Diagonalization without Diagonalization: A Direct Optimization Approach for Solid-State Density Functional Theory

Area of Science:

• Computational Physics
• Materials Science
• Quantum Chemistry

Background:

• Direct optimization in density functional theory (DFT) faces challenges with variable electron occupation numbers.
• Existing methods often require iterative self-consistent field (SCF) calculations.
• Accurate determination of electronic structure is crucial for predicting material properties.

Purpose of the Study:

• To develop a novel, direct optimization approach for DFT that addresses variable occupation number challenges.
• To introduce the concept of "self-diagonalization" for simplifying DFT calculations.
• To provide a fully differentiable, unconstrained optimization method solvable via gradient descent.

Main Methods:

• Parametrization of both eigenfunctions and the occupation matrix to minimize free energy.
• Leveraging stationary conditions for simultaneous diagonalization of occupation matrix and Kohn-Sham Hamiltonian.
• Implementing a gradient descent algorithm within the JAX framework.
• Incorporating physical constraints into the parametrization for an unconstrained problem.

Main Results:

• Demonstrated efficient "self-diagonalization" on aluminum and silicon test cases.
• Achieved correct Fermi-Dirac distribution for occupation numbers.
• Obtained band structures consistent with traditional SCF eigensolver methods (e.g., Quantum Espresso).

Conclusions:

• The novel parametrization and self-diagonalization approach offers an efficient alternative for DFT calculations.
• This method successfully handles variable occupation numbers and yields accurate electronic structures.
• The implementation in JAX provides a robust and scalable tool for computational materials science.