An explicit spectral collocation method using nonpolynomial basis functions for the time‐dependent Schrödinger equation

2018 ◽  
Vol 42 (1) ◽  
pp. 186-203 ◽  
Author(s):  
Jinwei Fang ◽  
Boying Wu ◽  
Wenjie Liu
Keyword(s):  
Schrödinger Equation ◽  
Collocation Method ◽  
Schrodinger Equation ◽  
Basis Functions ◽  
Time Dependent ◽  
Spectral Collocation Method ◽  
Spectral Collocation

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

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 253 ◽  
Author(s):  
Aditya Kamath ◽  
Sergei Manzhos
Keyword(s):  
Schrödinger Equation ◽  
Vibrational Spectrum ◽  
Collocation Method ◽  
Schrodinger Equation ◽  
Basis Functions ◽  
Collocation Points ◽  
Basis Size ◽  
Six Dimensions ◽  
Gaussian Basis Functions

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).

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

2018 ◽  
Author(s):  
Aditya Kamath ◽  
Sergei Manzhos
Keyword(s):  
Schrödinger Equation ◽  
Vibrational Spectrum ◽  
Collocation Method ◽  
Schrodinger Equation ◽  
Basis Functions ◽  
Collocation Points ◽  
Basis Size ◽  
Six Dimensions ◽  
Gaussian Basis Functions

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.

A Collocation Method Based on Jacobi and Fractional Order Jacobi Basis Functions for Multi-Dimensional Distributed-Order Diffusion Equations

2018 ◽  
Vol 19 (7-8) ◽  
pp. 781-792 ◽  
Author(s):  
M. A. Abdelkawy
Keyword(s):  
Collocation Method ◽  
Fractional Order ◽  
Jacobi Polynomials ◽  
Basis Functions ◽  
Spectral Collocation Method ◽  
Diffusion Equations ◽  
Spectral Collocation ◽  
Orthogonal Functions ◽  
Orthogonal Function ◽  
Distributed Order

AbstractIn this work, shifted fractional-order Jacobi orthogonal function in the interval $[0,\mathcal{T}]$ is outputted of the classical Jacobi polynomial (see Definition 2.3). Also, we list and derive some facts related to the shifted fractional-order Jacobi orthogonal function. Spectral collocation techniques are addressed to solve the multidimensional distributed-order diffusion equations (MDODEs). A mixed of shifted Jacobi polynomials and shifted fractional order Jacobi orthogonal functions are used as basis functions to adapt the spatial and temporal discretizations, respectively. Based on the selected basis, a spectral collocation method is listed to approximate the MDODEs. By means of the selected basis functions, the given conditions are automatically satisfied. We conclude with the application of spectral collocation method for multi-dimensional distributed-order diffusion equations.

Introduction

2018 ◽  
Author(s):  
Niels Engholm Henriksen ◽  
Flemming Yssing Hansen
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Degrees Of Freedom ◽  
State Level ◽  
Boltzmann Distribution ◽  
Theoretical Description ◽  
Time Dependent ◽  
Nuclear Dynamics ◽  
Molecular Reaction ◽  
Oppenheimer Approximation

This introductory chapter considers first the relation between molecular reaction dynamics and the major branches of physical chemistry. The concept of elementary chemical reactions at the quantized state-to-state level is discussed. The theoretical description of these reactions based on the time-dependent Schrödinger equation and the Born–Oppenheimer approximation is introduced and the resulting time-dependent Schrödinger equation describing the nuclear dynamics is discussed. The chapter concludes with a brief discussion of matter at thermal equilibrium, focusing at the Boltzmann distribution. Thus, the Boltzmann distribution for vibrational, rotational, and translational degrees of freedom is discussed and illustrated.

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