higher, thanobservedfrequencies. From the foregoing discussion it appears that the
fundamental reason ab initio frequencies are too high is because of the harmonic
approximation: equating of@^2 E=@q^2 with a force constant. There is no theoretical
reason why high-level calculations should converge toward the observed frequen-
cies; this statement applies to frequencies calculated, as is almost always the case,
by the harmonic approximation (above). Frequencies accurate to within about 1%
were obtained for a set of small molecules using high correlation levels and
medium-size basis sets [ 240 ].
Fortunately for us, we wish only to calculate IR spectra that resemble, or would
resemble, experimental ones, and for this there is a simple expedient. Calculated
and observed frequencies differ by a fairly constant factor, and ab initio (and other
theoretically-calculated) frequencies can be brought into reasonable agreement
with experiment by multiplying them by a correction factor. An extensive compari-
son by Scott and Radom of calculated and experimental frequencies [ 80 ] has
provided empirical correction factors for frequencies calculated by a variety of
methods. A few of the correction factors from this compilation are:
HF/3–21G() 0.9085
HF/6–31G 0.8953
HF/6–311G(df,p) 0.9054
MP2(fc)/6–31G* 0.9434
MP2(fc)/6–311G** 0.9496
The correction factors at the HF level with the three basis sets are similar,
0.90–0.91; the factors at the MP2 level are significantly closer to 1, but Scott and
Radom say that “MP2/6–31(d) does not appear to offer a significant improvement
in performance over HF/6–31G(d) and occasionally shows large errors”, and “The
most cost-effective procedures found in this study for predicting vibrational fre-
quencies are HF/6–31(d) and [certain density functional methods]”. Separate cor-
rection factors for zero-point vibrational energies were also given, and although it
was hitherto common practice to use the same correction factor for frequencies and
for ZPEs, the use of separate factors is now standard. Better agreement with
experiment can be obtained by using empirical correction factors for specific
kinds of vibrations (Scott and Radom give separate factors for low-frequency
vibrations, as opposed to the relatively high-frequency ones to which the factors
listed above refer), but this is rarely done.
5.5.3.2 Intensities of IR Bands
The bands in an IR spectrum have not justpositions(“frequencies”, denoted by
various wavenumbers), but alsointensities. IR intensities present considerably
more difficulties in their measurement and theoretical calculation than do frequen-
cies, and in fact experimental intensities are not routinely quantified, but are
commonly merely described as weak, medium, or strong. To calculate an IR
spectrum for visual comparison with experiment it is desirable to compute both
5.5 Applications of the Ab initio Method 335