Essentials of Computational Chemistry

9.3 SPECTROSCOPY OF NUCLEAR MOTION 341
factors for specific levels of theory based only on those molecules (srdata.nist.gov/cccbdb/).
One example of such an approach, albeit not using the NIST website, was provided by Yu,
Srinivas, and Schwartz (2003) who optimized scale factors just for the C−O stretch of metal
bound carbonyls.
Results from molecular mechanics can also be of reasonable accuracy, so long as the
molecules addressed contain only functionality well represented in the force field training
set. While extensive compilations of data are not available, Halgren has compared MM3 and
MMFF94 over a test set of 157 frequencies from organic molecules and found RMS errors
of 57 and 60 cm −1 , respectively.
An interesting alternative to scaling the frequencies is instead to scale the force constants
in the Hessian, which permits some sensitivity to different kinds of vibrations, e.g., stretches,
bends, and torsions (see, for example, Grunenberg and Herges 1997; Baker, Jarzecki, and
Pulay 1998; Arenas et al. 2000). Of course, as with any parameterization procedure, as the
number of parameters increases so too does the requirement for additional data to ensure
statistical reliability, and this approach has not yet seen wide application.
One final caveat with respect to comparing experimental IR spectra with theoretically
predicted frequencies is that the latter do not account for such experimental complications
as Fermi resonances (where two nearby fundamentals are shifted to higher and lower
frequencies, respectively), overtones, etc. Such details require case-by-case evaluation.
In comparing complete theoretical spectra to complete experimental spectra in molecules
of moderate to large size, there can be a large number of lines. To ensure proper correspondence
of the normal modes, it is helpful to compare not only the absorption frequencies
themselves but also the intensities of the absorptions. For a typical experimental spectrum,
such intensities are usually reported simply as strong, medium, or weak, although in careful
experiments absorption cross-sections can be measured accurately. From a computational
standpoint, the prediction of IR intensities can be accomplished using the mixed second
derivatives of the energy with respect to geometric motion and an external electric field
(thereby permitting estimation of the changes in the dipole moment as a function of the
vibrations, which is what IR intensities are proportional to). These mixed second derivatives
are available analytically for all levels of theory for which analytic second derivatives with
respect to the geometry are available, so it is a straightforward matter to compute IR intensities.
The actual computed values tend to be no better than qualitative in the absence of
using a very complete basis set and accounting for electron correlation, but insofar as most
experimental intensities are essentially qualitative, this is not typically much of a drawback.
Being able to line up strong absorptions in computed and experimental spectra is often quite
helpful for assessing the validity of the comparison.
An alternative experiment that measures the same vibrational fundamentals subject to
different selection rules is Raman spectroscopy. Raman intensities, however, are more difficult
to compute than IR intensities, as a mixed third derivative is required to approximate the
change in the molecular polarizability with respect to the vibration that is measured by the
experiment. The sensitivity of Raman intensities to basis set and correlation is even larger
than it is for IR intensities. However, Halls, Velkovski, and Schlegel (2001) have reported
good results from use of the large polarized valence-triple-ζ basis set of Sadlej (1992) and