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 $F\left[n\right]$, 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.

Figure 1

Electronic energy of the infinite chain from a model learned from extrapolated chain densities and energies. The accurate value was calculated with infinite DMRG (see text).

Figure 2

Binding curve for a 1D ${\mathrm{H}}_2$ molecule. Black: Highly accurate, converged DMRG results. Blue: LDA result restricted to a spin singlet [23].

Figure 3

Same as Fig. 2. The green curves are ML with $N_{\mathrm{T}}=5$ on both the exact (dashed) and ML-optimized (solid) densities. The red solid curve is the ML with $N_T=20$ (number of training points) on ML-optimized (solid) densities. Black dashed curve is the exact DMRG curve, matching nearly exactly the $N_T=20$ on ML line.

Figure 4

Optimal densities for 1D ${\mathrm{H}}_2$ molecule in the test set: DMRG (black dashed), ML with $N_T=5$ (orange solid), ML with $N_T=20$ (red solid).

Figure 5

Partition density of each H atom in ${\mathrm{H}}_8$.

Figure 6

Single H atom densities for H atoms in different chains and atomic distance (gray). The average density is plotted in red.

Figure 7

First seven principal components of the densities shown in Fig. 6, from top to bottom.

Figure 8

Learning curves for several 1D H chains. (a) ML using the total density. (b) ML using the bulk partition densities (see text).

Figure 9

Electronic energy per atom in the thermodynamic limit, both via DMRG chains (extrapolated to infinity) and using machine learning with 50 data points per chain.

Figure 10

For a given training set with $N_T$ training points, the functional driven error $\mathrm{\Delta }{F}_F$ per atom is shown in red (lower curve). The upper curve is the total energy error per atom evaluated self-consistently.