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

We present a novel approach to address the challenges of variable occupation numbers in direct optimization of density functional theory (DFT). By parameterizing both the eigenfunctions and the occupation matrix, our method minimizes the free energy with respect to these parameters. As the stationary conditions require the occupation matrix and the Kohn-Sham Hamiltonian to be simultaneously diagonalizable, this leads to the concept of ``self-diagonalization,'' where, by assuming a diagonal occupation matrix without loss of generality, the Hamiltonian matrix naturally becomes diagonal at stationary points. Our method incorporates physical constraints on both the eigenfunctions and the occupations into the parameterization, transforming the constrained optimization into an fully differentiable unconstrained problem, which is solvable via gradient descent. Implemented in JAX, our method was tested on aluminum and silicon, confirming that it achieves efficient self-diagonalization, produces the correct Fermi-Dirac distribution of the occupation numbers and yields band structures consistent with those obtained with SCF methods in Quantum Espresso.