Inverse Multiquadratic Functions as the Basis for the 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

“…There is no requirement that basis functions have amplitude where wave functions have amplitude. We have recently shown that collocation works well even when the basis functions do not satisfy the integrability condition Because energy levels can be computed with basis functions that have no amplitude at singularities (even when wave functions have significant amplitude at the singularities) and it is not necessary to approximate integrals, singularities cause no problems …”

“…There is no requirement that basis functions have amplitude where wave functions have amplitude. We have recently shown that collocation works well even when the basis functions do not satisfy the integrability condition …”

Computational vibrational spectroscopy of molecule–surface interactions: what is still difficult and what can be done about it

“…The viability of these approaches has been shown for free molecules 118,119 and they can have significant benefits for molecules on surfaces allowing further reduction of the required number of ab initio data. As collocation makes it easy to work with any basis functions (including non-integrable and non-smooth functions 120,121 ), innovations in basis functions could also be beneficial.…”

“…Because it is not necessary to compute integrals, collocation can be easily used with any coordinates and basis functions, even non-integrable or non-smooth functions -one simply does not place the collocation points at the singularities. 120,121 The basis need not have amplitude everywhere where the wave functions have significant amplitude. 118 Importantly, by applying the KEO numerically, the exact KEO can easily be used with any coordinates and basis functions.…”

Using Collocation to Solve the Schrödinger Equation

“…If the basis functions are functions of the coordinates in which the KEO is written, then in most cases applying derivative operators in the KEO to the basis functions can be done analytically and exactly. In several papers, − it has been shown that when using collocation it is straightforward to use a KEO in space-fixed Cartesian coordinates and basis functions that are functions of internal coordinates that specify only the shape of the molecule. This means that there is no need to know the KEO in terms of the coordinates of the basis functions.…”