Permutation-symmetry-adapted machine learning based on equivariant neural network

In quantum mechanics, the wave function should be permutation symmetric for bosons and permutation anti-symmetric for fermions. The latter is important in electronic structure calculation since the electron is one of fermion, and people usually use Slater determinants to preserve the anti-symmetry of the electronic wave function. However, the linear combination of determinants is necessary for high-accuracy electronic interaction calculation, which limits the application of high-level wave function methods such as full CI. Many researchers try to combine the neural network with Quantum Monte-Carlo (QMC) method to overcome both basis set limitation and exponent-growth determinants problem. Recently, David Pfau, et al. developed a promising model named FermiNet. They constructed generalized Slater determinants by Equivariant Neural Network (ENN) to describe electronic wave function and then optimized it by QMC. This model can significantly reduce determinants numbers and speed up QMC simulation. In our study, a more general permutation-symmetry-adapted formula has been introduced to mixed-symmetry systems, e.g., boson-fermion mixture system. We used ENN to preserve both permutation symmetry and permutation anti-symmetry of elements in the Diabatic Potential Energy Matrix (DPEM). We combined ENN model with the machine learning diabatization method, and construct the first global DPEM of the MgH₂ 1A' and 2A' state, which can be useful for non-adiabatic simulation in the future. Our Diabatization by Equivariant Neural Network (DENN) procedure provides a permutation-symmetry-adapted method for DPEM fitting and other symmetry preservation problems in complex systems.

Illustration of Diabatization by Equivariant Neural Network (DENN) procedure