2.3 Vibration-Rotation Spectra of Diatomic Molecules

The Rotating Oscillator

It is obvious that molecules in a gas are not prevented from rotating, nor is it realistic to expect a molecule that may be deformed by rotation not to vibrate, and indeed both occur simultaneously. To a good approximation, this can be coped with by amending the rotational constant to include the mean separation of the nuclei during the vibration (the period of vibration is small compared to that for rotation). In the same way a mean rotational constant may be defined for a given vibrational state. The effect of the interaction of vibration and rotation on energy levels is illustrated in figure 2-1 (Herzberg [1950]) for five vibrational levels.

An energy level diagram illustrating the fine structure of a vibration-rotation band of a diatomic molecule, such as hydrogen bromide (appendix A3), is shown in figure 2-2 (Herzberg [1950]).

2.4 Spectra of Other Polyatomic Molecules

In the foregoing only diatomic molecules have been considered; we shall now consider briefly and qualitatively some aspects of the spectra resulting from other polyatomic molecular configurations.

Figure 2-1. Energy level diagram for rotating oscillator (after Herzberg [1950]). Short horizontal lines represent some of the rotational levels for each of the first five vibrational levels indicated by the long lines (section 2.3).

Figure 2-2. Energy level diagram depicting fine structure of a vibration-rotation band (after Herzberg [1950]). In the main part, the vertical lines represent the allowed transitions. Below, (a) represents the spectrum that would result if interaction between vibration and rotation is allowed and (b) that which results in the absence of this interaction (section 2.3).

2.4.1 Rotational Spectra of Polyatomic Molecules

For spectroscopic purposes a molecule may be classified according to its moment of inertia in each of three orthogonal planes passing through its centre of mass (figure 2-3) as linear, spherical top, symmetric top, or asymmetric top; some of the basic properties are summarised below (data from Hollas [1995], Bernath [1995] and Herzberg [1950]).

Figure 2-3 Illustration of the planes of the moments of inertia of an oblate symmetric top molecule (section 2.4.1).

Linear Molecules

Moments of Inertia: I_a = 0, I_b = I_c.
By convention the moments of inertia are assigned in order of increasing magnitude in the three planes described in figure 2-3.
Rotational energy levels (rigid approximation): BJ(J+1).

Centrifugal Distortion Correction: -D[J(J+1)]².

It is conventional, and convenient to work in a rotating frame of reference and make use of such ‘fictitious’ parameters as centrifugal force that are not present in the laboratory frame.

D is the centrifugal distortion constant given by the Kratzer relationship (Bernath [1995]):

D = 4Be3/we2

(2.15)

where w is the vibrational frequency and the subscript e denotes an equilibrium value.

The above expression for the centrifugal distortion correction is a simplification; more rigorous treatment would include other higher order terms, but would be inappropriate here.

Spherical Top Molecules

Moments of Inertia: I_a = I_b = I_c.

Rotational energy levels (rigid approximation): BJ(J+1).

Centrifugal Distortion Correction: -D[J(J+1)]².

Symmetric Top Molecules

Moments of Inertia:
Prolate top: I_a < I_b = I_c.
Oblate top: I_a = I_b < I_c.

Rotational energy levels (rigid approximation): BJ(J+1) + (A-B)K².

Centrifugal Distortion Correction: -D_J[J(J+1)]² - D_KK⁴ - D_JKJ(J+1)K².

Asymmetric Top Molecules

Moments of Inertia:I_a ¹ I_b ¹ I_c.

Rotational energy levels: The Schrödinger equation for an asymmetric top molecule has no general analytical solution and so the energy levels in this case must be calculated numerically for any specific case of interest.

2.4.2 Vibrational Spectra of Polyatomic Molecules

Although there is an increase in complexity when moving from diatomic to polyatomic molecules, the modifications to the theory for diatomic molecules are logical and not great; as such developments are covered in standard texts only brief mention shall be made of some aspects of polyatomic vibration.

A diatomic molecule can have only one vibration mode but it can be shown (Banwell & McCash [1994]) that a molecule containing N atoms will have 3N-5 possible internal vibrations if it is linear and 3N-6 if it is non-linear. The difference between the linear and the non-linear cases arises from the fact that there is no discernible rotation about the bond axis in the case of linear molecules. The result of this is that bending modes in two orthogonal planes are indistinguishable. The non-linear case is illustrated in figure 2-4 which depicts the three (3N-6, N=3) normal vibration modes of water.

Figure 2-4 Schematic representation of the normal vibration modes of the water molecule (section 2.4.2).

It is a fundamental requirement for a transition to be visible in the infrared that there be a change in the electric dipole of the molecule (Banwell & McCash [1994]). In the case of a molecule such as carbon dioxide (figure 2-5), therefore, the symmetric stretch mode will be invisible in the infrared as it involves no change in the molecule’s dipole moment.

Figure 2-5 Schematic representation of the normal vibration modes of the carbon dioxide molecule. Only one of the two bending modes is shown on account of their degeneracy (section 2.4.2).

Only three modes are shown in figure 2-5 since although the foregoing insists that a linear triatomic molecule should exhibit four vibration modes, the two bending modes may occur in orthogonal planes which are indistinguishable, due to the symmetry of the molecule.

2.4.3 Vibration-Rotation Spectra of Polyatomic Molecules

Although the energies and selection rules for vibration-rotation transitions do depend on the shape of the molecule in question the selection rules for rotational transitions also depend on the orientation of the change in the dipole moment of the molecule due to vibration (Banwell & McCash [1994]). If this is parallel to the axis of symmetry of the molecule as is the case in the stretch modes of carbon dioxide and the bend and symmetric stretch modes of water, then the selection rules are identical to those for diatomic molecules, namely that DJ = ±1, or DJ = ±1, ±2, ±3,... for anharmonic motion. If the dipole moment oscillates perpendicular to the axis of the molecule then the selection rule for J is found to be DJ = 0, ±1. Transitions where DJ = 0 are purely vibrational and give rise to a Q-branch in the spectrum.

2.5 Some Other Aspects of Absorption Spectra

2.5.1 Line Strengths and Transition Probabilities

The foregoing discussion has been concerned merely with source and frequency structure of infrared molecular spectra with no reference to the relative strengths of any of the lines. The line strength, S, of any given line will depend on the probability of a transition between energy levels occurring and the relative energies and populations of those levels (Thorne [1988]). It may be stated as:

(2.16)

where k_s is the absorption coefficient, N₁ and N₂ are the number densities off atoms in the lower and upper levels of the transition in question and g₁ and g₂ are the degeneracies of those levels; s₀ is the frequency of the transition, B₁₂ is the Einstein coefficient describing the probability per unit time of absorption resulting in a transition from level one to level two and h and c have their usual meanings. If one assumes a Boltzmann distribution, the populations of the levels can be shown (Thorne [1988]) to vary as exp(-E/kT), where E is the energy of the level, T the temperature and k is Boltzmann’s constant. A detailed treatment of the subject may be found in texts such as Thorne [1988].