Chemistry Reference
In-Depth Information
way as the combination of all proper and improper rotations form the point group,
the combination of all allowed permutations and permutation-inversions defines the
molecular symmetry group
.
3.2 Historical Development of the Molecular Symmetry
Group Approach
The concept of the molecular symmetry group was first introduced by Longuet-
Higgins in 1963 [
246
]. Independently, Pechukas used pairs of rotation/reflection
operators and permutation operators to derive symmetry rules for transition states
of chemical reactions [
248
]. The paper by Hougen reviewed some of the spectro-
scopic applications and also discussed the advantages and disadvantages of the
point group treatment and of the permutation-inversion treatment [
247
]. Rodger
and Schipper developed symmetry selection rules for reaction mechanisms and
applied them to inorganic complexes [
249
-
251
]. Metropoulos translated these
results into the permutation-inversion and molecular symmetry group terminology
[
252
]. Bone, Rowlands, Handy, and Stone used the molecular symmetry groups to
find the transition states for rearrangements in non-rigid molecular complexes and
to deduce the number of such isomerization pathways [
253
]. In a second paper,
Bone used the complete nuclear permutation-inversion (CNPI) group to find the
symmetry of transition states and also discussed the interconversion of planar
structures via pairs of non-planar labeled-atom enantiomeric transition states
[
254
]. Permutation and inversion operators have also been used in the analysis of
permutational isomerization reactions of stereochemically non-rigid trigonal bipy-
ramidal and octahedral molecules where ligand substitution gives rise to constitu-
tional isomers [
255
-
259
]. The work by Klemperer and Ruch and H
ยจ
sselbarth on
isomer counting and the classification of modes of rearrangements is reviewed in
the topic by Brocas et al. [
260
]. It also includes an introduction into group theory
including a discussion of cosets and their properties. Furthermore, the application
of permutations in the field of dynamic NMR is discussed. Schaad and Hu have
extended the work of Pechukas, deducing a list of allowed point groups of transition
states of degenerate reactions for 49 common cases [
261
]. Bytautas and Klein noted
examples where the molecular symmetry group of non-rigid molecules is larger
than the Longuet-Higgins permutation/inversion group [
262
].
3.3 From the Complete Nuclear Permutation-Inversion
Group to Feasible Operations in the Molecular Symmetry
Group
The treatment of symmetry using permutation operators and permutation-inversion
operators may be derived from a quantum mechanical viewpoint: a molecule is an
