20.3 Homonuclear Diatomic Molecules 847
Term Symbols for Homonuclear Diatomic Molecules
As with atoms, term symbols are used to designate values of angular momentum quan-
tum numbers for homonuclear diatomic molecules. The scheme is slightly different
from the Russell–Saunders scheme that is used for atoms, sinceLis not a good quan-
tum number. The operator̂Lzdoes commute with the electronic Hamiltonian and an
energy eigenfunction can be an eigenfunction of̂Lz:
̂LzΨhM ̄ LΨ (20.3-18)whereMLis the same quantum number as in the atomic case. We define a non-negative
quantum numberΛequal to the magnitude ofML.Λ|ML| (20.3-19)Since there is no quantum numberL, the values ofMLare not limited.
The main part of the molecular term symbol is a capital Greek letter that specifies
the value ofΛ:Value ofΛ Symbol
0 Σ
1 Π
2 ∆
3 Φ
etc.Notice the similarity with the atomic term symbols with the letters S, P, D, and F.
The spin angular momentum operators commute with the electronic Hamiltonian
operator, so the energy eigenfunctions can be eigenfunctions of̂S^2 and̂Sz. The spin
quantum numbersSandMSfollow the same pattern as with atoms. A left superscript
equal to 2S+1 is used on the term symbol as with atoms, and the same terminology
is used (singlet, doublet, triplet, quartet, and so on). States with the same values ofΛ
andSconstitute an energy level, which is called aterm. In the ground state of H 2 both
of the electrons occupyσorbitals withm0 so thatML0 andΛ0. The ground
level of the H 2 molecule is nondegenerate and the term symbol is^1 Σ(singlet sigma).
Subscripts and superscripts are added to the term symbol to specify symmetry prop-
erties. The energy eigenfunctions of homonuclear diatomic molecules can be chosen
to be eigenfunctions of the symmetry operators that belong to the molecule. In an
orbital wave function, the symmetry operators must be applied to all of the orbitals. For
example, if the wave function contains an even number of orbitals with eigenvalues−1,
the eigenvalue of the wave function is+1. If the wave function is an eigenfunction of
the inversion operator with eigenvalue+1, a right subscriptgis attached to the term
symbol. If it is an eigenfunction of the inversion operator with eigenvalue−1, a right
subscriptuis attached. With sigma terms, if the wave function is an eigenfunction
of aσ̂voperator with eigenvalue+1 a right superscript+is added, and if it is an
eigenfunction of this operator with eigenvalue−1 a right superscript−is added. The
complete term symbol for the ground level of a hydrogen molecule is^1 Σ+g (“singlet
sigma plus g”). The excited electron configuration (σg 1 s)(σ∗u 1 s) of the H 2 molecule
can correspond to two different terms, a singlet and a triplet. The eigenvalue of the
inversion operator is−1 for both terms, because one orbital is g and the other is u.
The term symbols for this configuration are^1 Σ+uand^3 Σ+u.