Atomic Orbital - Shapes of Orbitals

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, however, 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 anywhere in space. Instead the diagrams are approximate representations of boundary or contour surfaces where the probability density |ψ(r,θ,φ)|2 has a constant value, chosen so that there is a certain probability (for example 90%) of finding the electron within the contour. Although |ψ|2 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 will be graphed to show its phases, rather than the |ψ(r,θ,φ)|2 which shows probability density but has no phases (which have been lost in the process of taking the absolute value, since ψ(r,θ,φ) is a complex number). |ψ(r,θ,φ)|2 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, in order to show wave function phases, shows mostly ψ(r,θ,φ) graphs.

The lobes can be viewed as interference patterns between the two counter rotating "m" and "-m" modes, with the projection of the orbital onto the xy plane having a resonant "m" wavelengths around the circumference. For each m there are two of these +<-m> and -<-m>. For the case where m=0 the orbital is vertical, counter rotating information is unknown, and the orbital is z-axis symmetric. For the case where l=0 there are no counter rotating modes. There are only radial modes and the shape is spherically symmetric. For any given n, the smaller l is, the more radial nodes there are. Loosely speaking n is energy, l is analogous to eccentricity, and m is orientation.

Generally speaking, the number n determines the size and energy of the orbital for a given nucleus: as n increases, the size of the orbital increases. However, in comparing different elements, the higher nuclear charge, Z, of heavier elements causes their orbitals to contract by comparison to lighter ones, so that the overall size of the whole atom remains very roughly constant, even as the number of electrons in heavier elements (higher Z) increases.

Also in general terms, determines an orbital's shape, and its orientation. However, since some orbitals are described by equations in complex numbers, the shape sometimes depends on also.

The single s-orbitals are shaped like spheres. For n = 1 the sphere is "solid" (it is most dense at the center and fades exponentially outwardly), but for n = 2 or more, each single s-orbital is composed 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).

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"). 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 .

Four of the five d-orbitals for n = 3 look similar, each with four pear-shaped lobes, each lobe tangent to two others, and the centers of all four lying in one plane, between a pair of axes. Three of these planes are the xy-, xz-, and yz-planes, and the fourth has the centres on the x and y axes. The fifth and final d-orbital consists of three regions of high probability density: a torus with two pear-shaped regions placed symmetrically on its z axis.

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

For each s, p, d, f and g set of orbitals, the set of orbitals which composes it forms a spherically symmetrical set of shapes. For non-s orbitals, which have lobes, the lobes point in directions so as to fill space as symmetrically as possible for number of lobes which exist for a set of orientations. For example, the three p orbitals have six lobes which are oriented to each of the six primary directions of 3-D space; for the 5 d orbitals, there are a total of 18 lobes, in which again six point in primary directions, and the 12 additional lobes fill the 12 gaps which exist between each pairs of these 6 primary axes.

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_x, p_y, and p_z are the same shape.