The infrared spectrum of gas samples shows the effect of rotational-energy changes along with the vibrational energy change.
As we know from the interpretations given to thermodynamic properties of gases, gas molecules are simultaneously rotating and vibrating. It follows that an absorption spectrum or the Raman spectrum of a gas might show the effects of changes in both rotational and vibrational energies. The expanded view of the infrared absorption band of HCl shows the band structure that must be attributed to the rotational energy changes accompanying this fundamental, v = 0 to v =1, vibrational transition.
The rigid-rotor and harmonic-oscillator expressions can be combined to give the rotation vibration allowed energy expression
ε_{rot-vib} = (v + 1/2)hv_{vib} + BJ(J + 1) v = 0, 1, 2 ...; J = 0, 1, 2 ....
The energy pattern that includes only the v = 0 and v = 1 levels and a few of the rotational levels for each vibrational state.
Analysis of the absorption spectrum for gas-phase molecules for which the energy expression applies can be made on the basis of this energy pattern and the selection rules that govern the transitions. Again the rules Δv = ±1 and ΔJ = ±1 hold, and while only Δv = ±1 and ΔJ = -1 are now possible. Transitions allowed by these rules are included.
The energies of the various transitions are determined by the rotation vibration energy expressions. In each case the vibrational contribution is given by hvvib. The rotational contributions are obtained and the J value in the v = 0 state, which determines the J value in thev = 1 state. Expressions for the energies of the components of an HCl type of rotation vibration band. Now, the transition lines have been placed on the diagram in order of increasing ε, or v?, from left to right. They thus are obtained as the expected components of a rotation-vibration band.
This analysis leads to a set of equally spaced components of the spectral band at either side of the band center. The spacing can be identified as 2B?. At the band center a gap equal to 4B?occurs. All these features are generally borne out by the rotation vibration band of HCl.
The lower frequency side of a rotation vibration band is known as the P branch, and the high frequency side as the R branch. In some cases, but not in the HCl type of spectrum, a central branch known as the Q branch occurs.
This analysis shows that a measurement of the spacing of the components of a value for the moment of inertia I. values so obtained are often not as precisely determined as those from microwave studies of pure rotational spectra, but the infrared region is more easily accessible. A similar analysis of the rotation vibration band structure could be carried through for Raman bands.
Dependence of bond length on vibrational state: now it is time to admit, that the spacing between the components of a rotation vibration band is not, in fact, constant. The rigid rotor harmonic oscillator model can be easily modified so that the observed spreading out of the P branch and closing up of the R branch can be accounted for. We recognize that the average bond length in the v = 1 vibrational state. (You might even anticipate that the average bond length in the v = 1vibrational state need not to be identical to the average bond length in the v = 0 state. If the bond length is different in different vibrational states, so also are the moment of inertia and the rotational constant B. let us denote the rotational constant for the ground v = 0 sate by B_{0} and that for the v = 1 state by B_{1}. Now the spacing between the components of a rotation vibration band depends on hvvib and on both B_{0} and B1. The energy change for the first line of the Rbranch, for example, is hv_{vib} + 2B_{1}, and that of the second line is hv_{vib} + 6B_{1} - 2B_{0}. Thus, the spacing between these lines is 2(2B_{1} - B_{0}). The spacing between the next two lines of the Rbranch is 2(3B_{1} - 2B_{0}). Thus, in general, the spacing between successive pairs of lines of the Rbranch and the P branch will depend on B_{1} and B_{0} and will be constant only if B_{1} = B_{0}. In spite of this added complexity, the value of B_{1} and B_{0} can be easily by modifying the graphical treatment.
The energy of difference between pairs of transitions that start at the same J level of the v = 0state can be used to deduce B_{1}, the rotational constant for the v = 1 state. The energy difference between pairs of transitions that end at the same J level of the v = 1 state that gives result for B_{1} and pairs that give results for B_{0}. Additional pairs can be treated, and average values of B_{1} and B_{0} can be obtained for all suitable component pairs. The validity of the model is determined by the constancy of the values for B_{1} and B_{0} obtained from the various component pairs.