Atomic Orbitals s, p, d

Every electron in an atom lives in a wave function $\psi_{n\ell m}(r,\theta,\phi)$ labeled by three integers: the principal quantum number $n$, the azimuthal (angular-momentum) quantum number $\ell$, and the magnetic quantum number $m_\ell$. The shape of the orbital — how the probability cloud is distributed in space — is controlled entirely by $\ell$.

The azimuthal quantum number $\ell$

For a given $n$, $\ell$ takes integer values $0, 1, 2, \ldots, n-1$. The letter names attached to these values come from old spectroscopic notation:

| $\ell$ | Name | Letter | |--------|------|--------| | 0 | sharp | s | | 1 | principal | p | | 2 | diffuse | d | | 3 | fundamental | f |

The magnitude of the orbital angular momentum is $L = \hbar\sqrt{\ell(\ell+1)}$, so an $s$ orbital ($\ell = 0$) carries zero orbital angular momentum; a $p$ orbital carries $L = \hbar\sqrt{2}$; a $d$ orbital carries $L = \hbar\sqrt{6}$.

s orbitals ($\ell = 0$)

When $\ell = 0$ there is only one allowed value of $m_\ell$: it must equal $0$. The corresponding spherical harmonic is the constant $Y_0^0 = 1/\sqrt{4\pi}$, so the angular part of the wave function is the same in every direction. The probability density $|\psi_{n00}|^2$ therefore depends only on the radial coordinate $r$: it is spherically symmetric.

Visually, an $s$ orbital looks like a sphere centered on the nucleus. The $1s$ wave function for hydrogen is

$\psi_{100}(r) = \frac{1}{\sqrt{\pi}}\left(\frac{1}{a_0}\right)^{3/2} e^{-r/a_0},$

where $a_0 \approx 0.0529\,\text{nm}$ is the Bohr radius. Higher $s$ orbitals ($2s$, $3s$, $\ldots$) remain spherically symmetric but develop concentric radial nodes — spherical shells on which the probability density vanishes. The $ns$ orbital has $n-1$ radial nodes.

p orbitals ($\ell = 1$)

For $\ell = 1$, $m_\ell$ can be $-1$, $0$, or $+1$, giving three distinct orbitals per $n$. The real linear combinations used in chemistry are labeled $p_x$, $p_y$, $p_z$ and are formed by taking sums and differences of the complex $m_\ell = \pm 1$ eigenstates. Each of these real combinations is proportional to one Cartesian coordinate divided by $r$: for example,

$p_z \propto \frac{z}{r} \cdot R_{n1}(r) = \cos\theta \cdot R_{n1}(r).$

The angular factor $\cos\theta$ vanishes on the plane $\theta = \pi/2$, i.e. the $xy$-plane. This gives $p_z$ its characteristic two-lobed dumbbell shape aligned along the $z$-axis, with a nodal plane through the nucleus. The $p_x$ and $p_y$ orbitals are identical in shape but oriented along their respective axes.

Every $p$ orbital therefore has one angular (planar) node passing through the nucleus, plus $n-2$ additional radial nodes for shell $n$.

d orbitals ($\ell = 2$)

For $\ell = 2$, $m_\ell \in \{-2,-1,0,1,2\}$, giving five orbitals per $n$. The real combinations carry the labels $d_{xy}$, $d_{xz}$, $d_{yz}$, $d_{x^2-y^2}$, and $d_{z^2}$.

The first four have four-leaf clover shapes: four lobes pointing between or along the Cartesian axes, separated by two nodal planes through the nucleus. For instance, $d_{xy}$ is proportional to $xy/r^2 \propto \sin^2\theta\,\sin 2\phi$ and has nodal planes $x = 0$ and $y = 0$.

The fifth, $d_{z^2}$, is the odd one out: it has two large lobes along $z$ plus a torus-shaped ring of probability density in the equatorial plane. Its angular factor is proportional to $(3\cos^2\theta - 1)$, which vanishes on two cones at $\theta = \arccos(\pm 1/\sqrt{3}) \approx 54.7°$ and $125.3°$ rather than on flat planes.

All five $d$ orbitals have two angular nodes; together with $n-3$ radial nodes for principal shell $n$ (where $n \geq 3$).

The node-counting rule

The total number of nodes in $\psi_{n\ell m}$ is always $n - 1$, distributed as $\ell$ angular nodes and $n - 1 - \ell$ radial nodes:

$\text{angular nodes} = \ell, \qquad \text{radial nodes} = n - 1 - \ell, \qquad \text{total} = n - 1.$

This pattern generalizes: an $f$ orbital ($\ell = 3$) has three angular nodes, and so on.

Summary

| Orbital | $\ell$ | $m_\ell$ count | Shape | Angular nodes | |---------|--------|----------------|-------|---------------| | $s$ | 0 | 1 | sphere | 0 | | $p$ | 1 | 3 | dumbbell (per axis) | 1 | | $d$ | 2 | 5 | clover / torus | 2 | | $f$ | 3 | 7 | complex multi-lobe | 3 |

The rule is simple: the letter tells you $\ell$, and $\ell$ tells you both the number of angular nodes and the multiplicity $2\ell + 1$ of orientations available within that shell.

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