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Using Collocation to Solve the Schrödinger Equation.

Sergei Manzhos1, Manabu Ihara1, Tucker Carrington2

  • 1School of Materials and Chemical Technology, Tokyo Institute of Technology, Ookayama 2-12-1, Meguro-ku, Tokyo 152-8552, Japan.

Journal of Chemical Theory and Computation
|March 28, 2023
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Summary
This summary is machine-generated.

The collocation method offers a flexible approach to solving the Schrödinger equation, particularly for complex molecular systems and surfaces where traditional methods struggle. Its adaptability in coordinate and basis function selection enhances computational efficiency.

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Area of Science:

  • Quantum Chemistry
  • Computational Physics
  • Materials Science

Background:

  • The Schrödinger equation is fundamental to understanding molecular behavior.
  • Traditional methods for solving the Schrödinger equation can be computationally intensive.
  • Challenges exist in modeling molecule-surface interactions and systems with complex potentials.

Purpose of the Study:

  • To review the collocation method for solving the Schrödinger equation.
  • To highlight its advantages, disadvantages, and interrelations with other methods.
  • To showcase its applications in electronic and vibrational problems, especially for molecule-surface systems.

Main Methods:

  • Review of the collocation approach and its rectangular formulation.
  • Discussion of coordinate and basis function flexibility, including non-integrable functions.
  • Exploration of handling potential singularities and tuning basis functions with machine learning.

Main Results:

  • Collocation allows for optimized coordinates and basis functions, simplifying singularity treatment.
  • Machine learning can optimize basis function shapes and point placement.
  • The method is advantageous for molecule-surface systems and when potential energy surfaces are unavailable.

Conclusions:

  • The collocation method provides a versatile and efficient alternative for solving the Schrödinger equation.
  • It offers significant benefits for challenging systems like molecules on surfaces.
  • Its adaptability makes it suitable for scenarios where standard quadrature grids are computationally prohibitive.