quality:corruption, linguistic:translation, information:filtering, information:highlighting
Shapes of orbitals

Simple pictures showing orbital shapes are intended to describe the angular forms of regions in space where the electrons occupying the orbital are likely to be found. The diagrams cannot show the entire region where an electron can be found, since according to quantum mechanics there is a non-zero probability of finding the electron (almost) anywhere in space. Instead the diagrams are approximate representations of boundary or contour surfaces where the probability density | ψ(r, θ, φ) | has a constant value, chosen so that there is a certain probability (for example 90%) of finding the electron within the contour. Although | ψ | as the square of an absolute value is everywhere non-negative, the sign of the wave function ψ(r, θ, φ) is often indicated in each subregion of the orbital picture.

Sometimes the ψ function is graphed to show its phases, rather than | ψ(r, θ, φ) | which shows probability density but has no phase (which is lost when taking absolute value, since ψ(r, θ, φ) is a complex number). | ψ(r, θ, φ) | orbital graphs tend to have less spherical, thinner lobes than ψ(r, θ, φ) graphs, but have the same number of lobes in the same places, and otherwise are recognizable. This article, to show wave function phase, shows mostly ψ(r, θ, φ) graphs.

The lobes can be seen as standing wave interference patterns between the two counter-rotating, ring-resonant traveling wave m and − m modes; the projection of the orbital onto the xy plane has a resonant m wavelength around the circumference. Although rarely shown, the traveling wave solutions can be seen as rotating banded tori; the bands represent phase information. For each m there are two standing wave solutions ⟨ m ⟩ + ⟨− m ⟩ and ⟨ m ⟩ − ⟨− m ⟩. If m = 0, the orbital is vertical, counter rotating information is unknown, and the orbital is z -axis symmetric. If ℓ = 0 there are no counter rotating modes. There are only radial modes and the shape is spherically symmetric.

Nodal planes and nodal spheres are surfaces on which the probability density vanishes. The number of nodal surfaces is controlled by the quantum numbers n and ℓ. An orbital with azimuthal quantum number ℓ has ℓ radial nodal planes passing through the origin. For example, the s orbitals (ℓ = 0) are spherically symmetric and have no nodal planes, whereas the p orbitals (ℓ = 1) have a single nodal plane between the lobes. The number of nodal spheres equals n-ℓ-1, consistent with the restriction ℓ ≤ n-1 on the quantum numbers. The principal quantum number controls the total number of nodal surfaces which is n-1. Loosely speaking, n is energy, ℓ is analogous to eccentricity, and m is orientation.

In general, n determines size and energy of the orbital for a given nucleus; as n increases, the size of the orbital increases. The higher nuclear charge Z of heavier elements causes their orbitals to contract by comparison to lighter ones, so that the size of the atom remains very roughly constant, even as the number of electrons increases.

Also in general terms, ℓ determines an orbital's shape, and m its orientation. However, since some orbitals are described by equations in complex numbers, the shape sometimes depends on m also. Together, the whole set of orbitals for a given ℓ and n fill space as symmetrically as possible, though with increasingly complex sets of lobes and nodes.

The single s orbitals (ℓ = 0 {\displaystyle \ell =0}) are shaped like spheres. For n = 1 it is roughly a solid ball (densest at center and fades outward exponentially), but for n ≥ 2, each single s orbital is made of spherically symmetric surfaces which are nested shells (i.e., the "wave-structure" is radial, following a sinusoidal radial component as well). See illustration of a cross-section of these nested shells, at right. The s orbitals for all n numbers are the only orbitals with an anti-node (a region of high wave function density) at the center of the nucleus. All other orbitals (p, d, f, etc.) have angular momentum, and thus avoid the nucleus (having a wave node at the nucleus). Recently, there has been an effort to experimentally image the 1s and 2p orbitals in a SrTiO crystal using scanning transmission electron microscopy with energy dispersive x-ray spectroscopy. Because the imaging was conducted using an electron beam, Coulombic beam-orbital interaction that is often termed as the impact parameter effect is included in the outcome (see the figure at right).

The shapes of p, d and f orbitals are described verbally here and shown graphically in the Orbitals table below. The three p orbitals for n = 2 have the form of two ellipsoids with a point of tangency at the nucleus (the two-lobed shape is sometimes referred to as a " dumbbell "—there are two lobes pointing in opposite directions from each other). The three p orbitals in each shell are oriented at right angles to each other, as determined by their respective linear combination of values of m. The overall result is a lobe pointing along each direction of the primary axes.

Four of the five d orbitals for n = 3 look similar, each with four pear-shaped lobes, each lobe tangent at right angles to two others, and the centers of all four lying in one plane. Three of these planes are the xy-, xz-, and yz-planes—the lobes are between the pairs of primary axes—and the fourth has the center along the x and y axes themselves. The fifth and final d orbital consists of three regions of high probability density: a torus in between two pear-shaped regions placed symmetrically on its z axis. The overall total of 18 directional lobes point in every primary axis direction and between every pair.

There are seven f orbitals, each with shapes more complex than those of the d orbitals.

Additionally, as is the case with the s orbitals, individual p, d, f and g orbitals with n values higher than the lowest possible value, exhibit an additional radial node structure which is reminiscent of harmonic waves of the same type, as compared with the lowest (or fundamental) mode of the wave. As with s orbitals, this phenomenon provides p, d, f, and g orbitals at the next higher possible value of n (for example, 3p orbitals vs. the fundamental 2p), an additional node in each lobe. Still higher values of n further increase the number of radial nodes, for each type of orbital.

The shapes of atomic orbitals in one-electron atom are related to 3-dimensional spherical harmonics. These shapes are not unique, and any linear combination is valid, like a transformation to cubic harmonics, in fact it is possible to generate sets where all the d's are the same shape, just like the p, p, and p are the same shape.

Although individual orbitals are most often shown independent of each other, the orbitals coexist around the nucleus at the same time. Also, in 1927, Albrecht Unsöld proved that if one sums the electron density of all orbitals of a particular azimuthal quantum number ℓ of the same shell n (e.g., all three 2p orbitals, or all five 3d orbitals) where each orbital is occupied by an electron or each is occupied by an electron pair, then all angular dependence disappears; that is, the resulting total density of all the atomic orbitals in that subshell (those with the same ℓ) is spherical. This is known as Unsöld's theorem.