We present a parameter-free method for an accurate determination of long-range van der Waals interactions from mean-field electronic structure calculations. Our method relies on the summation of interatomic C6 coefficients, derived from the electron density of a molecule or solid and accurate reference data for the free atoms. The mean absolute error in the C6 coefficients is 5.5% when compared to accurate experimental values for 1225 intermolecular pairs, irrespective of the employed exchange-correlation functional. We show that the effective atomic C6 coefficients depend strongly on the bonding environment of an atom in a molecule. Finally, we analyze the van der Waals radii and the damping function in the C6R(-6) correction method for density-functional theory calculations.

Deep learning has led to a paradigm shift in artificial intelligence, including web, text, and image search, speech recognition, as well as bioinformatics, with growing impact in chemical physics. Machine learning, in general, and deep learning, in particular, are ideally suitable for representing quantum-mechanical interactions, enabling us to model nonlinear potential-energy surfaces or enhancing the exploration of chemical compound space. Here we present the deep learning architecture SchNet that is specifically designed to model atomistic systems by making use of continuous-filter convolutional layers. We demonstrate the capabilities of SchNet by accurately predicting a range of properties across chemical space for molecules and materials, where our model learns chemically plausible embeddings of atom types across the periodic table. Finally, we employ SchNet to predict potential-energy surfaces and energy-conserving force fields for molecular dynamics simulations of small molecules and perform an exemplary study on the quantum-mechanical properties of C-fullerene that would have been infeasible with regular ab initio molecular dynamics.

We introduce a machine learning model to predict atomization energies of a diverse set of organic molecules, based on nuclear charges and atomic positions only. The problem of solving the molecular Schrödinger equation is mapped onto a non-linear statistical regression problem of reduced complexity. Regression models are trained on and compared to atomization energies computed with hybrid density-functional theory. Cross-validation over more than seven thousand small organic molecules yields a mean absolute error of ∼10 kcal/mol. Applicability is demonstrated for the prediction of molecular atomization potential energy curves.Solving the Schrödinger equation (SE), HΨ = EΨ, for assemblies of atoms is a fundamental problem in quantum mechanics. Alas, solutions that are exact up to numerical precision are intractable for all but the smallest systems with very few atoms. Hierarchies of approximations have evolved, usually trading accuracy for computational efficiency [1]. Conventionally, the external potential, defined by a set of nuclear charges {Z I } and atomic positions {R I }, uniquely determines the Hamiltonian H of any system, and thereby the potential energy by optimizingFor a diverse set of organic molecules, we show that one can use machine learning (ML) instead, {Z I , R I } ML −→ E. Thus, we circumvent the task of explicitly solving the SE by training once a machine on a finite subset of known solutions. Since many interesting questions in physics require to repeatedly solve the SE, the highly competitive performance of our ML approach may pave the way to large scale exploration of molecular energies in chemical compound space [3,4]. ML techniques have recently been used with success to map the problem of solving complex physical differential equations to statistical models. Successful attempts include solving Fokker-Planck stochastic differential equations [5], parameterizing interatomic force fields for fixed chemical composition [6,7], and the discovery of novel ternary oxides for batteries [8]. Motivated by these, and other related efforts [9-12], we develop a non-linear regression ML model for computing molecular atomization energies in chemical compound space [3]. Our model is based on a measure of distance in compound space that accounts for both stoichiometry and configurational variation. After training, energies are predicted for new (out-of-sample) molecular systems, differing in composition and geometry, at negligible computational cost, i.e. milli seconds instead of hours on a conventional CPU. While the model is trained and tested using atomization energies calculated at the hybrid density-functional theory (DFT) level [2,13,14], any other training set or level of theory could be used as a starting point for subsequent ML training. Cross-validation on 7165 molecules yields a mean absolute error of 9.9 kcal/mol, which is an order of magnitude more accurate than counting bonds or semi-empirical quantum chemistry.We use the GDB data base, a library of nearly one billion organic molecules that ...