Physical Chemistry , 1st ed.

than do the (v    1,J   1) transitions. All of the J  1 transitions

appear at higher energies than the pure vibrational transition (where Jwould

be 0), and all of the J1 transitions appear at lower energies than the pure

vibrational transition. The pure vibrational transition would fall in between

the two branches. The branch where J1 is called the P branch,and the

branch where J    1 is called the R branch.Figure 14.35 shows Pand R

branches for the rovibrational spectrum of HCl gas. Figure 14.36 shows a dia-

gram of the individual transitions in the rovibrational spectrum. In the P

branch, it can be seen that the quantum number Jdecreases by 1 for each tran-

sition. In the Rbranch,Jincreases by 1.

Rotational spectra of linear molecules can be related to a rotational constant

B, which in turn is related to the reduced mass and bond length of the mole-

cule. In rovibrational spectra, the excited vibrational state does not necessarily

have the same value for Bas the ground rotational state. We therefore need to

differentiate between B 0 and B 1 for the ground and excited vibrational states,

respectively. Also, there are anharmonicity and centrifugal distortion effects

(characterized by xeeand DJconstants, respectively) that will determine the

exact wavelength of light that a rovibrational transition will absorb. Such ef-

fects account for (1) the difference in spacing between the absorptions of the

Pbranch and the absorptions of the Rbranch, and (2) the small but observ-

able change in the separation of the absorptions within each branch. You

should be able to notice both effects in Figure 14.35. To a very good approxi-

mation, for diatomic molecules having rovibrational spectra, the lines in the

fundamental vibration spectrum (that is,v 0 →v1) can be predicted by

the following equations, which account for changes in v,J,B 0 and B 1 , and the

effects of anharmonicity and centrifugal distortion. For the Rbranch:

Eh 2 xee (B 1   B 0 )(Jlower 1) (B 1 B 0 )(Jlower 1)^2 (14.41)

 4 DJ(Jlower 1)^3

and for the Pbranch:

Eh 2 xee(B 1   B 0 )Jlower (B 1 B 0 )J^2 lower   4 DJJ^3 lower (14.42)

where Jlowerindicates that the equation uses the Jvalue of the lower rotational-

vibrational state. Notice the minor differences in signs and the terms in Jlower

in the two equations. These differences are enough to be noticeable in some

rovibrational spectra, like Figure 14.35. Although the above equations assume

the v 0 →v1 fundamental vibrational transition, they do not assume any

particular rotational state. Expressions like equations 14.41 and 14.42 are used

to calculate anharmonicities, rotational constants, and so on from experimen-

tal spectra, since in most cases there are a lot of absorptions to fit to the equa-

tions. Much of the data in Tables 14.2 and 14.4 was determined this way.

There is such a thing as a Q branch,where the change in the Jrotational

quantum number is zero; that is,J0. With no change in J, the only effects

on Efor the transition are vibrational, from the change in the harmonic vi-

brational frequency and the effects due to anharmonicity.Qbranches are

therefore much more compact than Por Rbranches. However, Figure 14.35

shows no visible Qbranch (which would be expected to occur right between

the Pand Rbranches). Recall the selection rule given in equation 14.18, which

is J1: a Jof 0 is usually forbidden, suggesting that Qbranches will not

be seen. This is true for most diatomic molecules. For polyatomic molecules,

linear or nonlinear,Qbranches may occur. However, there is no simple selec-

tion rule. Figure 14.37 shows the P,Q, and Rbranches for carbon dioxide, CO 2.

508 CHAPTER 14 Rotational and Vibrational Spectroscopy

P branch:

J   1

R branch:

J   1

  1

  0

J  4

J  0

J  3

J  2

J  1

J  0

J  1

J  4

J  3

J  2

Figure 14.36 Energy-level diagram showing
the origin ofPand Rbranches in a rovibrational
spectrum. For some molecules, a Qbranch, in
which J0, can also be seen.