Motion of the Proton in a Two-Dimensional Double Minimum Potential

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Description: A two dimensional double minimum potential energy surface is used to represent proton exchange in intramolecularly hydrogen bonded molecules, where the exchange is coupled to other internal motions of the molecule (as is found in malonaldehyde), or where there are two hydrogen bonded sites (as is found in 2,7-dimethylnapthazarin). The stationary states were found by the variational method using the eigenfunctions of the two dimensional harmonic oscillator for the two dimensional double minimum potential. The time dependence of the system was found by expanding the initial state of the proton's position at t=0 in terms of the stationary states. The expectation value of the proton's position does not follow the classical "reaction coordinate" but instead oscillates between the wells of the potential energy surface with a motion that can be described as the superposition of two anharmonic oscillations, one along a line connecting the wells, and one perpendicular to this line. The frequency of the oscillation along the line connecting the wells depends on the barrier, the interminimal distance, the curvature at the minima, the mass of the moving particle, and the energies of the saddle points. The amplitude of the oscillation along the y axis increases with the energy difference between the saddle points. A tunneling splitting frequency of 1.76 cm('-1) was found for malonaldehyde, which is an improvement over the value obtained using the classical one dimensional "reaction coordinate".
Language: English
Format: Degree Work