This paper proposes an efficient algorithm for solving the Hartree--Fock equation combining a multilevel correction scheme with an adaptive refinement technique to improve computational efficiency. The algorithm integrates a multilevel correction framework with an optimized implementation strategy. Within this framework, a series of linearized boundary value problems are solved, and their approximate solutions are corrected by solving small-scale Hartree--Fock equations in low-dimensional correction spaces. The correction space comprises a coarse space and the solution to the linearized boundary value problem, enabling high accuracy while preserving low-dimensional characteristics. The proposed algorithm efficiently addresses the inherent computational complexity of the Hartree--Fock equation. Innovative correction strategies eliminate the need for direct computation of large-scale nonlinear eigenvalue systems and dense matrix operations. Furthermore, optimization techniques based on precomputations within the correction space render the total computational workload nearly independent of the number of self-consistent field iterations. This approach significantly accelerates the solution process of the Hartree--Fock equation, effectively mitigating the traditional exponential scaling demands on computational resources while maintaining precision.

翻译：本文提出了一种求解Hartree-Fock方程的高效算法，该算法结合多层校正方案与自适应网格细化技术以提高计算效率。该算法将多层校正框架与优化的实施策略相结合。在此框架内，通过求解一系列线性化边值问题，并在低维校正空间中求解小规模Hartree-Fock方程来校正其近似解。校正空间由粗空间和线性化边值问题的解构成，从而在保持低维特性的同时实现高精度。所提算法有效应对了Hartree-Fock方程固有的计算复杂性。创新的校正策略避免了对大规模非线性特征值系统及稠密矩阵运算的直接计算。此外，基于校正空间内预计算的优化技术使得总计算工作量几乎与自洽场迭代次数无关。该方法显著加速了Hartree-Fock方程的求解过程，在保持精度的同时有效缓解了传统方法对计算资源的指数级增长需求。