Density Functional Theory: van der Waals corrections, symmetry-sphericity and machine learning

Abstract

van der Waals (vdW) or dispersion interaction dominates the weak bonding between the layers of a layered material or between a closed-shell molecule, monolayer, or multi-layered material and a solid surface. The computation of materials properties, including physical adsorption of closed-shell species on solid surfaces, must include a description of the intermediate and long-range parts of the vdW interaction. Calculation of the vdW interaction by correlated wave-functions or random-phase-approximation (RPA) methods is not computationally efficient enough to describe the large systems or supercells that arise in many adsorption problems. Standard semilocal density functional approximations (e.g., PBE and SCAN) to the exchange-correlation energy are more practical, but the long-range part is not included in them. The main purpose of this part of the research, and its main accomplishment, was to develop a computationally efficient intermediate- and long-range correction to semi-local density functionals that would include the needed level of detail. We introduced and applied a new approach, the damped Zaremba-Kohn (dZK) method, to semiconducting substrates of finite thickness. The method has been applied to the adsorption of a graphene monolayer on bulk graphite, bulk hexagonal boron nitride, and multilayer transition metal dichalcogenides (one to four layers of MoS_2, WS_2, MoSe_2, and WSe_2). Furthermore, we extended the model to molecules adsorbed on a curved cylindrical conducting surface and combined this model with semilocal density functionals; Calculations were made for the molecules: ammonia (NH_3) and nitrogen dioxide (NO_2) at two adsorption sites, using the PBE, SCAN, PBE+dZK, and SCAN+dZK methods.Turning now to the next part, we investigated several issues in density functional theory: the sensitivity to electron density of a hierarchy of nonempirical density functionals, the extent to which these density functionals approximate the exact functionals defined as constrained searches over wavefunctions versus ensembles, symmetry breaking and symmetry preservation, etc. We report results from several calculations with approximate density functionals which show that the total energies of non-spherical atoms are systematically lower than those for spherical atoms, a result which leads to appreciably improved molecular binding energies. In addition, we demonstrated that the energy consequences of the symmetry-breaking by self-consistent densities of open-shell atoms computed with approximate functionals are small, justifying their use to compute atomization energies of molecules and solids, and justifying the use of the results to test the accuracy of the approximations. Regarding the third part, our research presents a new approach to incorporate some exact constraints into machine learned density functionals (ML-DFT). ML-DFTs are one of the most exciting tools that have entered the material science toolbox in recent years. Recently, machine learning (ML) has been applied to parametrize exchange-correlation (XC) functionals without human domain knowledge by using kernel ridge regression (KRR), fully connected neural networks (NNs) and convolutional neural networks (CNNs). It is well-known that physical XC functionals must satisfy several exact conditions, such as coordinate scaling, spin scaling and derivative discontinuity. However, these exact conditions have not been incorporated implicitly into the machine learning modeling and pre-processing on large material datasets. In this work, we propose a schematic approach to incorporate a given physical constraint via contrastive learning. We then transfer the pretrained representation of electron density on an augmented molecular dataset which was generated by using the scaling property of exchange energy functionals based on the scaling factors chosen. The model with pretrained representation predicts exchange energies that satisfy the scaling relation, while the model trained on an unaugmented dataset gives poor predictions for the scaling-transformed electron density systems. Taken together, these findings suggest that pretraining task via contrastive learning can enhance the understanding of DFT theory for neural network models and generalize the application of NN-based XC functionals in a wide range of scenarios which are not always available experimentally but theoretically justified.