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Iterative Methods for Computing Vibrational Spectra
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Inverse Multiquadratic Functions as the Basis for the Rectangular Collocation Method to Solve the Vibrational Schrödinger Equation

Department of Mechanical Engineering, National University of Singapore, Block EA #07-08, 9 Engineering Drive 1, Singapore 117576, Singapore
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Mathematics 2018, 6(11), 253; https://doi.org/10.3390/math6110253
Received: 28 September 2018 / Revised: 12 November 2018 / Accepted: 13 November 2018 / Published: 15 November 2018
(This article belongs to the Special Issue Computational Spectroscopy)
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). View Full-Text
Keywords: Schrödinger equation; vibrational spectrum; collocation; inverse multiquadratic function; rectangular matrix Schrödinger equation; vibrational spectrum; collocation; inverse multiquadratic function; rectangular matrix
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Kamath, A.; Manzhos, S. Inverse Multiquadratic Functions as the Basis for the Rectangular Collocation Method to Solve the Vibrational Schrödinger Equation. Mathematics 2018, 6, 253.

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