Chemistry Reference
In-Depth Information
Chapter 5
What has Quantum Chemistry Got to Do
with It?
Abstract
The time has come to see how the concept of irreducible representations
ties in with quantum chemistry. After a brief introduction to the prequantum prin-
ciples of symmetry, we will show that eigenfunctions of the Hamiltonian are also
eigenfunctions of the symmetry operators that commute with the Hamiltonian. We
further analyze the concept of a degeneracy and show how the degenerate com-
ponents can be characterized by canonical symmetry relationships. The final sec-
tion will then provide a detailed account of the symmetry operations that leave the
Hamiltonian invariant.
Contents
5.1
ThePrequantumEra .................................
103
5.2
TheSchrödingerEquation ..............................
105
5.3
HowtoStructureaDegenerateSpace ........................
107
5.4
TheMolecularSymmetryGroup...........................
108
5.5
Problems .......................................
112
References...........................................
112
5.1 The Prequantum Era
Nature around us is full of disorder and chaos, yet it also offers intriguing examples
of perfect order and symmetry. Ever since prehistoric times, man has been admiring
the circular geometry of a full moon or the perfectly flat surface of a calm sea.
Crystals offer another example of almost ideal symmetrical shapes, and it is no
surprise that early recognition of the important role of symmetry in physics was
based on the study of properties of crystals. Two pioneers of the prequantum era,
Franz Neumann and Pierre Curie [
1
], stand out for their important conjectures.
Theorem 9
Neumann's principle states that the symmetry elements of any physical
property of a crystal must include all the symmetry elements of the point group of
the crystal
.
We can apply this directly to a molecule such as ammonia. Ammonia carries a
permanent dipole moment,
μ
z
, which is oriented along the threefold axis. In-plane
