Rotation-Vibration Transitions of a Parallel Band of a Symmetric Rotor

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This Demonstration shows the energy level transitions associated with the lines observed in the rotationally resolved infrared spectrum of a parallel band of a symmetric rotor. A symmetric rotor possesses a greater-than-two-fold rotation axis (symmetry element) and as a result there is an inherent degeneracy of the vibrational energy levels. This degeneracy, along with the Coriolis splitting of the vibrational levels, has been neglected in this Demonstration for clarity. The observed transitions are between non-degenerate vibrational energy levels and the fundamental vibrational transition is coupled with rotational transitions in which , .


Symmetric rotors, such as ammonia and benzene, are molecules containing two inertial axes with equal moments of inertia and a principal inertial axis with a unique moment of inertia. The vibrational selection rule is for an absorption spectrum. Since the change in the dipole moment is parallel to the principal axis for a parallel band spectrum of a symmetric rotor, the rotational selection rules are , if and , if , with the further restriction that . With two rotational quantum numbers, and , each value of is associated with a series of energy levels, as shown in the top graphic.

Because for a parallel band, a rotation-vibration transition can only occur between lower and excited state energy levels with the same value of . Transitions that satisfy the vibrational and rotational selection rules produce a spectrum of lines, as shown in the bottom graphic. The energy difference between the lower and excited state of a given transition determines the position of the line. The spectrum consists of a series of sub-bands, one for each value. Each sub-band is composed of a P branch (, , ), Q branch (, , , appears when ), and R branch (, , ). Since there are a decreasing number of lines at the beginning of each branch as the value of increases.

In this Demonstration, you can view the energy level transitions for the full spectrum or you can choose to focus on the individual branches. The transition label and energy can be found to the left within the controls area. The axis lower and upper limit controls will allow you to zoom in on any region of the spectrum.


Contributed by: Whitney R. Hess and Lisa M. Goss (March 2011)
(Idaho State University)
Open content licensed under CC BY-NC-SA



In this Demonstration, lower state constants are indicated by a superscript double prime (″) and excited state constants are indicated by a superscript prime (′).

The spectrum in the bottom graphic is simulated at a temperature of 50 Kelvin and it is assumed that the centrifugal distortion constants (, , and ) and the anharmonicity constant () are negligible. However, the interaction of rotation and vibration is considered by using unequal values for the lower and excited state rotational constants and (). To simulate the spectrum, , and were used.

For the top graphic, the axis in the energy level diagram is arbitrary. The arrows indicating transitions and the energy levels corresponding to each value are spread out only for clarity.

The transitions are labeled according to the designation , where , , , and are as follows:

: transitions with are designated with a superscript Q, and all transitions in a parallel band fall into this category

: P, Q, or R depending on whether , respectively

: the value of is indicated by the subscript

: the value of is enclosed in the parentheses

For example, indicates the line corresponding to the transition in the Q branch within the sub-band.

Snapshots 1, 2, 3, and 4: full spectrum, P branch, Q branch, R branch views, respectively


[1] P. Atkins and J. de Paula, Physical Chemistry, New York: Oxford University Press, 2006.

[2] G. Herzberg, Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules, Princeton, NJ: D. Van Nostrand Company, Inc., 1945.

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