Computational Chemistry

wavenumbers and intensities. The intensity of a vibration is determined by the
change in dipole moment accompanying the vibration. If a vibrational mode leads
to no change in dipole moment, the mode will, theoretically, not result in absorption
of an IR photon, because the oscillating electric fields of the radiation and the
vibrational mode will be unable to couple. Such a vibrational mode is said to beIR-
inactive, i.e. it should cause no observable band in the IR spectrum. Stretching
vibrations that, because of symmetry, are not accompanied by a change in dipole
moment, are expected to be IR-inactive. These occur mainly in homonuclear
molecules like O 2 and N 2 , and in linear molecules; thus the C/C triple bond stretch
in symmetrical akynes, and the symmetric OCO stretch in carbon dioxide, do not
engender bands in the IR spectrum. For Raman spectroscopy, in which one mea-
sures the scattered rather than the transmitted IR light, the requirement for observ-
ing a vibrational mode is that the vibration occur with a change in polarizability.
Raman spectra are routinely calculable (e.g. by the Gaussian programs [ 36 ]; the IR
and Ramanfrequencies, but not intensities, are the same) along with IR spectra. The
complementarity of IR and Raman spectra can be useful in studying the symmetry
of molecules. A band which should be IR-inactive or at least very weak can in fact
sometimes be seen because of coupling with other vibrational modes; thus the
triple-bond stretch of 1,2-benzyne (o-benzyne, dehydrobenzene, C 6 H 4 ) has been
observed [ 241 ], although it apparently should be accompanied by only a very small
change in dipole moment. Bands like this are expected to be, at best, weak.
As might be expected from the foregoing discussion, the intensity of an IR
normal mode can be calculated from the change in the dipole moment with the
change in geometry accompanying the vibration. The intensity is proportional to
the square of the change in dipole moment with respect to geometry:

I¼constant'

dm

dq

 2

ð 5 : 204 Þ

This can be used to calculate the relative intensities of IR bands (the calculation of
dipole moments is discussed in the next section). One way to calculate the derivative
is to approximate it as a ratio of finite increments (dbecomesD) and calculate the
change in dipole moment with a small change in geometry; there are also analytical
methods for calculating the derivative [ 242 ]. A book has been written on the subject
of vibrational intensities [ 243 ]. It has been reported that at the HF-level calculated IR-
band intensities often differ from experiment by a factor of over 100% but at the MP2
level are typically within 30% of experiment [ 244 ]. Schaefer and coworkers achieved
“Quantitative accord” between theory and experiment for six small molecules
using QCISD, CCSD, and CCSD(T) (Section 5.4.3) with Dunning’s aug-cc-pVTZ
(Section 5.3.1) basis sets [ 245 ], but these levels are currently too high for routine
optimization and frequencies on even medium-size (say about 10–20 heavy atoms)
molecules. With continued growth in computer power this situation will change. It
should be possible to increase the accuracy of predicted spectra empirically by per-
forming calculations on a series of known compounds and fitting the experimental to
the calculated wavenumbers, and perhaps intensities, to obtain empirical corrections
tailored specifically to the functional group of interest. Such painstaking work would

336 5 Ab initio Calculations