Physical Chemistry Third Edition

838 20 The Electronic States of Diatomic Molecules

PROBLEMS

Section 20.2: LCAOMOs. Approximate Molecular
Orbitals That Are Linear Combinations of Atomic
Orbitals

20.7 Sketch the orbital region of theσg 3 sLCAOMO.

20.8 Sketch the orbital regions for the six LCAO molecular

orbitals that can be formed from the 3patomic orbitals.

20.9 Sketch the orbital region of the following LCAOMO,

with the bond axis on thezaxis:

ψcAψ 1 sA+cBψ 2 pyB

Explain why this orbital is unusable ifcAandcBare both

nonzero.

20.10Sketch the orbital region of the following LCAOMO:

ψσg 3 pc(ψ 3 pzA+ψ 3 pzB)

20.11By inspection of the orbital regions, predict which

united-atom orbital will result from each of the following

LCAOMOs:

a.πu 2 px

b.πu 2 py

20.12By inspection of the orbital regions, predict which

united-atom orbital will result from each of the following

LCAOMOs:

a.σg 2 pz

b.π∗g 2 py

20.13Show that the normalization constant for theσ∗u 1 s

LCAOMO is

cu

1

√

2 − 2 S

(20.2-16)

20.3 Homonuclear Diatomic Molecules

Homonuclear diatomic molecules have two nuclei of the same element. We base our

discussion of these molecules on the LCAO molecular orbitals of the H+ 2 molecule

ion in much the same way as we based our discussion of multielectron atoms on the

hydrogen-like atomic orbitals.

Nucleus A

rAB

x

y

z

r1A

r2A

r1B

r2B

r 12

Nucleus B

Electron 1

Electron 2

Figure 20.10 The Hydrogen Molecule
System.

The Hydrogen Molecule

Figure 20.10 depicts the hydrogen molecule, consisting of two hydrogen nuclei at

locations A and B and two electrons at locations 1 and 2. The distances between the

particles are labeled in the figure. The Born–Oppenheimer Hamiltonian is

ĤBO−h ̄

2

2 m

(

∇^21 +∇ 22

)

+

e^2

4 πε 0

(

1

rAB

−

1

r1A

−

1

r1B

−

1

r2A

−

1

r2B

+

1

r 12

)

ĤHMI(1)+ĤHMI(2)+

e^2

4 πε 0 r 12

+

e^2

4 πε 0 rAB

Ĥel+

e^2

4 πε 0 rAB

̂Hel+Vnn (20.3-1)

whereĤHMI(1) and̂HHMI(2) are the hydrogen-molecule-ion electronic Hamiltonian

operators denoted byĤelin Eq. (20.1-4). The nucleus–nucleus repulsion termVnnis a

constant in the Born–Oppenheimer approximation, sincerABis fixed. We omit it during

the solution of the electronic Schrödinger equation and add it to the resulting electronic

energy eigenvalue to obtain the Born–Oppenheimer energy, as in Eq. (20.1-6).