  Orbital-free density functional theory (OF-DFT) for real-space systems has
historically depended on Lagrange optimization techniques, primarily due to the
inability of previously proposed electron density approaches to ensure the
normalization constraint. This study illustrates how leveraging contemporary
generative models, notably normalizing flows (NFs), can surmount this
challenge. We develop a Lagrangian-free optimization framework by employing
these machine learning models for the electron density. This diverse approach
also integrates cutting-edge variational inference techniques and equivariant
deep learning models, offering an innovative reformulation to the OF-DFT
problem. We demonstrate the versatility of our framework by simulating a
one-dimensional diatomic system, LiH, and comprehensive simulations of
hydrogen, lithium hydride, water, and four hydrocarbon molecules. The inherent
flexibility of NFs facilitates initialization with promolecular densities,
markedly enhancing the efficiency of the optimization process.
