A 1 ^14 (2sO 2 sO 2 sO 2 sO)
A 1 4 1 (2px,O 2 px,O 2 px,O 2 px,O)
A 1 ^14 (2py,O^2 py,O^2 py,O^2 py,O)
A 1 4 1 (2pz,O 2 pz,O 2 pz,O 2 pz,O)
The first and second combinations are the same (the terms are simply listed
in different order), and the fifth and sixth combinations are exactly zero. The
unique wavefunctions are, after some algebraic simplification:
A 1 ^12 (1sH1 1 sH2)
A 1 ^1 sO
A 1 2 sO
A 1 2 pz,O- Analogous steps will yield proper wavefunctions with A 2 ,B 1 , and B 2 sym-
metries. They ultimately yield, for the unique combinations,
B 1 ^12 (1sH1 1 sH2)
B 1 2 px,O
B 2 2 py,O
There are no nonzero linear combinations that can be labeled with the A 2
symmetry species. Although most of the molecular wavefunctions are repre-
sented by a single atomic wavefunction in this case, this will not always be so.
We get seven unique molecular orbitals from the seven atomic orbitals.
Electron spins are not addressed explicitly by MO theory, but they are
treated with respect to the Pauli principle just as atomic orbitals are: only two
electrons can occupy any one orbital, and their spins must be opposite. Just as
in atoms, electrons in molecules fill MOs starting with the lowest-energy MO,
and in order of increasing energy. If two or more MOs are degenerate, one
electron fills each MO before pairing of electrons in orbitals (Hund’s rule).13.10 Valence Bond Theory
Previously, we have treated orbitals as covering the molecule as a whole, and
have not from the start restricted the orbitals to any one atom. Many molecu-
lar orbitals can be approximated as linear combinations of atomic orbitals.
Another way to consider molecular wavefunctions is in terms ofproductsof
atomic orbitals. This is valence bond theory,and ultimately it is very useful
for describing the structures of molecules. Valence bond (or VB) theory dates
from 1927, when W. Heitler and F. W. London constructed the first successful
quantum-mechanical approximation of the hydrogen molecule, H 2. It was de-
veloped further by J. C. Slater (of Slater determinant fame) and Linus Pauling.
We will use the hydrogen molecule as an example. At infinite interatomic
separation, each hydrogen atom has its own independent wavefunction:
H1 1 sH1(1)
H2 1 sH2(2)446 CHAPTER 13 Introduction to Symmetry in Quantum Mechanics