Essentials of Computational Chemistry

SYMMETRY AND GROUP THEORY 561
B.3 Assigning Electronic State Symmetries
Individual molecular orbitals, which in symmetric systems may be expressed as symmetryadapted
combinations of atomic orbital basis functions, may be assigned to individual irreps.
The many-electron wave function is an antisymmetrized product of these orbitals, and thus
the assignment of the wave function to an irrep requires us to have defined mathematics for
taking the product between two irreps, e.g., a ′ ⊗ a ′′ in the Cs point group. These product
relationships may be determined from so-called character tables found in standard textbooks
on group theory. Tables B.1 through B.5 list the product rules for the simple point groups
Cs, Ci, C2, C2h, andC2v, respectively.
Assignment of an electronic wave function to an irrep is typically straightforward. All
doubly filled orbitals are ignored, as the product of all of them with one another is the
totally symmetric irrep, which is the multiplicative ‘one’ in all point groups. Thus, we need
only take the product of all of the singly occupied orbitals to determine the irrep of the wave
function. For a doublet, there is only one singly occupied orbital, so the irrep to which it
belongs determines the irrep of the wave function. Figure 6.9 illustrates this point for H2NO.
Note that, to distinguish it from orbital irreps, the wave-function state symmetry is usually
written with a capital letter. In triplets (and open-shell singlets), there are two singly occupied
Table B.1 Product rules for the Cs
point group a
⊗ a ′ a ′′
a ′ a ′ a ′′
a ′′ a ′′ a ′
a See text for irrep definitions.
Table B.2 Product rules for the Ci
point group a
⊗ ag au
ag ag au
au au ag
a Objects unchanged by inversion belong to
the ag irrep; objects that change phase on
inversion belong to the au irrep.
Table B.3 Product rules for the C2
point group a
⊗ a b
a a b
b b a
a Objects unchanged by rotation belong to
the a irrep; objects that change phase on
rotation belong to the b irrep.