Subject: The Symmetric Group in Quantum Chemistry

From: Pauncz Ruben <[email-masked]>
Date: Tue, 10 Oct 1995 10:17:49 +0200 (EET)

I would like to bring to your attention the following new book:

       The Symmetric Group in Quantum Chemistry

                Ruben Pauncz

   The use of the symmetric group in the quantum chemical treatment of the
structure of atoms, molecules and solids has  central importance. This is
the first book which gives a systematic treatment of the subject. It is
self-contained, it starts with the conventional Slater determinant approach,
it proceeds to the basic notions for the representation of the symmetric
group and the construction of spin-eigenfunctions. The heart of the book
is in the chapter dealing with spin-free quantum chemistry. It shows the
great interpretation value of this treatment and shows that computationally
this method is not inferior to the conventional treatment. The three last
chapters contain the unitary group approach, the symmetric group approach
and the spin-coupled valence bond method. A fairly extensive bibliography
concludes the book. 

   The book can be used as an textbook for a one semeter graduate  course 
or  for independent study by an advanced student of physical chemistry or 
physics.
             
            Table of contents

Chapter 1     The quantum mechanical background..............1

 1.1  Introduction...........................................1
 1.2  Spin-free Hamiltonian..................................1
 1.3  The antisymmetry principle.............................3
 1.4  Atomic and molecular orbitals..........................4
 1.5  Slater determinant.....................................5
 1.6  The self-consistent-field method.......................7
 1.7  Configuration interaction method.......................8
 1.8  Slater-Condon rules...................................10
 1.9  Lowdin rules..........................................12

Chapter 2     The Symmetric Group...........................19

 2.1  Introduction..........................................19
 2.2  Permutations..........................................19
 2.3  The symmetric group...................................20
 2.4  Cyclic permutation....................................20
 2.5  Classes of the symmetric group........................23
 2.6  Subgroups of the symmetric group......................25
 2.7  Double cosets.........................................27

Chapter 3     Representation of SN..........................31

 3.1  Reps of the symmetric group...........................31
 3.2  Young tableaux........................................32
 3.3  Young's orthogonal representation.....................36
 3.4  The branching law of the symmetric group..............38
 3.5  The conjugate representation..........................40
 3.6  The coset representation..............................42
 3.7  Decomposition of the coset representation.............46
 3.8  Characters of the symmetric group.....................48
 3.9  Calculation of the Characters.........................58
 3.10  The subgroup S2 x S_2 ......x S2.....................62
 3.11  Appendix.1.......................................... 63

Chapter  4    The symmetric group algebra...................67

 4.1  Algebraic notions.....................................67
 4.2  Class operators.......................................69
 4.3  Matric basis of the group algebra.....................72
 4.4  Matric basis for the centrum of the algebra...........75
 4.5  The Young operator basis..............................79

Chapter  5    Spin eigenfunctions...........................95

 5.1  Introduction..........................................95
 5.2  Construction of spin eigenfunctions...................95
 5.3  The genealogical construction.........................96
 5.4  The branching diagram.................................98
 5.5  Reps of the SN generated by the spinfns..............102
 5.6  Yamanouchi-Kotani method for the reps................105
 5.7  Branching diagram fns and Young tableaux.............108
 5.8  Serber spin functions................................112
 5.9  Projected spin eigenfunctions........................115
 5.10  Spin-paired spin eigenfunctions.....................123
 5.10.1   Correspondence between spin-paired functions and 
          Young tableaux...................................131

Chapter  6    Spatial functions............................135

 6.1  Antisymmetric wavefunction...........................135
 6.2  Decomposition of the wavefunction....................137
 6.3  Reps of  SN by the spatial functions.................138
 6.4  Branching diagram functions..........................139
 6.5  Serber wavefunction..................................141
 6.6  Projected wavefunction...............................149
 6.7  Valence bond wavefunction............................152

Chapter  7    Spin free quantum chemistry..................157

 7.1  Introduction.........................................157
 7.2  Orbital product functions............................157
 7.3  Invariance group of the primitive ket................158
 7.4  Spin free exclusion principle........................161
 7.5  Structure projections................................163
 7.5.1  The pair diagram...................................163
 7.5.2  The pair operators.................................165
 7.5.3  Spin free pair functions...........................167
 7.5.4  Pair projections in the function space.............168
 7.5.5  Spin free exclusion principle for structure 
        projections........................................169
 7.6  Spin free counterpart of  A Phi Vk...................170
 7.7  Spin free counterpart of the projected fn............175
 7.8  Gallup's tableau operators...........................180
 7.9  Calculation of the Pauling numbers...................183
 7.10  Li's algorithm......................................186

Chapter  8    Unitary group approach.......................195

 8.1  Introduction.........................................195
 8.2  Basic notions........................................195
 8.3  Tensor space.........................................201
 8.4  Model Hamiltonian....................................203
 8.5  Reps of the unitary group............................206
 8.6  The branching law of the unitary group...............209
 8.7  Representation matrices of the generators............210
 8.8  Weyl tableaux........................................217
 8.9  Electronic Gel'fand state............................219
 8.10  Paldus arrays.......................................221
 8.11  Graphical unitary group approach ...................233
 8.12  Direct configuration interaction method.............248

Chapter  9    Symmetric group approach in CI...............253

 9.1  Introduction.........................................253
 9.2  Representation matrices..............................254
 9.3  Sarma and Rettrup algorithm..........................258
 9.4  Duch and Karwowski algorithm.........................267
 9.5  Symmetric group graphical approach...................274

Chapter  10   Spin-coupled functions.......................281

 10.1  Introduction........................................281
 10.2  Historical development..............................282
 10.3  Spin-coupled wavefunctions..........................286
 10.4  SCVB functions......................................289
 10.5  Core-valence s-c wavefunction.......................290

 Bibliography..............................................295
 
 Author index..............................................321
 Subject index.............................................327

 CRC Press, Inc. Boca Raton, Florida     June 1995
 ISBN 0-8493-8291-2            330 pages

 U.S.  $ 110.00        Outside U.S.  $ 132.00

        *-------------------------------------------*
        | Dr. Ruben Pauncz, Professor Emeritus      |
        | Department of Chemistry                   |
        | Technion - Israel Institute of Technology |
        | Haifa 32000, Israel                       |
        |                                           |
        | [email-masked]           |
        | Telephone: +972-4-293743                  |
        | Fax:       +972-4-233735                  |
        *-------------------------------------------*