Physical Chemistry , 1st ed.

electron and nucleus 2, and a repulsive electrostatic potential between nucleus

1 and nucleus 2. The complete Hamiltonian for H 2 is

Hˆ

2



m

2

p

^2 p 1 

2



m

2

p

^2 p 2 

2



m

2

e

e^2 

4

e



2

0 r 1



4

e



2

0 r 2



4

e



2

0 R



(12.35)

where ^2 is the three-dimensional spatial second derivative for each of the

three particles (^2 p 1 refers to the first proton,^2 p 2 refers to the second proton,

and e^2 refers to the electron). The first two potential energy terms (the fourth

and fifth terms in equation 12.35) are the attractivepotential between the elec-

tron and proton 1 and proton 2, respectively (hence the negative signs), and

the final term is the repulsivepotential between the two nuclei (hence the pos-

itive sign). The mpand meare the masses of the proton and electron. The dis-

tances r 1 ,r 2 , and Rare as defined in Figure 12.11.

As might be expected, no known analytic wavefunctions are eigenfunctions

of the Hamiltonian operator in equation 12.35. Some simplifications are

needed in order to determine approximate solutions using perturbation or

variation theory. One of the complications of this system is that there are now

two nuclei, and a proper wavefunction should take into account not only the

behavior of the electron but also the behavior of the nuclei. It should be clear

that if the relative positions of the nuclei change (for example, during a vibra-

tion in which the nuclei are moving alternately closer and farther apart) then

the electronic motion will also change to compensate. Any true wavefunction

for electrons needs to consider nuclear behavior as well.

However, nuclei are much heavier than electrons (a proton has 1836 times

the mass of an electron), so it can be suggested that nuclei move much more

slowly than electrons. In fact, it can be assumed that the nuclei move so much

more slowly than electrons that for all intents and purposes the motion of an

electron can be approximated as if the nuclei were not moving.Although the

nuclei aremoving, we treat their motion independently from the motion of

the electrons. This statement is called the Born-Oppenheimer approximation

after Max Born (Figure 12.12) and J. Robert Oppenheimer (Figure 12.13).

Born and Oppenheimer’s statement is the ultimate basis for molecular quan-

tum mechanics.

Mathematically, the Born-Oppenheimer approximation is written as

molecule nuc  el (12.36)

which says that the complete molecular wavefunction is the product of a nu-

clear wavefunction and an electronic wavefunction. This treatment is reminis-

cent of how we solved the 3-D particle-in-a-box and 3-D rotational motion:

separation of variables. The complete Hamiltonian for H 2 can be approxi-

mated by two separate Schrödinger equations. The Schrödinger equation for

the electronic part is

 2



m

2

e

e^2 

4

e



2

0 r 1



4

e



2

0 r 2



4

e



2

0 R

elEelel (12.37)

where Ris the internuclear distance and is fixed at some value. Thus, the last

term in the parentheses represents, for a given value ofR,a fixedpotential en-

ergy value. The Schrödinger equation for the nuclei has the form

 2



m

2

e

^2 p 1 

2



m

2

e

^2 p 2 Eel(R)nucEnucnuc (12.38)

404 CHAPTER 12 Atoms and Molecules

Figure 12.13 J. Robert Oppenheimer (1904–
1967). With Born, Oppenheimer helped develop
quantum mechanics for application to molecules.
Oppenheimer is probably better known, however,
for leading the Manhattan Project, which devel-
oped the first atomic bombs, during World
War II.

Figure 12.12 Max Born (1882–1970). Not
only did he develop the probabilistic interpreta-
tion of the wavefunction, but he also devised a
quantum-mechanical description for molecules.