When a metal is subjected to the perturbation of an octahedral field, the energies of the d-orbitals split into two groups, the lower energy t_{2g}, at -0.4Δ_{o}, and the higher energy e_{g}, at 0.6Δ_{o}, where Δ_{o} is the ligand field splitting parameter.
When electrons are put into these orbitals, the orbitals which become occupied depend on the value of Δ_{o}. When there are x electrons in the t_{2g} orbitals, and y electrons in the e_{g} orbitals, the total energy of the electrons relative to the average energy of the electrons is known as the Ligand Field Stabilization Energy (LFSE):
LFSE = (-0.4x + 0.6y)ΔDo
The LFSE therefore depends on the number of electrons in the d-orbitals of the metal, x+y, the value of Δ_{o}, and the distribution of electrons between the t_{2g} and e_{g} levels.
When calculating the electronic configuration of a transition metal complex, we should note that the electrons always occupy the lowest energy configuration.
Electronic Configurations of d-metal complexes | ||
d^{1}: | t_{2g}^{1}:LFSE = -0.4Δ_{o} | |
d^{2}: | t_{2g}^{2}:LFSE = -0.8Δ_{o} | |
d^{3}: | t_{2g}^{3}:LFSE = -1.2Δ_{o} | |
d^{4}: | t_{2g}^{4}:LFSE = -1.6Δ_{o} | |
d^{4}: | t_{2g}^{3}e_{g}^{1}:LFSE = -0.6Δ_{o} |
As we can see, when the number of d-electrons is 0, 1, 2, or 3, there is no trouble assigning an electronic configuration, it being t_{2g}^{n}. However, when a 4th electron is added, there are two possible configurations: t_{2g}^{4} and t_{2g}^{3}e_{g}^{1}. These two configurations differ in LFSE by Δ_{o}, and so it would be predicted that the t_{2g}^{4} configuration would be lower in energy. However, this configuration involves the pairing of electrons in one of the t_{2g} orbitals, and the extra repulsion associated with paired electrons, with energy P, in the same orbital acts to destabilize the t_{2g}^{4} configuration.
We need to consider the overall stabilization energy, SE, which is the ligand field stabilization energy, LFSE, plus the pairing energy, PE.
Overall stabilization energy of a d^{4} complex | |
t_{2g}^{4}: | SE = LFSE + PE = -1.6Δ_{o} + P |
t_{2g}^{3}e_{g}^{1}: | SE = LFSE + PE = -0.6Δ_{o} + 0 = -1.6Δ_{o} + Δ_{o} |
The configuration adopted therefore depends upon the relative magnitude of the splitting parameter, Δ_{o}, and the pairing energy, P. If Δ_{o}<P, then the upper e_{g} orbital is occupied to minimize the pairing energy, whereas if Δ_{o}>P, the lower t_{2g} orbital is occupied to maximize the LFSE. P does not change, for a given element, and so the configuration is determined by the value of Δ_{o}. The first situation, with configuration t_{2g}^{3}e_{g}^{1} is known as the weak-field limit, and the second, with configuration t_{2g}^{4}, is known as the strong-field limit.
If we consider the MO diagrams for the two d^{4} complexes, we see that in the weak-field limit, all the electron spins are parallel, and the overall electron spin is 2 In the strong-field limit, two of the electrons are paired, and hence have antiparallel spins, so the overall electron spin is 1. When there is a choice of possible electronic configurations, the configuration with the lowest number of parallel electron spins is known as the low-spin configuration, and it corresponds to the strong field, and the configuration with the highest number of parallel electron spins is known at the high-spin configuration, and it corresponds to the weak-field limit.
Similar arguments can be constructed for d^{5}, d^{6}, and d^{7} complexes, but for d^{8}, d^{9}, and d^{10}, there is again only one possible configuration.
Ligand field transitions occur when an electron is excited from an orbital with one energy to an orbital with another energy. One example is the t_{2g}-to-e_{g} transition from which the LFSE, Δ_{o}, may be calculated. These will sometimes involve a change in the electron spin, and hence have an effect on the magnetic properties if the complex: the magnetic properties of the complex are determined by the number of unpaired electrons.