Therefore, the four sp^3 hybrid orbitals can bond with any other molecular or-
bital or set of molecular orbitals that have either A 1 or T 2 irreducible repre-
sentation labels in Tdsymmetry.The last statement of the above example mentioned a “set of orbitals.” A
similar procedure can be applied to a combination of atomic orbitals, just as
the four sp^3 hybrid orbitals above were treated in combination. For example,
in methane the four 1satomic orbitals of hydrogen, which the symmetry op-
erations in Tdmove onto each other, represent such a set of orbitals. Perhaps
not so surprisingly, the set of four 1satomic orbitals can also be labeled as
A 1 T 2 in the Tdpoint group.13.12 Summary
Symmetry is a powerful and useful tool in chemistry. In this chapter, we have
seen some examples of how symmetry ideas and group theory can be applied
to quantum mechanics. Other topics involving symmetry will be introduced in
future chapters. As illustrated in Example 13.12, symmetry considerations will
be very important in our consideration of the spectroscopy of atoms and mol-
ecules. Symmetry will also be important when considering crystals and sur-
faces, topics covered near the end of the text. Any advanced study of chemistry
must include symmetry and group theory, not only because it can be applied
to wavefunctions, as we did here, but to the three-dimensional structures of all
molecules.456 CHAPTER 13 Introduction to Symmetry in Quantum Mechanics