Because of the energy involved in vibrational transitions, almost all vibra-
tional spectra measured at room temperature are probing transitions of vibra-
tions from a v0 lowest-energy ground stateto the v1 first excited state.
Such transitions are called the fundamental vibrational transitions.In some
cases, higher vibrational states are significantly populated due to thermal en-
ergy, either because the vibrational energy state itself is small or the tempera-
ture is large. Under such conditions, transitions like v 1 →v2 or higher
vibrational levels can be probed. Such absorptions are called hot bands.
If the normal mode is acting as an ideal harmonic oscillator, then we can
use the quantum-mechanical expressions that describe its energy. Recall that
for an ideal harmonic oscillator,
E(v) h(v ^12 )
where vis the vibrational quantum number,is the classical frequency of the
oscillator, and his Planck’s constant. Therefore, it is easy to show that the
change in energy between adjacent energy levels is
E(v 1) E(v) Eh (14.34)
A vibrational spectrum is composed of absorptions that correspond to h,
where is the classical frequency of the vibration. Note the very broad applic-
ability of equation 14.34: it is independent of the quantum number v.For an
ideal harmonic oscillator, the allowed transitions occurring for any one normal
mode all have the same E, and so all will absorb the same frequency of light.
If a molecule shows transitions for v2,3,...,n, then it is easy to
show that for an n-fold change in the quantum number v,
Enh (14.35)
Changes in vibrational energy should be exact multiples of the v1 transi-
tion. However, real normal vibrations are not ideal (which is why such transi-
tions are observed occasionally in the first place), so absorptions due to over-
tone transitions are usually less than an integral number ofh. This deviation
is a measure of anharmonicity, which we will consider in the next section.
Table 14.3 lists the absorptions due to the fundamental and overtone vibra-
tional transitions for HCl (g). Also listed are the various multiples of the fun-
damental vibrational frequency, and the variance from the multiple as shown
by experiment. Note how the overtone absorptions get farther and farther
from ideal. The fact that v1 is possible (although to a much lesser extent
than v1) and the variance from exact multiples of the fundamental vi-
brational frequency are both reminders that molecules are not true harmonic
oscillators. They are anharmonic oscillators. The use of the ideal harmonic
oscillator system in describing molecular vibrations is an approximation—but
a good approximation.490 CHAPTER 14 Rotational and Vibrational Spectroscopy
Table 14.3 Fundamental and overtone vibrational absorptions of HCl (g)
Transition Frequency (cm^1 ) Fundamental multiple Variance
v 0 →v1 2,885.98 — —
v 0 →v2 5,667.98 2(2,885.98) 5,771.96 103.98
v 0 →v3 8,346.78 3(2,885.98) 8,657.94 311.16
v 0 →v4 10,922.81 4(2,885.98) 11,543.92 621.11
v 0 →v5 13,396.19 5(2,285.98) 14,429.90 1,033.71
Source:R. J. Sime,Physical Chemistry: Methods, Techniques, and Experiments,Saunders, Philadelphia, 1990. Referenced
there as D. H. Rank, D. P. Eastman, B. S. Rao, and T. A. Wiggins,J. Opt. Soc. Am.,1962, 52: 1–7.