Chemistry Reference
In-Depth Information
ditions. On the other hand, going beyond the Born
Oppenheimer approximation,
one may consider the dynamic states of ammonia that correspond to the tunnelling
of the nitrogen through the triangle of the hydrogens. For the tunnelling states, the
all-particle inversion operator,
E
∗
, is also a symmetry element, and the symmetry
group of the nonrigid ammonia thus attains the full
D
3
h
Longuet-Higgins group.
−
5.5 Problems
5.1 Prove that the electron-repulsion operator,
V
ee
, is invariant under the rotation
around the
z
-axis.
5.2 Construct a splitting field and ladder operators for the canonical components of
the icosahedral irreps in Appendix
D
.
5.3 Derive the permutation-inversion group for CH
3
BF
2
(methyl-boron-difluoride)
under the assumption that the methylgroup is almost freely rotating. This means
that the result of a permutation inversion can be rotated back by a bodily rota-
tion to a rotamer of the original structure. Determine the point group that is
isomorphic to the resulting dynamic symmetry group.
5.4 The barrier to rotation of the cylopentadienyl rings in Fe
(C
5
H
5
)
2
(ferrocene, see
Fig.
3.9
(a)), measured in the gas phase, is only a kcal/mol. Construct a dynamic
symmetry group for this molecule.
References
1. Katzir, S.: The emergence of the principle of symmetry in physics. Historical Studies in the
Physical Sciences
35
, 35 (2004)
2. Griffith, J.S.: The Theory of Transition-Metal Ions. Cambridge University Press, Cambridge
(1961)
3. Longuet-Higgins, H.C.: The symmetry group of non-rigid molecules. Mol. Phys.
6
, 445 (1963)
4. Bunker, P.R.: Molecular Symmetry and Spectroscopy. Academic Press, New York (1979)
