Permutation invariant multi-scale full quantum neural network wavefunction

Abstract

Solving the intricate quantum behavior of interacting particles is key to unlocking the mysteries of condensed matter, but capturing their complex correlations across different scales remains a monumental challenge. We introduce a neural network framework that overcomes this barrier by modeling the full quantum wavefunction of a system, including electrons, nuclei and muons, directly capturing the full quantum effects beyond the Born-Oppenheimer approximation. The neural network approximates joint wavefunction of different interacting particles with a rigorous handling of permutation invariance, enabling simultaneous treatment of nuclear quantum effects and electron-nucleus-muon couplings without explicit excited states. Validated on molecular systems, this approach offers a computationally feasible way to model full quantum phenomena in complex many-body systems, establishing a direct connection between fundamental particle properties and emergent material behavior.

Figures (4)

• Figure 1: Sketch of neural network architecture. Due to translation invariance, only the relative positions of particles are selected as input to the neural network. Drawing on the idea of the Born-Huang expansion, the overall network can be constructed by multiplying the wavefunction components of positively and negatively charged particles (similar to how the nuclear wavefunction is multiplied by the electron wavefunction). The determinant or permanent is used to maintain exchange symmetry, and a multi-determinant/permanent structure can enhance the representation capability of neural networks.
• Figure 2: (a) The training curve of isotopologues of the hydrogen molecule. The Jastrow factor is chosen to be of Gaussian type. A moving window average over 500 steps is applied. The dashed line represents the energy in BOA computed by FermiNet pfau2020ab at equilibrium configuration. (b) The linear fit of the average internuclear distance $\langle R \rangle$ for the hydrogen molecule and its isotopologues as a function of the inverse square root of the reduced mass $1/\sqrt{\mu}$ of the two-body system. (c) PES of hydrogen calculated using a model trained under different conditions. The red line is calculated by traditional basis method wolniewicz1995nonadiabatic. The green, yellow and blue line are model trained under different hydrogen mass with the GEM-type Jastrow factor described in Equation \ref{['eq:jastrow-plus']}. The horizontal axis coordinate $R$ represents the static nuclear distance as in the BOA. The vertical axis of the inset $\Delta E$ represents the energy difference relative to the traditional basis method (red line). The blue line exhibits non-physical divergence at the far end and is truncated here for clarity. For details, please refer to the SI.
• Figure 3: (a) Schematic diagram of ammonia molecule polarization under an external electric field. (b) Particle density distribution projected onto the $X$--$Y$ plane, where green, red, and blue denote hydrogen, nitrogen, and electrons, respectively, with darker shades indicating higher probability densities. (c) Training curve of the ammonia system under uniform electric fields of varying intensities $\mathcal{E}$. A Gaussian-type Jastrow factor is used, and a moving window average over 5000 optimization steps is applied. (d) Quadratic fit of the ground state energy $E$ of the ammonia system as a function of the intensity $\mathcal{E}$ of the applied uniform electric field. (e) Dipole moment of the ammonia molecule as a function of training steps, with nuclear mass reduced to $10\%$ (red line) and $1\%$ (green line) of its original value.
• Figure 4: (a) Training curve of the C$_2$H$_4$Mu system using PermNet (green line) and FermiNet (red line). The GEM-type Jastrow factor is used, and a moving window average over 5000 optimization steps is applied. (b) Contact spin density as a function of training steps, calculated via the third-order polynomial extrapolation. (c) Contact spin density calculated via different fitting order polynomial extrapolation at 500,000 training steps. (d) Particle density distribution of the C$_2$H$_4$Mu system in real space, where green, red, and blue denote hydrogen, carbon, and muon, respectively. (e) The radial density distributions of different particles as functions of the distance to the center of mass, where green denotes hydrogen, red dashed lines denote carbon, blue dashed lines denote the muon, and yellow dashed lines denote electrons.