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

Abstract

Machine learning (ML) algorithms have undergone an explosive development impacting every aspect of computational chemistry. To obtain reliable predictions, one needs to maintain a proper balance between the black-box nature of ML frameworks and the physics of the target properties. One of the most appealing quantum-chemical properties for regression models is the electron density, and some of us recently proposed a transferable and scalable model based on the decomposition of the density onto an atom-centered basis set. The decomposition, as well as the training of the model, is at its core a minimization of some loss function, which can be arbitrarily chosen and may lead to results of different quality. Well-studied in the context of density fitting (DF), the impact of the metric on the performance of ML models has not been analyzed yet. In this work, we compare predictions obtained using the overlap and the Coulomb-repulsion metrics for both decomposition and training. As expected, the Coulomb metric used as both the DF and ML loss functions leads to the best results for the electrostatic potential and dipole moments. The origin of this difference lies in the fact that the model is not constrained to predict densities that integrate to the exact number of electrons N. Since an a posteriori correction for the number of electrons decreases the errors, we proposed a modification of the model, where N is included directly into the kernel function, which allowed lowering of the errors on the test and out-of-sample sets.

Learning Electron Densities in the Condensed Phase

“…As a result, when approximating the density in this way it may be necessary to find a basis in which to expand the electron density which produces a tolerably low error not only in the density itself but also in some property of interest derived from this density, since the former does not guarantee the latter. 29 3.2. Predicting Electron Densities.…”

“…27,28 Different error metrics can be adopted to determine the expansion coefficients, which influence the accuracy that one is willing to achieve on prescribed classes of density-derived properties. 29 A Coulomb metric, for instance, is typically used to provide RI approximations that give minimal error in the Hartree energy. 30 In this work, we define the RI expansion coefficients as those which minimize the integral over a single unit cell of the square error in the density itself, i.e.,…”

“…Moreover, as clarified in ref 29, imposing a corresponding constraint on the machinelearning model would only limit the electronic charge conservation to those structures that are used for training, and including such a constraint inevitably leads to a breakdown in the stability of the machine-learning model as the number of training structures is increased. 29 Minimization of the RI error in eq 2 yields c S w…”

“…Different error metrics can be adopted to determine the expansion coefficients, which influence the accuracy that one is willing to achieve on prescribed classes of density-derived properties . A Coulomb metric, for instance, is typically used to provide RI approximations that give minimal error in the Hartree energy .…”

Learning the Exciton Properties of Azo-dyes

“…The computation of basis set projection and Coulomb integrals for all the fields included in this work was performed with PySCF. , The Coulomb metric was used both for the basis set decomposition with cc-pVQZ-JKFIT, and for the loss function of the machine learning model (see section S3). The tensorial λ-SOAP kernels were computed using an environment cutoff of 3 Å, Gaussian smearing of 0.3 Å, angular cutoff 6, radial cutoff 8, and environmental kernel exponent ζ = 2 (). Sparse regression was performed with a subset of M = 1000 reference environments to reduce the dimensionality of the regression problem.…”