Shapes of s, p and d Orbitals

The azimuthal quantum number $l$ fixes a characteristic boundary-surface shape that is the same for every orbital of a given subshell type, in every atom -- only the size of the orbital changes with the principal quantum number $n$.

s orbitals ($l = 0$). Every $s$ orbital is spherically symmetric: the probability of finding the electron is the same in every direction from the nucleus, at a given distance. A $1s$ orbital is a single, simple sphere centred on the nucleus. A $2s$ orbital is also spherical overall, but it contains, in addition, a spherical node -- a thin spherical shell on which the probability of finding the electron falls to exactly zero -- lying between an inner, smaller region of high probability and an outer, larger one; a $3s$ orbital, similarly spherical, contains two such nested spherical nodes. In general, an $ns$ orbital has $(n-1)$ radial nodes, but remains spherical throughout.

p orbitals ($l = 1$). Each $p$ subshell has three orbitals, oriented along the three Cartesian axes and denoted $p_x$, $p_y$ and $p_z$. Every $p$ orbital has the same characteristic dumbbell shape: two lobes of high probability, of equal size, lying on either side of the nucleus along its axis, touching at the nucleus itself. Exactly at the nucleus, and everywhere on the plane through the nucleus perpendicular to that axis, the probability of finding the electron is zero -- this is called a nodal plane. The three $p$ orbitals of a given shell are identical in shape and size and differ only in their orientation in space.

d orbitals ($l = 2$). Each $d$ subshell has five orbitals. Four of them -- $d_{xy}$, $d_{yz}$, $d_{xz}$ and $d_{x^2-y^2}$ -- have an identical four-lobed ("cloverleaf") shape: $d_{xy}$, $d_{yz}$ and $d_{xz}$ each have their four lobes lying between a pair of axes, in the plane named by the subscript (for example, $d_{xy}$'s lobes point into the four quadrants between the $x$ and $y$ axes), while $d_{x^2-y^2}$ has its four lobes pointing directly along the $x$ and $y$ axes. The fifth orbital, $d_{z^2}$, looks different from the other four: it has two lobes along the $z$-axis, similar to a $p_z$ orbital, plus an additional donut-shaped ring of electron density (a torus) encircling the nucleus in the $xy$-plane. All five $d$ orbitals of a given shell are equal in energy in an isolated atom (they are said to be degenerate) and differ only in their spatial orientation and, in the case of $d_{z^2}$, in their overall shape. …