Orbitals

Intuition

A hydrogen state is a radial function times a spherical harmonic, and its probability density is a cloud in space. s orbitals are spherical; p orbitals are two lobes on either side of a nodal plane; d orbitals have four lobes or two lobes and a ring. Chemists draw the real combinations — $p_{x}$ , $p_{y}$ , $p_{z}$ — because they point along axes and form bonds, while physicists often use the states of definite $m$ . Each orbital has $n-1$ nodal surfaces in all: $n-\ell-1$ spheres from the radial part and $\ell$ from the angular part.

A standing wave on a drum has lines where the skin does not move; an orbital has surfaces where the electron is never found. Counting them is counting the wave’s wiggles in each direction.

Contours of the density of the $2p_{z}$ orbital in a plane containing the $z$ axis, which points up, with lengths in Bohr radii: two lobes, densest at $2a_{0}$ above and below the nucleus, and a nodal plane between them where the electron is never found.

A hydrogen orbital is $\psi_{n\ell m}=R_{n\ell}(r)\,Y_{\ell m}(\theta,\varphi)$ . Real combinations of , and of , give orbitals that point along axes.

\begin{aligned}p_{z}&\propto r\cos\theta\,e^{-r/2}=z\,e^{-r/2}\\ p_{x}&\propto\tfrac{1}{\sqrt2}\bigl(Y_{1,-1}-Y_{11}\bigr)\propto x\,e^{-r/2}\end{aligned}

Properties

$p_{x}$ , $p_{y}$ and $p_{z}$ are , and times : the same shape pointing along three axes.

Counting nodal surfaces

\begin{aligned}&\psi_{n\ell m}=R_{n\ell}Y_{\ell m}\ \Longrightarrow\\ &(n-\ell-1)+\ell=n-1\ \text{nodal surfaces}\end{aligned}

A product vanishes where either factor does. The radial function vanishes on $n-\ell-1$ spheres. A spherical harmonic of order $\ell$ vanishes on $\ell$ planes and cones through the nucleus, a property of the Legendre functions used here without proof. Together they make $n-1$ .

Proof steps

\psi=R_{n\ell}(r)\,Y_{\ell m}(\theta,\varphi)=0\iff R=0\ \text{or}\ Y=0

A product vanishes where one of its factors does.

R_{n\ell}=0\ \text{on}\ n-\ell-1\ \text{spheres}

The polynomial of degree $n-\ell-1$ has that many positive roots.

Y_{\ell m}=0\ \text{on}\ \ell\ \text{planes or cones}

$\ell-|m|$ cones of constant $\theta$ and $|m|$ planes of constant $\varphi$ for the real orbitals — quoted from the theory of Legendre functions.

(n-\ell-1)+\ell=n-1

The total depends only on $n$ .

Applications

Practice