Chapter 7
Spherical Symmetry and Spins
Abstract A brief excursion is made into the concept of continuous groups, with an
example of the rotation groups. The purpose is to familiarize the reader with the
concept of electron spin. The coupling of spins is discussed. Applications are taken
from Crystal-Field Theory and Electron Spin Resonance.
Contents
7 . 1
T h eS p h e r i c a l - S y m m e t r yG r o u p ........................... 1 6 3
7 . 2
A p p l i c a t i o n :C r y s t a l - F i e l dP o t e n t i a l s......................... 1 6 7
7.3
Interactions of a Two-Component Spinor ...................... 1 7 0
7 . 4
T h eC o u p l i n go fS p i n s ................................ 1 7 3
7.5
Double Groups .................................... 1 7 5
7 . 6
K r a m e r sD e g e n e r a c y................................. 1 8 0
T i m e - R e v e r s a lS e l e c t i o nR u l e s............................ 1 8 2
7 . 7
A p p l i c a t i o n :S p i nH a m i l t o n i a nf o rt h eO c t a h e d r a lQ u a r t e tS t a t e .......... 1 8 4
7 . 8
P r o b l e m s ....................................... 1 8 9
R e f e r e n c e s........................................... 1 9 0
7.1 The Spherical-Symmetry Group
Consider the row vector of coordinate functions (|x|y|z) in 3D space. A rotation
must conserve the norm of this function space. As we have seen in Chap. 2, this can
be realized by a unitary matrix transformation. For real functions, the unitary transformation is reduced to an orthogonal one. A matrix is orthogonal if the following
condition is fulfilled (where T denotes transposition):
D
T D = I = DD
T
(7.1)
The set of all ortho-gonal 3 × 3 matrices forms a group, which is called the orthogonal group in three dimensions, O(3). The order of this group is infinite. It can
be shown that this group is isomorphic with the complete group of all proper and
improper rotations in three dimensions. The structure that embodies this symmetry is the sphere. Hence, O(3) is the symmetry group of the sphere. The determinants of the matrices must be real with modulus one, which means that they can
A.J. Ceulemans, Group Theory Applied to Chemistry, Theoretical Chemistry and
Computational Modelling, DOI 10.1007/978-94-007-6863-5_7,
© Springer Science+Business Media Dordrecht 2013
163
Spherical Symmetry and Spins
Abstract A brief excursion is made into the concept of continuous groups, with an
example of the rotation groups. The purpose is to familiarize the reader with the
concept of electron spin. The coupling of spins is discussed. Applications are taken
from Crystal-Field Theory and Electron Spin Resonance.
Contents
7 . 1
T h eS p h e r i c a l - S y m m e t r yG r o u p ........................... 1 6 3
7 . 2
A p p l i c a t i o n :C r y s t a l - F i e l dP o t e n t i a l s......................... 1 6 7
7.3
Interactions of a Two-Component Spinor ...................... 1 7 0
7 . 4
T h eC o u p l i n go fS p i n s ................................ 1 7 3
7.5
Double Groups .................................... 1 7 5
7 . 6
K r a m e r sD e g e n e r a c y................................. 1 8 0
T i m e - R e v e r s a lS e l e c t i o nR u l e s............................ 1 8 2
7 . 7
A p p l i c a t i o n :S p i nH a m i l t o n i a nf o rt h eO c t a h e d r a lQ u a r t e tS t a t e .......... 1 8 4
7 . 8
P r o b l e m s ....................................... 1 8 9
R e f e r e n c e s........................................... 1 9 0
7.1 The Spherical-Symmetry Group
Consider the row vector of coordinate functions (|x|y|z) in 3D space. A rotation
must conserve the norm of this function space. As we have seen in Chap. 2, this can
be realized by a unitary matrix transformation. For real functions, the unitary transformation is reduced to an orthogonal one. A matrix is orthogonal if the following
condition is fulfilled (where T denotes transposition):
D
T D = I = DD
T
(7.1)
The set of all ortho-gonal 3 × 3 matrices forms a group, which is called the orthogonal group in three dimensions, O(3). The order of this group is infinite. It can
be shown that this group is isomorphic with the complete group of all proper and
improper rotations in three dimensions. The structure that embodies this symmetry is the sphere. Hence, O(3) is the symmetry group of the sphere. The determinants of the matrices must be real with modulus one, which means that they can
A.J. Ceulemans, Group Theory Applied to Chemistry, Theoretical Chemistry and
Computational Modelling, DOI 10.1007/978-94-007-6863-5_7,
© Springer Science+Business Media Dordrecht 2013
163