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Highly Accurate Real-space Electron Densities with Neural Networks

Lixue Cheng, P. Bernát Szabó, Zeno Schätzle, Derk P. Kooi, Jonas Köhler, Klaas J. H. Giesbertz, Frank Noé, Jan Hermann, Paola Gori-Giorgi, Adam Foster

TL;DR

The electron density is considered as a central observable in quantum chemistry and a novel method to obtain accurate densities from real-space many-electron wave functions by representing the density with a neural network that captures known asymptotic properties and is trained from the wave function by score matching and noise-contrastive estimation is introduced.

Abstract

Variational ab-initio methods in quantum chemistry stand out among other methods in providing direct access to the wave function. This allows in principle straightforward extraction of any other observable of interest, besides the energy, but in practice this extraction is often technically difficult and computationally impractical. Here, we consider the electron density as a central observable in quantum chemistry and introduce a novel method to obtain accurate densities from real-space many-electron wave functions by representing the density with a neural network that captures known asymptotic properties and is trained from the wave function by score matching and noise-contrastive estimation. We use variational quantum Monte Carlo with deep-learning ansätze (deep QMC) to obtain highly accurate wave functions free of basis set errors, and from them, using our novel method, correspondingly accurate electron densities, which we demonstrate by calculating dipole moments, nuclear forces, contact densities, and other density-based properties.

Highly Accurate Real-space Electron Densities with Neural Networks

TL;DR

The electron density is considered as a central observable in quantum chemistry and a novel method to obtain accurate densities from real-space many-electron wave functions by representing the density with a neural network that captures known asymptotic properties and is trained from the wave function by score matching and noise-contrastive estimation is introduced.

Abstract

Variational ab-initio methods in quantum chemistry stand out among other methods in providing direct access to the wave function. This allows in principle straightforward extraction of any other observable of interest, besides the energy, but in practice this extraction is often technically difficult and computationally impractical. Here, we consider the electron density as a central observable in quantum chemistry and introduce a novel method to obtain accurate densities from real-space many-electron wave functions by representing the density with a neural network that captures known asymptotic properties and is trained from the wave function by score matching and noise-contrastive estimation. We use variational quantum Monte Carlo with deep-learning ansätze (deep QMC) to obtain highly accurate wave functions free of basis set errors, and from them, using our novel method, correspondingly accurate electron densities, which we demonstrate by calculating dipole moments, nuclear forces, contact densities, and other density-based properties.
Paper Structure (19 sections, 19 equations, 9 figures, 2 tables)

This paper contains 19 sections, 19 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Variational and deep quantum Monte Carlo
  4. Electron density
  5. Kato's cusp condition
  6. Exponential decay and tail convergence
  7. Previous developments on classical VMC density estimators
  8. Machine-learning energy based models
  9. Electron Density Estimation
  10. Designing an EBM for the electron density
  11. Score matching
  12. Noise contrastive estimation
  13. Results
  14. Small atoms
  15. LiH dissociation
  16. ...and 4 more sections

Figures (9)

  • Figure 1: Schematic diagram for training of NERDs. For an input molecular geometry a highly accurate wave function is obtained with deep QMC. Samples from the wave function and associated gradients serve as training data in a combined score matching and noise contrastive estimation approach to optimizing the NERD.
  • Figure 2: Comparison between densities from the neural network model and FCI/aug-cc-pVQZ for (a) Li atom and (b) Be atom. Both nuclei are located at the origin. Inset plots are displayed to better visualize the cusps around nuclei.
  • Figure 3: Comparison between radial derivatives of log of spin-up densities from the neural network model, FCI/aug-cc-pVQZ and converting literature ionization potential values via Eq. \ref{['eq:ip']} for (a) Li atom and (b) Be atom. Both nuclei are located at the origin.
  • Figure 4: Comparison between the total densities from the neural network model and literature for (a) Be atom and (b) Ne atom. Both nuclei are located at the origin and indicated by the gray dashed lines. The y-axis is plotted on a log scale. The literature densities are computed using VMC and obtained from Filippi et al. filippi1996generalized
  • Figure 5: (a) NERD and FCI/aug-cc-pVTZ densities along the dissociation of the LiH molecule, (b) effective potentials (see eq. \ref{['eq:veff']}) extracted from the densities. The densities in (a) are plotted on a logarithm scale for y-axis. Darker shades of blue correspond to longer bond lengths, while the FCI/aug-cc-pVTZ calculation was carried out at equilibrium bond length only.
  • ...and 4 more figures