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Eigenvalues of the Potential Function V=z4Bz2 and the Effect of Sixth Power Terms

Volume 24, Number 1 (Feb. 1970) Page 73-80

Laane, Jaan


The one-dimensional Schrödinger equation in reduced form is solved for the potential function V=z4+Bz2 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 Cz6 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.