Essentials of Computational Chemistry

562 APPENDIX B
Table B.5 Product rules for the C2h point group a
⊗ ag au bg bu
ag ag au bg bu
au au ag bu bg
bg bg bu ag au
bu bu bg au ag
a Objects unchanged by rotation belong to a type irreps, while objects changing
phase on rotation belong to b type irreps; objects unchanged by reflection through
the horizontal σ belong to −g type irreps, while objects changing phase on reflection
belong to −u type irreps.
Table B.4 Product rules for the C2v point group a
⊗ a1 a2 b1 b2
a1 a1 a2 b1 b2
a2 a2 a1 b2 b1
b1 b1 b2 a1 a2
b2 b2 b1 a2 a1
a See text for irrep definitions.
orbitals, so their product must be taken to determine the state symmetry. Figure 14.4 provides
examples of this process. In more complex open-shell systems, the sequential product of all
of the singly occupied orbitals determines the electronic state symmetry.
Some complications can arise. Although in many point groups the product of any two
irreps is another irrep (as is true for the examples in Tables B.1 through B.5), in some cases
the product of two irreps can only be expressed as a linear combination of two or more
different irreps. A determinant that does not belong to a single irrep is not a true wave
function, but must be combined with other determinants to construct a wave function having
a pure state symmetry. Such situations are beyond the scope of this text.
B.4 Symmetry in the Evaluation of Integrals and Partition
Functions
Mathematical functions and operators can be assigned to irreps just as orbitals can be. This
has enormous implications for practical computations because the integral over all space of
any product that does not contain the totally symmetric representation vanishes, i.e., there
is no point evaluating it. The Fock and Hamiltonian operators both belong to the totally
symmetric irrep of any point group, because they depend only on interparticle distances and
the ∇2 operator, and these quantities are unaffected by changes in the coordinate system
brought about by rotations, reflections, etc. Thus, in evaluating Fock matrix elements of
the form
Fµν = φµ(1)F φν(2)dr(1)dr(2) (B.1)