Wolfram Demonstrations Project

This Demonstration shows the rotation-vibration energy level transitions that correspond to the lines observed in a rotationally resolved infrared spectrum of a perpendicular band of a symmetric rotor. The inherent degeneracy of the vibrational levels (since a symmetric rotor possesses a greater-than-twofold rotation axis) has been neglected in order to maintain clarity in this Demonstration. The transitions occur between nondegenerate vibrational energy levels, and the fundamental vibrational transition is accompanied by rotational transitions in which

Symmetric rotors, such as benzene and ammonia, are molecules that possess a principal inertial axis with a unique moment of inertia and two other inertial axes with equal moments of inertia. A perpendicular band spectrum of a symmetric rotor results when the change in the dipole moment is perpendicular to the principal axis. The vibrational selection rule is

for an absorption spectrum, and for a perpendicular band the rotational selection rules are

, and the allowed transitions are further restricted by

Because

for a perpendicular band, rotation-vibration transitions can only occur between lower and excited state energy levels with different values of

. The series of sub-bands corresponding to

are the "positive" sub-bands and the series of sub-bands corresponding to

are the "negative" sub-bands. As can be seen from the energy level diagram (top graphic), the transition energies for the lines in the "negative" sub-bands are smaller than the transition energies for the lines in the "positive" sub-bands, resulting in the "negative" sub-bands being shifted to lower wavenumbers in relation to the positions of the "positive" sub-bands in the simulated spectrum (bottom graphic). Each sub-band consists of a

branch (

), and

branch (

). There is no

"negative" sub-band, and due to the restriction of

, there is an increasing number of lines missing from the beginning of each branch as

increases.

In this Demonstration, you can explore the energy level transitions within the "positive" and "negative" sub-bands of a particular value of

, and the

axis lower and upper boundary controls let you zoom in on any region of the spectrum. The color coding, transition labels, and transition energies can be found to the left and right within the controls area.

Contributed by: Whitney R. Hess and Lisa M. Goss
(Idaho State University)

Slider Zoom
Gamepad Controls
Automatic Animation

In this Demonstration, a superscript double prime (″) indicates lower state constants and quantum numbers and a superscript prime (′) indicates excited state constants and quantum numbers.

The spectrum in the bottom graphic is simulated at a temperature of 50 Kelvin and with the assumptions that the centrifugal distortion constants and anharmonicity are negligible. Unequal values for the lower and excited state rotational constants

and

were used in order to account for the interaction of rotation and vibration

. The values of

and

were used to simulate the spectrum.

The

axis in the top graphic is arbitrary. The arrows indicating transitions and the energy levels corresponding to each value of

are spread out along the

axis only for clarity.

The transition labels are written as

, where

, and

are as follows:

: transitions with

("positive" sub-bands) are designated with a superscript

and transitions with

("negative" sub-bands) are designated with a superscript

;

, or

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

branch within the

"negative" sub-band.

Snapshots 1 and 2: full spectrum (does not utilize the Manipulate functionality) and sub-band views, respectively

Snapshot 3: using the

axis lower and upper boundary controls will zoom in on a region of the spectrum and if a transition of interest is outside of the chosen range, the message "transition out of

axis range" will appear

Snapshot 4: the separation between the "positive" and "negative" sub-bands becomes larger as a result of the greater difference in the transition energies between the sub-bands as

increases

References

[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.