Impact of quantum-chemical metrics on the machine learning prediction of electron density.

Machine learning (ML) algorithms are currently spreading to almost every aspect of computational chemistry, and to obtain reliable results it is important to maintain the balance between the intrinsic black-box nature of ML frameworks and physical assumptions about the properties of the system under study. One of the most appealing quantum-chemical properties for the regression models is the electron density, and recently some of us have proposed a transferable and scalable machine learning model based on decomposition of the density field onto an atom-centered basis set. The decomposition, as well as the training of the model, is at its core a minimization of some quadratic loss function, which can be arbitrarily chosen and may lead to results of different quality. Well-studied in context of density fitting (DF) within the density functional theory, the impact of this choice on the performance of machine learning models has not been analyzed yet. In this work, we used the overlap and the Coulomb repulsion metrics in both -- decomposition and training -- roles and compared the electrostatic potential and dipole moments predicted by these four models. As expected, the Coulomb metric used as both the DF and ML loss function leads to the best results. The main source of this difference is the fact that the model is not constrained to predict densities which integrates to the exact number of electrons $N$, and the Coulomb metric tends to yield more accurate $N$. Since an \textit{a posteriori} correction of the number of electrons decreases the errors, we proposed a modification of the model where $N$ is included directly into the ML kernel function, which allowed to decrease noticeably the errors on the test and out-of-sample sets.