Why are d orbitals divided into a set of five?

Why are d orbitals divided into a set of five?

In quantum mechanics, electrons in atoms reside in regions of space called orbitals, which are categorized by the angular momentum quantum number ($l$). The type of orbital is denoted by letters: $s$ ($l=0$), $p$ ($l=1$), $d$ ($l=2$), and $f$ ($l=3$).

The Quantum Mechanical Origin

For any given subshell with an angular momentum quantum number $l$, the number of individual orbitals is determined by the magnetic quantum number ($m_l$). The value of $m_l$ can range from $-l$ to $+l$, including zero.

For $d$ orbitals, $l = 2$. Therefore, the possible values for $m_l$ are:

$$\text{m}_l = -2, -1, 0, +1, +2$$

Counting these up, there are $2(2) + 1 = 5$ distinct integer values. Each of these values corresponds to a unique quantum state and a specific spatial orientation of the orbital in three-dimensional space.

Spatial Orientation and Shapes

Because $l = 2$ defines the $d$ subshell, the electron wavefunctions yield five geometrically distinct shapes:

1. $d_{xy}$
2. $d_{yz}$
3. $d_{xz}$
4. $d_{x^2 - y^2}$
5. $d_{z^2}$

The first four consist of four-lobed cloverleaf shapes lying between the Cartesian axes, while the fifth ($d_{z^2}$) has a unique dumbbell shape with a donut-like ring around its center. These five orientations allow up to 10 electrons (two per orbital with opposite spins) to occupy the $d$ subshell while minimizing electrostatic repulsion between them.