**Crystal field stabilization energy** is the gain in energy achieved by the preferential filling up of orbitals by electrons. In other words, the reduction of a transition metal ion’s energy in a certain ligand environment is called crystal field stabilization energy (CFSE). The CFSE depends on factors like geometry, spin pairing energy, the number of d electrons, and ligand character (spectrochemical series), among others.

## Crystal Field Stabilization Energy

**Crystal field stabilization energy** is defined as the change in energy due to the splitting of the d-orbitals of metal cation under the influence of ligand field in a complex. The stability of the complex increases as the amount of CFSE increases. Consequently, it is the magnitude of the energy difference between the two sets (t_{2g} and e_{g} ) orbitals.

## Crystal field stabilization energy formula

The formula for CFSE in an **octahedral complex** with a d^{p+q} configuration with p electrons in t_{2g} and q electrons in e_{g} orbitals is given by:

CFSE = (-0.4p + 0.6q)Δ_{o} + mP

or, CFSE = (-4p + 6q)Dq + mP

Where m = number of paired electrons, and P = mean pairing energy

According to Hund’s rule, the electrons in the d^{1}, d^{2}, and d^{3} cases occupy various degenerate t_{2g} orbitals to stay unpaired. As a result, the CFSE values are 4, 8, and 12 Dq, respectively. They are said to be stabilized to this extent as compared to the hypothetical spherical field.

There are two potential outcomes in the d^{4} case: either the electron may enter the higher e_{g} or may pair in t_{2g} orbitals. The actual configuration will of course be the lowest energy one and will depend on the relative magnitude of Δ_{o} and P. The fourth electron in the d^{4} case prefers to enter the e_{g} orbital in the presence of a weak field or a high spin state, and the configuration changes to t_{2g}^{3} e_{g}^{1}, So

CFSE = (3× -0.4 + 1× 0.6) Δo = -0.6Δo = -6Dq

In the case of weak field d^{5} and d^{10 }cases, the CFSE becomes zero because of the^{ }exact balance of stabilization of t_{2g} by e_{g} set. Therefore, subshells that are partially filled and fully filled are spherically symmetric, and no stabilization by the addition of external/additional ligands takes place.

In the case of strong field or low spin cases Δ_{o} > P the electrons prefer to stay coupled in the t2g level, whereas the e_{g} level remains unoccupied from d^{1} to d^{6} ions. As a result, for strong field cases compared to weak field cases, the CFSEs of complexes with 4 to 7 electrons will be higher.

For example, for the d^{4 }case, the strong field (low spin) configuration will be t_{2g}^{4} giving CFSE of 1.6Δ_{o }(16Dq) as compared to 6Dq (0.6Δo) for the weak field case. But the CFSEs for d^{8}, d^{9}, and d^{10} cases

are the same irrespective of field strength, which is 12, 6, and 0 Dq respectively. Moreover, crystal field effects for weak and strong octahedral fields are illustrated below:

In the case of **tetrahedral complexes**, pairing energy is larger than Δ_{t }or 10Dq. So electrons enter t_{2g} orbitals as soon as e_{g} orbitals are singly occupied. Therefore, only weak field or high spin cases are considered.

CFSE for d^{3} tetrahedral case, the configuration is e_{g}^{2}, t_{2g}^{1}.

CFSE = (-0.6q + 0.4P) = (2×-0.6 + 1×0.4) = -0.8Δt = -8Dq

## Significance of CFSE

- Determination of complexes crystal structures.
- Prediction of the stabilization of oxidation state.
- It is useful to understand the relative stability of a given complex’s stereochemistry.
- It helps to explain the heat of hydration of transition elements.