Computational Chemistry

The characterization of stationary points by the number of imaginary frequen-
cies was discussed inChapter 2, and zero-point energies inChapter 2and earlier
sections of this chapter. Here we will examine the utility of ab initio calculations for
the prediction of IR spectra [ 235 ]. It is important to remember that frequencies
should be calculated at the same level (e.g. HF/3–21G(), MP2/6–31G, ...) as was
used for the geometry optimization (Section 2.4). This is because accurate calcula-
tion of the curvature of the PES at a stationary point requires that the second
derivatives∂^2 E/∂qi∂qjbe found at the same level as was used to create the surface
on which the point sits.

5.5.3.1 Positions (Frequencies) of IR Bands

In Section 2.5, we saw that diagonalization of the force constant matrix gives an
eigenvector matrix whose elements are the “direction vectors” of the normal-mode
vibrations, and an eigenvalue matrix whose elements are the force constants of
these vibrations. “Mass-weighting” the force constants gives the wavenumbers
(“frequencies”) of the normal-mode vibrations, and their motions can be identified
by using the direction vectors to animate the vibrations. So we can calculate the
wavenumbers of IR bands and associate each band with some particular vibrational
mode. The wavenumbers (“frequencies”) from ab initio calculations are larger than
the experimental ones, i.e. the frequencies are too high. There are two reasons why
this might be so: the principle of equating second derivatives of energy (with
respect to geometry changes) with force constants might be at fault, or the basis
set and/or correlation level might be deficient.
The principle of equating a second derivative with a stretching or bending force
constant is not exactly correct. A second derivative@^2 E=@q^2 would be strictly equal
to a force constant only if the energy were a quadratic function of the geometry, i.e.
if a graph ofEversusqwere a parabola. However vibrational curves are not exactly
parabolas (Fig.5.32). For a parabolicE/qrelationship, and considering a diatomic
molecule for simplicity, we would have:

E¼

k

2

ðq#qeqÞ^2 ð 5 : 203 Þ

whereqeqis the equilibrium geometry. Herekis by definition the force constant, the
second derivative ofE, and@^2 E=@q^2 ¼k. For a real molecule, however, theE/q
relationship is more complicated, being a power series inq^2 ,q^3 , etc., terms, and
there is not just one constant. Equation (5.203) holds for what is calledsimple
harmonic motion, and the coefficients of the higher-power terms in the more
accurate equation are calledanharmonicity corrections. Assuming that bond vibra-
tions are simple harmonic is theharmonic approximation.
For small molecules it is possible to calculate from the experimental IR spectrum
the simple harmonic force constantkand the anharmonicity corrections. Usingk,
theoreticalharmonic frequencies can be calculated [ 236 ]. These correspond to a

5.5 Applications of the Ab initio Method 333