Physical Chemistry , 1st ed.

Example 12.14

Using R1.32 Å, evaluate S 12 ,H 12 ,E 1 ,E 2 , and the wavefunctions for H 2 .

Use 13.60 eV as the value for the energy of a 1shydrogen electron, and ex-

press answers in units of eV. (An electron volt, or eV, equals 1.602    10 ^19 J

and is a useful unit for atomic-scale values of energy.)

Solution

Since both Rand a 0 are in units of Å, it is unnecessary to consider any unit

conversions. Using the expressions above,

S 12 e1.32Å/0.529Å    1 

0

1

.

.

5

3

2

2

9

Å

Å



3

(

(

1

0

.

.

3

5

2

29

Å

Å

)^2

)^2

 0.459

H 12 (13.60 eV)(0.459) 2(13.60 eV) e1.32Å/0.529Å 1 

0

1

.

.

5

3

2

2

9

Å

Å



14.08 eV

E 1 18.97 eV

E 2 0.887 eV

The wavefunctions that have these energies are

H 2 ,10.585(H(1)H(2) )

H 2 ,20.961(H(1)H(2))

12.12 Properties of Molecular Orbitals

Consider the wavefunctions determined for the MOs of H 2 . Although they

are very simple molecular orbitals, they do have certain characteristics that can

be used to describe all molecular orbitals. Figure 12.18 shows a representation

of the sum and difference of the two atomic H orbitals and their squares. Since

the probability that the electron will exist in a region is proportional to the

square of the wavefunction, Figure 12.18b indicates the probability for an elec-

tron existing in the molecule. Since the system is cylindrical, so is the proba-

bility (this is similar to our discussion of radial shell probabilities for the hy-

drogen atom). In the lower-energy wavefunction (Figures 12.18a and b), the

probability of the electron in the cylindrical volume betweenthe two nuclei has

increased relative to the original, separate atomic wavefunctions. Since the two

positive nuclei would otherwise repel each other, this increase in electron prob-

ability or electron density serves to lower the repulsion between the nuclei and

stabilize the entire molecular system; that is, it lowers the energy. Any molec-

ular orbital whose energy is lower than the energy of the separated atomic or-

bitals is called a bonding orbital.

The higher-energy molecular orbital (Figures 12.18c and d), on the other

hand, concentrates more electron probability in a cylindrical volume outside

the two nuclei. An electron in this orbital would therefore have a decreased

probability of being found between the nuclei, and the repulsion between pos-

itively charged nuclei would increase, destabilizing the overall system. Any

molecular orbital whose energy is higher than the energy of the separated

atomic orbitals is called an antibonding orbital.This antibonding orbital has a

13.60 eV (14.08 eV)

1 0.459

13.60 eV 14.08 eV



1 0.459

12.12 Properties of Molecular Orbitals 409

(H+ 2 ), 2

(H+ 2 ), 1

H

H

H

(d)

H

(H 2 +), 1^2

H
(b)

H

(a)

(c)

(H 2 +), 2^2

HH

Figure 12.18 Radial plots of the molecular

orbitals and their probabilities for H 2 . For the

bonding orbital (a), there is an increased proba-

bility for the electrons between the nuclei (b). For

the antibonding orbital (c), there is a decreased

probability between the nuclei (d). The anti-

bonding orbital shows a node between the two

nuclei.