Fig. 6. Multi-valued dipole moments in liquid water.

Comparison of x component of dipole moments (μ_x) calculated with force-only training (FIREANN-wF) and brute-force dipole moment training (FIREANN-wD) based on the field-induced recursively embedded atom neural network (FIREANN) with density functional theory (DFT) and corrected DFT data (AIMD-correction) along the ab initio molecular dynamics (AIMD) trajectory and as a function of time, for a zero field and b an electric field along x axis (ε_x) with the intensity of 0.6 V Å⁻¹. Note that the FIREANN-wF predicted dipole moment, which is deduced by the energy gradient to the electric field vector introduces an undetermined constant factor, is shifted by a constant to match the DFT value. Source data are provided as a Source Data file.