purpose. All the levels predict the correct stability order in these four cases, but are
vulnerable to varying extents to a narrowing of energy differences, as indicated in
the following brief analysis. The HF/3–21G()level can overestimate (cyclopro-
pane/propene) or underestimate (dimethyl ether/ethanol) the enthalpy difference
very considerably. The HF/6–31G level does better, but considerably underesti-
mates the difference for dimethyl ether/ethanol. The MP2 level does fairly
well semiquantitatively, its energy difference being within experimental by about
10–15 kJ mol#^1 , except possibly for fulvene/benzene, where the difference could
be ca. 20 kJ mol#^1 if the reported experimental heat of formation of fulvene is in
error ([ 229 ] and the note on Table5.13). The G3(MP2) method does well; even the
apparent 10 kJ mol#^1 discrepancy for fulvene/benzene could be due to possible
experimental error for fulvene [ 229 ]. Of course the G3(MP2) values shown here are
differencesin heats of formation, and may benefit from cancellation of errors in
calculated 298 K enthalpies. We saw in Section 5.5.2.2c the calculation of “abso-
lute” heats of formation.
5.5.3 Frequencies and Vibrational Spectra.............................
The calculation of normal-mode frequencies (Section 2.5) is important because:
- The number of imaginary frequencies of a molecular species tells us the quali-
tative nature of the curvature of the potential energy surface at that particular
stationary point: whether an optimized structure (i.e. a stationary point-species)
is a minimum, a transition state (a first-order saddle point), or a higher-order
saddle point. Note that frequency calculations are normally valid only for
stationary points; this rule is knowingly violated occasionally, e.g. when techni-
cally invalid but useful force constants or frequencies are calculated as aids to an
algorithmic process like geometry optimization (Section 2.4)or following an
IRC (Sections 2.2, 2.5 and 2.6). Routinely checking optimized structures with a
frequency calculation is a good idea, if the size of the job does not make this
impractical (frequencies take longer than optimizations). - The frequencies must be calculated to get the zero point energy of the molecule.
This is needed for accurate energy comparisons (Section 2.5). - The normal-mode vibrational frequencies of a molecule correspond, with qua-
lifications, to the bands seen in the infrared (IR) spectrum of the substance.
Discrepancies may arise from overtone and combination bands in the experi-
mental IR, and from problems in accurate calculation of relative intensities (less
so, probably, from problems in calculation of frequency positions). Thus the IR
spectrum of a substance that has never been made can be calculated to serve as a
guide for the experimentalist. Unidentified IR bands observed in an experiment
can sometimes be assigned to a particular substance on the basis of the calcu-
lated spectrum of a suspect; if the spectra of the usual suspects are not available
from experiment (they might be extremely reactive, transient species), we can
calculate them.
332 5 Ab initio Calculations