A Generalized Grid-Based Fast Multipole Method for Integrating Helmholtz Kernels.

Area of Science:

• Computational Chemistry
• Quantum Chemistry
• Materials Science

Background:

• Accurate calculation of molecular orbitals is crucial for understanding chemical properties.
• Existing methods can be computationally intensive for large systems.
• Developing efficient algorithms for electronic structure calculations is an ongoing challenge.

Purpose of the Study:

• To develop and implement a grid-based fast multipole method (GB-FMM) for optimizing 3D numerical molecular orbitals.
• To generalize a previously published GB-FMM approach for electrostatic potentials and two-electron interactions.
• To enable accurate and efficient electronic structure calculations.

Main Methods:

• Integration of the Helmholtz kernel in a double basis (bubbles and cube).
• Representation of steep functions near nuclei using one-center bubble functions.
• Expansion of the remaining part on a 3D grid.
• Utilizing one-center expansions with spherical harmonics and Bessel functions for bubble integration.
• Implementation of massively parallel algorithms for Graphics Processing Units (GPGPU).

Main Results:

• Successful development and implementation of the GB-FMM for molecular orbital optimization.
• Demonstrated accuracy of 10^-4 to 10^-7 E_h in Hartree-Fock self-consistent-field (HF-SCF) calculations.
• Validation on H2, H2O, and CO molecules, confirming correctness and accuracy.
• Efficient parallelization for GPGPU enabling large-scale computations.

Conclusions:

• The developed GB-FMM provides an accurate and efficient method for 3D molecular orbital optimization.
• This approach generalizes previous GB-FMM applications to electronic structure calculations.
• The GPGPU implementation facilitates high-performance computing for quantum chemistry.
• The method achieves high accuracy, suitable for various molecular systems.