无需对角化即可实现DFT优化，直接求解电子占据分布。
Diagonalization without Diagonalization: A Direct Optimization Approach for Solid-State Density Functional Theory
- 同时参数化波函数与占据矩阵，通过梯度下降优化自由能。
- 在稳态下自动实现哈密顿矩阵对角化，获得正确费米-狄拉克分布。
- 适用于金属与半导体体系，适合需要高效DFT计算的研究者。
我们提出一种新方法，解决密度泛函理论（DFT）直接优化中占据数可变的挑战。通过同时参数化本征函数和占据矩阵，该方法在这些参数上最小化自由能。由于驻点条件要求占据矩阵与郭恩-沙姆哈密顿量同时可对角化，由此引出“自对角化”概念：假设占据矩阵为对角形式（不失一般性），则哈密顿矩阵在驻点处自然对角化。该方法将波函数与占据的物理约束融入参数化过程，将带约束优化转化为全可微的无约束问题，可通过梯度下降求解。在JAX中实现，已在铝和硅体系上测试，结果表明该方法能高效实现自对角化，获得正确的费米-狄拉克占据分布，并得到与Quantum Espresso中自洽场（SCF）方法一致的能带结构。
原文摘要 · Abstract (English)
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.
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