High accuracy interpolation of diatomic pseudo-potentials with high order Hermite splines

作者：Schmidling David^*

来源：Computational and Theoretical Chemistry, 2011, 963(1): 76-84.

DOI：10.1016/j.comptc.2010.10.002

• Hermite spline
• Interpolation
• Diatomic
• Pseudo-potential

Pseudo-potentials for diatomic molecules can be interpolated to high accuracy with a high order Hermite spline via the function Y = -[(E(S) - E(U))/E(U)]E(E) [(1 - z)/2](2) where E(S) is the energy of the separated atoms, E(U) the united atom energy, EE the electronic energy and z = (R - R(a))/(R + R(a)), where R is the inter-nuclear distance and Ra is an adjustable parameter. Both Y and its derivative with respect to z are very smooth, which facilitates the interpolation process.
Limiting laws are used at each end to provide spline constraints and also to replace the spline in the regions beyond the data, giving a hybrid result. The low R limiting law is of the form E(E) = E(U) + Sigma(i=2) A(i) R(i). The A coefficients are obtained by least-squares. The high R limiting law is of the form E = E(S) - Sigma C(i)R(-i); the C coefficients can be obtained from perturbation theory, least-squares or in combination.
Test data with a pseudo-potential similar to that of most diatomic molecules with added noise was interpolated with an order 12 spline at data intervals of 0.05, 0.1 and 0.2 Bohr. The resulting maximum errors in EE and vibrational energy levels were no more than several times a noise level of 10(-17), 10(-15) and 10(-10), respectively. The lowest noise level at which the maximum errors were acceptable decreased with data range, spline order and smoothness of the function. The interpolation procedure was successfully applied to the H(2) X, C and a states, for which abundant data in the form of tables of energy and gradient are available.

• 出版日期2011-1