Physical Chemistry Third Edition

22.2 The Nonelectronic States of Diatomic Molecules 919

b. Find the change in energy if one of the translational

quantum numbers is increased by unity from its

value in part a. Find the ratio of this change to the

translational energy. Compare your values with those

of Example 22.2.

c.The first excited electronic level of xenon is 8.315 eV

above the ground level. Find the ratio of the energy

change in part b to this energy difference.

22.2 The Nonelectronic States of Diatomic

Molecules

We assume that the Born–Oppenheimer Schrödinger equation for the electrons of

a diatomic molecule has been solved repeatedly for different internuclear distances,

giving the electronic energy,Eel, as a function of the internuclear distance,rAB. The

Born–Oppenheimer energy is the sum of the electronic energy and the nuclear–nuclear

repulsion energy,Vnn:

EBOEel+Vnn (22.2-1)

For a diatomic molecule, the nuclear–nuclear repulsion energy is given by

Vnn

e^2 ZAZB

4 πε 0 rAB

Vnn(rAB) (22.2-2)

whereeis the charge on a proton,ZAis the number of protons in nucleus A,ZBis

the number of protons in nucleus B, and where the internuclear distance is given by a

three-dimensional version of the theorem of Pythagoras:

rAB[(xB−xA)^2 +(yB−yA)^2 +(zB−zA)^2 ]^1 /^2 (22.2-3)

The Born–Oppenheimer energy depends only onrAB

EBOEel(rAB)+Vnn(rAB)V(rAB)

BecauseEBOdepends only on nuclear position, it acts like a potential energy for nuclear

motion, and we denote it asV(rAB).

The Hamiltonian operator for nuclear motion is now

Ĥnuc−h ̄

2

2 mA

∇A^2 +

−h ̄^2

2 mB

∇B^2 +V(rAB)+Vext (22.2-4)

where∇A^2 and∇B^2 are the Laplacian operators for the nuclear coordinates. The electrons

in the molecule have been removed from the problem and act only as a source of the

potential energy functionV(rAB). If the molecule is not in any containerVextis equal

to a constant that we can set equal to zero. If the molecule is in a container the collision

of a molecule with the container wall will be slightly different for different rotational

and vibrational states of the molecule. If the container is large compared with the size

of the molecule, it should be a good approximation to ignore this fact and to assume

thatVextis equal to zero if the center of mass of the molecule is inside the container

and approaches infinity outside of the container.

SinceVextdepends only on the center-of-mass coordinates,̂Hnuccan be separated

into a center-of-mass term and a relative term in the same way as is discussed in