  Reduced density matrix functional theory (RDMFT) calculations are usually
implemented in a decoupled manner, where the orbital and occupation
optimizations are repeated alternately. Typically, orbital updates are
performed using the unitary optimization method, while occupations are
optimized through the explicit-by-implicit (EBI) method. The EBI method
addresses explicit constraints by incorporating implicit functions, effectively
transforming constrained optimization scenarios into unconstrained
minimizations. Although the unitary and EBI methods individually achieve robust
performance in optimizing orbitals and occupations, respectively, the decoupled
optimization methods often suffer from slow convergence and require dozens of
alternations between the orbital and occupation optimizations. To address this
issue, this work proposes a coupled optimization method that combines unitary
and EBI optimizations to update orbitals and occupations simultaneously at each
step. To achieve favorable convergence in coupled optimization using a simple
first-order algorithm, an effective and efficient preconditioner and line
search are further introduced. The superiority of the new method is
demonstrated through numerous tests on different molecules, random initial
guesses, different basis sets and different functionals. It outperforms all
decoupled optimization methods in terms of convergence speed, convergence
results and convergence stability. Even a large system like $\mathrm{C_{60}}$
can converge to $10^{-8}$ au in 154 iterations, which shows that the coupled
optimization method can make RDMFT more practical and facilitate its wider
application and further development.
