# How many d orbitals can there be in one energy level?

Jul 20, 2016

This depends on the number of magnetic quantum numbers ${m}_{l}$ that correspond to a single angular momentum quantum number $l$.

• $n$ is the principal quantum number (the energy level) going as $1 , 2 , 3 , . . . , N$, and $N$ is a large integer. For one energy level, $n$ is only one number at a time.
• $l$ is the angular momentum quantum number, and it goes as $0 , 1 , 2 , . . . , n - 1$. For one subshell, $l$ is only one number at a time, and $l = 0 , 1 , 2 , 3 , 4 . . .$ corresponds to the $s , p , d , f , g , . . .$ subshells. However, there can be more than one $l$ for the same energy level.
• ${m}_{l}$ is the magnetic quantum number, and takes on all numbers in the set $\left\{- l , - l + 1 , . . . , 0 , . . . , + l - 1 , + l\right\}$. It corresponds to how many orbitals are in a subshell.

For an energy level to validly have $d$ orbitals, $n \ge 3$. Any smaller $n$, and the number of $d$ orbitals is $0$.

For any $n$, $l \le \left(n - 1\right)$ is allowed, and for $d$ orbitals, $l = 2$ (of course, $2 = 3 - 1$, so that's why $n \ge 3$ for $d$ orbitals). Therefore, what we have for ${m}_{l}$ is:

$\textcolor{b l u e}{{m}_{l} = \left\{- 2 , - 1 , 0 , + 1 , + 2\right\}}$

So, there are $2 l + 1 = \setminus m a t h b f \left(5\right)$ different (but degenerate, same in energy) $d$ atomic orbitals in the same $n d$ subshell.

Example

$3 {d}_{x y}$, $3 {d}_{x z}$, $3 {d}_{y z}$, $3 {d}_{{x}^{2} - {y}^{2}}$, $3 {d}_{{z}^{2}}$: