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.

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