Page 271 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

254 V. BARONE ET AL.
equation formally equivalent to that governing the motion of a particle with a unit mass in a
one dimensional space. This model is intimately linked to the use of local vibrational basis
functions centered at different points along the path. In our approach [16,26,27], cubic
splines are used to interpolate the potential along the path and to generate a larger set of
equispaced points on which cubic splines are also used as basis functions. This kind of
treatment avoids any modelling of the ab-initio data and involves only analytical integrals.
Although the size of the basis set is larger than the one necessary when employing Hermite,
Morse or Gaussian functions, the spline approach remains competitive since the matrices to
be diagonalized are banded with a constant width of 7. Furthermore, no new integrals are
introduced by the computation of expectation values of observables (also represented by
spline fittings), and the additional computational effort depends on the number of
eigenstates to be taken into account, rather than on the dimension of the primitive spline
basis set. The expectation value of a given observable in the eigenstate j
corresponding to the eigenvalue is given by

The temperature dependence of the observable is obtained by assuming a Boltzmann
population of the vibrational levels, so that

3. Results

Full geometry optimizations and calculations of harmonic force constants were performed at
the UMP2/6-311G** level. Although this is not the main concern of this study, it is
noteworthy that the relatively unexpansive theoretical treatment we have developped
provides structural and spectroscopic parameters in close agreement with experiment (see
Table 1). More precisely, the harmonic approximation seems quite adequate for
whereas strong anharmonicities affect the CH stretchings and the out-of-plane motion of
The wave number of this latter vibration is increased to (in much better
agreement with the experimental value of by our one-dimensional anharmonic
treatment. Such a strong positive correction is in agreement with experimental estimates
[29]. From another point of view, the two radicals are well suited to point out the influence
of the shape of the potential well on vibrational effects: a simple well and a double-
well with an high inversion barrier
The influence of out-of-plane bending on geometrical parameters, electronic energy and
coupling constants is shown in Figures 2-4. The linear relationship between the s
coordinate and the angle is well evidenced in Figure 2a. We recall that in our approach,
although the geometries in internal coordinates used to build the path are mass independent,
the arc length s varies with the atomic masses, whereas the reduced mass governing the
motion always remains unitary. The larger mass of fluorine versus hydrogen then explains
the lower slope of the curve versus s for than for Also noteworthy is the
increase of the CH and CF bond lengths upon inversion (Figure 2b).