Toward the Exact Exchange–Correlation Potential: A Three-Dimensional Convolutional Neural Network Construct

A deep neural network is constructed to yield in principle exact exchange−correlation potential. It requires merely the electron densities of small molecules and ions and yet can determine the exchange−correlation potentials of large molecules. We train and validate the neural network based on the data for H2 and HeH + and subsequently determine the ground-state electron density of stretched HeH, linear H3 , and H−He−He−H. Moreover, the deep neural network is proven to model the van der Waals interaction by being trained and validated on a data set containing He2. Comparisons to B3LYP are given to illustrate the accuracy and transferability of the neural network. D theory (DFT) has become the most widely used quantum mechanical simulation method. Despite its success, DFT suffers from chronic deficiencies such as the self-interaction error, chemical accuracy issue for thermochemistry, poor treatment of van der Waals interaction, inaccurate reaction barrier, hydrogen bond, charge-transfer, delocalization error, and static correlation and strong correlation error. Medvedev et al. showed that although the accuracy in energy of DFT methods had been improving over the past decade or two, the calculated electron density distribution had become less accurate. Despite the continuing efforts to find more accurate exchange−correlation (xc) functionals, the improvement in calculated accuracy has slowed down. Alternatives beyond the traditional approaches must be sought. Efforts have been made to construct high-quality xc potentials for given electron densities via optimization procedures. Modern machine learning techniques have also been incorporated in the DFT method aiming at boosting calculations, calibrating results, or improving the xc functional. In 2004 we proposed a neural network-based B3LYP xc functional. The three hybrid parameters in the B3LYP xc functional, a0, ax, and ac, are in principle the functionals of the ground-state electron density function and thus system-dependent. Their values were evaluated via a neural network which had been trained and tested against the experimental data. The input descriptors of the neural network were the number of electrons, dipole moment, quadrupole moment, kinetic energy, and spin multiplicity of the system, which are the functionals of the ground-state electron density function of the molecule. In 2012, Burke and co-workers proposed a machine learning-based density functional for the kinetic energy of one-dimensional electron gas, which was later extended for real molecules. All these machine learning-based functionals require the knowledge of the global electron density function. For small molecules or ions, it is straightforward to obtain their electron density functions and the corresponding energies using highly accurate quantum chemistry methods such as the coupled-cluster method, quantum Monte Carlo, and the density matrix renormalization group technique. However, electron density functions and energies of larger molecules or ions are difficult to determine, if not impossible. As a result, no or little data are available for large molecular systems. This problem is difficult to circumvent and prevents all the existing machine learningbased methods for the DFT xc functional from being applicable for large molecular systems. The xc potential vxc(r) of the density-functional theory 2