Preprint Article Version 1 This version is not peer-reviewed

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

Version 1 : Received: 28 September 2018 / Approved: 28 September 2018 / Online: 28 September 2018 (10:15:42 CEST)

A peer-reviewed article of this Preprint also exists.

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. 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.

Journal reference: Mathematics 2018, 6, 253
DOI: 10.3390/math6110253

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.

Subject Areas

Schrödinger equation; vibrational spectrum; collocation; inverse multiquadratic function; rectangular matrix

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