Computing energy and forces directly from machine learned wavefuntion/density

Question

I have been reading up on methods where electron density was estimated directly from some machine learning method, followed by evaluation of energy and forces[1,2]. My understanding is that this is achieved by

1. predicting the wavefunction coefficients (i.e. $C_i$ in $\sum C_i G_i$, where $G_i$ is gaussian/planewave the basis set
2. computing energy and forces directly from this wavefunction/density

I was trying to replicate similar approach to better understand it, but by using simple Hartree Fock method. I tried to compute energy of such an ML determined wave function directly. However to compute the energy of a wavefunction I needed the Fock matrix, which would need not only the determined wavefunction, but also other orbitals.

That is, the Fock matrix (Szabo and Ostlund, 3.154):

$$ F_{\mu\nu} = H_{\mu\nu} + \sum_{ij}P_{ij}(\mu\nu|ij) - 0.5 (\mu i|j\nu) $$

requires contributions $P_{ij}$ from all orbitals.

So is similar approach impossible for HF and only feasible for DFT? Or do I need to predict the full density matrix for HF?

References

1. Fiedler, L., Modine, N.A., Schmerler, S. et al. Predicting electronic structures at any length scale with machine learning. npj Comput Mater 9, 115 (2023). DOI

2. Rackers, J. A.; Tecot, L.; Geiger, M.; Smidt, T. E. Cracking the Quantum Scaling Limit with Machine Learned Electron Densities. arXiv.org. DOI

Answer 1

You can't do Hartree-Fock with densities, only. You need density matrices to evaluate exact exchange.

However, you can do semilocal density functional calculations with knowledge of the electron density, only. A commonly used way to learn the electron density is through the auxiliary basis expansion: the electron density, which reads in the atomic orbital basis as $n({\bf r}) = \sum_{\mu \nu} P_{\mu\nu} \chi_\mu({\bf r}) \chi_\nu({\bf r})$ is expressed in an auxiliary basis familiar from density fitting methods as $n({\bf r}) \approx \sum_A c_A \chi_A({\bf r})$. Machine learning (ML) is then done on the coefficients $c_A$.

Knowing the ML prediction, one can compute the Fock matrix in the orbital basis following density fitting techniques. The density functional is evaluated by quadrature on a grid, and the Coulomb matrix is computed with three-center two-electron integrals as $J_{\mu \nu} = \sum_{A}(\mu \nu|A)c_A$.