Using collocation and a hierarchical basis to solve the vibrational Schrödinger equation.

Area of Science:

• Quantum Chemistry
• Computational Chemistry
• Molecular Spectroscopy

Background:

• Calculating vibrational energy levels of polyatomic molecules is crucial for understanding molecular behavior.
• Traditional methods can be computationally expensive due to the high dimensionality of the problem.

Purpose of the Study:

• To develop and validate a novel collocation method for computing molecular vibrational energy levels.
• To simplify the implementation of collocation methods for molecular systems.

Main Methods:

• Utilized a pruned basis set of one-dimensional harmonic oscillator functions.
• Employed a Smolyak grid constructed from symmetrized Leja points.
• Implemented hierarchical 1-D basis functions to facilitate matrix inversion.

Main Results:

• Successfully computed 100 vibrational energy levels for CH2NH.
• Demonstrated the efficiency of the pruned basis and Smolyak grid approach.
• Showcased a simplified derivation of the collocation equation.

Conclusions:

• The developed collocation method offers an efficient and implementable approach for calculating vibrational energy levels.
• The use of pruned basis sets and specialized grids overcomes limitations of tensor product methods.
• This method provides a valuable tool for theoretical molecular spectroscopy.