Chemistry Reference
In-Depth Information
Chapter 6
Interactions
Abstract
In quantum mechanics the observable phenomena are interactions, ex-
pressed as matrix elements of operators in a function space. These spaces and oper-
ators are like communicating vessels, reality is neither the operation nor the repre-
sentation, but the interaction. The evaluation of the corresponding matrix elements
requires the coupling of representations, and can be factorized into an intrinsic scalar
quantity that contains the physics of the interaction, and a tensorial coupling coeffi-
cient that contains its symmetry. This factorization is first illustrated for the case of
overlap integrals, where the operator is just the unit operator, and then extended to
the case of non-trivial operators, such as the Hamiltonian, and electric and magnetic
dipole operators. The Wigner-Eckart theorem is introduced, together with the sym-
metry selection rules, both at the level of representations and subrepresentations.
The results are applied to chemical reaction theory, and to the theory of the Jahn-
Teller effect. Selection rules are illustrated for linear and circular dichroism. Finally,
the polyhedral Euler theorem is introduced and applied to valence-bond theory for
clusters.
Contents
6.1
OverlapIntegrals..................................
114
6.2
The Coupling of Representations
. ........................
115
6.3
Symmetry Properties of the Coupling Coefficients
................
117
6.4
Product Symmetrization and the Pauli Exchange-Symmetry
...........
122
6.5
MatrixElementsandtheWigner-EckartTheorem ................
126
6.6
Application: The Jahn-Teller Effect ........................
128
6.7
Application: Pseudo-Jahn-Teller interactions . ..................
134
6.8
Application:LinearandCircularDichroism ...................
138
LinearDichroism .................................
139
CircularDichroism.................................
144
6.9
Induction Revisited: The Fibre Bundle
......................
148
6.10
Application: Bonding Schemes for Polyhedra . ..................
150
Edge Bonding in Trivalent Polyhedra .......................
155
FrontierOrbitalsinLeapfrogFullerenes .....................
156
6.11
Problems......................................
159
References...........................................
160
