112
5 What has Quantum Chemistry Got to Do with It?
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, ˆ
E ∗ , is also a symmetry element, and the symmetry
group of the nonrigid ammonia thus attains the full D 3h 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 rotation 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)
5 What has Quantum Chemistry Got to Do with It?
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, ˆ
E ∗ , is also a symmetry element, and the symmetry
group of the nonrigid ammonia thus attains the full D 3h 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 rotation 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)