Wave function represents the quantum state of an atom, including the position and movement states of the nucleus and electrons. For decades researchers have struggled to determine the exact wave function when analyzing a normal chemical molecule system, which has its nuclear position fixed and electrons spinning. Fixing wave function has proven problematic even with help from the Schrödinger equation.

Previous research in this field used a Slater-Jastrow Ansatz application of quantum Monte Carlo (QMC) methods, which takes a linear combination of Slater determinants and adds the Jastrow multiplicative term to capture the close-range correlations.

Now, a group of DeepMind researchers have brought QMC to a higher level with the Fermionic Neural Network — or Fermi Net — a neural network with more flexibility and higher accuracy. Fermi Net takes the electron information of the molecules or chemical systems as inputs and outputs their estimated wave functions, which can then be used to determine the energy states of the input chemical systems.

Global architecture of the Fermionic Neural Network

Major features that Fermi Net takes in are positional differences and absolute distances — either between the electron and the nuclear as a single electron stream, or between the electron and another electron as a paired electron stream. The intermediate layers within the neural network take the mean of the activation functions from each stream and concatenate the input for a final layer, which applies a spin-dependent linear transformation, weighs the desired outputs as the solution wave function, and enforces the boundary conditions.

In their experiments the research group proved the superior accuracy of Fermi Net compared to other QMC alternatives, especially the well-known and maturely constructed Slater-Jastrow Ansatz. Fermi Net shows high accuracy in single atom determination systems of all first-row elements, CO systems, and N2 systems.

Fermi Net also showed impressive performance in predicting the dissociation curve of a nitrogen molecule and hydrogen chain when compared to CCSD(T) methods which are considered the gold standard for quantum chemistry calculation.

Furthermore, Fermi Net also addresses the bias set extrapolation problem and error importing, since in each layer of the neural network the bias terms are computed rather than preset.

The paper’s lead author David Pfau and his colleagues believe that in addition to the addressed application in QMC, their Fermionic Neural Network can also be applied to other fields beyond Variational Monte Carlo (VMC), for example as a trial wave function for projector QMC methods, and to accelerate the progress of quantum chemistry.

The paper Ab-Initio Solution of the Many-Electron Schrödinger Equation with Deep Neural Networks is on arXiv.