Using rectangular collocation with finite difference derivatives to solve electronic Schrödinger equation.

Area of Science:

• Computational Chemistry
• Quantum Mechanics
• Theoretical Physics

Background:

• Solving the electronic Schrödinger equation (ESE) is fundamental to understanding molecular electronic structure.
• Traditional methods often face limitations with basis set choices and point distributions.
• Accurate solutions are crucial for predicting chemical properties and reactions.

Purpose of the Study:

• To introduce and validate a rectangular collocation method for solving the ESE.
• To demonstrate the method's effectiveness for atomic and molecular systems.
• To highlight the advantages of collocation over traditional variational methods.

Main Methods:

• A rectangular collocation approach, evaluating matrix elements via a quadrature-like scheme.
• Utilizing more collocation points than basis functions.
• Testing with the H atom, H2+ cation, CO, and H2O using Slater-type orbital-like basis functions.

Main Results:

• Achieved millihartree accuracy for all tested systems (H atom, H2+, CO, H2O).
• Demonstrated that collocation points can be chosen flexibly, not requiring specific distributions.
• Showed reduced sensitivity of results to point set choice with improved basis sets.

Conclusions:

• The rectangular collocation method is an effective and accurate approach for solving the ESE and Kohn-Sham equations.
• This method allows the use of diverse basis functions, including Slater-type orbitals, expanding computational possibilities.
• The flexibility and accuracy make collocation a promising alternative for electronic structure calculations.