Shapes of Atomic Orbitals

The probability of finding an electron at a given point in an atom is proportional to $\psi^2$ at that point — a quantity Max Born interpreted as the probability DENSITY of the electron at that point. This can be pictured in two related ways. A probability-density (or 'electron cloud') diagram shades a region more densely where $\psi^2$ is larger, giving the orbital a cloud-like appearance; the 2s orbital's cloud, for instance, shows one spherical NODE — a thin region of nearly-zero probability density, where the underlying wave function also changes sign — separating an inner and an outer denser region. Alternatively, a boundary-surface diagram draws a single surface enclosing the region within which $\psi^2$ stays roughly constant and the total enclosed probability of finding the electron is typically taken as about 90%; since $\psi^2$ never actually reaches zero at any finite distance from the nucleus, a boundary enclosing 100% of the probability could only be drawn at infinity, so 90% is used as the practical convention. Under this convention, s orbitals (1s, 2s, ...) are simple spheres centred on the nucleus, growing larger as n increases — meaning the electron is, on average, found farther from the nucleus as n grows. The three p orbitals of a given shell (e.g. $2p_x$, $2p_y$, $2p_z$) are each dumbbell-shaped, with two lobes lying on opposite sides of a nodal plane that passes through the nucleus (a plane where $\psi^2$ is close to zero); the three p orbitals of one shell are identical in size and energy, differing only in which axis their lobes point along. The five d orbitals of a given shell ($d_{xy}$, $d_{yz}$, $d_{xz}$, $d_{x^2-y^2}$, $d_{z^2}$) have more elaborate shapes — four of them are four-lobed (clover-leaf) shapes oriented differently i …