Physical Chemistry Third Edition

824 20 The Electronic States of Diatomic Molecules

20.1 The Born–Oppenheimer Approximation

and the Hydrogen Molecule Ion

The Schrödinger equation of a two-particle system such as the hydrogen atom can

be solved mathematically by separating the center-of-mass motion from the rela-

tive motion. For atoms with more than one electron, we resorted to approximations,

including the assumption that the nucleus was stationary when studying the elec-

tronic motion. Our study of the electronic motion in molecules is based on a similar

assumption, theBorn–Oppenheimerapproximation.^1 This approximation corresponds

to assuming that the nuclei are stationary while the electrons move around them. It

is a good approximation, because electrons move much more rapidly than the nuclei

and adapt quickly to new nuclear positions. For example, as the nuclei of a diatomic

molecule carry out one period of vibration, the electrons might orbit the nuclei 500 or

1000 times.

Max Born, 1882–1970, was a
German-British physicist who
participated in the early mathematical
development of quantum mechanics.

J. Robert Oppenheimer, 1904–1967,
was an American physicist who made
important contributions to nuclear
physics, including work on the
Manhattan Project, which developed
the first nuclear weapons during World
War II.

If the electronic Schrödinger equation is solved with stationary nuclei the energy

eigenvalue is called the Born–Oppenheimer energy. If it is solved repeatedly with

different fixed values of the nuclear positions, a different value of the energy eigen-

value is obtained for each set of nuclear positions. For a diatomic molecule, the Born–

Oppenheimer energy depends only on the internuclear distance. A smooth curve can

be drawn through a plot representing the Born–Oppenheimer energy as a function of

the internuclear distance. Figure 20.1 shows such a curve. The minimum in the curve

corresponds to the equilibrium internuclear distance.

EBO

rAB

Figure 20.1 Born–Oppenheimer
Energy as a Function of Internuclear
Distance for a Diatomic Molecule
(Schematic).

x

z

y

Nucleus B

Electron

Nucleus A

Figure 20.2 The Hydrogen Molecule
Ion (H 2 +) System.

The Hydrogen Molecule Ion

The hydrogen molecule ion, H+ 2 , is the simplest possible molecule. It consists of two

nuclei and a single electron, as depicted in Figure 20.2. It is highly reactive, but it

is chemically bonded and has been observed spectroscopically in the gas phase. We

apply the Born–Oppenheimer approximation and place our coordinate system with the

nuclei on thezaxis and the origin of coordinates midway between the nuclei. One

nucleus is at position A and the other nucleus is at position B. The Born–Oppenheimer

Hamiltonian for the hydrogen molecule ion is

ĤBO−h ̄

2

2 m

∇^2 +

e^2

4 πε 0

(

1

rAB

−

1

rA

−

1

rB

)

(20.1-1)

where∇^2 is the Laplacian operator for the electron’s coordinates,mis the electron mass,

rABis the internuclear distance,rAis the distance from the electron to the nucleus at

position A, andrBis the distance from the electron to the nucleus at position B. There

are no kinetic energy terms for the nuclei because they are fixed. We denote the potential

energy of internuclear repulsion byVnn. It is equal to a constant because the nuclei are

fixed:

e^2

4 πε 0 rAB

Vnnconstant (20.1-2)

(^1) M. Born and J. R. Oppenheimer,Ann. Phys., 84 , 457 (1927).