Shapes of s, p, and d Orbitals

Crystal- and ligand-field theory in coordination chemistry is built entirely on the real $d$-orbital shapes: which lobes point directly at ligands ($d_{z^2}, d_{x^2-y^2}$, the $e_g$ set in octahedral geometry) versus between them ($d_{xy}, d_{xz}, d_{yz}$, the $t_{2g}$ set) determines electrostatic/covalent destabilization patterns and hence color, magnetism, and reactivity of transition-metal complexes used in catalysis, MRI contrast agents, and battery cathode materials. Organic pericyclic reaction selection rules (Woodward–Hoffmann) are direct bookkeeping of $p$-orbital sign patterns around a ring during bond reorganization. Scanning tunneling microscopy and orbital-resolved photoemission spectroscopy now image molecular orbital lobes and their sign changes directly on surfaces, providing literal experimental pictures of exactly the shapes derived from $Y_l^m$. Density functional theory codes used across drug design and materials discovery still expand molecular orbitals in real atomic-orbital basis sets constructed as these exact axis-aligned $s,p,d$ combinations (Gaussian-type or Slater-type orbitals built on the real spherical harmonics), making this recombination the computational backbone of essentially all electronic-structure software in use in chemistry and materials science.

Ask the tutor: How exactly do these same real $p$ and $d$ shapes get reused, with an added radial shielding twist, when moving from hydrogen to multi-electron atoms?

Every hydrogen-like orbital is the product of a radial function and an angular function,

The radial function $R_{nl}(r)$ depends on the principal quantum number $n$ and angular momentum quantum number $l$; it controls the size of the orbital and how many concentric spherical nodes it has. The angular function $Y_l^m(\theta,\phi)$ depends only on $l$ and the magnetic quantum number $m$; it controls the shape — the number of lobes, their orientation, and the location of angular (planar or conical) nodes. Crucially, $Y_l^m$ does not depend on $n$ at all. A $2p_z$, $3p_z$, and $4p_z$ orbital are all shaped like an identical dumbbell along $z$; they differ only in radial size and radial nodes. The complex spherical harmonics that solve $\hat L^2$ and $\hat L_z$ simultaneously are, for $l=1$,

$Y_1^0$ is already real and becomes $p_z$ directly. But $Y_1^{+1}$ and $Y_1^{-1}$ are complex, and neither one individually looks like the familiar $p_x$ or $p_y$ dumbbell. Since $Y_1^{+1}$ and $Y_1^{-1}$ are degenerate eigenstates of the same energy (both have $l=1$, same $n$), any linear combination of them is an equally valid stationary state. Chemists pick the two combinations that are real and point along the $x$- and $y$-axes:

This is the origin of every "orbital shape" picture in general chemistry: not a direct plot of an eigenfunction of $\hat L_z$, but a real recombination of a degenerate pair, engineered to align with Cartesian axes because that is what overlaps cleanly with neighboring atoms in a molecule or a ligand field. The same recombination trick, applied to the five $l=2$ harmonics, produces the five real $d$ orbitals: $d_{z^2}$, $d_{xz}$, $d_{yz}$, $d_{xy}$, and $d_{x^2-y^2}$. Four of them (all but $d_{z^2}$) have the four-lobed cloverleaf shape with two perpendicular nodal planes; $d_{z^2}$ looks different — a dumbbell along $z$ girdled by a torus — because it is built from $Y_2^0$ alone plus a symmetric combination of $Y_2^{+2}$ and $Y_2^{-2}$ is used for $d_{x^2-y^2}$, leaving $d_{z^2}$ as the odd one out algebraically even though it is fully equivalent in node count and symmetry class.

Reading these shapes correctly means treating the coloring/shading as sign of $\psi$, not electron density (which is $|\psi|^2$, always positive, and looks like the same lobes but without any coloring distinction). The sign matters enormously for bonding: two lobes of the same sign overlapping in space give constructive interference and a bonding interaction; lobes of opposite sign overlapping give destructive interference, a node between the atoms, and an antibonding interaction. This is the entire content of "orbital symmetry" arguments used in explaining $\pi$ bonding, pericyclic selection rules, and crystal-field splitting.

The full hydrogen-atom eigenfunction is

where: $n=1,2,3,\dots$ is the principal quantum number, $l = 0,1,\dots,n-1$ is the orbital angular momentum quantum number, $m = -l,\dots,+l$ is the magnetic quantum number, $R_{nl}(r)$ is the radial function, and $Y_l^m(\theta,\phi)$ is a spherical harmonic. Reading: the wavefunction is a product of a radial "how far" factor and an angular "what shape" factor, and they can be analyzed completely independently. Node counting follows directly from the mathematical form of $R_{nl}$ and $Y_l^m$:

where: $l$ fixes how many nodal planes/cones the angular part has, $n-l-1$ fixes how many nodal spheres the radial part has, and their sum is always $n-1$, independent of $l$. Reading: a $3s$ orbital ($l=0$) has $0$ angular nodes and $3-0-1=2$ radial nodes (two nested spherical nodes); a $3p$ orbital ($l=1$) has $1$ angular node and $3-1-1=1$ radial node; a $3d$ orbital ($l=2$) has $2$ angular nodes and $3-2-1=0$ radial nodes. All three still sum to $n - 1 = 2$. The real $p$ orbitals are built from complex harmonics by

where: $Y_1^0, Y_1^{\pm1}$ are the complex spherical harmonics for $l=1$, and the numerical prefactors $1/\sqrt2$, $i/\sqrt2$ are chosen to keep each combination normalized and real. Reading: only the $m=0$ harmonic is already real (hence $p_z$ needs no recombination); the $m=\pm1$ pair must be added and subtracted, with an $i$ inserted in one combination to cancel the residual imaginary phase, in order to produce two real functions that point along $x$ and $y$. The real $d$ orbitals follow the same logic for $l=2$:

where: each pair $(Y_2^{-m}, Y_2^{+m})$ combines into two real, mutually perpendicular cloverleaf orbitals. Reading: five real $d$ orbitals emerge from five complex ones in exactly the same one-real-plus-two-recombined-pairs pattern as the $p$ set, generalized to $m = 0, \pm1, \pm2$.

Example 1 — Foundational: node count and sketch for the $4d_{xz}$ orbital. Identify $n=4$, $l=2$ (all $d$ orbitals have $l=2$). Angular nodes: $l = 2$. Radial nodes: $n-l-1 = 4-2-1 = 1$. Total nodes: $n-1=3$, and indeed $2+1=3$. ✓ The angular part is the same cloverleaf shape as every $d_{xz}$ orbital of any $n$ (four lobes, alternating sign, lying in the $xz$-plane, nodal at $x=0$ and $z=0$ planes — i.e., nodes along the $y$-axis direction and along the plane bisecting $x$ and $z$... more precisely $d_{xz}\propto xz/r^2$, vanishing whenever $x=0$ (the $yz$-plane) or $z=0$ (the $xy$-plane), giving the required two angular nodal planes). Additionally, because $n-l-1=1$, there is one spherical radial node: a sphere of some radius $r_0>0$ at which the radial wavefunction changes sign, nested inside the outer lobes, invisible in the usual "clover" cartoon but very real in a proper contour plot and directly responsible for the subtle differences between $3d$ and $4d$ radial distribution functions used in transition-metal spectroscopy.

Example 2 — Applied: sign matching in a $\pi$-bond between two $2p_z$ orbitals of a hypothetical asymmetric diatomic, X–Y, with unequal orbital coefficients. Consider forming a $\pi$ molecular orbital from a carbon $2p_z$ and an adjacent nitrogen $2p_z$ (as in an imine C=N unit), with LCAO coefficients $c_{\rm C}=0.62$ and $c_{\rm N}=0.79$ (unequal, since nitrogen is more electronegative and contributes more to the bonding MO):

Both atomic $2p_z$ orbitals, drawn with the same sign convention (say, "+" lobe pointing up in $+z$ on both atoms, since both atoms sit in the same local frame with $z$ perpendicular to the molecular plane), overlap lobe-to-lobe with matching signs above the internuclear axis and matching signs below it. Because $c_{\rm C}$ and $c_{\rm N}$ are both positive here, the combination is bonding: constructive interference above the plane, constructive below, and a node exactly in the nodal plane of each $p_z$ (the molecular plane itself) — consistent with $\pi$ symmetry, which always has a nodal plane containing the internuclear axis. Now flip the relative phase, forming the antibonding combination:

(coefficients swap in magnitude for the antibonding partner, as required by orthogonality to $\psi_\pi$, with nitrogen — the more electronegative, lower-energy AO — now contributing less to this higher-energy MO). Here the two $2p_z$ lobes point oppositely at the internuclear midpoint, producing an additional node perpendicular to the axis, between the two nuclei, on top of the original in-plane node — the signature of any antibonding MO: one more node than its bonding partner. Orthogonality can be checked directly:

in the simplified case of unit overlap $S=1$ along the shared axis assumption used for illustration (a genuine LCAO treatment uses the true overlap integral $S<1$, but the sign cancellation pattern is the chemically important point here).

Example 3 — Exam/competition level: how many nodal cones does $Y_3^{1}$ (an $f$-orbital angular function) have, and what is $d_{z^2}$'s node arithmetic by contrast? For $l=3, m=1$: total angular nodes $=l=3$. Of these, $|m|=1$ nodal plane comes from the $\phi$-dependence ($\cos\phi$ vanishing along one plane through the $z$-axis), leaving $l - |m| = 3-1=2$ conical nodes from the $\theta$-dependence (two cones of half-angle $\theta_1<\theta_2$ symmetric about the $z$-axis, on which $P_3^1(\cos\theta)=0$). Total: $1+2=3=l$. ✓ Now compare $d_{z^2}$ ($l=2, m=0$): by the same accounting, $|m|=0$ contributes zero planar nodes, so all $l-|m|=2$ angular nodes are conical — and indeed $d_{z^2}$'s angular node structure is famously two cones (not a plane at all), which is exactly why its shape (dumbbell + torus) looks so different from the four-lobed clover of $d_{xz}, d_{yz}, d_{xy}, d_{x^2-y^2}$, even though all five $d$ orbitals share $l=2$ and therefore exactly 2 angular nodes. The apparent "special" look of $d_{z^2}$ is not a violation of any rule — it is simply the $m=0$ member of the same $l=2$ family, with both of its angular nodes taking conical rather than planar form. This distinction is exactly why crystal-field diagrams always draw $d_{z^2}$ separately from the other four $d$ orbitals even while treating all five as degenerate in a free ion.

The real (as opposed to complex) atomic orbital pictures used throughout chemistry crystallized through the work of chemists translating Schrödinger's and Pauli's mathematics into pictures usable at the bench — most visibly in the systematic tabulations and diagrams popularized in the 1920s–30s that fixed the $s,p,d,f$ naming and lobe-shape conventions still taught today.