[Paper Review] Observations on Gaussian bases for Schrodinger's equation

Ilan Degani|ArXiv.org|Jul 31, 2007

Spectroscopy and Quantum Chemical Studies19 references3 citations

TL;DR

This paper proposes a modified Gaussian basis method for solving multi-dimensional Schrödinger eigenproblems, improving accuracy by avoiding overly wide and sparse Gaussians in steep potential regions—particularly near high potential barriers. The approach enhances collocation and Galerkin methods, with numerical results showing dramatic error reduction, especially when using a Hamiltonian trace minimization criterion for width scaling over orthogonality deviation.

ABSTRACT

One of the few methods for generating efficient function spaces for multi-D Schrodinger eigenproblems is given by Garashchuk and Light in J.Chem.Phys. 114 (2001) 3929. Their Gaussian basis functions are wider and sparser in high potential regions, and narrower and denser in low ones. We suggest a modification of their approach based on the following observation: In very steep potential regions, wide, sparse, Gaussians should be avoided even if their centers have high potential values. Our numerical results illustrate that a dramatic improvement in accuracy may be obtained in this way. We also compare the errors of collocation to those of a Galerkin approach, test a criterion for scaling Gaussian widths based on deviation from orthogonality of collocation eigenfunctions, and suggest a criterion for scaling Gaussian widths based on Hamiltonian trace minimization.

Motivation & Objective

• To address the inefficiency of existing Gaussian basis methods in steep potential regions where wide, sparse Gaussians degrade accuracy.
• To improve the accuracy of both collocation and Galerkin discretization methods for multi-dimensional Schrödinger eigenproblems.
• To develop and test new criteria for scaling Gaussian widths, particularly one based on Hamiltonian trace minimization.
• To evaluate the relative performance of collocation versus Galerkin methods when basis functions are optimized for each method.
• To assess the utility of eigenfunction orthogonality deviation as a guide for width selection in collocation schemes.

Proposed method

• Modifies the Gaussian basis construction of Garashchuk and Light by restricting wide, sparse Gaussians in high-potential, steep-gradient regions.
• Applies both Galerkin and collocation methods to 1D Morse potential problems using adaptively scaled Gaussian bases.
• Uses a global width parameter c to control basis function spread, optimizing it to minimize mean eigenfunction error.
• Introduces a new width scaling criterion based on minimizing the trace of the Hamiltonian matrix, improving stability and accuracy.
• Employs the deviation from orthogonality of collocation-calculated eigenfunctions as a heuristic to guide width selection.
• Compares condition numbers of the overlap matrix S and the basis function matrix Φ to assess numerical stability.

Experimental results

Research questions

• RQ1 Can avoiding wide, sparse Gaussians in steep potential regions significantly improve the accuracy of Gaussian basis methods for multi-dimensional Schrödinger problems?
• RQ2 How does the performance of collocation compare to Galerkin when basis functions are optimized separately for each method?
• RQ3 Can the deviation from orthogonality of collocation eigenfunctions serve as a reliable indicator for optimal Gaussian width selection?
• RQ4 Does a width scaling criterion based on Hamiltonian trace minimization yield better accuracy than existing methods?
• RQ5 What is the impact of basis function distribution and width on the condition numbers of the system matrices in collocation and Galerkin schemes?

Key findings

• The modified Gaussian basis method, which avoids wide Gaussians in steep potential regions, yields a dramatic improvement in accuracy compared to the original Garashchuk-Light approach.
• When basis widths are optimized separately for each method, Galerkin consistently outperforms collocation in terms of eigenvalue and eigenfunction error, with the Galerkin/basis set 3 combination achieving the lowest errors.
• The Hamiltonian trace minimization criterion for width scaling produced superior accuracy and better-conditioned matrices compared to the orthogonality deviation heuristic.
• For the Morse A potential, the Galerkin method with basis set 3 achieved mean eigenfunction error below 10^-10 at optimal width, with condition numbers of S and Φ below 10^10.
• Collocation performance was significantly improved by optimizing widths for that method, but remained inferior to Galerkin, especially in high-accuracy regimes.
• The condition number of the overlap matrix S was reduced by orders of magnitude when using the trace minimization criterion, indicating enhanced numerical stability.