Character table for dirhenium decacarbonyl of full non-rigid molecule group

E. Suleiman and MI. Bello / Science Forum (Journal of Pure and Applied Sciences) 16 (2019) 1 – 4
2
Before we start our calculation, one has defined precisely the symmetry group of a nonrigid molecule and shown how to use its character table in solving physical problems. This is the purpose of this paper. But one must begin by studying closely the familiar point groups of rigid molecules, and the following remarks owe much to a paper by J. T. Hougen (
Longuet-Higgins, A molecule is an collection of atomic nuclei and electrons. Its Hamiltonian is invariant under the following types of transformation:
a) Any permutation of the positions and spins of the electrons.
b) Any rotation of the positions and spins of all the particles (electrons and nuclei) about an axis through the center of mass.
c) Any overall translation in spaced) The reversal of all particle momenta and spins.
e) The simultaneous inversion of the positions of all particles in the center of mass.
f) Any permutation of the positions and spins of any set of identical nuclei (
Longuet-Higgins, The complete group of the Hamiltonian is thus the direct product of several groups. But for most molecules, not all of the elements of the complete group need betaken into account. This is because the timescale of a given laboratory experiment maybe too short to allow certain nuclear permutations ever to occur. One therefore, restricts attention to feasible transformations, those which can be achieved without passing over an insuperable energy barrier. The molecular symmetry
group is then composed of feasible elements only, but it does not comprise all the feasible elements of the complete group of the Hamiltonian, Its clearest and most useful definition seems to be as follows:
Let P be any permutation of the positions and spins of identical nuclei or any product of such permutations. Let E be the identity, E
−1
the inversion of all particle positions, and P
−1
the product PE
−1
= E
−1
P. Then the molecular symmetry group is the set of i. all feasible P, including E,
ii. all feasible P
−1
, not necessarily, including E
−1
We shall now show that this definition accords with our intuitive ideas of molecular symmetry, except for linear molecules, which would require special discussion.
First, we must reconcile the present definition with the customary assertion that molecular symmetry groups comprise rotations, reflections etc. A moment’s thought will convince one that this assertion cannot betaken literally.
The bodily rotation of a molecule can make no difference to its internal coordinates—the distances between the constituent particles—whereas vibrational symmetry coordinates and electronic wave functions are not
necessarily invariant under the elements of the symmetry group. If they were, the symmetry group would be quite useless. No, it is not the positions of the particles which are, rotated but their relative coordinates, and this result is certainly achieved by permuting the positions of identical nuclei. Even the electronic wave function is susceptible to nuclear permutations since it is a function of the coordinates of the electrons relative to
the nuclear framework, not of their absolute positions in space. However, an example is likely to be more convincing than any general remarks, so we consider the particular case of H
2
O (
Longuet-Higgins, We give the final remark, before leaving our discussion on rigid molecules. It may not always be obvious what permutation or permutation-inversion is to be associated with a given rotation or reflection belonging to the point group of a rigid molecule. The example of H
2
O may again be helpful here. If one were to rotate the (undistorted) water molecule about its twofold symmetry axis one would interchange nuclei
1 and 2; "C
2
" is indeed the permutation (12). Again, reflecting all the particle positions in the xy plane would not permute any nuclei, but merely reverse the sense of the molecular configuration "ó
xy
" is understood to mean E
−1
. Finally, a reflection in the yz plane would interchange 1 and 2 and also reverse the sense "ó
yx
" is to be interpreted as (12)
−1
. This example may help the reader to avoid the pitfall of supposing that E
−1
has any direct connection with the inversion belonging to the point group of a centrosymmetric molecule −H
2
O is not centrosymmetric (
Longuet-Higgins, 1963
).