Crystal Field Theory (CFT)

The Basic Idea of CFT

Crystal Field Theory is an electrostatic model. It considers the metal–ligand bond to be ionic, arising purely from electrostatic interactions between the central metal ion and the surrounding ligands. The ligands are not treated as atoms with their own orbitals; they are reduced to point charges (for anionic ligands such as $\ce{Cl-}$ and $\ce{CN-}$) or point dipoles (for neutral ligands such as $\ce{NH3}$ and $\ce{H2O}$, whose negative ends face the metal).

The starting point is the five d orbitals of an isolated gaseous metal ion. These are degenerate — all of identical energy. If we imagine the metal ion surrounded by a perfectly spherical shell of negative charge, all five orbitals are raised in energy by the same amount and remain degenerate. The crucial step comes when that negative field is supplied by ligands placed at definite positions: the field becomes asymmetric, the degeneracy is lifted, and the d orbitals split into sets of unequal energy. The exact pattern of this splitting depends on the geometry of the ligand field.

The whole edifice of CFT rests on one physical fact: a d orbital that points towards an approaching ligand is repelled strongly and rises in energy, while a d orbital that points between the ligands is repelled less and is lowered relative to the average. Everything that follows — colour, magnetism, stability — is a consequence of how the d electrons distribute themselves over these split levels.

Splitting in an Octahedral Field

In an octahedral coordination entity, six ligands approach the metal ion along the $\pm x$, $\pm y$ and $\pm z$ axes. The repulsion between the d electrons and the ligand charges is greater for orbitals that point along these axes. The $d_{x^2-y^2}$ and $d_{z^2}$ orbitals point directly at the ligands and are therefore raised in energy; collectively they form the higher-energy $e_g$ set. The $d_{xy}$, $d_{yz}$ and $d_{xz}$ orbitals point between the axes, away from the ligands, and are lowered; they form the lower-energy $t_{2g}$ set.

The energy separation between the $t_{2g}$ and $e_g$ sets is the crystal field splitting energy, written $\Delta_o$ (the subscript o for octahedral). The splitting conserves the average energy (the "barycentre"): the two $e_g$ orbitals rise by $\tfrac{3}{5}\Delta_o$ each, while the three $t_{2g}$ orbitals fall by $\tfrac{2}{5}\Delta_o$ each, so the total energy is unchanged.

Spectrochemical Series & Δo

The size of $\Delta_o$ is not fixed; it depends on the field produced by the ligand and on the charge on the metal ion. Some ligands produce strong fields and a large splitting; others produce weak fields and a small splitting. Arranging ligands in order of increasing field strength gives the spectrochemical series, an experimentally determined ordering based on the light absorbed by complexes:

$\ce{I- < Br- < SCN- < Cl- < S^2- < F- < OH- < C2O4^2- < H2O < NCS- < edta^4- < NH3 < en < CN- < CO}$

Ligands at the left (the halides, $\ce{S^2-}$) are weak-field; those at the right ($\ce{NH3}$, $\ce{en}$, $\ce{CN-}$, $\ce{CO}$) are strong-field. The position of a ligand in this series controls $\Delta_o$, which in turn controls colour and magnetism. Note that a higher metal oxidation state also increases $\Delta_o$.

High-Spin vs Low-Spin

For configurations $d^1$, $d^2$ and $d^3$ there is no ambiguity: the electrons enter the three $t_{2g}$ orbitals singly, obeying Hund's rule. The interesting choice appears at $d^4$. The fourth electron has two options. It can pair up in a $t_{2g}$ orbital, paying the price of the pairing energy $P$, or it can jump to the higher $e_g$ level, paying the price of $\Delta_o$. Which route is cheaper decides the spin state:

When $\Delta_o$ is small (weak-field ligand), it costs less to promote the electron to $e_g$ than to pair it, so the electrons spread out — a high-spin complex with the maximum number of unpaired electrons. When $\Delta_o$ is large (strong-field ligand), pairing in the $t_{2g}$ set is the cheaper option, giving a low-spin complex. NCERT notes that for the same reason, $d^4$ to $d^7$ entities are more stable in the strong-field case.

Electron Filling d¹–d¹⁰

The table below summarises how the $t_{2g}$ and $e_g$ sets fill in an octahedral field. For $d^1$–$d^3$ and $d^8$–$d^{10}$ the weak-field and strong-field arrangements are identical; only $d^4$–$d^7$ branch into distinct high-spin and low-spin patterns.

Crystal Field Stabilisation Energy

Because the $t_{2g}$ orbitals lie below the barycentre, every electron placed there lowers the energy of the system, while electrons in $e_g$ raise it. The net energy lowering relative to the unsplit (spherical) field is the Crystal Field Stabilisation Energy (CFSE). Counting $-\tfrac{2}{5}\Delta_o$ for each $t_{2g}$ electron and $+\tfrac{3}{5}\Delta_o$ for each $e_g$ electron gives:

$\text{CFSE} = \left[-\tfrac{2}{5}\,n(t_{2g}) + \tfrac{3}{5}\,n(e_g)\right]\Delta_o + mP$

where $n(t_{2g})$ and $n(e_g)$ are the electron counts in each set and $mP$ accounts for any extra electron pairs forced by a strong field. CFSE explains why $d^4$–$d^7$ complexes gain extra stability in strong fields, and it underlies trends in lattice energies, hydration enthalpies and the preference of certain ions for octahedral over tetrahedral sites. For NEET, the key takeaway is the sign convention: $t_{2g}$ electrons stabilise, $e_g$ electrons destabilise.

Splitting in a Tetrahedral Field

A tetrahedral field inverts the octahedral picture. Here only four ligands surround the metal, and none of them points directly at any d orbital. The orbitals that come closer to the ligand directions — $d_{xy}$, $d_{yz}$, $d_{xz}$ (the $t_2$ set) — are now raised in energy, while $d_{x^2-y^2}$ and $d_{z^2}$ (the $e$ set) are lowered. Because the tetrahedron has no centre of symmetry, the "g" subscript is dropped: we write $e$ and $t_2$, not $e_g$ and $t_{2g}$.

Crucially, the tetrahedral splitting is much smaller. For the same metal, the same ligands and the same metal–ligand distance it can be shown that $\Delta_t = \tfrac{4}{9}\,\Delta_o$. This splitting is so small that it is almost never larger than the pairing energy, so tetrahedral complexes are nearly always high spin — low-spin tetrahedral complexes are rarely observed.

Colour by d–d Transitions

One of the most distinctive properties of transition-metal complexes is their wide range of colours, and CFT explains it directly. When white light passes through a complex, light of energy equal to $\Delta$ is absorbed and promotes an electron from the lower d set to the higher one — a d–d transition. The wavelengths not absorbed are transmitted, so the observed colour is the complementary colour of the light absorbed. If a complex absorbs green light it appears red.

The classic example is $\ce{[Ti(H2O)6]^3+}$, a $3d^1$ system. Its single electron sits in the $t_{2g}$ level in the ground state. Absorption of blue-green light excites the transition $t_{2g}^{1}e_g^{0} \rightarrow t_{2g}^{0}e_g^{1}$, and the complex appears violet. The energy relationship is simply $\Delta = h\nu = \dfrac{hc}{\lambda}$, so a larger $\Delta$ means a higher energy absorbed and therefore a shorter absorbed wavelength.

Magnetic Behaviour

Because CFT fixes the number of unpaired electrons through the high-spin/low-spin choice, it predicts the magnetic behaviour of a complex. A complex with one or more unpaired electrons is paramagnetic; one with all electrons paired is diamagnetic. The spin-only magnetic moment follows $\mu = \sqrt{n(n+2)}\ \text{BM}$, where $n$ is the number of unpaired electrons.

This is exactly where weak and strong fields diverge. Both $\ce{[Fe(H2O)6]^3+}$ (weak field, high spin, $t_{2g}^{3}e_g^{2}$, five unpaired) and $\ce{[Fe(CN)6]^3-}$ (strong field, low spin, $t_{2g}^{5}$, one unpaired) are $d^5$ iron(III), yet their moments are $5.92$ BM and $1.73$ BM respectively. The same logic explains why $\ce{[CoF6]^3-}$ is paramagnetic while $\ce{[Co(NH3)6]^3+}$ is diamagnetic. Magnetic susceptibility measurements are therefore a direct experimental probe of the spin state predicted by CFT.

Limitations of CFT

CFT is remarkably successful at explaining the formation, structures, colour and magnetic properties of coordination compounds. But its founding assumption — that ligands are pure point charges — has consequences that disagree with experiment. If the interaction were purely electrostatic, anionic ligands should produce the greatest splitting; yet anionic ligands such as the halides actually sit at the low (weak-field) end of the spectrochemical series, while neutral $\ce{CO}$ and $\ce{CN-}$ produce the largest splitting.

The deeper flaw is that CFT ignores the covalent character of the metal–ligand bond. It cannot account for overlap between metal and ligand orbitals or for $\pi$-bonding effects, both of which are needed to rationalise the order of the spectrochemical series. These shortcomings are addressed by Ligand Field Theory and Molecular Orbital Theory, which restore the covalent contribution that CFT discards.