Animated Vibrational Modes of Triatomic Molecules

Giles Henderson and Christine Liberatore
Eastern Illinois University, Charleston, IL 61920

The complex vibrational motions of non-linear molecules can be resolved into 3N-6 normal modes (3N-5 in the case of linear molecules) in which each atom moves with simple harmonic motion and at the same frequency as all of the other atoms. Accurately scaled, digital animations of these motions require mathematical descriptions of both the frequency and the amplitude of the atomic displacements. The procedure for calculating these parameters from atomic masses and a given molecular structure and force field is called a normal coordinate analysis (1).

In this example, we employ a simple matrix algorithm described by Gwin (2) to characterize molecular vibrations for both linear (carbon dioxide) and non-linear (sulfur dioxide) triatomic molecules. A valence force model (3) is used to describe the molecular force field. Bond stretching and bending force constants are used to construct an energy matrix in mass weighted Cartesian displacement coordinates. An orthogonal transformation is then carried out to diagonalize this matrix to obtain vibrational frequencies and atomic displacements for each vibrational mode from their respective eigenvalues and eigenvectors. These results are then used with graphics software to generate animated composite portraits for both molecules.

Sulfur dioxide exhibits 3N-6 = 3 vibrational modes: symmetric and asymmetric stretching of the O-S-O bonds and bending of the O-S-O bond angle. Computer animations of these individual modes and their composite superposition can be seen in Figure 1.

Figure 1. Vibrational motion of SO₂.

Here we note that the stretching modes occur at higher frequency than bending. This behavior is consistent with a well established trend: stretching bonds requires more energy than bond bending. The composite superposition exhibits complex and perhaps unexpected features. We note one S-O bond undergoes large amplitude vibrations while the other S-O bond length remains nearly constant. The dynamics of the S-O bond lengths periodically alternate with the phase evolution of symmetric stretching and the slightly higher frequency, asymmetric stretching modes.

Carbon dioxide exhibits 3N-5 = 4 vibrational modes: symmetric and asymmetric stretching of the C-O bonds and two degenerate and orthogonal O-C-O bending modes. A linear combination of the degenerate bending modes gives rise to vibrational angular momentum (4). An animation of this process along with a complete set of normal modes and composite superposition are presented in Figure 2. This animation reveals how the missing rotational degree of freedom is reallocated to vibration. We have selected an oblique viewing angle to reveal the out-of-plane components of this angular motion.

Figure 2. Vibrational motion of CO₂.

The opportunity to simultaneously view individual normal modes greatly enhances our understanding of the complex dynamics of composite zero point vibrational motions. These simple examples suggest that larger molecules might periodically exhibit unexpected, non-equilibrium structures with enhanced chemical reactivity. Reactive geometries brought about by linear combinations of normal mode vibrations can only exist when certain critical phase requirements are fulfilled. The concepts illustrated here provide insight on how reaction rate constants depend on the dynamics of molecular vibrations.

References

1. Wilson, E. B.; Decius, J. C.; Cross, P. C. Molecular Vibrations, McGraw-Hill: New York, 1955 (Dover, New York: 1980).
2. Gwin, W. D. J. Chem. Phys. 1971, 55, 477.
3. Herzberg , G. Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand Reinhold: New York, 1945; 131-201.
4. Herzberg , G. Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand Reinhold: New York, 1945; 75.

Abstract
	J. Chem. Educ. 1998, 75, 779.