A deep potential model with long-range electrostatic interactions

Machine learning models for the potential energy of multi-atomic systems, such as the deep potential (DP) model, make molecular simulations with the accuracy of quantum mechanical density functional theory possible at a cost only moderately higher than that of empirical force fields. However, the majority of these models lack explicit long-range interactions and fail to describe properties that derive from the Coulombic tail of the forces. To overcome this limitation, we extend the DP model by approximating the long-range electrostatic interaction between ions (nuclei + core electrons) and valence electrons with that of distributions of spherical Gaussian charges located at ionic and electronic sites. The latter are rigorously defined in terms of the centers of the maximally localized Wannier distributions, whose dependence on the local atomic environment is modeled accurately by a deep neural network. In the DP long-range (DPLR) model, the electrostatic energy of the Gaussian charge system is added to short-range interactions that are represented as in the standard DP model. The resulting potential energy surface is smooth and possesses analytical forces and virial. Missing effects in the standard DP scheme are recovered, improving on accuracy and predictive power. By including long-range electrostatics, DPLR correctly extrapolates to large systems the potential energy surface learned from quantum mechanical calculations on smaller systems. We illustrate the approach with three examples: the potential energy profile of the water dimer, the free energy of interaction of a water molecule with a liquid water slab, and the phonon dispersion curves of the NaCl crystal.

用于多原子系统势能的机器学习模型，例如深度势能（DP）模型，使得以仅略高于经验力场的成本进行具有量子力学密度泛函理论精度的分子模拟成为可能。然而，这些模型中的大多数缺乏明确的长程相互作用，并且无法描述由力的库仑尾产生的性质。为了克服这一限制，我们通过用位于离子和电子位点的球形高斯电荷分布来近似离子（原子核 + 核心电子）和价电子之间的长程静电相互作用来扩展DP模型。后者是根据最大局域化万尼尔分布的中心严格定义的，其对局部原子环境的依赖性通过深度神经网络精确建模。在DP长程（DPLR）模型中，高斯电荷系统的静电能被添加到如标准DP模型中所表示的短程相互作用中。所得的势能面是光滑的，并具有解析力和维里。标准DP方案中缺失的效应得以恢复，提高了准确性和预测能力。通过包含长程静电作用，DPLR能将从小系统的量子力学计算中学到的势能面正确外推到大系统。我们用三个例子来说明该方法：水二聚体的势能曲线、一个水分子与液态水层相互作用的自由能以及氯化钠晶体的声子色散曲线。