Pure density functional for strong correlation and the thermodynamic limit from machine learning

We use the density-matrix renormalization group, applied to a one-dimensional model of continuum Hamiltonians, to accurately solve chains of hydrogen atoms of various separations and numbers of atoms. We train and test a machine-learned approximation to $F[n]$, the universal part of the electronic density functional, to within quantum chemical accuracy. We also develop a data-driven, atom-centered basis set for densities which greatly reduces the computational cost and accurately represents the physical information in the machine-learning calculation. Our calculation (a) bypasses the standard Kohn-Sham approach, avoiding the need to find orbitals, (b) includes the strong correlation of highly stretched bonds without any specific difficulty (unlike all standard DFT approximations), and (c) is so accurate that it can be used to find the energy in the thermodynamic limit to quantum chemical accuracy.