Physical Chemistry Third Edition

23.3 Rotational and Vibrational Spectra of Diatomic Molecules 965

whereμ(0) is the value of the dipole moment atx0 and where the subscript 0 on the

derivative means that it is evaluated atx0. Using harmonic oscillator wave functions

and the two terms included explicitly in Eq. (23.3-8), the selection rule is

∆v0,±1 for nonzero dipole moment (23.3-9a)

All∆vforbidden for zero dipole moment (23.3-9b)

Since∆v0 is allowed, transitions in which only the rotational quantum number

changes are allowed, giving the pure rotational spectrum in the microwave region that

we have discussed. Transitions for which∆v±1 give spectra in the infrared region

in which both rotational and vibrational quantum numbers change.

It is possible to interpret the selection rule of Eq. (23.3-9b) classically. In order for

a vibrating diatomic molecule to interact with electromagnetic radiation, the molecule

must present a fluctuating dipole to the radiation as the molecule vibrates with the

correct frequency. The molecule must have a permanent dipole moment to do this.

EXAMPLE23.5

A permanent dipole moment corresponds to a constant nonzero value ofμ(0) in Eq. (23.3-8).

Using thev0 harmonic oscillator function of Eq. (15.4-10), show that a nonzero value of

μ(0) leads to a nonzero value of the transition dipole moment for thev0tov0 transition.

Solution

From Eqs. (15.4-10) and (23.3-8):

(μx) 00 

∞∫

−∞

ψ 0 ∗̂μ(x)ψ 0 dx

(a

π

) 1 / 2 ∞∫

−∞

e−ax

2 / 2

[μ(0)+

(

dμ

dx

)

0

x]e−ax

2 / 2

dx



(a

π

) 1 / 2 ∞∫

−∞

e−ax

(^2) / 2
μ(0)e−ax
(^2) / 2
dx+
(a
π
) 1 / 2 ∞∫
−∞
e−ax
(^2) / 2
(
dμ
dx
)
0
xe−ax
(^2) / 2
dx

(a
π
) 1 / 2
μ(0)
∞∫
−∞
e−ax
2
dx+
(a
π
) 1 / 2 (dμ
dx
)
0
∞∫
−∞
xe−ax
2
dx
μ(0)+ 0 μ(0) 0
Exercise 23.6
Use thev0 function and thev1 function from Eq. (15.4-11) to show that a nonzero value of
the second term on the right-hand side of the equation leads to a nonzero value of the transition
dipole moment for thev0tov1 transition.
Vibrational transitions do not occur without rotational transitions, because∆J 0
is forbidden for diatomic molecules. Figure 23.8 shows the allowed transitions that
occur between the ground vibrational state (v0) and the first excited vibrational
state (v1) of a diatomic molecule. The resulting set of spectral lines is called the