Page 274 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

VIBRATIONAL MODULATION EFFECTS ON EPR SPECTRA 257
The ground vibrational wave function of planar systems is peaked at the planar
structure. Vibrational averaging then changes the coupling constants toward values which
would be obtained for an angle in a static description. The wave function of the
ground vibrational state being symmetrically spread around introduces contributions
of pyramidal configurations. This results in a noticeable increase of the absolute values of
the coupling constants, which are minimal at planar structures (see Figure 3). Vibrational
averaging then provides hyperfine coupling constants in close agreement with experiment.
The effect is even more pronounced in the first excited vibrational state, whose wave
function has a node at the planar structure and is more delocalized than the fundamental
one, thus giving increased weight to pyramidal structures.
For radicals characterized by a double-well potential the vibrational effect acts in
an opposite direction, bringing the coupling constants to values which would be obtained
for The ground state vibrational wave function is now more localized inside the
potential well, even under the barier, than outside. So it introduces more contributions of
internal points. Vibrational effects, while still operative, are less apparent in this case since
high energy barriers imply high vibrational frequencies with the consequent negligible
population of excited vibrational states and smaller displacements around the equilibrium
positions. This explains the good agreement between experimental and static theoretical
computations.

Let us now turn to the second parameter, namely the shape of the "property surface".
Around reference configurations, the dependence of the hyperfine coupling constants on the
inversion motion is well represented by:

The average value of a can be written as:

The mean and mean square values of the LA coordinate s represent the principal
anharmonic and harmonic vibrational contributions, respectively [3].
In the case of a planar equilibrium structure, the lineaar term is absent since symmetry
constraints impose that Since, in our case, hyperfine coupling

constants reach a minimum value at the planar reference structure (Figures 3b and 4b), the
third term is always positive. Vibrational frequencies of this class of molecules are, of
course small (Table 1), leading to large mean square amplitudes and consequently,
to significant corrections to static values computed at the reference structure.
Unless Boltzmann averaging gives significant weight to vibrational states above the barrier,
strongly pyramidal molecules like can be effectively treated as systems governed by a
single well potential unsymmetrically rising on the two sides of the minimum energy
configuration. If we shift s so that now s = 0 at the equilibrium structure, the difference
with the previous case resides in the presence of the linear term in Eq.(8). This is due to the