, when both are iron(II) wearing six ligands? The shape alone says nothing about color or magnetism. To get those, we have to ask what the ligands do to the metal's d electrons — and the first model that did this well is almost insultingly simple.
[[crystal-field-theory|Crystal field theory]] makes one brazen assumption: forget that the metal–ligand bond is a real bond at all. Pretend each ligand is nothing but a tiny point of negative charge — a frozen lone pair, a naked minus sign — sitting where the donor atom would be. The metal ion is a positive charge at the center surrounded by these point charges. There are no shared electrons, no orbitals overlapping, no covalency. It is a purely electrostatic picture, the kind you could compute with a high-school formula for the repulsion between charges. Born around 1929 in Hans Bethe's and John Van Vleck's work on ions in crystals, the model was built to describe a transition-metal ion sitting in a solid lattice, surrounded by the negative ions of the crystal — hence the name 'crystal field'.
Five d orbitals, two shapes
The whole drama plays out among the metal's five d orbitals, so recall their shapes from the atomic-structure rung. Three of them — called d(xy), d(xz), and d(yz) — are four-lobed clovers that lie *between* the Cartesian axes, their lobes pointing into the diagonal gaps. The other two are different. The d(x2-y2) orbital is a four-lobed clover whose lobes point straight *along* the x and y axes. And d(z2) is the odd one out, a fat lobe up and down the z axis with a little doughnut around its waist; for our purposes treat it as pointing along the z axis. The key fact: two of the five orbitals aim directly along the axes, and three of them aim into the gaps between axes.
Out in a free, isolated metal ion floating in a vacuum, all five d orbitals have exactly the same energy — they are degenerate. An electron is equally happy in any of them, because there is nothing around to tell one direction from another; space is symmetric. That degeneracy is the calm before the storm. The moment we surround the ion with ligands, the directions stop being equivalent, and the five orbitals can no longer all share one energy. How they split apart depends entirely on *where* we put the point charges — which is to say, on the [[coordination-geometry|coordination geometry]].
The octahedral case, worked carefully
Take the most common geometry of all: six ligands at the corners of an octahedron. Place them on the coordinate axes — one ligand each at +x and -x, +y and -y, +z and -z. Now bring in those six negative point charges and watch what happens to a d electron. A d orbital is a region where the metal's d electron likes to sit. Slide a negative point charge straight into a lobe of that orbital and the electron, also negative, is repelled — its energy goes up, because it is being squeezed against an unwelcome neighbor. Slide the charge into the empty gap *between* lobes instead, and the electron barely notices; its energy is hardly raised at all.
Now overlay the two orbital shapes onto the six ligands sitting on the axes. The d(x2-y2) and d(z2) orbitals point *straight at* the ligands — d(x2-y2) along x and y, d(z2) along z. Their lobes ram head-on into the point charges, so an electron in either of them is strongly repelled and pushed *up* in energy. These two orbitals form a matched pair called the eg set. The other three — d(xy), d(xz), d(yz) — point into the diagonal gaps, sliding neatly between the ligands. An electron in them feels far less repulsion, so they sink *down* in energy relative to the average. These three are the t2g set. The single degenerate level of the free ion has split into a lower triplet and an upper doublet.
The energy gap between the lower t2g set and the upper eg set is the single most important number in this whole subject: the octahedral splitting parameter, written delta-o (the o is for octahedral; some books write 10Dq). Here is a subtlety worth getting right. The splitting is reckoned around a weighted average called the *barycenter*, the energy the orbitals would have if the ligand charge were smeared evenly over a sphere. Because there are two eg orbitals going up and three t2g orbitals going down, conservation of that average forces an uneven share: the two eg orbitals each rise by +0.6 delta-o, while the three t2g orbitals each drop by -0.4 delta-o. Check it: 2 x (+0.6) + 3 x (-0.4) = +1.2 - 1.2 = 0. The center of gravity is preserved, exactly as it must be. This is the [[octahedral-field-splitting|octahedral field splitting]], and it is the heart of crystal field theory.
free ion (vacuum) octahedral field
all 5 d degenerate
___ ___ eg (dz2, dx2-y2) +0.6 delta_o
/ up: lobes point AT ligands
- - - - - - - - - barycenter (weighted average)
===== ===== \
\___ ___ ___ t2g (dxy, dxz, dyz) -0.4 delta_o
down: lobes point BETWEEN ligands
gap (eg - t2g) = delta_o check: 2(+0.6) + 3(-0.4) = 0Filling the orbitals: high spin or low spin
A splitting diagram is just an empty stage until you populate it with the metal's d electrons. First nail the [[d-electron-count|d-electron count]]: strip the metal back to its ion in its oxidation state and count its remaining d electrons. Iron(III), Fe3+, is a d5 ion; iron(II), Fe2+, is d6; cobalt(III), Co3+, is d6; titanium(III), Ti3+, is d1. Now feed those electrons into the t2g and eg levels from the bottom up, obeying the same rules you learned for atoms — lowest energy first, one electron per orbital before pairing, parallel spins where you can.
For d1, d2, d3 there is no dilemma — the electrons go one each into the three t2g orbitals, all spins parallel. The interesting fork appears at d4 through d7. Consider d6 like Co3+. The fourth, fifth, and sixth electrons face a genuine choice. They could pair up inside the already-occupied, lower t2g orbitals — but pairing two electrons into one orbital costs energy, the [[electron-pairing-energy|pairing energy]] P, the electrostatic price of forcing two like charges into the same small space. Or they could stay unpaired by climbing into the empty, higher eg orbitals — but that costs delta-o, the price of the rent upstairs. Nature picks whichever is cheaper, and that single comparison, delta-o versus P, decides everything.
If the field is strong and the gap is wide — delta-o larger than P — pairing downstairs is the bargain, so the electrons cram into t2g and leave eg empty. For d6 that gives the configuration t2g^6 eg^0: all six electrons paired, *zero* unpaired electrons. This is the low-spin case. If instead the field is weak and the gap is narrow — delta-o smaller than P — climbing upstairs is cheaper, so electrons spread out to stay unpaired. For d6 that gives t2g^4 eg^2 with four unpaired electrons: the high-spin case. This is exactly the iron(II) puzzle from the opening. [Fe(CN)6]4- is low-spin (cyanide makes a large delta-o), every electron paired, so it is [[diamagnetism-and-paramagnetism|diamagnetic]] — not drawn into a magnet. [Fe(H2O)6]2+ is high-spin (water makes a smaller delta-o), four unpaired electrons, so it is paramagnetic — pulled into a magnetic field. Same metal, same oxidation state, opposite magnetism, purely because the ligands set different gaps.
Other geometries, and what to trust
Change the geometry and you change the pattern, because you change which orbitals point at the charges. In a [[tetrahedral-field-splitting|tetrahedral field]] — four ligands at alternating corners of a cube — none of the d orbitals points straight at a ligand, but now it is d(xy), d(xz), d(yz) that come closest to them, while d(x2-y2) and d(z2) lie furthest. So the pattern flips upside down: the three-orbital set rises and the two-orbital set falls, the opposite of the octahedron. And because only four ligands push, and none head-on, the gap is much smaller — as a rule of thumb delta-tet is roughly 4/9 of delta-o for the same metal and ligands. That tiny gap is almost always smaller than the pairing energy, which is why tetrahedral complexes are essentially always high-spin.
This flip is the source of the opening color riddle. Pink [Co(H2O)6]2+ is octahedral; deep-blue [CoCl4]2- is tetrahedral. Same cobalt(II), but the two geometries give different splitting patterns and gap sizes, so they absorb different colors of light and show different colors to your eye — and remember, the color you see is the [[complementary-color|complement]] of the light the complex absorbs, not the light it absorbs. A square-planar field (think Pt2+ and Ni2+ complexes) splits the orbitals into four distinct levels and is best seen as an octahedron with the two z-axis ligands pulled away to infinity; that is a story for a later guide.
Hold the model's honest limits in mind. The point charges are fiction; real bonds are partly covalent, and that covalency is exactly what ligand field theory will restore. The high-spin/low-spin choice rests on delta versus the pairing energy P, not on the field alone — get either number wrong and you predict the wrong magnetism. The t2g-below-eg picture is specific to the octahedron and *inverts* in the tetrahedron, so never quote the pattern without naming the geometry. And what makes delta-o large or small in the first place — why cyanide is a strong-field ligand and water a weak one — crystal field theory cannot really say; for that you need the [[high-spin-and-low-spin|spectrochemical series]] and the covalent insight of the guides ahead. But as a first lens that turns Werner's bare shapes into color and magnetism, this deliberately crude electrostatic model is one of the best bargains in all of chemistry.