Physical Chemistry Third Edition

22.2 The Nonelectronic States of Diatomic Molecules 927

The term containing the parameterDcauses the corrected levels to be more closely

spaced for larger values of the rotational quantum numberJthan for smaller values

ofJ. It corresponds to centrifugal stretching of the molecule, which increases the value

of the moment of inertia for larger values ofJ. This effect is typically small, and the

term inDcan be neglected except for highly accurate work. The term containing the

parameterxecauses the corrected energy levels to be more closely spaced for larger

values of the vibrational quantum numberνand is a correction for the anharmonicity

of the potential energy function. The term containing the parameterαcontains both the

vibrational and rotational quantum numbers, and expresses the interaction of vibration

and rotation. The origin of this interaction can be seen in Figure 22.3b. For larger values

ofνthe classically allowed region of the vibrational coordinate moves to the right in

the figure, so that the moment of inertia is larger for larger values ofν, lowering the

rotational energy below that of the uncorrected expression.

EXAMPLE22.5

Calculate the energy of theν2,J2 level of the CO molecule using the values of the

parameters:νe 6. 5049 × 1013 s−^1 ,xe 6. 124 × 10 −^3 ,Be 5. 7898 × 1010 s−^1 ,D

1. 83516 × 105 s−^1 ,α 5. 24765 × 108 s−^1.

Solution

E/hνe(5/2)−νexe(5/2)^2 +Be(2)(3)−D(36)−α(5/2)(6)

(6. 5049 × 1013 s−^1 )(5/2)−(3. 9836 × 1011 s−^1 )(5/2)^2

+(5. 7898 × 1010 s−^1 )(6)−(1. 83516 × 105 s−^1 )(36)

−(5. 24765 × 108 s−^1 )(5/2)(6)

 1. 6262 × 1014 s−^1 − 2. 4897 × 1012 s−^1 + 3. 4739 × 1011 s−^1

− 6. 6066 × 106 s−^1 − 7. 8715 × 109 s−^1

 1. 6047 × 1014 s−^1

E(6. 6261 × 10 −^34 J s)(1. 6047 × 1014 s−^1 ) 1. 0633 × 10 −^19 J

Theαterm is fairly small and theDterm is insignificant to five significant digits.

Theνexeterm is large compared with the main rotational term (theJterm), but is fairly

small compared with theνeterm.

The energy levels are often given in terms of energies divided byhc, wherecis the

speed of light andhis Planck’s constant. This quantity has the dimensions of reciprocal

wavelength, and its difference for two levels is equal to the reciprocal of the wavelength

of the photon emitted or absorbed in a transition between these levels. It is sometimes

called thetermof the energy level, and denoted byT.

TvJ

EvJ

hc

 ̃ve

(

v+

1

2

)

+ ̃vexe

(

v+

1

2

) 2

+ ̃BeJ(J+ 1 )

− ̃DJ^2 (J+ 1 )^2 − ̃α

(

v+

1

2

)

J(J+ 1 ) (22.2-50)