Two new numerical methods, the log derivative and the renormalized Numerov, are developed and applied to the calculation of bound‐state solutions of the one‐dimensional Schroedinger equation. They are efficient and stable; no convergence difficulties are encountered with double minimum potentials. A useful interpolation formula for calculating eigenfunctions at nongrid points is also derived. Results of example calculations are presented and discussed.

New numerical methods applied to solving the one‐dimensional eigenvalue problem

Disagreement between calculated vibrational eigenvalues for the E, F 1Σ+g state of H2 obtained by three different numerical techniques has been interpreted in the literature in terms of inherent inadequacies of these techniques. We have tested these three numerical procedures by using each to obtain the eigenvalues of an unsymmetrical double minimum potential. An analytical potential was used to eliminate uncertainties introduced by interpolation. All three techniques are shown to give accurate results when properly applied.

Comparison of three numerical techniques for calculating eigenvalues of an unsymmetrical double minimum oscillator

On quantal bound state solutions and potential energy surface fitting. A comparison of the finite element, Numerov-Cooley and finite difference methods

For single and double minimum model potentials, energy eigenvalues are calculated using the Froman and Froman phase‐integral approximations and compared with exact (numerical) quantum mechanical results For the double minimum potential, results are obtained both from the correct phase‐integral quantization condition including quantum effects near the barrier maximum (the σ term) up to and including the fifth‐order approximation, and from the quantization condition with the σ term neglected up to and including the 13th‐order approximation When the σ term is included, the third‐ and fifth‐order eigenvalues are accurate enough for all practical purposes, even close to the barrier maximum, but if the σ term is neglected, the phase‐integral quantization condition breaks down near the barrier maximum, and this breakdown becomes more dramatic with increasing order This property is used to discuss the question of which order of phase‐integral approximation will give the optimum result For a single minimum LJ(

Reliability of high-order phase-integral eigenvalues for single and double minimum potentials

On the numerical integration of the Schrödinger equation with a double-minimum potential

The finite‐difference boundary value method is applied to the calculation of Born–Oppenheimer vibrational energies and expectation values of R−2 for an excited state of H2. We estimate the accuracy attainable by this method, point out a systematic error in the previous calculations of Tobin and Hinze, and correct several unjustified statements in the literature. Finally we point out that there is a large uncertainty in the final results due to choice of interpolation scheme.

Application of the finite‐difference boundary value method to the calculation of Born–Oppenheimer vibrational eigenenergies for the double‐minimum E,F1Σ+g state of the hydrogen molecule

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Application of the finite‐difference boundary value method to the calculation of Born–Oppenheimer vibrational eigenenergies for the double‐minimum E,F1Σ+g state of the hydrogen molecule