Physical Chemistry , 1st ed.

electronic wavefunctions are. Electrons in molecules are described approxi-

mately with orbitals just like electrons in atoms are described by orbitals. We

have seen how quantum mechanics treats atomic orbitals. How does quantum

mechanics treat molecular orbitals? Molecular orbital theory is the most pop-

ular way to describe electrons in molecules. Rather than being localized on in-

dividual atoms, an electron in a molecule has a wavefunction that extends over

the entire molecule. There are several mathematical procedures for describing

molecular orbitals, one of which we consider in this section. (Another per-

spective on molecular orbitals, called valence bond theory, will be discussed in

Chapter 13. Valence bond theory focuses on electrons in the valence shell.)

Consider what happens when a molecule is formed: two (or more) atoms

combine to make a molecular system. The individual orbitals of the separate

atoms combine to make orbitals that span the entire molecule. Why not use

this description as a basis for defining molecular orbitals? This is exactly what

is done. By using linear variation theory, one can take linear combinations of

occupied atomic orbitals and mathematically construct molecular orbitals.

This defines the linear combination of atomic orbitals—molecular orbitals

(LCAO-MO) theory, sometimes referred to simply as molecular orbital theory.

In the case of H 2 , the molecular orbitals can be expressed in terms of the

ground-state atomic orbitals of each hydrogen atom:

H 2 c 1 H(1)c 2 H(2) (12.39)

where H(1)refers to a ground-state (that is, 1s) atomic wavefunction from hy-

drogen 1, and H(2)refers to a 1satomic wavefunction from hydrogen 2.

Because both hydrogen atoms participate equally in the molecule, it can be ar-

gued that the two constants c 1 and c 2 have the same magnitude. It can also be

argued that there are two linear combinations are possible, a sum and a dif-

ference of the two atomic orbitals (AOs). Therefore, the two atomic orbitals are

combining to make two molecular orbitals(MOs) having the forms

H 2 ,1c 1 (H(1)H(2))

H 2 ,2c 2 (H(1)H(2))

(12.40)

Remember that when natomic orbitals are used, there will be nlinearly inde-

pendent combinations to describe nmolecular orbitals. At this point we can-

not assume that c 1 c 2. A graphical representation of the sum and difference

of the two atomic orbitals is shown in Figure 12.16. Each hydrogen wavefunc-

tion is spherically symmetric, although the combination of two atoms makes

a system that is no longer spherically symmetric. However, we note that it does

have cylindricalsymmetry, so Figure 12.16 actually represents the magnitudes

of the wavefunctions along the axis of a cylinder, which is the internuclear axis

of the molecule.

Just as in linear variation theory, the coefficients can be determined using

a secular determinant. But unlike the earlier examples using secular determi-

nants, in this case some of the integrals are not identically zero or 1 due to

orthonormality. In cases where there is an integral in terms of*H(1)H(2)or

vice versa,we cannot assume that the integral is identically zero. This is because

the wavefunctions are centered on different atoms. The orthonormality condi-

tions to this point are only strictly applicable to wavefunctions of the same sys-

tem. Since the ideal wavefunctions H(1)and H(2)are centered on different

nuclei, the orthonormality condition does not automatically apply. The solu-

tion to the secular determinant will therefore be slightly more complicated.

406 CHAPTER 12 Atoms and Molecules

H1 H2

(H 2 +), 2

(H 2 +), 1

H H

±

H

H

HH

or

Figure 12.16 Representations of H 2 molec-
ular orbitals from linear combinations of hydro-
gen atomic orbitals. The MO in the middle plot
is the sum of the two AOs at the top, with elec-
tron density concentrated between the nuclei. The
lower MO is the difference of the two AOs, with
electron density concentrated more outside the
two nuclei.