Stark Effects on Rigid Rotor Wave Functions:
A Quantum Description of Dipolar Rotors Trapped in Electric Fields as Pendulum Oscillators

Brad Logsdon and Giles Henderson
Eastern Illinois University, Charleston, IL 61920

The authors (1) developed a quantum mechanical description of how the Stark effect influences molecular orientation. The variational method was employed to calculate rotational wave functions and quantum probability functions for a linear rigid rotor, perturbed by the interaction of its dipole moment with an applied electric field. The full three-dimensional description of this problem requires two spherical polar angles to specify the orientation of the rotor in a laboratory frame. However, the calculations were greatly simplified by restricting the rotor to a plane in which the orientation was specified by a single angle. Moreover, this constraint allows students to better relate the zero-to-high field correlations to their previous experience with one dimensional harmonic oscillators.

Computer graphics were used to illustrate the shifting and splitting of the rotor's degenerate energy levels as a function of the Stark field. It was shown that the lowest lying energy levels are approximately described by a near harmonic librator model in which the rotor is trapped by the Stark field and undergoes pendulum-like motions.

Computer graphics were also used to show how the rotational wave functions evolve with increasing fields from a free rotor to a near harmonic librator. The corresponding orientational probability functions were presented and compared using both polar and Cartesian representations. The transformations induced in the rotational wave functions by the laboratory field were found to be similar to the corresponding hybridization of atomic orbitals induced by the fields of a bonding neighboring atom.

It is instructive to use classical mechanics to animate the effect of an electric field on the molecules' angular motion. The classical trajectories can then be compared to the quantum descriptions. Figure 1 shows an animation that correlates the classical motion with quantized energy levels and orientational probability functions.

Figure 1. Incremented Stark Field.

Figure 1 depicts a dipolar molecule interacting with four levels of electric field. As the field increases, the animated classical motion evolves from a free rotor, to a hindered rotor, to a large amplitude pendulum oscillator, and finally to a nearly harmonic librator. The rotational kinetic energy of the molecule is initialized at a value corresponding to the energy of the m = 1, (tagged) quantum level. The quantized energy of the m = 0 and 1 levels are shown in the upper right panel. Both polar and Cartesian plots of the orientational quantum probability function of the "tagged" state are shown in the lower panels. The power supply up-indicator is illuminated as the field in incremented-to-higher values, as displayed by the power supply volt meter. The upper right panel also displays the potential energy curve which results from the interaction of the molecular dipole with the applied electric field. The quantum descriptions of the tagged level are initially color-coded in blue to indicate that the energy of the rotor is larger than the rotational potential barrier. As the field increases to a point where the tagged level is trapped by the potential barrier, the rotational motion becomes a "pendulum oscillator" or librator motion and the energy and probability plots are color-coded in yellow.

Figure 2. Incremented Eigenstates.

Rather than describing the evolution of a single eigenstate as a function of increasing field, Figure 2 animates the classical motion for each of the first four quantum levels of a dipolar rotor interacting with a fixed electric (Stark) field. All of the graphic conventions and features described above for Figure 1 are also employed in this animation.

In addition to their pedological value, these graphics and animations suggest that it is possible to orient rotationally-cold molecules produced by supersonic expansions in a molecular beam. These circumstances will undoubtedly be exploited to control the orientation of polar molecules colliding with neutral atoms or nonpolar molecules. In addition, spectroscopy and electron diffraction studies might also be carried out on oriented gas-phase molecules rather than the usual random orientations of a static gas.

Literature Cited

1. Henderson, Giles; Logsdon, Brad; J. Chem. Educ. 1995, 72, 1021-1024.

Abstract of Original Article
	J. Chem. Educ. 1995, 72, 1021-1024
Original Article
	J. Chem. Educ. 1995, 72, 1021-1024. (PDF file; requires Adobe Acrobat Reader)