Degeneration of kernel regression with Matern kernels into low-order polynomial regression in high dimension.

Kernel methods such as kernel ridge regression and Gaussian process regression with Matern-type kernels have been increasingly used, in particular, to fit potential energy surfaces (PES) and density functionals, and for materials informatics. When the dimensionality of the feature space is high, these methods are used with necessarily sparse data. In this regime, the optimal length parameter of a Matern-type kernel may become so large that the method effectively degenerates into a low-order polynomial regression and, therefore, loses any advantage over such regression. This is demonstrated theoretically as well as numerically in the examples of six- and fifteen-dimensional molecular PES using squared exponential and simple exponential kernels. The results shed additional light on the success of polynomial approximations such as PIP for medium-size molecules and on the importance of orders-of-coupling-based models for preserving the advantages of kernel methods with Matern-type kernels of on the use of physically motivated (reproducing) kernels.

Neural Network Potential Energy Surfaces for Small Molecules and Reactions.

We review progress in neural network (NN)-based methods for the construction of interatomic potentials from discrete samples (such as ab initio energies) for applications in classical and quantum dynamics including reaction dynamics and computational spectroscopy. The main focus is on methods for building molecular potential energy surfaces (PES) in internal coordinates that explicitly include all many-body contributions, even though some of the methods we review limit the degree of coupling, due either to a desire to limit computational cost or to limited data. Explicit and direct treatment of all many-body contributions is only practical for sufficiently small molecules, which are therefore our primary focus. This includes small molecules on surfaces. We consider direct, single NN PES fitting as well as more complex methods that impose structure (such as a multibody representation) on the PES function, either through the architecture of one NN or by using multiple NNs. We show how NNs are effective in building representations with low-dimensional functions including dimensionality reduction. We consider NN-based approaches to build PESs in the sums-of-product form important for quantum dynamics, ways to treat symmetry, and issues related to sampling data distributions and the relation between PES errors and errors in observables. We highlight combinations of NNs with other ideas such as permutationally invariant polynomials or sums of environment-dependent atomic contributions, which have recently emerged as powerful tools for building highly accurate PESs for relatively large molecular and reactive systems.

Machine learning in computational chemistry: interplay between (non)linearity, basis sets, and dimensionality.

Machine learning (ML) based methods and tools have now firmly established themselves in physical chemistry and in particular in theoretical and computational chemistry and in materials chemistry. The generality of popular ML techniques such as neural networks or kernel methods (Gaussian process and kernel ridge regression and their flavors) permitted their application to diverse problems from prediction of properties of functional materials (catalysts, solid state ionic conductors, etc.) from descriptors to the building of interatomic potentials (where ML is currently routinely used in applications) and electron density functionals. These ML techniques are assumed to have superior expressive power of nonlinear methods, and are often used "as is", with concepts such as "non-parametric" or "deep learning" used without a clear justification for their need or advantage over simpler and more robust alternatives. In this Perspective, we highlight some interrelations between popular ML techniques and traditional linear regressions and basis expansions and demonstrate that in certain regimes (such as a very high dimensionality) these approximations might collapse. We also discuss ways to recover the expressive power of a nonlinear approach and to help select hyperparameters with the help of high-dimensional model representation and to obtain elements of insight while preserving the generality of the method.

Clarifying the effects of nanoscale porosity of silicon on the bandgap and alignment: a combined molecular dynamics-density functional tight binding computational study.

Porous silicon (pSi) has been studied for its applications in solar cells, in particular in silicon-silicon tandem solar cells. It is commonly believed that porosity leads to an expansion of the bandgap due to nano-confinement. Direct confirmation of this proposition has been elusive, as experimental band edge quantification is subject to uncertainties and effects of impurities, while electronic structure calculations on relevant length scales are still outstanding. Passivation of pSi is another factor affecting the band structure. We present a combined force field-density functional tight binding study of the effects of porosity of silicon on its band structure. We thus perform electron structure-level calculations for the first time on length scales (several nm) that are relevant to real pSi, and consider multiple nanoscale geometries (pores, pillars, and craters) with key geometrical features and sizes of real porous Si. We consider the presence of a bulk-like base with a nanostructured top layer. We show that the bandgap expansion is not correlated with the pore size but with the size of the Si framework. Significant band expansion would require features of silicon (as opposed to pore sizes) to be as small as 1 nm, while the nanosizing of pores does not induce gap expansion. We observe a graded junction-like behavior of the band gap as a function of Si feature sizes as one moves from the bulk-like base to the nanoporous top layer.

Neural Network with Optimal Neuron Activation Functions Based on Additive Gaussian Process Regression.

Feed-forward neural networks (NNs) are a staple machine learning method widely used in many areas of science and technology, including physical chemistry, computational chemistry, and materials informatics. While even a single-hidden-layer NN is a universal approximator, its expressive power is limited by the use of simple neuron activation functions (such as sigmoid functions) that are typically the same for all neurons. More flexible neuron activation functions would allow the use of fewer neurons and layers and thereby save computational cost and improve expressive power. We show that additive Gaussian process regression (GPR) can be used to construct optimal neuron activation functions that are individual to each neuron. An approach is also introduced that avoids nonlinear fitting of neural network parameters by defining them with rules. The resulting method combines the advantage of robustness of a linear regression with the higher expressive power of an NN. We demonstrate the approach by fitting the potential energy surfaces of the water molecule and formaldehyde. Without requiring any nonlinear optimization, the additive-GPR-based approach outperforms a conventional NN in the high-accuracy regime, where a conventional NN suffers more from overfitting.

The loss of the property of locality of the kernel in high-dimensional Gaussian process regression on the example of the fitting of molecular potential energy surfaces.

Kernel-based methods, including Gaussian process regression (GPR) and generally kernel ridge regression, have been finding increasing use in computational chemistry, including the fitting of potential energy surfaces and density functionals in high-dimensional feature spaces. Kernels of the Matern family, such as Gaussian-like kernels (basis functions), are often used which allow imparting to them the meaning of covariance functions and formulating GPR as an estimator of the mean of a Gaussian distribution. The notion of locality of the kernel is critical for this interpretation. It is also critical to the formulation of multi-zeta type basis functions widely used in computational chemistry. We show, on the example of fitting of molecular potential energy surfaces of increasing dimensionality, the practical disappearance of the property of locality of a Gaussian-like kernel in high dimensionality. We also formulate a multi-zeta approach to the kernel and show that it significantly improves the quality of regression in low dimensionality but loses any advantage in high dimensionality, which is attributed to the loss of the property of locality.

A controlled study of the effect of deviations from symmetry of the potential energy surface (PES) on the accuracy of the vibrational spectrum computed with collocation.

Symmetry, in particular permutational symmetry, of a potential energy surface (PES) is a useful property in quantum chemical calculations. It facilitates, in particular, state labelling and identification of degenerate states. In many practically important applications, however, these issues are unimportant. The imposition of exact symmetry and the perception that it is necessary create additional methodological requirements narrowing or complicating algorithmic choices that are thereby biased against methods and codes that by default do not incorporate symmetry, including most off-the-shelf machine learning methods that cannot be directly used if exact symmetry is demanded. By introducing symmetric and unsymmetric errors into the PES of H2CO in a controlled way and computing the vibrational spectrum with collocation using symmetric and nonsymmetric collocation point sets, we show that when the deviations from an ideal PES are random, imposition of exact symmetry does not bring any practical advantages. Moreover, a calculation ignoring symmetry may be more accurate. We also compare machine-learned PESs with and without symmetrization and demonstrate that there is no advantage of imposing exact symmetry for the accuracy of the vibrational spectrum.

Machine learning of kinetic energy densities with target and feature smoothing: Better results with fewer training data.

Machine learning (ML) of kinetic energy functionals (KEFs), in particular kinetic energy density (KED) functionals, is a promising way to construct KEFs for orbital-free density functional theory (DFT). Neural networks and kernel methods including Gaussian process regression (GPR) have been used to learn Kohn-Sham (KS) KED from density-based descriptors derived from KS DFT calculations. The descriptors are typically expressed as functions of different powers and derivatives of the electron density. This can generate large and extremely unevenly distributed datasets, which complicates effective application of ML techniques. Very uneven data distributions require many training datapoints, can cause overfitting, and can ultimately lower the quality of an ML KED model. We show that one can produce more accurate ML models from fewer data by working with smoothed density-dependent variables and KED. Smoothing palliates the issue of very uneven data distributions and associated difficulties of sampling while retaining enough spatial structure necessary for working within the paradigm of KEDF. We use GPR as a function of smoothed terms of the fourth order gradient expansion and KS effective potential and obtain accurate and stable (with respect to different random choices of training points) kinetic energy models for Al, Mg, and Si simultaneously from as few as 2000 samples (about 0.3% of the total KS DFT data). In particular, accuracies on the order of 1% in a measure of the quality of energy-volume dependence B'=EV0-ΔV-2EV0+E(V0+ΔV)ΔV/V02 (where V0 is the equilibrium volume and ΔV is a deviation from it) are obtained simultaneously for all three materials.

Computational vibrational spectroscopy of molecule-surface interactions: what is still difficult and what can be done about it.

Interactions of molecules with solid surfaces are responsible for key functionalities in a range of currently actively pursued technologies, including heterogeneous catalysis for synthesis or decomposition of molecules, sensitization, surface functionalization and molecular doping, etc. Modeling of such interactions is important, in particular, for the assignment of species and assignment and design of reaction pathways and ultimately for rational design of better functional materials for a range of applications. Key types of calculations involve calculations of adsorption structures and energies, electronic structures, charge transport, vibrational and optical spectra, and reaction dynamics. While some of these calculations are routinely doable for small-size models, other types of calculations, including anharmonic vibrational spectroscopy, accurate optical spectroscopy, quantum reaction dynamics, and calculations on large systems, are still too difficult to be routinely doable. In this Perspective, we specifically focus on issues related to computing accurate vibrational spectra including quantum effects and anharmonicity and coupling of key degrees of freedom. Vibrational spectroscopies are widely used for species assignment on surfaces but computations of vibrational spectra are still dominated by the harmonic approximation. We discuss approaches that can make accurate quantum anharmonic computational spectroscopy easier and enable its wider deployment in applications. We describe advantages and disadvantages of different techniques including perturbation theory, variational, vibrational self-consistent field and vibrational configuration interaction, as well as collocation which we argue has significant potential in this application, allowing computing accurate spectra directly from non-expensive ab initio data and with modest CPU cost. Examples of applications of anharmonic techniques to molecule-surface systems are given.

The Role of Local DFT+U Minima in the First-Principles Modeling of the Metal-Insulator Transition in Vanadium Dioxide.

The DFT+U method is frequently employed to improve the first-principles description of strongly correlated materials. However, it is prone to deliver metastable electronic minima. While these local minima of the DFT+U method are often considered to be computational artifacts, their physical meaning and relationship to true excited states remains unclear. In this work, the possibility of theoretically modeling transformations in the solid state that require thermal or optical excitations of electrons is explored, taking into account the metastable states of the computationally undemanding DFT+U formalism. For this purpose, we choose to examine the example of the VO2 metal-insulator transition. Metastable states that are located on different electronic potential energy surfaces are found to correspond to experimentally observed VO2 phases. The identified metastable electronic states can be used to model the collapse of the VO2 band gap at elevated temperatures and upon photoexcitation as well as other monoclinic-monoclinic phase transformations. The results suggest that local DFT+U minima can indeed carry physical meaning, while they remain under-reported in theoretical literature on transition metal oxides like VO2.

Density-Based Descriptors of Redox Reactions Involving Transition Metal Compounds as a Reality-Anchored Framework: A Perspective.

Description of redox reactions is critically important for understanding and rational design of materials for electrochemical technologies, including metal-ion batteries, catalytic surfaces, or redox-flow cells. Most of these technologies utilize redox-active transition metal compounds due to their rich chemistry and their beneficial physical and chemical properties for these types of applications. A century since its introduction, the concept of formal oxidation states (FOS) is still widely used for rationalization of the mechanisms of redox reactions, but there exists a well-documented discrepancy between FOS and the electron density-derived charge states of transition metal ions in their bulk and molecular compounds. We summarize our findings and those of others which suggest that density-driven descriptors are, in certain cases, better suited to characterize the mechanism of redox reactions, especially when anion redox is involved, which is the blind spot of the FOS ansatz.

Synergistic Effect of Dielectric Property and Energy Transfer on Charge Separation in Non-Fullerene-Based Solar Cells.

In non-fullerene-based photovoltaic devices, it is unclear how excitons efficiently dissociate into charge carriers under small driving force. Here, we developed a modified method to estimate dielectric constants of PM6 donor and non-fullerene acceptors. Surprisingly, most non-fullerene acceptors and blend films showed higher dielectric constants. Moreover, they exhibited larger dielectric constants differences at the optical frequency. These results are likely bound to reduced exciton binding energy and bimolecular recombination. Besides, the overlap between the emission spectrum of donor and absorption spectra of non-fullerene acceptors allowed the energy transfer from donor to acceptors. Hence, based on the synergistic effect of dielectric property and energy transfer resulting in efficient charge separation, our finding paves an alternative path to elucidate the physical working mechanism in non-fullerene-based photovoltaic devices.

Reply to the 'Comment on "Revisiting π backbonding: the influence of d orbitals on metal-CO bonds and ligand red shifts"' by G. Frenking and S. Pan, Phys. Chem. Chem. Phys., 2019, 22, DOI: 10.1039/C9CP05951B.

We respond to the comment by Pan and Frenking with regard to our investigation on transition and alkaline earth metal d orbital influence on their bonding to carbonyl ligands to clarify misconceptions. We do not consider the points raised in the comment as affecting our conclusions.

Reply to the 'Comment on "Revisiting π backbonding: the influence of d orbitals on metal-CO bonds and ligand red shifts"' by G. Frenking and S. Pan, Phys. Chem. Chem. Phys., 2019, 22, DOI.

We respond to the comment by Pan and Frenking with regard to our investigation on transition and alkaline earth metal d orbital influence on their bonding to carbonyl ligands to clarify misconceptions. We do not consider the points raised in the comment as affecting our conclusions.

Intramolecular mode coupling of the isotopomers of water: a non-scalar charge density-derived perspective.

The H2O/D2O/HDO isotopomers of water are presented in terms that enable bond-flexing, bond-twist and bond-anharmonicity to be quantified during the bending (Q1), symmetric-stretch (Q2) and anti-symmetric-stretch (Q3) normal modes of vibration. Bond-flexing was detected by the presence of curved bonding and the bond-anharmonicity was detected by the presence of motion of the bond critical point (BCP) relative to the oxygen atom. These 2-D scalar measures are unable to fully describe the 3-D nature of the normal modes of vibration and therefore any susceptibility towards normal mode coupling, or to fully distinguish the three isotopomers. To detect bond-twist a vector-based measure was used, in the form of the bond critical point (BCP) trajectory, constructed in terms of preferred directions of electronic motion, defined by the variation of the position of the BCP during the normal modes of vibration. The BCP trajectories describe the coupling of the intramolecular bending and symmetric-stretch normal modes as well as distinguishing all three isotopomers within the harmonic approximation. The coupling of the bending and symmetric-stretch normal modes are suggested to be facilitated by the absence of bond-twist that would disrupt the coupling between Sigma O-H bonds and hydrogen-bonding. Partial coupling was found for the mixed isotopomer HDO.

Nonparametric Local Pseudopotentials with Machine Learning: A Tin Pseudopotential Built Using Gaussian Process Regression.

We present novel nonparametric representation math for local pseudopotentials (PP) based on Gaussian Process Regression (GPR). Local pseudopotentials are needed for materials simulations using Orbital-Free Density Functional Theory (OF-DFT) to reduce computational cost and to allow kinetic energy functional (KEF) application only to the valence density. Moreover, local PPs are important for the development of accurate KEFs for OF-DFT, but they are only available for a limited number of elements. We optimize local PPs of tin (Sn) represented with GPR to reproduce the experimental lattice constants of α- and ß-Sn and the energy difference between these two phases as well as their electronic structure and charge density distributions which are obtained with Kohn-Sham Density Functional Theory employing semilocal PPs. The use of a nonparametric GPR-based PP representation avoids difficulties associated with the use of parametrized functions and has the potential to construct an optimal local PP independent of prior assumptions. The GPR-based Sn local PP results in well-reproduced bulk properties of α- and ß-tin and electronic valence densities similar to those obtained with semilocal PP.

Random Sampling High Dimensional Model Representation Gaussian Process Regression (RS-HDMR-GPR) for Multivariate Function Representation: Application to Molecular Potential Energy Surfaces.

We present an approach combining a representation of a multivariate function using subdimensional functions with machine learning based representation of component functions: Random sampling high dimensional model representation Gaussian process regression (RS-HDMR-GPR). The use of Gaussian process regressions to represent component functions allows nonparametric (unbiased) representation and the possibility to work only with functions of desired dimensionality, obviating the need to build an expansion over orders of coupling. All component functions are determined from a single set of samples. The method is tested by fitting six- and 15-dimensional potential energy surfaces (PES) of polyatomic molecules as well as by computing vibrational spectra for a four-atomic molecule.

Data-driven kinetic energy density fitting for orbital-free DFT: Linear vs Gaussian process regression.

We study the dependence of kinetic energy densities (KEDs) on density-dependent variables that have been suggested in previous works on kinetic energy functionals for orbital-free density functional theory. We focus on the role of data distribution and on data and regressor selection. We compare unweighted and weighted linear and Gaussian process regressions of KEDs for light metals and a semiconductor. We find that good quality linear regression resulting in good energy-volume dependence is possible over density-dependent variables suggested in previous literature studies. This is achieved with weighted fitting based on the KED histogram. With Gaussian process regressions, excellent KED fit quality well exceeding that of linear regressions is obtained as well as a good energy-volume dependence, which was somewhat better than that of best linear regressions. We find that while the use of the effective potential as a descriptor improves linear KED fitting, it does not improve the quality of the energy-volume dependence with linear regressions but substantially improves it with Gaussian process regression. Gaussian process regression is also able to perform well without data weighting.

Aggregate-State Effects in the Atomistic Modeling of Organic Materials for Electrochemical Energy Conversion and Storage Devices: A Perspective.

Development of new functional materials for novel energy conversion and storage technologies is often assisted by ab initio modeling. Specifically, for organic materials, such as electron and hole transport materials for perovskite solar cells, LED (light emitting diodes) emitters for organic LEDs (OLEDs), and active electrode materials for organic batteries, such modeling is often done at the molecular level. Modeling of aggregate-state effects is onerous, as packing may not be known or large simulation cells may be required for amorphous materials. Yet aggregate-state effects are essential to estimate charge transport rates, and they may also have substantial effects on redox potentials (voltages) and optical properties. This paper summarizes recent studies by the author's group of aggregation effects on the electronic properties of organic materials used in optoelectronic devices and in organic batteries. We show that in some cases it is possible to understand the mechanism and predict specific performance characteristics based on simple molecular models, while in other cases the inclusion of effects of aggregation is essential. For example, it is possible to understand the mechanism and predict the overall shape of the voltage-capacity curve for insertion-type organic battery materials, but not the absolute voltage. On the other hand, oligomeric models of p-type organic electrode materials can allow for relatively reliable estimates of voltages. Inclusion of aggregate state modeling is critically important for estimating charge transport rates in materials and interfaces used in optoelectronic devices or when intermolecular charge transfer bands are important. We highlight the use of the semi-empirical DFTB (density functional tight binding) method to simplify such calculations.

Pyrrolo[3,2-b]pyrrole-1,4-dione (IsoDPP) End Capped with Napthalimide or Phthalimide: Novel Small Molecular Acceptors for Organic Solar Cells.

We introduce two novel solution-processable electron acceptors based on an isomeric core of the much explored diketopyrrolopyrrole (DPP) moiety, namely pyrrolo[3,2-b]pyrrole-1,4-dione (IsoDPP). The newly designed and synthesized compounds, 6,6'-[(1,4-bis{4-decylphenyl}-2,5-dioxo-1,2,4,5-tetrahydropyrrolo[3,2-b]pyrrole-3,6-diyl)bis(thiophene-5,2-diyl)]bis[2-(2-butyloctyl)-1H-benzo[de]isoquinoline-1,3(2H)-dione] (NAI-IsoDPP-NAI) and 5,5'-[(1,4-bis{4-decylphenyl}-2,5-dioxo-1,2,4,5-tetrahydropyrrolo[3,2-b]pyrrole-3,6-diyl)bis(thiophene-5,2-diyl)]bis[2-(2-butyloctyl)isoindoline-1,3-dione] (PI-IsoDPP-PI) have been synthesized via Suzuki couplings using IsoDPP as a central building block and napthalimide or phthalimide as end-capping groups. The materials both exhibit good solubility in a wide range of organic solvents including chloroform (CF), dichloromethane (DCM), and tetrahydrofuran (THF), and have a high thermal stability. The new materials absorb in the wavelength range of 300-600 nm and both compounds have similar electron affinities, with the electron affinities that are compatible with their use as acceptors in donor-acceptor bulk heterojunction (BHJ) organic solar cells. BHJ devices comprising the NAI-IsoDPP-NAI acceptor with poly(3-n-hexylthiophene) (P3HT) as the donor were found to have a better performance than the PI-IsoDPP-PI containing cells, with the best device having a VOC of 0.92 V, a JSC of 1.7 mAcm-2, a FF of 63%, and a PCE of 0.97%.