discuss the electromagnetic spectrum and see how the different forms of spec-
troscopy relate to the energies of photons.
Rotational spectroscopy arises from the quantized rotations of molecules in
three-dimensional space. Atoms do not have rotational spectra. However, di-
atomic molecules have a relatively simple rotational spectrum, since they can
rotate in only two dimensions (a “rotation” about the internuclear axis will not
be observed) and their behavior of rotation is the same for both directions.
Nonlinear polyatomic molecules have one (for highly symmetric molecules) to
three (for most, less-symmetric species) different rotations in space, compli-
cating a rotational spectrum.
Vibrational spectroscopy arises from the vibrations of the individual atoms in
molecules with respect to each other. The atoms are stretching, bending, or twist-
ing about an imagined equilibrium position. Usually such motions occur at higher
frequency than rotations, so vibrational spectra are observed using higher-energy
light than in rotational spectroscopy. As with rotations, individual atoms do not
have vibrational spectra, because two or more atoms bonded together are re-
quired for a vibration. Like rotations, vibrations occur in certain patterns. However,
the symmetry of the molecules has a greater influence on the number of transi-
tions observed in a vibrational spectrum. Transitions between vibrational energy
levels also follow selection rules, which are different from those for rotations.
Spectroscopy is a powerful tool for studying matter. The treatment in this
text cannot do the topic justice—series of books are written on just the topics
in the next three chapters. However, the following material should give you
some idea of how spectroscopy aids our understanding of atoms and molecules.14.2 Selection Rules
In spectroscopy, an atomic or molecular system having a certain wavefunction
and energy absorbs or emits energy, usually in the form of light, and in doing
so becomes described by a different wavefunction and energy. In all forms of
spectroscopy, it is the difference in energies of the wavefunctions that is the
primary observable of interest. (Hence its overwhelming importance in quan-
tum mechanics.) The law of conservation of energy requires that the light,
usually in the form of a single photon, have the same energy as the difference
in energy of the initial and final states. That is,
E(final) E(initial) Ephoton
Eh (14.1)
Equation 14.1 is written in the original form of the Bohr frequency condition:
the difference in energy of the two quantum states equals the energy of the
photon, which equals h.
Remember, however, that wavefunctions have symmetry, and so do opera-
tors. The light that causes the system to go from one state to another (either
by absorption or emission) can be assigned an irreducible representation from
the point group of the system of interest. Quantum mechanics defines a spe-
cific expression, called a transition moment,to which the irreducible represen-
tations can be applied. For an absorption or emission of a photon, the transi-
tion moment Mis defined asM*finalˆinitiald (14.2)
ˆˆx ˆy ˆz
# ofei(xi yi zi)
particles462 CHAPTER 14 Rotational and Vibrational Spectroscopy