Sparse Projected-Gradient Method As a Linear-Scaling Low-Memory Alternative to Diagonalization in Self-Consistent...

Area of Science:

• Computational Chemistry
• Materials Science
• Quantum Mechanics

Background:

• Large-scale electronic structure calculations frequently encounter significant nonlinear eigenvalue problems.
• Traditional methods often rely on computationally expensive eigenvalue decompositions of the Fock matrix, limiting scalability.

Purpose of the Study:

• To present an efficient alternative to eigenvalue decomposition for solving large nonlinear eigenvalue problems in electronic structure calculations.
• To develop a method that preserves matrix sparsity and exhibits linear scaling in memory and computational time.

Main Methods:

• A novel algorithm based on projected gradient iteration is applied to the constraint fulfillment problem.
• The method avoids explicit eigenvalue decomposition of the Fock matrix throughout the iterative process.

Main Results:

• The algorithm maintains sparsity of matrices at each iteration, achieving linear memory scaling with system size.
• Computational time scales approximately linearly with the number of atoms or non-null matrix elements.
• The method demonstrates superior performance over standard eigenvalue decomposition for sparse matrices exceeding 50,000 non-null elements and accurately reproduces semiempirical SCF iteration sequences.

Conclusions:

• The proposed projected gradient iteration method provides an efficient and scalable solution for large-scale electronic structure calculations.
• This approach significantly reduces computational cost and memory requirements compared to traditional eigenvalue decomposition techniques.