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Inverse Multiquadratic Functions as the Basis for the Rectangular Collocation Method to Solve the Vibrational Schrödinger Equation

Author

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  • Aditya Kamath

    (Department of Mechanical Engineering, National University of Singapore, Block EA #07-08, 9 Engineering Drive 1, Singapore 117576, Singapore)

  • Sergei Manzhos

    (Department of Mechanical Engineering, National University of Singapore, Block EA #07-08, 9 Engineering Drive 1, Singapore 117576, Singapore)

Abstract

We explore the use of inverse multiquadratic (IMQ) functions as basis functions when solving the vibrational Schrödinger equation with the rectangular collocation method. The quality of the vibrational spectrum of formaldehyde (in six dimensions) is compared to that obtained using Gaussian basis functions when using different numbers of width-optimized IMQ functions. The effects of the ratio of the number of collocation points to the number of basis functions and of the choice of the IMQ exponent are studied. We show that the IMQ basis can be used with parameters where the IMQ function is not integrable. We find that the quality of the spectrum with IMQ basis functions is somewhat lower that that with a Gaussian basis when the basis size is large, and for a range of IMQ exponents. The IMQ functions are; however, advantageous when a small number of functions is used or with a small number of collocation points (e.g., when using square collocation).

Suggested Citation

  • Aditya Kamath & Sergei Manzhos, 2018. "Inverse Multiquadratic Functions as the Basis for the Rectangular Collocation Method to Solve the Vibrational Schrödinger Equation," Mathematics, MDPI, vol. 6(11), pages 1-9, November.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:11:p:253-:d:183007
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    Cited by:

    1. Xiaodong Lv & Guangming Zhang & Mingxiang Zhu & Huimin Ouyang & Zhihan Shi & Zhiqing Bai & Igor V. Alexandrov, 2022. "Adaptive Neural Network Global Nonsingular Fast Terminal Sliding Mode Control for a Real Time Ground Simulation of Aerodynamic Heating Produced by Hypersonic Vehicles," Energies, MDPI, vol. 15(9), pages 1-25, April.

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