We present the Electronic Tensor Reconstruction Algorithm (ELECTRA) - an equivariant model for predicting electronic charge densities using "floating" orbitals. Floating orbitals are a long-standing idea in the quantum chemistry community that promises more compact and accurate representations by placing orbitals freely in space, as opposed to centering all orbitals at the position of atoms. Finding ideal placements of these orbitals requires extensive domain knowledge though, which thus far has prevented widespread adoption. We solve this in a data-driven manner by training a Cartesian tensor network to predict orbital positions along with orbital coefficients. This is made possible through a symmetry-breaking mechanism that is used to learn position displacements with lower symmetry than the input molecule while preserving the rotation equivariance of the charge density itself. Inspired by recent successes of Gaussian Splatting in representing densities in space, we are using Gaussians as our orbitals and predict their weights and covariance matrices. Our method achieves a state-of-the-art balance between computational efficiency and predictive accuracy on established benchmarks.

Electron density $\rho(\vec{r})$ is the fundamental variable in the calculation of ground state energy with density functional theory (DFT). Beyond total energy, features in $\rho(\vec{r})$ distribution and modifications in $\rho(\vec{r})$ are often used to capture critical physicochemical phenomena in functional materials and molecules at the electronic scale. Methods providing access to $\rho(\vec{r})$ of complex disordered systems with little computational cost can be a game changer in the expedited exploration of materials phase space towards the inverse design of new materials with better functionalities. We present a machine learning framework for the prediction of $\rho(\vec{r})$. The model is based on equivariant graph neural networks and the electron density is predicted at special query point vertices that are part of the message passing graph, but only receive messages. The model is tested across multiple data sets of molecules (QM9), liquid ethylene carbonate electrolyte (EC) and LixNiyMnzCo(1-y-z)O2 lithium ion battery cathodes (NMC). For QM9 molecules, the accuracy of the proposed model exceeds typical variability in $\rho(\vec{r})$ obtained from DFT done with different exchange-correlation functional and show beyond the state of the art accuracy. The accuracy is even better for the mixed oxide (NMC) and electrolyte (EC) datasets. The linear scaling model's capacity to probe thousands of points simultaneously permits calculation of $\rho(\vec{r})$ for large complex systems many orders of magnitude faster than DFT allowing screening of disordered functional materials.

Data-driven methods based on machine learning have the potential to accelerate analysis of atomic structures. However, machine learning models can produce overconfident predictions and it is therefore crucial to detect and handle uncertainty carefully. Here, we extend a message passing neural network designed specifically for predicting properties of molecules and materials with a calibrated probabilistic predictive distribution. The method presented in this paper differs from the previous work by considering both aleatoric and epistemic uncertainty in a unified framework, and by re-calibrating the predictive distribution on unseen data. Through computer experiments, we show that our approach results in accurate models for predicting molecular formation energies with calibrated uncertainty in and out of the training data distribution on two public molecular benchmark datasets, QM9 and PC9. The proposed method provides a general framework for training and evaluating neural network ensemble models that are able to produce accurate predictions of properties of molecules with calibrated uncertainty.