Shapes of Atomic Orbitals

James Richard Fromm

The quantum-mechanical treatment of electrons in atoms gives a clear picture of the energy levels associated with every atomic orbital which can be defined by a set of four quantum numbers. The question of where in space the electron may be is a somewhat more ambiguous question and receives a somewhat more ambiguous answer. Solutions of the Schroedinger equation correspond to waves rather than to particles, and at least in principle all electromagnetic waves extend throughout all space! This interesting result is not helpful in locating the physical electron on a given atom.

The location of an electron can be described in either of two equivalent ways using the quantum-mechanical results. If the electron is visualized as a real very small object moving very rapidly, then the space it occupies can be described in terms of the probability of finding the electron at a given point or within a given space at any instant. If the desired probability is set at 99%, or 95%, a physical space occupied by the electron can be calculated on this time-average basis. If, on the other hand, the electron is visualized as an electromagnetic wave, then the amplitude of that wave, or the wave function, will be greater at some locations than at others. The space the electron occupies can be considered to be the space within which the amplitude of its wave function is greater than 1%, or 5%, of its maximum amplitude. Again, a three-dimensional space "containing" the electron will be defined.

The electron can be described equally well in either way and the three-dimensional spaces defined as "containing" the electron as a wave and as a particle are the same. Chemists find it convenient to describe the location of electrons in atoms and molecules in terms of this type of shape. These representations of orbital shapes are those which are said to contain 99% or 95% of the electron density of the orbital.

In a previous section, the shapes of the orbitals were assumed to be spherical and atomic and ionic radii could be calculated on that basis. In the quantum-mechanical treatment of atoms, ions, and molecules, many of the orbitals are not found to be spherical in shape. Moreover, the different orbitals on the same atom do interpenetrate each other and the electronic structure of the outer atom is to some extent a composite of several orbitals. For these reasons the atomic radius and ionic radius are now viewed as useful empirical measurements of the sizes of atoms and ions rather than as properties with fundamental significance. Even so, practicing chemists still learn much about the structures of compounds using molecular models made up of scale models of spherical atoms.

The values of all four quantum numbers influence the location of an electron, or in the terminology just introduced the distribution of electron density in space or the shape of an orbital, but the effects of the four different numbers are not the same. The principal quantum number n affects primarily the size of the orbital and has a lesser influence on its shape. The subshell quantum number l affects primarily the shape of the orbital. The magnetic quantum number m affects primarily the orientation of the orbital in three-dimensional space. The spin quantum number s has little effect upon the location of the orbitals of an isolated atom, but does have an influence on orbital interactions when the orbitals of different atoms impinge upon each other.

s Orbitals

Orbitals with subshell quantum number l = 0 are called s orbitals. All s orbitals are spherical in shape and have spherical symmetry. This means that the wave function will depend only on the distance from the nucleus and not on the direction. A plot of the wave function (psi)2against radial distance from the nucleus is shown for hydrogen s orbitals in the Figure below. This plot, which represents the relative probability of finding the electron per unit volume, is difficult to visualize physically because there is much less volume close to the nucleus than there is further out. It is easier to visualize where the electron may be by plotting the radial probability density 4(pi)r2(psi)2, rather than (psi)2, against the distance from the nucleus. In any atom, the size of the s orbital increases as the principal quantum number of the orbital increases but the geometry remains spherical. The electron density also tends to extend further. Other orbitals behave in the same way as the principal quantum numbers of the orbitals increase.

p Orbitals

Orbitals with subshell quantum number l = 1 are called p orbitals. Since the magnetic quantum number m can be -1, 0, or +1 when the value of the subshell quantum number l is one, p orbitals come in sets of three. In each set, one of the orbitals is aligned along each of the three mutually perpendicular axes of the atom; these axes are traditionally designated x, y, and z. The three 2p orbitals are correspondingly designated 2px, 2py, and 2pz. The p orbitals either as a set or individually do not have spherical symmetry and so a simple plot of radial probability density cannot be made for them. If, however, the distance from the nucleus is taken along any one of the three axes and the orbital is that along the same axis, then a suitable plot can be made.

d Orbitals

Orbitals with subshell quantum number l = 2 are called d orbitals; since m can be -2, -1, 0, +1, or +2 when l is two, d orbitals come in sets of five. The d orbitals, and the more complex f orbitals, are usually visualized in three-dimensional representations, even if these have to be shown on a two-dimensional page.

f Orbitals

Orbitals with subshell quantum number l = 3 are called f orbitals. These orbitals are found only in the lanthanide and actinide elements. Since m can be -3, -2, -1, 0, +1, +2, or +3 when l has the value 3, f orbitals come in sets of seven. The f orbitals are rarely of direct chemical interest because they tend to be buried deep within the electronic cloud of an atom, but they do play a major role in the spectroscopy of the lanthanides and actinides. The f orbitals are the most complex orbitals with which most chemists are concerned, even though 5g orbitals, with quantum number l = 4, are known to exist.

Copyright 1997 James R. Fromm