Essentials of Computational Chemistry

9.3 SPECTROSCOPY OF NUCLEAR MOTION 343
To illustrate a case where this is not true, consider the methyl radical CH3 ž . The equilibrium
structure for this system is planar, and the unpaired electron occupies the out-of-plane p z
orbital on carbon. Because this orbital has a node at the carbon atom, in the absence of
polarization the 13 C isotropic hyperfine splitting should be zero (only spin polarization makes
it non-zero). However, one of the normal modes of the methyl radical is the so-called
‘umbrella’ mode that simultaneously bends all the hydrogen atoms to one side of the plane
or the other. This motion rehybridizes the singly occupied molecular orbital (SOMO) so that it
includes some s character, and thus the 13 C h.f.s. value should become increasingly positive.
Moreover, this is true irrespective of the side to which the umbrella motion takes place.
That is, if we now take A to be the change in 13 C h.f.s., it is an even function about the
equilibrium structure (Figure 9.8). As such, we expect from Eq. (9.51) that the expectation
value of the h.f.s. splitting over the umbrella mode vibrational wave function should be
significantly different from the value at the equilibrium position (the other vibrational modes
of CH3 ž do not cause the molecule to deviate from planarity, so they have minimal impact
on the expectation value).
Table 9.4 compares to experiment the isotropic h.f.s. values computed for 13 Cand 1 Hin
the methyl radical at the UMP2/6-311G(d,p) level both (i) at the UHF/6-31G(d) equilibrium
geometry and (ii) as the expectation value over the umbrella mode vibrational wave function
computed at this level. Also included are data for the monofluoromethyl radical CH2F ž ,
which is even more affected by vibrational averaging because it has a very shallow doublewell
potential along the umbrella mode (i.e., the equilibrium structure is pyramidal, but the
barrier to inversion is less than 1 kcal mol −1 ), so that its vibrational wave function has large
amplitude around a planar structure with smaller 13 C h.f.s. than for the equilibrium structure.
Energy
umbrella angle
q, q = f − 90°
f
f
C
H
H
f
H
qeq = 0
q
Figure 9.8 Potential energy and zeroth vibrational level with associated wave function (reference
left ordinate) and 13 C h.f.s. (reference right ordinate) as a function of umbrella angle for CH3 ž
13 C h.f.s.