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

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.

Machine Learning Optimization of the Collocation Point Set for Solving the Kohn–Sham Equation

“…We have recently shown that collocation works well even when the basis functions do not satisfy the integrability condition. 60 (iii) Because energy levels can be computed with basis functions that have no amplitude at singularities (even when wavefunctions have significant amplitude at the singularities) and it is not necessary to approximate integrals, singularities cause no problems. 31 (iv) The basis is deemed complete if, at the M>N collocation points,…”

Rectangular eigenvalue problems

“…With expansion functions that satisfy the differential equation but not the boundary conditions, one gets series methods [33] or the Method of Fundamental Solutions (MFS) [3,13] or lightning or log-lightning methods for PDE problems with corner singularities [16,26]. Related expansions that do not satisfy the differential equation and hence need fitting in the interior of a domain, not just on the boundary, lead to least-squares methods for radial basis functions (RBFs) or other kernels [10,14,21,29]. RBF methods are an example of the broad category of meshfree methods.…”

“…Our plan is to set forth some of the simplest methods for solving rectangular eigenproblems and to illustrate them with a sequence of examples. The closest previous contributions we know of on this topic are by Manzhos and coauthors, who have developed what they call "rectangular collocation" methods for eigenvalue problems in quantum chemistry [21,22], and by the first two authors [17]. The emphasis in [17] is on spectral methods for ODEs, and the linear algebra is carried out by the method of Ito and Murota [19], involving the singular value decomposition (SVD) of a matrix with twice as many columns as there are basis functions.…”

Rectangular eigenvalue problems

“…With expansion functions that satisfy the differential equation but not the boundary conditions, one gets series methods [32] or the Method of Fundamental Solutions (MFS) [3,13] or lightning or log-lightning methods for PDE problems with corner singularities [16,26]. Related expansions that do not satisfy the differential equation and hence need fitting in the interior of a domain, not just on the boundary, lead to least-squares methods for radial basis functions (RBFs) or other kernels [10,14,21,29]. RBF methods are an example of the broad category of meshfree methods.…”

“…Our plan is to set forth some of the simplest methods for solving rectangular eigenproblems and illustrate them with a sequence of examples. The closest previous contributions we know of on this topic are by Manzhos and coauthors, who have developed what they call "rectangular collocation" methods for eigenvalue problems in quantum chemistry [21,22], and by the first two authors [17]. The emphasis in [17] is on spectral methods for ODEs, and the linear algebra is carried out by the method of Ito and Murota [19], involving the singular value decomposition (SVD) of a matrix with twice as many columns as there are basis functions.…”