Abstract

The one-dimensional Schrödinger equation in reduced form is solved for the potential function <i>V=z</i>⁴+<i>Bz</i>² where <i>B</i> may be positive or negative. The first 17 eigenvalues are reported for 58 values of <i>B</i> in the range −50 ≤ <i>B</i> ≤ 100. The interval of <i>B</i> between the tabulated values is sufficiently small so that the eigenvalues for any <i>B</i> in this range can be found by interpolation. At the limits of the range of <i>B</i> the potential function approaches that of a harmonic oscillator with only small anharmonicity. The effect of a small <i>Cz</i>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 <i>B.</i> 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.

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