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

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

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Answer 1 · 2016-07-20T19:37:53

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

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

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

For any $n$ $n$ , $l \leq (n - 1)$ $l <= (n-1)$ is allowed, and for $d$ $d$ orbitals, $l = 2$ $l = 2$ (of course, $2 = 3 - 1$ $2 = 3 - 1$ , so that's why $n \geq 3$ $n >= 3$ for $d$ $d$ orbitals). Therefore, what we have for $m_{l}$ $m_l$ is:

$m_{l} = {- 2, - 1, 0, + 1, + 2}$ $color(blue)(m_l = {-2,-1,0,+1,+2})$

So, there are $2 l + 1 = 5$ $2l+1 = \mathbf(5)$ different (but degenerate, same in energy) $d$ $d$ atomic orbitals in the same $n d$ $nd$ subshell.

Example

$3 d_{x y}$ $3d_(xy)$ , $3 d_{x z}$ $3d_(xz)$ , $3 d_{y z}$ $3d_(yz)$ , $3 d_{x^{2} - y^{2}}^{2}$ $3d_(x^2-y^2)$ , $3 d_{z^{2}}^{2}$ $3d_(z^2)$ :

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

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