Electron density learning of non-covalent systems

Artificial Neural Networks as Mappings between Proton Potentials, Wave Functions, Densities, and Energy Levels

Artificial neural networks (ANNs) have become important in quantum chemistry. Herein, applications to nuclear quantum effects, such as zero-point energy, vibrationally excited states, and hydrogen tunneling, are explored. ANNs are used to solve the time-independent Schrödinger equation for single- and double-well potentials representing hydrogen-bonded molecular systems capable of proton transfer. ANN mappings are trained to predict the lowest five proton vibrational energies, wave functions, and densities from the proton potentials and to predict the excited state proton vibrational energies and densities from the proton ground state density. For the inverse problem, ANN mappings are trained to predict the proton potential from the proton vibrational energy levels or the proton ground state density. This latter mapping is theoretically justified by the first Hohenberg-Kohn theorem establishing a one-to-one correspondence between the external potential and the ground state density. ANNs for two- and three-dimensional systems are also presented to illustrate the straightforward extension to higher dimensions.

Peptide-based electrochemical biosensors utilized for protein detection

Proteins are generally detected as biomarkers for tracing or determining various disorders in organisms. Biomarker proteins can be tracked in samples with various origins and in different concentrations, revealing whether an organism is in a healthy or unhealthy state. In regard to detection, electrochemical biosensors are a potential fusion of electronics, chemistry, and biology, allowing for fast and early point-of-care detection from a biological sample with the advantages of high sensitivity, simple construction, and easy operation. Peptides present a promising approach as a biorecognition element when connected with electrochemical biosensors. The benefits of short peptides lie mainly in their good stability and selective affinity to a target analyte. Therefore, peptide-based electrochemical biosensors (PBEBs) represent an alternative approach for the detection of different protein biomarkers. This review provides a summary of the past decade of recently proposed PBEBs designed for protein detection, dividing them according to different protein types: (i) enzyme detection, including proteases and kinases; (ii) antibody detection; and (iii) other protein detection. According to these protein types, different sensing mechanisms are discussed, such as the peptide cleavage by a proteases, phosphorylation by kinases, presence of antibodies, and exploiting of affinities; furthermore, measurements are obtained by different electrochemical methods. A discussion and comparison of various constructions, modifications, immobilization strategies and different sensing techniques in terms of high sensitivity, selectivity, repeatability, and potential for practical application are presented.

A Step toward the Quantification of Noncovalent Interactions in Large Biological Systems: The Independent Gradient Model-Extremely Localized Molecular Orbital Approach

The independent gradient model (IGM) is a recent electron density-based computational method that enables to detect and quantify covalent and noncovalent interactions. When applied to large systems, the original version of the technique still relies on promolecular electron densities given by the sum of spherically averaged atomic electron distributions, which leads to approximate evaluations of the inter- and intramolecular interactions occurring in systems of biological interest. To overcome this drawback and perform IGM analyses based on quantum mechanically rigorous electron densities also for macromolecular systems, we coupled the IGM approach with the recently constructed libraries of extremely localized molecular orbitals (ELMOs) that allow fast and reliable reconstructions of polypeptide and protein electron densities. The validation tests performed on small polypeptides and peptide dimers have shown that the novel IGM-ELMO strategy provides results that are systematically closer to the fully quantum mechanical ones and outperforms the IGM method based on the crude promolecular approximation, but still keeping a quite low computational cost. The results of the test calculations carried out on proteins have also confirmed the trends observed for the IGM analyses conducted on small systems. This makes us envisage the future application of the novel IGM-ELMO approach to unravel complicated noncovalent interaction networks (e.g., in protein-protein contacts) or to rationally design new drugs through molecular docking calculations and virtual high-throughput screenings.

Pure non-local machine-learned density functional theory for electron correlation

Density-functional theory (DFT) is a rigorous and (in principle) exact framework for the description of the ground state properties of atoms, molecules and solids based on their electron density. While computationally efficient density-functional approximations (DFAs) have become essential tools in computational chemistry, their (semi-)local treatment of electron correlation has a number of well-known pathologies, e.g. related to electron self-interaction. Here, we present a type of machine-learning (ML) based DFA (termed Kernel Density Functional Approximation, KDFA) that is pure, non-local and transferable, and can be efficiently trained with fully quantitative reference methods. The functionals retain the mean-field computational cost of common DFAs and are shown to be applicable to non-covalent, ionic and covalent interactions, as well as across different system sizes. We demonstrate their remarkable possibilities by computing the free energy surface for the protonated water dimer at hitherto unfeasible gold-standard coupled cluster quality on a single commodity workstation.

Semilocal density functionals, while computationally efficient, do not account for non-local correlation. Here, the authors propose a machine-learning approach to DFT that leads to non-local and transferable functionals applicable to non-covalent, ionic and covalent interactions across system of different sizes.

From the Linnett-Gillespie model to the polarization of the spin valence shells of metals in complexes

In this paper, we present a novel approach to track the origin of the metal complex structure from the topology of the α and β spin densities as an extension of the Linnett-Gillespie model. Usually, the theories that explain the metal-ligand interactions consider the disposition and the relative energies of the empty or occupied set of d orbitals, ignoring the spin contribution explicitly. Our quantum topological approach considers the spatial distribution of the α and β spin valence shells, and the energy interaction between them. We used the properties of the atomic graph, a topological object that summarises the charge concentrations and depletions on the valence shell of an atom in a molecule, and the interacting quantum atoms (IQA) energy partition scheme. Unlike the Linnett-Gillespie model, which is based on electron-electron repulsion, our approach states that the ligands provoke a redistribution of the electron density to maximize the nuclear-electron interactions in each spin valence shell to bypass the concentration of electron-electron interactions, resulting in a polarization pattern which determines the position of the ligands.

An Efficient Deep Learning Scheme To Predict the Electronic Structure of Materials and Molecules: The Example of Graphene-Derived Allotropes

Computations based on density functional theory (DFT) are transforming various aspects of materials research and discovery. However, the effort required to solve the central equation of DFT, namely the Kohn-Sham equation, which remains a major obstacle for studying large systems with hundreds of atoms in a practical amount of time with routine computational resources. Here, we propose a deep learning architecture that systematically learns the input-output behavior of the Kohn-Sham equation and predicts the electronic density of states, a primary output of DFT calculations, with unprecedented speed and chemical accuracy. The algorithm also adapts and progressively improves in predictive power and versatility as it is exposed to new diverse atomic configurations. We demonstrate this capability for a diverse set of carbon allotropes spanning a large configurational and phase space. The electronic density of states, along with the electronic charge density, may be used downstream to predict a variety of materials properties, bypassing the Kohn-Sham equation, leading to an ultrafast and high-fidelity DFT emulator.

Quantum chemical accuracy from density functional approximations via machine learning

Kohn-Sham density functional theory (DFT) is a standard tool in most branches of chemistry, but accuracies for many molecules are limited to 2-3 kcal ⋅ mol-1 with presently-available functionals. Ab initio methods, such as coupled-cluster, routinely produce much higher accuracy, but computational costs limit their application to small molecules. In this paper, we leverage machine learning to calculate coupled-cluster energies from DFT densities, reaching quantum chemical accuracy (errors below 1 kcal ⋅ mol-1) on test data. Moreover, density-based Δ-learning (learning only the correction to a standard DFT calculation, termed Δ-DFT ) significantly reduces the amount of training data required, particularly when molecular symmetries are included. The robustness of Δ-DFT  is highlighted by correcting "on the fly" DFT-based molecular dynamics (MD) simulations of resorcinol (C6H4(OH)2) to obtain MD trajectories with coupled-cluster accuracy. We conclude, therefore, that Δ-DFT  facilitates running gas-phase MD simulations with quantum chemical accuracy, even for strained geometries and conformer changes where standard DFT fails.

Machine Learning Approaches toward Orbital-free Density Functional Theory: Simultaneous Training on the Kinetic Energy Density Functional and Its Functional Derivative

Orbital-free approaches might offer a way to boost the applicability of density functional theory by orders of magnitude in system size. An important ingredient for this endeavor is the kinetic energy density functional. Snyder et al. [Phys. Rev. Lett.2012, 108, 25300223004593] presented a machine learning approximation for this functional achieving chemical accuracy on a one-dimensional model system. However, a poor performance with respect to the functional derivative, a crucial element in iterative energy minimization procedures, enforced the application of a computationally expensive projection method. In this work we circumvent this issue by including the functional derivative into the training of various machine learning models. Besides kernel ridge regression, the original method of choice, we also test the performance of convolutional neural network techniques borrowed from the field of image recognition.

Machine learning accurate exchange and correlation functionals of the electronic density

Density functional theory (DFT) is the standard formalism to study the electronic structure of matter at the atomic scale. In Kohn–Sham DFT simulations, the balance between accuracy and computational cost depends on the choice of exchange and correlation functional, which only exists in approximate form. Here, we propose a framework to create density functionals using supervised machine learning, termed NeuralXC. These machine-learned functionals are designed to lift the accuracy of baseline functionals towards that provided by more accurate methods while maintaining their efficiency. We show that the functionals learn a meaningful representation of the physical information contained in the training data, making them transferable across systems. A NeuralXC functional optimized for water outperforms other methods characterizing bond breaking and excels when comparing against experimental results. This work demonstrates that NeuralXC is a first step towards the design of a universal, highly accurate functional valid for both molecules and solids.

Increasing the non-locality of the exchange and correlation functional in DFT theory comes at a steep increase in computational cost. Here, the authors develop NeuralXC, a supervised machine learning approach to generate density functionals close to coupled-cluster level of accuracy yet computationally efficient.

Exploring chemical compound space with quantum-based machine learning

Rational design of compounds with specific properties requires understanding and fast evaluation of molecular properties throughout chemical compound space - the huge set of all potentially stable molecules. Recent advances in combining quantum-mechanical calculations with machine learning provide powerful tools for exploring wide swathes of chemical compound space. We present our perspective on this exciting and quickly developing field by discussing key advances in the development and applications of quantum-mechanics-based machine-learning methods to diverse compounds and properties, and outlining the challenges ahead. We argue that significant progress in the exploration and understanding of chemical compound space can be made through a systematic combination of rigorous physical theories, comprehensive synthetic data sets of microscopic and macroscopic properties, and modern machine-learning methods that account for physical and chemical knowledge.

Simulation vs. Understanding: A Tension, in Quantum Chemistry and Beyond. Part B. The March of Simulation, for Better or Worse

In the second part of this Essay, we leave philosophy, and begin by describing Roald's being trashed by simulation. This leads us to a general sketch of artificial intelligence (AI), Searle's Chinese room, and Strevens' account of what a go-playing program knows. Back to our terrain-we ask "Quantum Chemistry, † ca. 2020?" Then we move to examples of Big Data, machine learning and neural networks in action, first in chemistry and then affecting social matters, trivial to scary. We argue that moral decisions are hardly to be left to a computer. And that posited causes, even if recognized as provisional, represent a much deeper level of understanding than correlations. At this point, we try to pull the reader up, giving voice to the opposing view of an optimistic, limitless future. But we don't do justice to that view-how could we, older mammals on the way to extinction that we are? We try. But then we return to fuss, questioning the ascetic dimension of scientists, their romance with black boxes. And argue for a science of many tongues.

Monte Carlo Simulations of Au38(SCH3)24 Nanocluster Using Distance-Based Machine Learning Methods

We present an implementation of distance-based machine learning (ML) methods to create a realistic atomistic interaction potential to be used in Monte Carlo simulations of thermal dynamics of thiolate (SR) protected gold nanoclusters. The ML potential is trained for Au₃₈(SR)₂₄ by using previously published, density functional theory (DFT) based, molecular dynamics (MD) simulation data on two experimentally characterized structural isomers of the cluster and validated against independent DFT MD simulations. This method opens a door to efficient probing of the configuration space for further investigations of thermal-dependent electronic and optical properties of Au₃₈(SR)₂₄. Our ML implementation strategy allows for generalization and accuracy control of distance-based ML models for complex nanostructures having several chemical elements and interactions of varying strength.

Comparison of the Performance of Machine Learning Models in Representing High-Dimensional Free Energy Surfaces and Generating Observables

Free energy surfaces of chemical and physical systems are often generated using a popular class of enhanced sampling methods that target a set of collective variables (CVs) chosen to distinguish the characteristic features of these surfaces. While some of these approaches are typically limited to low (∼1-3)-dimensional CV subspaces, methods such as driven adiabatic free-energy dynamics/temperature-accelerated molecular dynamics have been shown to be capable of generating free energy surfaces of quite high dimension by sampling the associated marginal probability distribution via full sweeps over the CV landscape. These approaches repeatedly visit conformational basins, producing a scattering of points within the basins on each visit. Consequently, they are particularly amenable to synergistic combination with regression machine learning methods for filling in the surfaces between the sampled points and for providing a compact and continuous (or semicontinuous) representation of the surfaces that can be easily stored and used for further computation of observable properties. Given the central role of machine learning techniques in this combined approach, it is timely to provide a detailed comparison of the performance of different machine learning strategies and models, including neural networks, kernel ridge regression, support vector machines, and weighted neighbor schemes, for their ability to learn these high-dimensional surfaces as a function of the amount of sampled training data and, once trained, to subsequently generate accurate ensemble averages corresponding to observable properties of the systems. In this article, we perform such a comparison on a set of oligopeptides, in both gas and aqueous phases, corresponding to CV spaces of 2-10 dimensions and assess their ability to provide a global representation of the free energy surfaces and to generate accurate ensemble averages.

Quantum Chemistry in the Age of Machine Learning

As the quantum chemistry (QC) community embraces machine learning (ML), the number of new methods and applications based on the combination of QC and ML is surging. In this Perspective, a view of the current state of affairs in this new and exciting research field is offered, challenges of using machine learning in quantum chemistry applications are described, and potential future developments are outlined. Specifically, examples of how machine learning is used to improve the accuracy and accelerate quantum chemical research are shown. Generalization and classification of existing techniques are provided to ease the navigation in the sea of literature and to guide researchers entering the field. The emphasis of this Perspective is on supervised machine learning.

Modified Density Functional Dispersion Correction for Inorganic Layered MFX Compounds (M = Ca, Sr, Ba, Pb and X = Cl, Br, I)

MFX (M = Ca, Ba, Sr, Pb and X = Cl, Br, I) compounds have received considerable attention due to their technological application as X-ray detectors, pressure sensors, and optical data storage materials, when doped with rare-earth ions. MFX compounds belong to the class of layered materials with a tetragonal Matlockite crystal structure, characterized by weakly stacked double-halide layers along the crystallographic c-axis. These layers predominantly determine phase transitions, elastic, and mechanical properties. However, the correct description of the lattice parameter c is a challenge for most standard DFT functionals, which tend to overestimate the lattice parameter c. Because of the weak interactions between the halide layers, dispersion-corrected functionals seem to be a better choice. We investigated 11 different inorganic layered MFX compounds for which experimental data are available, with standard and dispersion-corrected functionals to assess their performance in reproducing the lattice parameter c, structural, and vibrational properties of the MFX compounds. Our results revealed that these functionals do not describe the weak interactions between the halide layers in a balanced way. Therefore, we modified Grimme's popular DFT-D2 dispersion correction scheme in two different ways by (i) replacing the dispersion coefficients and van der Waals radii with those of noble gas atoms or (ii) increasing the van der Waals radii of the MFX atoms up to 40%. Comparison with the available experimental data revealed that the latter approach applied to the PBE (Perdew-Burke-Ernzerhof)-D2 functional with 30% increased van der Waals radii, which we coined PBE-D2* (S_rvdw 1.30) is best suited to fine-tune the description of the weak interlayer interactions in MFX compounds, thus significantly improving the description of their structural, vibrational, and mechanical properties. Work is in progress applying this new, computationally inexpensive scheme to other inorganic layered compounds and periodic systems with weakly stacked layers.

Gaussian Process-Based Refinement of Dispersion Corrections

We employ Gaussian process (GP) regression to adjust for systematic errors in D3-type dispersion corrections. We refer to the associated, statistically improved model as D3-GP. It is trained on differences between interaction energies obtained from PBE-D3(BJ)/ma-def2-QZVPP and DLPNO-CCSD(T)/CBS calculations. We generated a data set containing interaction energies for 1248 molecular dimers, which resemble the dispersion-dominated systems contained in the S66 data set. Our systems represent not only equilibrium structures but also dimers with various relative orientations and conformations at both shorter and longer distances. A reparametrization of the D3(BJ) model based on 66 of these dimers suggests that two of its three empirical parameters, a₁ and s₈, are zero, whereas a₂ = 5.6841 bohr. For the remaining 1182 dimers, we find that this new set of parameters is superior to all previously published D3(BJ) parameter sets. To train our D3-GP model, we engineered two different vectorial representations of (supra-)molecular systems, both derived from the matrix of atom-pairwise D3(BJ) interaction terms: (a) a distance-resolved interaction energy histogram, histD3(BJ), and (b) eigenvalues of the interaction matrix ordered according to their decreasing absolute value, eigD3(BJ). Hence, the GP learns a mapping from D3(BJ) information only, which renders D3-GP-type dispersion corrections comparable to those obtained with the original D3 approach. They improve systematically if the underlying training set is selected carefully. Here, we harness the prediction variance obtained from GP regression to select optimal training sets in an automated fashion. The larger the variance, the more information the corresponding data point may add to the training set. For a given set of molecular systems, variance-based sampling can approximately determine the smallest subset being subjected to reference calculations such that all dispersion corrections for the remaining systems fall below a predefined accuracy threshold. To render the entire D3-GP workflow as efficient as possible, we present an improvement over our variance-based, sequential active-learning scheme [ J. Chem. Theory Comput. 2018 , 14 , 5238 ]. Our refined learning algorithm selects multiple (instead of single) systems that can be subjected to reference calculations simultaneously. We refer to the underlying selection strategy as batchwise variance-based sampling (BVS). BVS-guided active learning is an essential component of our D3-GP workflow, which is implemented in a black-box fashion. Once provided with reference data for new molecular systems, the underlying GP model automatically learns to adapt to these and similar systems. This approach leads overall to a self-improving model (D3-GP) that predicts system-focused and GP-refined D3-type dispersion corrections for any given system of reference data.