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The static dielectric constant ϵ_0 and its temperature dependence for liquid water is investigated using neural network quantum molecular dynamics (NNQMD). We compute the exact dielectric constant in canonical ensemble from NNQMD trajectories using fluctuations in macroscopic polarization computed from maximally localized Wannier functions (MLWF). Two deep neural networks are constructed. The first, NNQMD, is trained on QMD configurations for liquid water under a variety of temperature and density conditions to learn potential energy sur
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