Eigenvalues of the Potential Function V=z 4±Bz 2 and the Effect of Sixth Power Terms

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Abstract:

The one-dimensional Schrödinger equation in reduced form is solved for the potential function V=z 4+Bz 2 where B may be positive or negative. The first 17 eigenvalues are reported for 58 values of B in the range −50 ≤ B ≤ 100. The interval of B between the tabulated values is sufficiently small so that the eigenvalues for any B in this range can be found by interpolation. At the limits of the range of B the potential function approaches that of a harmonic oscillator with only small anharmonicity. The effect of a small Cz 6 term on this potential is studied and it is concluded that a previously reported approximation formula is quite applicable but only for positive values of B. The success of the quartic–harmonic potential function for the analysis of the ring-puckering vibration is shown; it is also demonstrated that the same potential serves as a useful approximation for many other systems, especially those of the double minimum type.

Keywords: Computer applications; Far infrared; Inversion, internal rotation, and double minimum type vibrations; Potential functions for ring-puckering

Document Type: Research Article

DOI: http://dx.doi.org/10.1366/000370270774372047

Affiliations: Department of Chemistry, Texas A&M University, College Station, Texas 77843

Publication date: January 1, 1970

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