Self-Consistent Convolutional Density Functional Approximations: Application to Adsorption at Metal Surfaces

The exchange-correlation (XC) functional in density functional theory is used to approximate multi-electron interactions. A plethora of different functionals are available, but nearly all are based on the hierarchy of inputs commonly referred to as "Jacob's ladder." This paper introduces an approach to construct XC functionals with inputs from convolutions of arbitrary kernels with the electron density, providing a route to move beyond Jacob's ladder. We derive the variational derivative of these functionals, showing consistency with the generalized gradient approximation (GGA), and provide equations for variational derivatives based on multipole features from convolutional kernels. A proof-of-concept functional, PBEq, which generalizes the PBE α ${\alpha }$ framework with α ${\alpha }$ being a spatially-resolved function of the monopole of the electron density, is presented and implemented. It allows a single functional to use different GGAs at different spatial points in a system, while obeying PBE constraints. Analysis of the results underlines the importance of error cancellation and the XC potential in data-driven functional design. After testing on small molecules, bulk metals, and surface catalysts, the results indicate that this approach is a promising route to simultaneously optimize multiple properties of interest.

Model-Based Design of Experiments for Temporal Analysis of Products (TAP): A Simulated Case Study in Oxidative Propane Dehydrogenation

Temporal analysis of products (TAP) reactors enable experiments that probe numerous kinetic processes within a single set of experimental data through variations in pulse intensity, delay, or temperature. Selecting additional TAP experiments often involves an arbitrary selection of reaction conditions or the use of chemical intuition. To make experiment selection in TAP more robust, we explore the efficacy of model-based design of experiments (MBDoE) for precision in TAP reactor kinetic modeling. We successfully applied this approach to a case study of synthetic oxidative propane dehydrogenation (OPDH) that involves pulses of propane and oxygen. We found that experiments identified as optimal through the MBDoE for precision generally reduce parameter uncertainties to a higher degree than alternative experiments. The performance of MBDoE for model divergence was also explored for OPDH, with the relevant active sites (catalyst structure) being unknown. An experiment that maximized the divergence between the three proposed mechanisms was identified and provided evidence that improved the mechanism discrimination. However, reoptimization of kinetic parameters eliminated the ability to discriminate between models. The findings yield insight into the prospects and limitations of MBDoE for TAP and transient kinetic experiments.

Kohn-Sham accuracy from orbital-free density functional theory via Δ-machine learning

We present a Δ-machine learning model for obtaining Kohn-Sham accuracy from orbital-free density functional theory (DFT) calculations. In particular, we employ a machine-learned force field (MLFF) scheme based on the kernel method to capture the difference between Kohn-Sham and orbital-free DFT energies/forces. We implement this model in the context of on-the-fly molecular dynamics simulations and study its accuracy, performance, and sensitivity to parameters for representative systems. We find that the formalism not only improves the accuracy of Thomas-Fermi-von Weizsäcker orbital-free energies and forces by more than two orders of magnitude but is also more accurate than MLFFs based solely on Kohn-Sham DFT while being more efficient and less sensitive to model parameters. We apply the framework to study the structure of molten Al0.88Si0.12, the results suggesting no aggregation of Si atoms, in agreement with a previous Kohn-Sham study performed at an order of magnitude smaller length and time scales.

Formation of Carbon-Induced Nitrogen-Centered Radicals on Titanium Dioxide under Illumination

Titanium dioxide is the most studied photocatalytic material and has been reported to be active for a wide range of reactions, including the oxidation of hydrocarbons and the reduction of nitrogen. However, the molecular-scale interactions between the titania photocatalyst and dinitrogen are still debated, particularly in the presence of hydrocarbons. Here, we used several spectroscopic and computational techniques to identify interactions among nitrogen, methanol, and titania under illumination. Electron paramagnetic resonance spectroscopy (EPR) allowed us to observe the formation of carbon radicals upon exposure to ultraviolet radiation. These carbon radicals are observed to transform into diazo- and nitrogen-centered radicals (e.g., CHxN2• and CHxNHy•) during photoreaction in nitrogen environment. In situ infrared (IR) spectroscopy under the same conditions revealed C-N stretching on titania. Furthermore, density functional theory (DFT) calculations revealed that nitrogen adsorption and the thermodynamic barrier to photocatalytic nitrogen fixation are significantly more favorable in the presence of hydroxymethyl or surface carbon. These results provide compelling evidence that carbon radicals formed from the photooxidation of hydrocarbons interact with dinitrogen and suggest that the role of carbon-based "hole scavengers" and the inertness of nitrogen atmospheres should be reevaluated in the field of photocatalysis.

Screening and Discovery of Metal Compound Active Sites for Strong and Selective Adsorption of N2 in Air

Photocatalytic nitrogen fixation has the potential to provide a greener route for producing nitrogen-based fertilizers under ambient conditions. Computational screening is a promising route to discover new materials for the nitrogen fixation process, but requires identifying "descriptors" that can be efficiently computed. In this work, we argue that selectivity toward the adsorption of molecular nitrogen and oxygen can act as a key descriptor. A catalyst that can selectively adsorb nitrogen and resist poisoning of oxygen and other molecules present in air has the potential to facilitate the nitrogen fixation process under ambient conditions. We provide a framework for active site screening based on multifidelity density functional theory (DFT) calculations for a range of metal oxides, oxyborides, and oxyphosphides. The screening methodology consists of initial low-fidelity fixed geometry calculations and a second screening in which more expensive geometry optimizations were performed. The approach identifies promising active sites on several TiO2 polymorph surfaces and a VBO4 surface, and the full nitrogen reduction pathway is studied with the BEEF-vdW and HSE06 functionals on two active sites. The findings suggest that metastable TiO2 polymorphs may play a role in photocatalytic nitrogen fixation, and that VBO4 may be an interesting material for further studies.

Automated Generation of Microkinetics for Heterogeneously Catalyzed Reactions Considering Correlated Uncertainties

The study presents an ab-initio based framework for the automated construction of microkinetic mechanisms considering correlated uncertainties in all energetic parameters and estimation routines. 2000 unique microkinetic models were generated within the uncertainty space of the BEEF-vdW functional for the oxidation reactions of representative exhaust gas emissions from stoichiometric combustion engines over Pt(111) and compared to experiments through multiscale modeling. The ensemble of simulations stresses the importance of considering uncertainties. Within this set of first-principles-based models, it is possible to identify a microkinetic mechanism that agrees with experimental data. This mechanism can be traced back to a single exchange-correlation functional, and it suggests that Pt(111) could be the active site for the oxidation of light hydrocarbons. The study provides a universal framework for the automated construction of reaction mechanisms with correlated uncertainty quantification, enabling a DFT-constrained microkinetic model optimization for other heterogeneously catalyzed systems.

Robust and scalable uncertainty estimation with conformal prediction for machine-learned interatomic potentials

Uncertainty quantification (UQ) is important to machine learning (ML) force fields to assess the level of confidence during prediction, as ML models are not inherently physical and can therefore yield catastrophically incorrect predictions. Established a-posteriori UQ methods, including ensemble methods, the dropout method, the delta method, and various heuristic distance metrics, have limitations such as being computationally challenging for large models due to model re-training. In addition, the uncertainty estimates are often not rigorously calibrated. In this work, we propose combining the distribution-free UQ method, known as conformal prediction (CP), with the distances in the neural network’s latent space to estimate the uncertainty of energies predicted by neural network force fields. We evaluate this method (CP+latent) along with other UQ methods on two essential aspects, calibration, and sharpness, and find this method to be both calibrated and sharp under the assumption of independent and identically-distributed (i.i.d.) data. We show that the method is relatively insensitive to hyperparameters selected, and test the limitations of the method when the i.i.d. assumption is violated. Finally, we demonstrate that this method can be readily applied to trained neural network force fields with traditional and graph neural network architectures to obtain estimates of uncertainty with low computational costs on a training dataset of 1 million images to showcase its scalability and portability. Incorporating the CP method with latent distances offers a calibrated, sharp and efficient strategy to estimate the uncertainty of neural network force fields. In addition, the CP approach can also function as a promising strategy for calibrating uncertainty estimated by other approaches.

A Universal Framework for Featurization of Atomistic Systems

Machine-learning force fields have become increasingly popular because of their balance of accuracy and speed. However, a significant limitation is the use of element-specific features, leading to poor scalability with the number of elements. This work introduces the Gaussian multipole (GMP) featurization scheme that utilizes physically relevant multipole expansions of the electron density around atoms to yield feature vectors that interpolate between element types and have a fixed dimension regardless of the number of elements present. We combine GMP with neural networks and apply these models to the MD17 and QM9 data sets, revealing high computational efficiency, systematically improvable accuracy, and the ability to make reasonable predictions on elements not included in the training set. Finally, we test GMP-based models for the OCP data set, demonstrating comparable performance to graph-convolutional models. The results indicate that this featurization scheme fills a critical gap in the construction of efficient and transferable machine-learned force fields.

Gaussian approximation of dispersion potentials for efficient featurization and machine-learning predictions of metal–organic frameworks

Energy-related descriptors in machine learning are a promising strategy to predict adsorption properties of metal–organic frameworks (MOFs) in the low-pressure regime. Interactions between hosts and guests in these systems are typically expressed as a sum of dispersion and electrostatic potentials. The energy landscape of dispersion potentials plays a crucial role in defining Henry’s constants for simple probe molecules in MOFs. To incorporate more information about this energy landscape, we introduce the Gaussian-approximated Lennard-Jones (GALJ) potential, which fits pairwise Lennard-Jones potentials with multiple Gaussians by varying their heights and widths. The GALJ approach is capable of replicating information that can be obtained from the original LJ potentials and enables efficient development of Gaussian integral (GI) descriptors that account for spatial correlations in the dispersion energy environment. GI descriptors would be computationally inconvenient to compute using the usual direct evaluation of the dispersion potential energy surface. We show that these new GI descriptors lead to improvement in ML predictions of Henry’s constants for a diverse set of adsorbates in MOFs compared to previous approaches to this task.

Heterogeneity in susceptibility dictates the order of epidemic models

The fundamental models of epidemiology describe the progression of an infectious disease through a population using compartmentalized differential equations, but typically do not incorporate population-level heterogeneity in infection susceptibility. Here we combine a generalized analytical framework of contagion with computational models of epidemic dynamics to show that variation strongly influences the rate of infection, while the infection process simultaneously sculpts the susceptibility distribution. These joint dynamics influence the force of infection and are, in turn, influenced by the shape of the initial variability. We find that certain susceptibility distributions (the exponential and the gamma) are unchanged through the course of the outbreak, and lead naturally to power-law behavior in the force of infection; other distributions are often sculpted towards these "eigen-distributions" through the process of contagion. The power-law behavior fundamentally alters predictions of the long-term infection rate, and suggests that first-order epidemic models that are parameterized in the exponential-like phase may systematically and significantly over-estimate the final severity of the outbreak. In summary, our study suggests the need to examine the shape of susceptibility in natural populations as part of efforts to improve prediction models and to prioritize interventions that leverage heterogeneity to mitigate against spread.

The Role of Adventitious Carbon in Photo-catalytic Nitrogen Fixation by Titania

Photo-catalytic fixation of nitrogen by titania catalysts at ambient conditions has been reported for decades, yet the active site capable of adsorbing an inert N₂ molecule at ambient pressure and the mechanism of dissociating the strong dinitrogen triple bond at room temperature remain unknown. In this work in situ near-ambient-pressure X-ray photo-electron spectroscopy and density functional theory calculations are used to probe the active state of the rutile (110) surface. The experimental results indicate that photon-driven interaction of N₂ and TiO₂ is observed only if adventitious surface carbon is present, and computational results show a remarkably strong interaction between N₂ and carbon substitution (C*) sites that act as surface-bound carbon radicals. A carbon-assisted nitrogen reduction mechanism is proposed and shown to be thermodynamically feasible. The findings provide a molecular-scale explanation for the long-standing mystery of photo-catalytic nitrogen fixation on titania. The results suggest that controlling and characterizing carbon-based active sites may provide a route to engineering more efficient photo(electro)-catalysts and improving experimental reproducibility.

To address surface reaction network complexity using scaling relations machine learning and DFT calculations

Surface reaction networks involving hydrocarbons exhibit enormous complexity with thousands of species and reactions for all but the very simplest of chemistries. We present a framework for optimization under uncertainty for heterogeneous catalysis reaction networks using surrogate models that are trained on the fly. The surrogate model is constructed by teaching a Gaussian process adsorption energies based on group additivity fingerprints, combined with transition-state scaling relations and a simple classifier for determining the rate-limiting step. The surrogate model is iteratively used to predict the most important reaction step to be calculated explicitly with computationally demanding electronic structure theory. Applying these methods to the reaction of syngas on rhodium(111), we identify the most likely reaction mechanism. Propagating uncertainty throughout this process yields the likelihood that the final mechanism is complete given measurements on only a subset of the entire network and uncertainty in the underlying density functional theory calculations.

Analyzing the Case for Bifunctional Catalysis

Bifunctional coupling of two different catalytic site types has often been invoked to explain experimentally observed enhanced catalytic activities. We scrutinize such claims with generic scaling-relation-based microkinetic models that allow exploration of the theoretical limits for such a bifunctional gain for several model reactions. For sites at transition-metal surfaces, the universality of the scaling relations between adsorption energies largely prevents any improvements through bifunctionality. Only the consideration of systems that involve the combination of different materials, such as metal particles on oxide supports, offers hope for significant bifunctional gains.

Catalysis. Assessing the reliability of calculated catalytic ammonia synthesis rates

We introduce a general method for estimating the uncertainty in calculated materials properties based on density functional theory calculations. We illustrate the approach for a calculation of the catalytic rate of ammonia synthesis over a range of transition-metal catalysts. The correlation between errors in density functional theory calculations is shown to play an important role in reducing the predicted error on calculated rates. Uncertainties depend strongly on reaction conditions and catalyst material, and the relative rates between different catalysts are considerably better described than the absolute rates. We introduce an approach for incorporating uncertainty when searching for improved catalysts by evaluating the probability that a given catalyst is better than a known standard.

Ultra fast and parsimonious materials screening for polymer solar cells using differentially pumped slot-die coating

We present a technique that enables the probing of the entire parameter space for each parameter with good statistics through a simple roll-to-roll processing method where gradients of donor, acceptor, and solvent are applied by differentially pumped slot-die coating. We thus demonstrate how the optimum donor-acceptor ratio and device film thickness can be determined with improved accuracy by varying the composition in small steps. We give as an example P3HT-PCBM devices and vary the composition between P3HT and PCBM in steps of 0.5-1% giving 100-200 individual solar cells. The coating experiment itself takes less than 4-8 min and requires 15-30 mg each of donor and acceptor material. The optimum donor-acceptor composition of P3HT and PCBM was found to be a broad maximum centered on a 1:1 ratio. We demonstrate how the optimal thickness of the active layer can be found by the same method and materials usage by variation of the layer thickness in small steps of 1.5-4 nm. Contrary to expectation we did not find oscillatory variation of the device performance with device thickness because of optical interference. We ascribe this to the nature of the solar cell type explored in this example that employs nonreflective or semitransparent printed electrodes. We further found that very thick active layers on the order of 1 μm can be prepared without loss in performance and estimate the active layer thickness could easily approach 4-5 μm while maintaining photovoltaic properties.

Grid-connected polymer solar panels: initial considerations of cost, lifetime, and practicality

Large solar panels were constructed from polymer solar cell modules prepared using full roll-to-roll (R2R) manufacture based on the previously published ProcessOne. The individual flexible polymer solar modules comprising multiple serially connected single cell stripes were joined electrically and laminated between a 4 mm tempered glass window and black Tetlar foil using two sheets of 0.5 mm thick ethylene vinyl acetate (EVA). The panels produced up to 8 W with solar irradiance of ~960 Wm⁻², and had outer dimensions of 1 m x 1.7 m with active areas up to 9180 cm². Panels were mounted on a tracking station and their output was grid connected between testing. Several generations of polymer solar cells and panel constructions were tested in this context to optimize the production of polymer solar panels. Cells lacking a R2R barrier layer were found to degrade due to diffusion of oxygen after less than a month, while R2R encapsulated cells showed around 50% degradation after 6 months but suffered from poor performance due to de-lamination during panel production. A third generation of panels with various barrier layers was produced to optimize the choice of barrier foil and it was found that the inclusion of a thin protective foil between the cell and the barrier foil is critical. The findings provide a preliminary foundation for the production and optimization of large-area polymer solar panels and also enabled a cost analysis of solar panels based on polymer solar cells.