Recursive Factorization of the Inverse Overlap Matrix in Linear-Scaling Quantum Molecular Dynamics Simulations

Area of Science:

• Computational Chemistry
• Materials Science
• Quantum Mechanics

Background:

• Solving the generalized eigenvalue problem is crucial for quantum-based molecular dynamics (MD) simulations.
• Traditional methods involving overlap matrix diagonalization can be computationally expensive, limiting simulation size and speed.

Purpose of the Study:

• To present a reduced complexity algorithm for computing inverse overlap factors.
• To enable faster and more stable quantum-based MD simulations.

Main Methods:

• Recursive, iterative refinement of an initial guess for the inverse square root of the overlap matrix (Z).
• Utilizing approximate divide-and-conquer or dynamical methods for initial Z estimation.
• Employing sparse matrix algebra with ELLPACK-R format for linear-scaling performance and parallelization.

Main Results:

• Achieved long-term stability and energy conservation with approximate iterative refinement of Z.
• Demonstrated linear-scaling performance and efficient shared-memory parallelization.
• Obtained an average speedup factor of 122 for Z computation in a large polyalanine system.

Conclusions:

• The developed algorithm substantially accelerates quantum-based simulations, even for intermediate-sized molecular structures.
• The method offers a significant speedup over conventional diagonalization techniques.
• This advancement facilitates larger and more complex quantum-based MD simulations.