How to know how many electrons are there in $T_{2g}$ and $E_g$ orbitals?

I think rather than a broad answer, an example would help. So, consider the compound $\ce{Ni(CO)4}$ and how to calculate CFSE for it?


Firstly, what geometry does the complex have?

Once you know the geometry, find the spin-state of the $d$-electrons, as well as the number of $d$-electrons applicable.

Find the crystal field diagram relevant to your geometry. Fill in the $d$-electrons according to the Aufbau principle and to Pauli's exclusion principle.

For your example, tetracarbonylnickel has a tetrahedral geometry and is a $d$10 compound. Carbon monoxide is a strong field ligand causing a high spin complex (note that since $d$ = 10, all of the $t_{2}$ and $e$ orbitals are full and hence spin-state is irrelevant).

As per the Aufbau principle, you start building up from the bottom, placing two electrons in the bottom $t_2$ orbitals and three in the top three $e$ orbitals, giving:

$e$: ↑ ↑ ↑

$t_2$: ↑ ↑

The remaining five electrons go in each half-filled orbital to give 2 filled $t_2$ orbitals and 3 filled $e$ orbitals:

$e$: ↑↓ ↑↓ ↑↓

$t_2$: ↑↓ ↑↓

To find the CSFE, the $t_2$ orbitals each stabilise by $\frac{3}{5}$ and the $e$ destabilize by $\frac{2}{5}$. Since $2 * \frac{3}{5} - 3 * \frac{2}{5}$ = 0, the CFSE = 0.

For octahedral complexes, the $e_g$ and $t_{2g}$ orbital populations can be used analogously to find the stabilisation energy.

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