**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.

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)^{2}against 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)r^{2}(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.

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 2p_{x}, 2p_{y}, and 2p_{z}. 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.

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.

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