Unifying machine learning and quantum chemistry with a deep neural network for molecular wavefunctions

Abstract

Machine learning advances chemistry and materials science by enabling large-scale exploration of chemical space based on quantum chemical calculations. While these models supply fast and accurate predictions of atomistic chemical properties, they do not explicitly capture the electronic degrees of freedom of a molecule, which limits their applicability for reactive chemistry and chemical analysis. Here we present a deep learning framework for the prediction of the quantum mechanical wavefunction in a local basis of atomic orbitals from which all other ground-state properties can be derived. This approach retains full access to the electronic structure via the wavefunction at force-field-like efficiency and captures quantum mechanics in an analytically differentiable representation. On several examples, we demonstrate that this opens promising avenues to perform inverse design of molecular structures for targeting electronic property optimisation and a clear path towards increased synergy of machine learning and quantum chemistry.

Fig. 1

Synergy of quantum chemistry and machine learning. a Forward model: ML predicts chemical properties based on reference calculations. If another property is required, an additional ML model has to be trained. b Hybrid model: ML predicts the wavefunction. All ground state properties can be calculated and no additional ML is required. The wavefunctions can act as an interface between ML and QM

Fig. 2

Prediction of electronic properties with SchNOrb. a Illustration of the network architecture. The neural network architecture consists of three steps (grey boxes) starting from initial representations of atom types and positions (top), continuing with the construction of representations of chemical environments of atoms and atom pairs (middle) before using these to predict energy and Hamiltonian matrix respectively (bottom). The left path through the network to the energy prediction E is rotationally invariant by design, while the right pass to the Hamiltonian matrix H allows for a maximum angular momentum L of predicted orbitals by employing a multiplicative construction of the basis ω_ij using sequential interaction passes l = 0…2L. The onsite and offsite blocks of the Hamiltonian matrix are treated separately. The prediction of overlap matrix S is performed analogously. b Illustration of the SchNet interaction block. c Illustration of SchNorb interaction block. The pairwise representation ${\mathbf{h}}_{ij}^l$ of atoms i, j is constructed by a factorised tensor layer f_tensor from atomic representations as well as the interatomic distance. Using this, rotationally invariant interaction refinements ${\mathbf{v}}_i^m$ and basis coefficients ${\mathbf{p}}_{ij}^l$ are computed. d Loewdin population analysis for uracil based on the density matrix calculated from the predicted Hamiltonian and overlap matrices. e Mean abs. errors of lowest 20 orbitals (13 occupied + 7 virtual) of ethanol for Hartree–Fock and DFT@PBE. f The predicted (solid black) and reference (dashed grey) orbital energies of an ethanol molecule for DFT. Shown are the last four occupied and first four unoccupied orbitals, including HOMO and LUMO. The associated predicted and reference molecular orbitals are compared for four selected energy levels

Fig. 3

Proton transfer in malondialdehyde. a Excerpt of the MD trajectory showing the proton transfer, the electron density as well as the relevant MOs HOMO-2 and HOMO-3 for three configurations (I, II, III). b Forces exerted by the MOs on the transferred proton for configurations I and II. c Density of states broadened across the proton transfer trajectory. MO energies of the equilibrium structure are indicated by grey dashed lines. The inset shows a zoom of HOMO-2 and HOMO-3

Fig. 4

Applications of SchNOrb. a Optimisation of the HOMO-LUMO gap. HOMO and LUMO with energy levels are shown for a randomly drawn configuration of the malonaldehyde dataset (centre) as well as for configurations that were obtained from minimising or maximising the HOMO-LUMO gap prediction using SchNOrb (left and right, respectively). For the optimised configurations, the difference of the orbitals are shown in green (increase) and violet (decrease). The dominant geometrical change is indicated by the black arrows. b The predicted MO coefficients for the uracil configurations from the test set are used as a wavefunction guess to obtain accurate solutions from DFT at a reduced number of self-consistent-field (SCF) iterations. This reduces the required SCF iterations by an average of 77% using a Newton solver. In terms of runtime, it is more efficient to use SOSCF, even though this saves only 15% of iterations for uracil