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Optimal grids for generalized finite basis and discrete variable representations: definition and method of
1Crystal Physics Laboratory, Research Institute for Solid State Physics and Optics, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary. viktor@mail.szfki.hu
The Journal of Chemical Physics
|October 25, 2006
Summary
This study introduces optimal generalized finite basis and discrete variable representations (FBR and DVR) for enhanced accuracy in quantum mechanics calculations. Optimized grids significantly improve eigenvalue calculations, outperforming traditional variational methods.
Area of Science:
- Quantum mechanics
- Computational chemistry
- Numerical analysis
Background:
- Standard finite basis and discrete variable representations (FBR and DVR) rely on Gaussian quadrature grids and polynomial bases.
- Generalizations are needed for non-standard grids and bases in quantum calculations.
Purpose of the Study:
- To develop and demonstrate an optimal generalized FBR/DVR method using optimized grids.
- To show how to obtain optimal grid points for a given truncated basis.
- To compare the accuracy of optimized grids against standard methods.
Main Methods:
- Generalized FBR/DVR method for arbitrary grids and bases.
- Definition of basis set optimized and potential optimized grids.
- Minimization of a function related to the Hamiltonian operator.
- Solving systems of nonlinear equations for grid optimization.
Main Results:
- Optimal grids minimize a derived function, leading to improved accuracy.
- Basis set optimized grids are equivalent to Gaussian quadrature grids.
- Potential optimized grids yield orders of magnitude higher accuracy for eigenvalue calculations compared to variational methods.
- Optimal generalized FBR quadrature rules offer superior numerical integration accuracy.
Conclusions:
- The optimal generalized FBR/DVR method provides a powerful framework for accurate quantum mechanical calculations.
- Optimized grids are crucial for maximizing the efficiency and accuracy of FBR/DVR methods.
- The study extends concepts of Gaussian quadrature to more general basis functions.
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