Chapter 4
Representations
Abstract Having made acquaintance with the basic properties of groups, we now
turn our attention to the structure of the matrices that represent the group action in
a function space. It turns out that there exists only a limited set of standard patterns.
These are called the irreducible representations. They form the principal mathematical concept on which this monograph is based, and much care is devoted to acquire
a gradual understanding of what this concept really means. Then the character theorem, matrix theorems, and projection operators are introduced. The concepts of
subduction and induction relate the representations of subgroups and those of their
parent groups. The chapter also offers a detailed group-theoretical analysis of three
chemical applications: the tetrahedral hybridization of carbon, the molecular vibrations of UF 6 , and the electronic structure of conjugated hydrocarbons, according to
the Hückel model and the method of the London model of gauge-invariant atomic
orbitals.
Contents
4.1
Symmetry-Adapted Linear Combinations of Hydrogen Orbitals in Ammonia . .
52
4 . 2
C h a r a c t e rT h e o r e m s ................................
5 6
4 . 3
C h a r a c t e rT a b l e s..................................
6 2
4 . 4
M a t r i xT h e o r e m ..................................
6 3
4 . 5
P r o j e c t i o nO p e r a t o r s................................
6 4
4.6
Subduction and Induction .............................
6 9
4.7
Application: The sp 3 H y b r i d i z a t i o no fC a r b o n ..................
7 6
4.8
Application: The Vibrations of UF 6 ........................
7 8
4 . 9
A p p l i c a t i o n :H ü c k e lT h e o r y............................
8 4
C y c l i cP o l y e n e s ..................................
8 5
P o l y h e d r a lH ü c k e lS y s t e m so fE q u i v a l e n tA t o m s.................
9 1
T r i p h e n y l m e t h y lR a d i c a la n dH i d d e nS y m m e t r y .................
9 5
4 . 1 0
P r o b l e m s......................................
9 9
R e f e r e n c e s........................................... 1 0 1
A.J. Ceulemans, Group Theory Applied to Chemistry, Theoretical Chemistry and
Computational Modelling, DOI 10.1007/978-94-007-6863-5_4,
© Springer Science+Business Media Dordrecht 2013
51
Representations
Abstract Having made acquaintance with the basic properties of groups, we now
turn our attention to the structure of the matrices that represent the group action in
a function space. It turns out that there exists only a limited set of standard patterns.
These are called the irreducible representations. They form the principal mathematical concept on which this monograph is based, and much care is devoted to acquire
a gradual understanding of what this concept really means. Then the character theorem, matrix theorems, and projection operators are introduced. The concepts of
subduction and induction relate the representations of subgroups and those of their
parent groups. The chapter also offers a detailed group-theoretical analysis of three
chemical applications: the tetrahedral hybridization of carbon, the molecular vibrations of UF 6 , and the electronic structure of conjugated hydrocarbons, according to
the Hückel model and the method of the London model of gauge-invariant atomic
orbitals.
Contents
4.1
Symmetry-Adapted Linear Combinations of Hydrogen Orbitals in Ammonia . .
52
4 . 2
C h a r a c t e rT h e o r e m s ................................
5 6
4 . 3
C h a r a c t e rT a b l e s..................................
6 2
4 . 4
M a t r i xT h e o r e m ..................................
6 3
4 . 5
P r o j e c t i o nO p e r a t o r s................................
6 4
4.6
Subduction and Induction .............................
6 9
4.7
Application: The sp 3 H y b r i d i z a t i o no fC a r b o n ..................
7 6
4.8
Application: The Vibrations of UF 6 ........................
7 8
4 . 9
A p p l i c a t i o n :H ü c k e lT h e o r y............................
8 4
C y c l i cP o l y e n e s ..................................
8 5
P o l y h e d r a lH ü c k e lS y s t e m so fE q u i v a l e n tA t o m s.................
9 1
T r i p h e n y l m e t h y lR a d i c a la n dH i d d e nS y m m e t r y .................
9 5
4 . 1 0
P r o b l e m s......................................
9 9
R e f e r e n c e s........................................... 1 0 1
A.J. Ceulemans, Group Theory Applied to Chemistry, Theoretical Chemistry and
Computational Modelling, DOI 10.1007/978-94-007-6863-5_4,
© Springer Science+Business Media Dordrecht 2013
51