Neural network iterative diagonalization method to solve eigenvalue problems in quantum mechanics.

Area of Science:

• Computational Chemistry
• Quantum Mechanics
• Numerical Linear Algebra

Background:

• Calculating eigenvalues and eigenvectors of large, sparse, complex symmetric or Hermitian matrices is computationally intensive.
• Existing methods may struggle with dense spectrum regions or interior eigenvalue problems.

Purpose of the Study:

• To propose a novel multi-layer feed-forward neural network iterative diagonalization method (NNiDM).
• To efficiently compute eigenvalues and eigenvectors of large sparse complex symmetric or Hermitian matrices.
• To demonstrate the method's capability in finding interior eigenstates within dense spectrum regions.

Main Methods:

• The NNiDM algorithm integrates the complex guided spectral transform Lanczos (cGSTL) method, a thick restart technique, and a multi-layered basis contraction scheme.
• Artificial neurons are defined by orthogonal Lanczos polynomials, with dynamically determined weights and biases through iterative diagonalizations.
• The method employs a spectral transform technique and linear transform diagonalization for eigenvalue and eigenvector computation.

Main Results:

• The NNiDM algorithm successfully computes eigenvalues and eigenvectors near a specified reference value.
• It demonstrates effectiveness in identifying interior eigenstates, even in crowded spectral regions.
• The method was applied to calculate energies, widths, and wavefunctions for HO2 and CH4 molecules.

Conclusions:

• The NNiDM provides an effective approach for eigenvalue and eigenvector problems in large sparse matrices.
• Its ability to compute interior eigenstates makes it valuable for quantum mechanical applications.
• The method shows promise for accurate molecular property calculations.