Enhancing Reduced\nDensity Matrix Functional Theory\nCalculations by Coupling Orbital and Occupation Optimizations

Reduced density matrix functional theory (RDMFT) calculations\nare\nusually implemented in a decoupled manner, where the orbital and occupation\noptimizations are repeated alternately. Typically, orbital updates\nare performed using the unitary optimization method, while occupations\nare optimized through the explicit-by-implicit (EBI) method. The EBI\nmethod addresses explicit constraints by incorporating implicit functions,\neffectively transforming constrained optimization scenarios into unconstrained\nminimizations. Although the unitary and EBI methods individually achieve\nrobust performance in optimizing orbitals and occupations, respectively,\nthe decoupled optimization methods often suffer from slow convergence\nand require dozens of alternations between the orbital and occupation\noptimizations. To address this issue, this work proposes a coupled\noptimization method that combines unitary and EBI optimizations to\nupdate orbitals and occupations simultaneously at each step. To achieve\nfavorable convergence in coupled optimization using a simple first-order\nalgorithm, an effective and efficient preconditioner and line search\nare further introduced. The superiority of the new method is demonstrated\nthrough numerous tests on different molecules, random initial guesses,\ndifferent basis sets, and different functionals. It outperforms all\ndecoupled optimization methods in terms of convergence speed, convergence\nresults, and convergence stability. Even a large system like C<sub>60</sub> can converge to 10<sup>–8</sup> au in 154 iterations,\nwhich shows that the coupled optimization method can make RDMFT more\npractical and facilitate its wider application and further development.