The ONETEP linear-scaling density functional theory program

Orbital-Resolved DFT+U for Molecules and Solids

We present an orbital-resolved extension of the Hubbard U correction to density-functional theory (DFT). Compared to the conventional shell-averaged approach, the prediction of energetic, electronic and structural properties is strongly improved, particularly for compounds characterized by both localized and hybridized states in the Hubbard manifold. The numerical values of all Hubbard parameters are readily obtained from linear-response calculations. The relevance of this more refined approach is showcased by its application to bulk solids pyrite (FeS2) and pyrolusite (β-MnO2), as well as to six Fe(II) molecular complexes. Our findings indicate that a careful definition of Hubbard manifolds is indispensable for extending the applicability of DFT+U beyond its current boundaries. The present orbital-resolved scheme aims to provide a computationally undemanding yet accurate tool for electronic structure calculations of charge-transfer insulators, transition-metal (TM) complexes and other compounds displaying significant orbital hybridization.

Noise reduction of stochastic density functional theory for metals

Density Functional Theory (DFT) has become a cornerstone in the modeling of metals. However, accurately simulating metals, particularly under extreme conditions, presents two significant challenges. First, simulating complex metallic systems at low electron temperatures is difficult due to their highly delocalized density matrix. Second, modeling metallic warm-dense materials at very high electron temperatures is challenging because it requires the computation of a large number of partially occupied orbitals. This study demonstrates that both challenges can be effectively addressed using the latest advances in linear-scaling stochastic DFT methodologies. Despite the inherent introduction of noise into all computed properties by stochastic DFT, this research evaluates the efficacy of various noise reduction techniques under different thermal conditions. Our observations indicate that the effectiveness of noise reduction strategies varies significantly with the electron temperature. Furthermore, we provide evidence that the computational cost of stochastic DFT methods scales linearly with system size for metal systems, regardless of the electron temperature regime.

Tuning the work function of graphite nanoparticles via edge termination

Graphite nanoparticles are important in energy materials applications such as lithium-ion batteries (LIBs), supercapacitors and as catalyst supports. Tuning the work function of the nanoparticles allows local control of lithiation behaviour in LIBs, and the potential of zero charge of electrocatalysts and supercapacitors. Using large scale density functional theory (DFT) calculations, we find that the surface termination of multilayer graphene nanoparticles can substantially modify the work function. Calculations in vacuum and in electrolyte show that manipulating the edge termination substantially modifies the potential not only around the edge, but also on the basal plane. Termination with hydrogen or oxygen completely reverses the potential distribution surrounding the basal plane and edges. The trends can be explained based on the work function differences of the edges dependent on termination, and that of the basal plane. Electronic equilibration between different surfaces at the nanoscale allows manipulation of the work function. We demonstrate a link between the area of the graphite basal plane via changing the nanoparticle size, and the work function. We expect that these insights can be utilised for local control of electrochemical functions of graphite nanoparticles prepared under oxidising or reducing conditions.

The OpenMMPol library for polarizable QM/MM calculations of properties and dynamics

We present a new library designed to provide a simple and straightforward way to implement QM/AMOEBA (Atomic Multipole Optimized Energetics for Biomolecular Applications) and other polarizable QM/MM (Molecular Mechanics) methods based on induced point dipoles. The library, herein referred to as OpenMMPol, is free and open-sourced and is engineered to address the increasing demand for accurate and efficient QM/MM simulations. OpenMMPol is specifically designed to allow polarizable QM/MM calculations of ground state energies and gradients and excitation properties. Key features of OpenMMPol include a modular architecture facilitating extensibility, parallel computing capabilities for enhanced performance on modern cluster architectures, a user-friendly interface for intuitive implementation, and a simple and flexible structure for providing input data. To show the capabilities offered by the library, we present an interface with PySCF to perform QM/AMOEBA molecular dynamics, geometry optimization, and excited-state calculation based on (time-dependent) density functional theory.

Temperature-transferable tight-binding model using a hybrid-orbital basis

Finite-temperature calculations are relevant for rationalizing material properties, yet they are computationally expensive because large system sizes or long simulation times are typically required. Circumventing the need for performing many explicit first-principles calculations, tight-binding and machine-learning models for the electronic structure emerged as promising alternatives, but transferability of such methods to elevated temperatures in a data-efficient way remains a great challenge. In this work, we suggest a tight-binding model for efficient and accurate calculations of temperature-dependent properties of semiconductors. Our approach utilizes physics-informed modeling of the electronic structure in the form of hybrid-orbital basis functions and numerically integrating atomic orbitals for the distance dependence of matrix elements. We show that these design choices lead to a tight-binding model with a minimal amount of parameters that are straightforwardly optimized using density functional theory or alternative electronic-structure methods. The temperature transferability of our model is tested by applying it to existing molecular-dynamics trajectories without explicitly fitting temperature-dependent data and comparison with density functional theory. We utilize it together with machine-learning molecular dynamics and hybrid density functional theory for the prototypical semiconductor gallium arsenide. We find that including the effects of thermal expansion on the onsite terms of the tight-binding model is important in order to accurately describe electronic properties at elevated temperatures in comparison with experiment.

Perspectives Toward Damage-Tolerant Nanostructure Ceramics

Advanced ceramic materials and devices call for better reliability and damage tolerance. In addition to their strong bonding nature, there are examples demonstrating superior mechanical properties of nanostructure ceramics, such as damage-tolerant ceramic aerogels that can withstand high deformation without cracking and local plasticity in dense nanocrystalline ceramics. The recent progresses shall be reviewed in this perspective article. Three topics including highly elastic nano-fibrous ceramic aerogels, load-bearing nanoceramics with improved mechanical properties, and implementing machine learning-assisted simulations toolbox in understanding the relationship among structure, deformation mechanisms, and microstructure-properties shall be discussed. It is hoped that the perspectives present here can help the discovery, synthesis, and processing of future structural ceramic materials that are insensitive to processing flaws and local damages in service.

A Critical Assessment on Calculating Vibrational Spectra in Nanostructured Materials

Vibrational spectroscopy is an omnipresent spectroscopic technique to characterize functional nanostructured materials such as zeolites, metal-organic frameworks (MOFs), and metal-halide perovskites (MHPs). The resulting experimental spectra are usually complex, with both low-frequency framework modes and high-frequency functional group vibrations. Therefore, theoretically calculated spectra are often an essential element to elucidate the vibrational fingerprint. In principle, there are two possible approaches to calculate vibrational spectra: (i) a static approach that approximates the potential energy surface (PES) as a set of independent harmonic oscillators and (ii) a dynamic approach that explicitly samples the PES around equilibrium by integrating Newton's equations of motions. The dynamic approach considers anharmonic and temperature effects and provides a more genuine representation of materials at true operating conditions; however, such simulations come at a substantially increased computational cost. This is certainly true when forces and energy evaluations are performed at the quantum mechanical level. Molecular dynamics (MD) techniques have become more established within the field of computational chemistry. Yet, for the prediction of infrared (IR) and Raman spectra of nanostructured materials, their usage has been less explored and remain restricted to some isolated successes. Therefore, it is currently not a priori clear which methodology should be used to accurately predict vibrational spectra for a given system. A comprehensive comparative study between various theoretical methods and experimental spectra for a broad set of nanostructured materials is so far lacking. To fill this gap, we herein present a concise overview on which methodology is suited to accurately predict vibrational spectra for a broad range of nanostructured materials and formulate a series of theoretical guidelines to this purpose. To this end, four different case studies are considered, each treating a particular material aspect, namely breathing in flexible MOFs, characterization of defects in the rigid MOF UiO-66, anharmonic vibrations in the metal-halide perovskite CsPbBr3, and guest adsorption on the pores of the zeolite H-SSZ-13. For all four materials, in their guest- and defect-free state and at sufficiently low temperatures, both the static and dynamic approach yield qualitatively similar spectra in agreement with experimental results. When the temperature is increased, the harmonic approximation starts to fail for CsPbBr3 due to the presence of anharmonic phonon modes. Also, the spectroscopic fingerprints of defects and guest species are insufficiently well predicted by a simple harmonic model. Both phenomena flatten the potential energy surface (PES), which facilitates the transitions between metastable states, necessitating dynamic sampling. On the basis of the four case studies treated in this Review, we can propose the following theoretical guidelines to simulate accurate vibrational spectra of functional solid-state materials: (i) For nanostructured crystalline framework materials at low temperature, insights into the lattice dynamics can be obtained using a static approach relying on a few points on the PES and an independent set of harmonic oscillators. (ii) When the material is evaluated at higher temperatures or when additional complexity enters the system, e.g., strong anharmonicity, defects, or guest species, the harmonic regime breaks down and dynamic sampling is required for a correct prediction of the phonon spectrum. These guidelines and their illustrations for prototype material classes can help experimental and theoretical researchers to enhance the knowledge obtained from a lattice dynamics study.

Deterministic/Fragmented-Stochastic Exchange for Large-Scale Hybrid DFT Calculations

We develop an efficient approach to evaluate range-separated exact exchange for grid- or plane-wave-based representations within the generalized Kohn-Sham-density functional theory (GKS-DFT) framework. The Coulomb kernel is fragmented in reciprocal space, and we employ a mixed deterministic-stochastic representation, retaining long-wavelength (low-k) contributions deterministically and using a sparse ("fragmented") stochastic basis for the high-k part. Coupled with a projection of the Hamiltonian onto a subspace of valence and conduction states from a prior local-DFT calculation, this method allows for the calculation of the long-range exchange of large molecular systems with hundreds and potentially thousands of coupled valence states delocalized over millions of grid points. We find that even a small number of valence and conduction states is sufficient for converging the HOMO and LUMO energies of the GKS-DFT. Excellent tuning of long-range separated hybrids (RSH) is easily obtained in the method for very large systems, as exemplified here for the chlorophyll hexamer of Photosystem II with 1320 electrons.

A Workflow for Identifying Viable Crystal Structures with Partially Occupied Sites Applied to the Solid Electrolyte Cubic Li7La3Zr2O12

To date, experimental and theoretical works have been unable to uncover the ground-state configuration of the solid electrolyte cubic Li7La3Zr2O12 (c-LLZO). Computational studies rely on an initial low-energy structure as a reference point. Here, we present a methodology for identifying energetically favorable configurations of c-LLZO for a crystallographically predicted structure. We begin by eliminating structures that involve overlapping Li atoms based on nearest neighbor counts. We further reduce the configuration space by eliminating symmetry images from all remaining structures. Then, we perform a machine learning-based energetic ordering of all remaining structures. By considering the geometrical constraints that emerge from this methodology, we determine that a large portion of previously reported structures may not be feasible or stable. The method developed here could be extended to other ion conductors. We provide a database containing all of the generated structures with the aim of improving accuracy and reproducibility in future c-LLZO research.

A Perspective on Sustainable Computational Chemistry Software Development and Integration

The power of quantum chemistry to predict the ground and excited state properties of complex chemical systems has driven the development of computational quantum chemistry software, integrating advances in theory, applied mathematics, and computer science. The emergence of new computational paradigms associated with exascale technologies also poses significant challenges that require a flexible forward strategy to take full advantage of existing and forthcoming computational resources. In this context, the sustainability and interoperability of computational chemistry software development are among the most pressing issues. In this perspective, we discuss software infrastructure needs and investments with an eye to fully utilize exascale resources and provide unique computational tools for next-generation science problems and scientific discoveries.

Treating Semiempirical Hamiltonians as Flexible Machine Learning Models Yields Accurate and Interpretable Results

Quantum chemistry provides chemists with invaluable information, but the high computational cost limits the size and type of systems that can be studied. Machine learning (ML) has emerged as a means to dramatically lower the cost while maintaining high accuracy. However, ML models often sacrifice interpretability by using components such as the artificial neural networks of deep learning that function as black boxes. These components impart the flexibility needed to learn from large volumes of data but make it difficult to gain insight into the physical or chemical basis for the predictions. Here, we demonstrate that semiempirical quantum chemical (SEQC) models can learn from large volumes of data without sacrificing interpretability. The SEQC model is that of density-functional-based tight binding (DFTB) with fixed atomic orbital energies and interactions that are one-dimensional functions of the interatomic distance. This model is trained to ab initio data in a manner that is analogous to that used to train deep learning models. Using benchmarks that reflect the accuracy of the training data, we show that the resulting model maintains a physically reasonable functional form while achieving an accuracy, relative to coupled cluster energies with a complete basis set extrapolation (CCSD(T)*/CBS), that is comparable to that of density functional theory (DFT). This suggests that trained SEQC models can achieve a low computational cost and high accuracy without sacrificing interpretability. Use of a physically motivated model form also substantially reduces the amount of ab initio data needed to train the model compared to that required for deep learning models.

The central role of density functional theory in the AI age

Density functional theory (DFT) plays a pivotal role in chemical and materials science because of its relatively high predictive power, applicability, versatility, and computational efficiency. We review recent progress in machine learning (ML) model developments, which have relied heavily on DFT for synthetic data generation and for the design of model architectures. The general relevance of these developments is placed in a broader context for chemical and materials sciences. DFT-based ML models have reached high efficiency, accuracy, scalability, and transferability and pave the way to the routine use of successful experimental planning software within self-driving laboratories.

Finite Difference Interpolation for Reduction of Grid-Related Errors in Real-Space Pseudopotential Density Functional Theory

The real-space pseudopotential approach is a well-known method for large-scale density functional theory (DFT) calculations. One of its main limitations, however, is the introduction of errors associated with the positioning of the underlying real-space grid, a phenomenon usually known as the "egg-box" effect. The effect can be controlled by using a finer grid, but this raises the cost of the calculations or even undermines their feasibility altogether. Therefore, there is ongoing interest in the reduction of the effect per a given real-space grid. Here, we present a finite difference interpolation of electron orbitals as a means of exploiting the high resolution of the pseudopotential to reduce egg-box effects systematically. We implement the method in PARSEC, a finite difference real-space pseudopotential DFT code, and demonstrate error mitigation and improved convergence at a low additional computational cost.

What is the Optimal Dipole Moment for Nonpolarizable Models of Liquids?

In classical nonpolarizable models, electrostatic interactions are usually described by assigning fixed partial charges to interaction sites. Despite the multitude of methods and theories proposed over the years for partial charge assignment, a fundamental question remains─what is the correct degree of polarization that a fixed-charge model should possess to provide the best balance of interactions (including induction effects) and yield the best description of the potential energy surface of a liquid phase? We address this question by approaching it from two separate and independent viewpoints: the QUantum mechanical BEspoke (QUBE) approach, which assigns bespoke force field parameters for individual molecules from ab initio calculations with minimal empirical fitting, and the Polarization-Consistent Approach (PolCA) force field, based on empirical fitting of force field parameters with an emphasis on transferability by rigorously accounting for polarization effects in the parameterization process. We show that the two approaches yield consistent answers to the above question, namely, that the dipole moment of the model should be approximately halfway between those of the gas and the liquid phase. Crucially, however, the reference liquid-phase dipole needs to be estimated using methods that explicitly consider both mean-field and local contributions to polarization. In particular, continuum dielectric models are inadequate for this purpose because they cannot account for local effects and therefore significantly underestimate the degree of polarization of the molecule. These observations have profound consequences for the development, validation, and testing of nonpolarizable models.

Atomistic simulations on the carbidisation processes in Pd nanoparticles

The formation of interstitial PdC _x nanoparticles (NPs) is investigated through DFT calculations. Insights on the mechanisms of carbidisation are obtained whilst the material's behaviour under conditions of increasing C-concentration is examined. Incorporation of C atoms in the Pd octahedral interstitial sites is occurring through the [111] facet with an activation energy barrier of 19.3-35.7 kJ mol^-1 whilst migration through the [100] facet corresponds to higher activation energy barriers of 124.5-127.4 kJ mol^-1. Furthermore, interstitial-type diffusion shows that C will preferentially migrate and reside at the octahedral interstitial sites in the subsurface region with limited mobility towards the core of the NP. For low C-concentrations, migration from the surface into the interstitial sites of the NPs is thermodynamically favored, resulting in the formation of interstitial carbide. Carbidisation reaction energies are exothermic up to 11-14% of C-concentration and slightly vary depending on the shape of the structure. The reaction mechanisms turn to endothermic for higher concentration levels showing that C will preferentially reside on the surface making the interstitial carbide formation unfavorable. As experimentally observed, our simulations confirm that there is a maximum concentration of C in Pd carbide NPs opening the way for further computational investigations on the activity of Pd carbides in directed catalysis.

Using molecular dynamics to simulate realistic structures of nitrocellulose of different nitration levels

Nitrocellulose is a reactive derivative of cellulose, one of the most commonly occurring natural materials. Nitration of cellulose decreases the stability of the structure, meaning less is understood about its structure and reactions. Although cellulose is often found in fully crystalline forms, nitrocellulose is more commonly paracrystalline, or amorphous. We present a protocol based on molecular dynamics simulations for creating realistic structures of nitrocellulose, particularly focusing on the crystallinity of the systems being created. We will also provide a detailed analysis of the geometric and dynamical parameters used to quantify the degree of crystallinity for the structures created here, with nitration levels varying from 0-14.14 wt% nitrogen content. Paracrystalline cellulose was not created using the protocol designed here, although it was found that the more nitrated a nitrocellulose system, the more the structure tends to paracrystallinity. This is due to a decrease in the number of hydrogen bonds present, and an increase in the size of the functional groups pushing the chains apart and weakening the interactions between the chains of the structure. The structures created are representative of realistic systems, which in the future will be able to be used to build further understanding of long-term storage of nitrocellulose.

Conformational dynamics and putative substrate extrusion pathways of the N-glycosylated outer membrane factor CmeC from Campylobacter jejuni

The outer membrane factor CmeC of the efflux machinery CmeABC plays an important role in conferring antibiotic and bile resistance to Campylobacter jejuni. Curiously, the protein is N-glycosylated, with the glycans playing a key role in the effective function of this system. In this work we have employed atomistic equilibrium molecular dynamics simulations of CmeC in a representative model of the C. jejuni outer membrane to characterise the dynamics of the protein and its associated glycans. We show that the glycans are more conformationally labile than had previously been thought. The extracellular loops of CmeC visit the open and closed states freely suggesting the absence of a gating mechanism on this side, while the narrow periplasmic entrance remains tightly closed, regulated via coordination to solvated cations. We identify several cation binding sites on the interior surface of the protein. Additionally, we used steered molecular dynamics simulations to elucidate translocation pathways for a bile acid and a macrolide antibiotic. These, and additional equilibrium simulations suggest that the anionic bile acid utilises multivalent cations to climb the ladder of acidic residues that line the interior surface of the protein.

A novel fold for acyltransferase-3 (AT3) proteins provides a framework for transmembrane acyl-group transfer

Acylation of diverse carbohydrates occurs across all domains of life and can be catalysed by proteins with a membrane bound acyltransferase-3 (AT3) domain (PF01757). In bacteria, these proteins are essential in processes including symbiosis, resistance to viruses and antimicrobials, and biosynthesis of antibiotics, yet their structure and mechanism are largely unknown. In this study, evolutionary co-variance analysis was used to build a computational model of the structure of a bacterial O-antigen modifying acetyltransferase, OafB. The resulting structure exhibited a novel fold for the AT3 domain, which molecular dynamics simulations demonstrated is stable in the membrane. The AT3 domain contains 10 transmembrane helices arranged to form a large cytoplasmic cavity lined by residues known to be essential for function. Further molecular dynamics simulations support a model where the acyl-coA donor spans the membrane through accessing a pore created by movement of an important loop capping the inner cavity, enabling OafB to present the acetyl group close to the likely catalytic resides on the extracytoplasmic surface. Limited but important interactions with the fused SGNH domain in OafB are identified, and modelling suggests this domain is mobile and can both accept acyl-groups from the AT3 and then reach beyond the membrane to reach acceptor substrates. Together this new general model of AT3 function provides a framework for the development of inhibitors that could abrogate critical functions of bacterial pathogens.

Large Scale Quantum Chemistry with Tensor Processing Units

We demonstrate the use of Googles cloud-based Tensor Processing Units (TPUs) to accelerate and scale up conventional (cubic-scaling) density functional theory (DFT) calculations. Utilizing 512 TPU cores, we accomplish the largest such DFT computation to date, with 247848 orbitals, corresponding to a cluster of 10327 water molecules with 103270 electrons, all treated explicitly. Our work thus paves the way toward accessible and systematic use of conventional DFT, free of any system-specific constraints, at unprecedented scales.

Accurate Hellmann-Feynman forces from density functional calculations with augmented Gaussian basis sets

The Hellmann-Feynman (HF) theorem provides a way to compute forces directly from the electron density, enabling efficient force calculations for large systems through machine learning (ML) models for the electron density. The main issue holding back the general acceptance of the HF approach for atom-centered basis sets is the well-known Pulay force which, if naively discarded, typically constitutes an error upward of 10 eV/Å in forces. In this work, we demonstrate that if a suitably augmented Gaussian basis set is used for density functional calculations, the Pulay force can be suppressed, and HF forces can be computed as accurately as analytical forces with state-of-the-art basis sets, allowing geometry optimization and molecular dynamics to be reliably performed with HF forces. Our results pave a clear path forward for the accurate and efficient simulation of large systems using ML densities and the HF theorem.

Computational design of magnetic molecules and their environment using quantum chemistry, machine learning and multiscale simulations

Having served as a playground for fundamental studies on the physics of d and f electrons for almost a century, magnetic molecules are now becoming increasingly important for technological applications, such as magnetic resonance, data storage, spintronics and quantum information. All of these applications require the preservation and control of spins in time, an ability hampered by the interaction with the environment, namely with other spins, conduction electrons, molecular vibrations and electromagnetic fields. Thus, the design of a novel magnetic molecule with tailored properties is a formidable task, which does not only concern its electronic structures but also calls for a deep understanding of the interaction among all the degrees of freedom at play. This Review describes how state-of-the-art ab initio computational methods, combined with data-driven approaches to materials modelling, can be integrated into a fully multiscale strategy capable of defining design rules for magnetic molecules.

Exploration and validation of force field design protocols through QM-to-MM mapping

The scale of the parameter optimisation problem in traditional molecular mechanics force field construction means that design of a new force field is a long process, and sub-optimal choices made in the early stages can persist for many generations. We hypothesise that careful use of quantum mechanics to inform molecular mechanics parameter derivation (QM-to-MM mapping) should be used to significantly reduce the number of parameters that require fitting to experiment and increase the pace of force field development. Here, we design and train a collection of 15 new protocols for small, organic molecule force field derivation, and test their accuracy against experimental liquid properties. Our best performing model has only seven fitting parameters, yet achieves mean unsigned errors of just 0.031 g cm^-3 and 0.69 kcal mol^-1 in liquid densities and heats of vaporisation, compared to experiment. The software required to derive the designed force fields is freely available at https://github.com/qubekit/QUBEKit.

Modelling Interfaces in Thin-Film Photovoltaic Devices

Developing effective device architectures for energy technologies—such as solar cells, rechargeable batteries or fuel cells—does not only depend on the performance of a single material, but on the performance of multiple materials working together. A key part of this is understanding the behaviour at the interfaces between these materials. In the context of a solar cell, efficient charge transport across the interface is a pre-requisite for devices with high conversion efficiencies. There are several methods that can be used to simulate interfaces, each with an in-built set of approximations, limitations and length-scales. These methods range from those that consider only composition (e.g. data-driven approaches) to continuum device models (e.g. drift-diffusion models using the Poisson equation) and ab-initio atomistic models (developed using e.g. density functional theory). Here we present an introduction to interface models at various levels of theory, highlighting the capabilities and limitations of each. In addition, we discuss several of the various physical and chemical processes at a heterojunction interface, highlighting the complex nature of the problem and the challenges it presents for theory and simulation.

Kinetic Monte Carlo modeling of oxide thin film growth

In spite of the increasing interest in and application of ultrathin film oxides in commercial devices, the understanding of the mechanisms that control the growth of these films at the atomic scale remains limited and scarce. This limited understanding prevents the rational design of novel solutions based on precise control of the structure and properties of ultrathin films. Such a limited understanding stems in no minor part from the fact that most of the available modeling methods are unable to access and robustly sample the nanosecond to second timescales required to simulate both atomic deposition and surface reorganization at ultrathin films. To contribute to this knowledge gap, here we have combined molecular dynamics and adaptive kinetic Monte Carlo simulations to study the deposition and growth of oxide materials over an extended timescale of up to ∼0.5 ms. In our pilot studies, we have examined the growth of binary oxide thin films on oxide substrates. We have investigated three scenarios: (i) the lattice parameter of both the substrate and thin film are identical, (ii) the lattice parameter of the thin film is smaller than the substrate, and (iii) the lattice parameter is greater than the substrate. Our calculations allow for the diffusion of ions between deposition events and the identification of growth mechanisms in oxide thin films. We make a detailed comparison with previous calculations. Our results are in good agreement with the available experimental results and demonstrate important limitations in former calculations, which fail to sample phase space correctly at the temperatures of interest (typically 300–1000 K) with self-evident limitations for the representative modeling of thin films growth. We believe that the present pilot study and proposed combined methodology open up for extended computational support in the understanding and design of ultrathin film growth conditions tailored to specific applications.

Transition-Based Constrained DFT for the Robust and Reliable Treatment of Excitations in Supramolecular Systems

Despite the variety of available computational approaches, state-of-the-art methods for calculating excitation energies, such as time-dependent density functional theory (TDDFT), are computationally demanding and thus limited to moderate system sizes. Here, we introduce a new variation of constrained DFT (CDFT), wherein the constraint corresponds to a particular transition (T), or a combination of transitions, between occupied and virtual orbitals, rather than a region of the simulation space as in traditional CDFT. We compare T-CDFT with TDDFT and ΔSCF results for the low-lying excited states (S1 and T1) of a set of gas-phase acene molecules and OLED emitters and with reference results from the literature. At the PBE level of theory, T-CDFT outperforms ΔSCF for both classes of molecules, while also proving to be more robust. For the local excitations seen in the acenes, T-CDFT and TDDFT perform equally well. For the charge transfer (CT)-like excitations seen in the OLED molecules, T-CDFT also performs well, in contrast to the severe energy underestimation seen with TDDFT. In other words, T-CDFT is equally applicable to both local excitations and CT states, providing more reliable excitation energies at a much lower computational cost than TDDFT cost. T-CDFT is designed for large systems and has been implemented in the linear-scaling BigDFT code. It is therefore ideally suited for exploring the effects of explicit environments on excitation energies, paving the way for future simulations of excited states in complex realistic morphologies, such as those which occur in OLED materials.

Machine Learning and Optoelectronic Materials Discovery: A Growing Synergy

Novel optoelectronic materials have the potential to revolutionize the ongoing green transition by both providing more efficient photovoltaic (PV) devices and lowering energy consumption of devices like LEDs and sensors. The lead candidate materials for these applications are both organic semiconductors and more recently perovskites. This Perspective illustrates how novel machine learning techniques can help explore these materials, from speeding up ab initio calculations toward experimental guidance. Furthermore, based on existing work, perspectives around machine-learned molecular dynamics potentials, physically informed neural networks, and generative methods are outlined.

Efficiently Computing Excitations of Complex Systems: Linear-Scaling Time-Dependent Embedded Mean-Field Theory in Implicit Solvent

Quantum embedding schemes have the potential to significantly reduce the computational cost of first-principles calculations while maintaining accuracy, particularly for calculations of electronic excitations in complex systems. In this work, I combine time-dependent embedded mean field theory (TD-EMFT) with linear-scaling density functional theory and implicit solvation models, extending previous work within the ONETEP code. This provides a way to perform multilevel calculations of electronic excitations on very large systems, where long-range environmental effects, both quantum and classical in nature, are important. I demonstrate the power of this method by performing simulations on a variety of systems, including a molecular dimer, a chromophore in solution, and a doped molecular crystal. This work paves the way for high accuracy calculations to be performed on large-scale systems that were previously beyond the reach of quantum embedding schemes.

Understanding Adsorption of Organics on Pt(111) in the Aqueous Phase: Insights from DFT Based Implicit Solvent and Statistical Thermodynamics Models

Adsorption of organics in the aqueous phase is an area which is experimentally difficult to measure, while computational techniques require extensive configurational sampling of the solvent and adsorbate. This is exceedingly computationally demanding, which excludes its routine use. If implicit solvent could be applied instead, this would dramatically reduce the computational cost as configurational sampling of solvent is not needed. Here, using statistical thermodynamic arguments and DFT calculations with implicit solvent models, we show that semiquantitative values for the free energy and entropy change of adsorption in the aqueous phase (ΔG_ads^solv and ΔS_ads^solv) for small organics can be calculated, for a range of coverages. We parametrize the soft sphere based solute dielectric cavity to an approximated free energy of solvation for a single Pt atom at the (111) facet, forming upper and lower bounds based on the entropy of water at the aqueous metal interface (ΔG_solv(Pt) = -4.35 to -7.18 kJ mol^-1). This captures the decrease in ΔG_ads^solv compared to the free energy of adsorption in the vacuum phase (ΔG_ads^vac), while solvent models with electron density based cavities fail to do so. For a range of oxygenated aromatics, the adsorption energetics using horizontal gas phase geometries significantly overestimate ΔG_ads^solv compared to experiment by ∼100 kJ mol^-1, but they agree with ab initio MD simulations using similar geometries. This suggests oxygenated aromatic compounds adsorb perpendicular to the metallic surface, while the ΔG_ads^solv for vertical geometries of furfural and cyclohexanol agree to within 20 kJ mol^-1 of experimental studies. The proposed techniques provide an inexpensive toolset for validation and prediction of adsorption energetics on solvated metallic surfaces, which could be further validated by the future availability of more experimental measurements for the aqueous entropy/free energy of adsorption.

Forces from Stochastic Density Functional Theory under Nonorthogonal Atom-Centered Basis Sets

We develop a formalism for calculating forces on the nuclei within the linear-scaling stochastic density functional theory (sDFT) in a nonorthogonal atom-centered basis set representation (Fabian et al. Wiley Interdiscip. Rev.: Comput. Mol. Sci.2019, 9, e1412, 10.1002/wcms.1412) and apply it to the Tryptophan Zipper 2 (Trp-zip2) peptide solvated in water. We use an embedded-fragment approach to reduce the statistical errors (fluctuation and systematic bias), where the entire peptide is the main fragment and the remaining 425 water molecules are grouped into small fragments. We analyze the magnitude of the statistical errors in the forces and find that the systematic bias is of the order of 0.065 eV/Å (∼1.2 × 10^–3E_h/a₀) when 120 stochastic orbitals are used, independently of system size. This magnitude of bias is sufficiently small to ensure that the bond lengths estimated by stochastic DFT (within a Langevin molecular dynamics simulation) will deviate by less than 1% from those predicted by a deterministic calculation.

Ultrafast transient absorption spectroelectrochemistry: femtosecond to nanosecond excited-state relaxation dynamics of the individual components of an anthraquinone redox couple

Many photoactivated processes involve a change in oxidation state during the reaction pathway and formation of highly reactive photoactivated species. Isolating these reactive species and studying their early-stage femtosecond to nanosecond (fs-ns) photodynamics can be challenging. Here we introduce a combined ultrafast transient absorption-spectroelectrochemistry (TA-SEC) approach using freestanding boron doped diamond (BDD) mesh electrodes, which also extends the time domain of conventional spectrochemical measurements. The BDD electrodes offer a wide solvent window, low background currents, and a tuneable mesh size which minimises light scattering from the electrode itself. Importantly, reactive intermediates are generated electrochemically, via oxidation/reduction of the starting stable species, enabling their dynamic interrogation using ultrafast TA-SEC, through which the early stages of the photoinduced relaxation mechanisms are elucidated. As a model system, we investigate the ultrafast spectroscopy of both anthraquinone-2-sulfonate (AQS) and its less stable counterpart, anthrahydroquinone-2-sulfonate (AH2QS). This is achieved by generating AH2QS in situ from AQS via electrochemical means, whilst simultaneously probing the associated early-stage photoinduced dynamical processes. Using this approach we unravel the relaxation mechanisms occurring in the first 2.5 ns, following absorption of ultraviolet radiation; for AQS as an extension to previous studies, and for the first time for AH2QS. AQS relaxation occurs via formation of triplet states, with some of these states interacting with the buffered solution to form a transient species within approximately 600 ps. In contrast, all AH2QS undergoes excited-state single proton transfer with the buffered solution, resulting in formation of ground state AHQS- within approximately 150 ps.

BRD4: quantum mechanical protein–ligand binding free energies using the full-protein DFT-based QM-PBSA method

Fully quantum mechanical approaches to calculating protein–ligand free energies of binding have the potential to reduce empiricism and explicitly account for all physical interactions responsible for protein–ligand binding.

Applications of density functional theory in COVID-19 drug modeling

The rapidly evolving Coronavirus 2019 (COVID-19) pandemic has led to millions of deaths around the world, highlighting the pressing need to develop effective antiviral pharmaceuticals. Recent efforts with computer-aided rational drug discovery have allowed detailed examination of drug–macromolecule interactions primarily by molecular mechanics (MM) techniques. Less widely applied in COVID-19 drug modeling is density functional theory (DFT), a quantum mechanics (QM) method that enables electronic structure calculations and elucidations of reaction mechanisms. Here, we review recent advances in applying DFT in molecular modeling studies of COVID-19 pharmaceuticals. We start by providing an overview of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) drugs and targets, followed by a brief introduction to DFT. We then provide a discussion of different approaches by which DFT has been applied. Finally, we discuss essential factors to consider when incorporating DFT in future drug modeling research.

Implicit Solvation Methods for Catalysis at Electrified Interfaces

Implicit solvation is an effective, highly coarse-grained approach in atomic-scale simulations to account for a surrounding liquid electrolyte on the level of a continuous polarizable medium. Originating in molecular chemistry with finite solutes, implicit solvation techniques are now increasingly used in the context of first-principles modeling of electrochemistry and electrocatalysis at extended (often metallic) electrodes. The prevalent ansatz to model the latter electrodes and the reactive surface chemistry at them through slabs in periodic boundary condition supercells brings its specific challenges. Foremost this concerns the difficulty of describing the entire double layer forming at the electrified solid–liquid interface (SLI) within supercell sizes tractable by commonly employed density functional theory (DFT). We review liquid solvation methodology from this specific application angle, highlighting in particular its use in the widespread ab initio thermodynamics approach to surface catalysis. Notably, implicit solvation can be employed to mimic a polarization of the electrode’s electronic density under the applied potential and the concomitant capacitive charging of the entire double layer beyond the limitations of the employed DFT supercell. Most critical for continuing advances of this effective methodology for the SLI context is the lack of pertinent (experimental or high-level theoretical) reference data needed for parametrization.

Massively parallel linear-scaling Hartree-Fock exchange and hybrid exchange-correlation functionals with plane wave basis set accuracy

We extend our linear-scaling approach for the calculation of Hartree-Fock exchange energy using localized in situ optimized orbitals [Dziedzic et al., J. Chem. Phys. 139, 214103 (2013)] to leverage massive parallelism. Our approach has been implemented in the onetep (Order-N Electronic Total Energy Package) density functional theory framework, which employs a basis of non-orthogonal generalized Wannier functions (NGWFs) to achieve linear scaling with system size while retaining controllable near-complete-basis-set accuracy. For the calculation of Hartree-Fock exchange, we use a resolution-of-identity approach, where an auxiliary basis set of truncated spherical waves is used to fit products of NGWFs. The fact that the electrostatic potential of spherical waves (SWs) is known analytically, combined with the use of a distance-based cutoff for exchange interactions, leads to a calculation cost that scales linearly with the system size. Our new implementation, which we describe in detail, combines distributed memory parallelism (using the message passing interface) with shared memory parallelism (OpenMP threads) to efficiently utilize numbers of central processing unit cores comparable to, or exceeding, the number of atoms in the system. We show how the use of multiple time-memory trade-offs substantially increases performance, enabling our approach to achieve superlinear strong parallel scaling in many cases and excellent, although sublinear, parallel scaling otherwise. We demonstrate that in scenarios with low available memory, which preclude or limit the use of time-memory trade-offs, the performance degradation of our algorithm is graceful. We show that, crucially, linear scaling with system size is maintained in all cases. We demonstrate the practicability of our approach by performing a set of fully converged production calculations with a hybrid functional on large imogolite nanotubes up to over 1400 atoms. We finish with a brief study of how the employed approximations (exchange cutoff and the quality of the SW basis) affect the calculation walltime and the accuracy of the obtained results.

Computational and data driven molecular material design assisted by low scaling quantum mechanics calculations and machine learning

Electronic structure methods based on quantum mechanics (QM) are widely employed in the computational predictions of the molecular properties and optoelectronic properties of molecular materials. The computational costs of these QM methods, ranging from density functional theory (DFT) or time-dependent DFT (TDDFT) to wave-function theory (WFT), usually increase sharply with the system size, causing the curse of dimensionality and hindering the QM calculations for large sized systems such as long polymer oligomers and complex molecular aggregates. In such cases, in recent years low scaling QM methods and machine learning (ML) techniques have been adopted to reduce the computational costs and thus assist computational and data driven molecular material design. In this review, we illustrated low scaling ground-state and excited-state QM approaches and their applications to long oligomers, self-assembled supramolecular complexes, stimuli-responsive materials, mechanically interlocked molecules, and excited state processes in molecular aggregates. Variable electrostatic parameters were also introduced in the modified force fields with the polarization model. On the basis of QM computational or experimental datasets, several ML algorithms, including explainable models, deep learning, and on-line learning methods, have been employed to predict the molecular energies, forces, electronic structure properties, and optical or electrical properties of materials. It can be conceived that low scaling algorithms with periodic boundary conditions are expected to be further applicable to functional materials, perhaps in combination with machine learning to fast predict the lattice energy, crystal structures, and spectroscopic properties of periodic functional materials.

Atomistic level characterisation of ssDNA translocation through the E. coli proteins CsgG and CsgF for nanopore sequencing

Two proteins of the Escherichia coli membrane protein complex, CsgG and CsgF, are studied as proteinaceous nanopores for DNA sequencing. It is highly desirable to control the DNA as it moves through the pores, this requires characterisation of DNA translocation and subsequent optimization of the pores. In order to inform protein engineering to improve the pores, we have conducted a series of molecular dynamics simulations to characterise the mechanical strength and conformational dynamics of CsgG and the CsgG-CsgF complex and how these impact ssDNA, water and ion movement. We find that the barrel of CsgG is more susceptible to damage from external electric fields compared to the protein vestibule. Furthermore, the presence of CsgF within the CsgG-CsgF complex enables the complex to withstand higher electric fields. We find that the eyelet loops of CsgG play a key role in both slowing the translocation rate of DNA and modulating the conductance of the pore. CsgF also impacts the DNA translocation rate, but to a lesser degree than CsgG.

Learning Electron Densities in the Condensed Phase

We introduce a local machine-learning method for predicting the electron densities of periodic systems. The framework is based on a numerical, atom-centered auxiliary basis, which enables an accurate expansion of the all-electron density in a form suitable for learning isolated and periodic systems alike. We show that, using this formulation, the electron densities of metals, semiconductors, and molecular crystals can all be accurately predicted using symmetry-adapted Gaussian process regression models, properly adjusted for the nonorthogonal nature of the basis. These predicted densities enable the efficient calculation of electronic properties, which present errors on the order of tens of meV/atom when compared to ab initio density-functional calculations. We demonstrate the key power of this approach by using a model trained on ice unit cells containing only 4 water molecules to predict the electron densities of cells containing up to 512 molecules and see no increase in the magnitude of the errors of derived electronic properties when increasing the system size. Indeed, we find that these extrapolated derived energies are more accurate than those predicted using a direct machine-learning model. Finally, on heterogeneous data sets SALTED can predict electron densities with errors below 4%.

Approaching the basis set limit in Gaussian-orbital-based periodic calculations with transferability: Performance of pure density functionals for simple semiconductors

Simulating solids with quantum chemistry methods and Gaussian-type orbitals (GTOs) has been gaining popularity. Nonetheless, there are few systematic studies that assess the basis set incompleteness error (BSIE) in these GTO-based simulations over a variety of solids. In this work, we report a GTO-based implementation for solids and apply it to address the basis set convergence issue. We employ a simple strategy to generate large uncontracted (unc) GTO basis sets that we call the unc-def2-GTH sets. These basis sets exhibit systematic improvement toward the basis set limit as well as good transferability based on application to a total of 43 simple semiconductors. Most notably, we found the BSIE of unc-def2-QZVP-GTH to be smaller than 0.7 mEh per atom in total energies and 20 meV in bandgaps for all systems considered here. Using unc-def2-QZVP-GTH, we report bandgap benchmarks of a combinatorially designed meta-generalized gradient approximation (mGGA) functional, B97M-rV, and show that B97M-rV performs similarly (a root-mean-square-deviation of 1.18 eV) to other modern mGGA functionals, M06-L (1.26 eV), MN15-L (1.29 eV), and Strongly Constrained and Appropriately Normed (SCAN) (1.20 eV). This represents a clear improvement over older pure functionals such as local density approximation (1.71 eV) and Perdew-Burke-Ernzerhof (PBE) (1.49 eV), although all these mGGAs are still far from being quantitatively accurate. We also provide several cautionary notes on the use of our uncontracted bases and on future research on GTO basis set development for solids.

Hybrid Quantum Mechanics/Molecular Mechanics (QM/MM) Simulation: A Tool for Structure-based Drug Design and Discovery

Quantum mechanics (QM) is physics based theory which explains the physical properties of nature at the level of atoms and sub-atoms. Molecular mechanics (MM) construct molecular systems through the use of classical mechanics. So, hybrid quantum mechanics and molecular mechanics (QM/MM) when combined together can act as computer-based methods which can be used to calculate structure and property data of molecular structures. Hybrid QM/MM combines the strengths of QM with accuracy and MM with speed. QM/MM simulation can also be applied for the study of chemical process in solutions as well as in the proteins, and has a great scope in structure-based drug design (CADD) and discovery. Hybrid QM/MM also applied to HTS, to derive QSAR models and due to availability of many protein crystal structures; it has a great role in computational chemistry, especially in structure- and fragment-based drug design. Fused QM/MM simulations have been developed as a widespread method to explore chemical reactions in condensed phases. In QM/MM simulations, the quantum chemistry theory is used to treat the space in which the chemical reactions occur; however the rest is defined through molecular mechanics force field (MMFF). In this review, we have extensively reviewed recent literature pertaining to the use and applications of hybrid QM/MM simulations for ligand and structure-based computational methods for the design and discovery of therapeutic agents.

Single Indium Atoms and Few-Atom Indium Clusters Anchored onto Graphene via Silicon Heteroatoms

Single atoms and few-atom nanoclusters are of high interest in catalysis and plasmonics, but pathways for their fabrication and placement remain scarce. We report here the self-assembly of room-temperature-stable single indium (In) atoms and few-atom In clusters (2-6 atoms) that are anchored to substitutional silicon (Si) impurity atoms in suspended monolayer graphene membranes. Using atomically resolved scanning transmission electron microscopy (STEM), we find that the symmetry of the In structures is critically determined by the three- or fourfold coordination of the Si "anchors". All structures are produced without electron-beam induced materials modification. In turn, when activated by electron beam irradiation in the STEM, we observe in situ the formation, restructuring, and translation of the Si-anchored In structures. Our results on In-Si-graphene provide a materials system for controlled self-assembly and heteroatomic anchoring of single atoms and few-atom nanoclusters on graphene.

Gaussian Process Regression for Materials and Molecules

We provide an introduction to Gaussian process regression (GPR) machine-learning methods in computational materials science and chemistry. The focus of the present review is on the regression of atomistic properties: in particular, on the construction of interatomic potentials, or force fields, in the Gaussian Approximation Potential (GAP) framework; beyond this, we also discuss the fitting of arbitrary scalar, vectorial, and tensorial quantities. Methodological aspects of reference data generation, representation, and regression, as well as the question of how a data-driven model may be validated, are reviewed and critically discussed. A survey of applications to a variety of research questions in chemistry and materials science illustrates the rapid growth in the field. A vision is outlined for the development of the methodology in the years to come.

Accurate and Efficient Computation of Optical Absorption Spectra of Molecular Crystals: The Case of the Polymorphs of ROY

When calculating the optical absorption spectra of molecular crystals from first principles, the influence of the crystalline environment on the excitations is of significant importance. For such systems, however, methods to describe the excitations accurately can be computationally prohibitive due to the relatively large system sizes involved. In this work, we demonstrate a method that allows optical absorption spectra to be computed both efficiently and at high accuracy. Our approach is based on the spectral warping method successfully applied to molecules in solvent. It involves calculating the absorption spectrum of a supercell of the full molecular crystal using semi-local time-dependent density functional theory (TDDFT), before warping the spectrum using a transformation derived from smaller-scale semi-local and hybrid TDDFT calculations on isolated dimers. We demonstrate the power of this method on three polymorphs of the well-known color polymorphic compound ROY and find that it outperforms both small-scale hybrid TDDFT dimer calculations and large-scale semi-local TDDFT supercell calculations, when compared to the experiment.

Electrochemistry from first-principles in the grand canonical ensemble

Progress in electrochemical technologies, such as automotive batteries, supercapacitors, and fuel cells, depends greatly on developing improved charged interfaces between electrodes and electrolytes. The rational development of such interfaces can benefit from the atomistic understanding of the materials involved by first-principles quantum mechanical simulations with Density Functional Theory (DFT). However, such simulations are typically performed on the electrode surface in the absence of its electrolyte environment and at constant charge. We have developed a new hybrid computational method combining DFT and the Poisson-Boltzmann equation (P-BE) capable of simulating experimental electrochemistry under potential control in the presence of a solvent and an electrolyte. The charged electrode is represented quantum-mechanically via linear-scaling DFT, which can model nanoscale systems with thousands of atoms and is neutralized by a counter electrolyte charge via the solution of a modified P-BE. Our approach works with the total free energy of the combined multiscale system in a grand canonical ensemble of electrons subject to a constant electrochemical potential. It is calibrated with respect to the reduction potential of common reference electrodes, such as the standard hydrogen electrode and the Li metal electrode, which is used as a reference electrode in Li-ion batteries. Our new method can be used to predict electrochemical properties under constant potential, and we demonstrate this in exemplar simulations of the differential capacitance of few-layer graphene electrodes and the charging of a graphene electrode coupled to a Li metal electrode at different voltages.

Implementation of the QUBE Force Field in SOMD for High-Throughput Alchemical Free-Energy Calculations

The quantum mechanical bespoke (QUBE) force-field approach has been developed to facilitate the automated derivation of potential energy function parameters for modeling protein-ligand binding. To date, the approach has been validated in the context of Monte Carlo simulations of protein-ligand complexes. We describe here the implementation of the QUBE force field in the alchemical free-energy calculation molecular dynamics simulation package SOMD. The implementation is validated by demonstrating the reproducibility of absolute hydration free energies computed with the QUBE force field across the SOMD and GROMACS software packages. We further demonstrate, by way of a case study involving two series of non-nucleoside inhibitors of HIV-1 reverse transcriptase, that the availability of QUBE in a modern simulation package that makes efficient use of graphics processing unit acceleration will facilitate high-throughput alchemical free-energy calculations.

Vibronic and Environmental Effects in Simulations of Optical Spectroscopy

Including both environmental and vibronic effects is important for accurate simulation of optical spectra, but combining these effects remains computationally challenging. We outline two approaches that consider both the explicit atomistic environment and the vibronic transitions. Both phenomena are responsible for spectral shapes in linear spectroscopy and the electronic evolution measured in nonlinear spectroscopy. The first approach utilizes snapshots of chromophore-environment configurations for which chromophore normal modes are determined. We outline various approximations for this static approach that assumes harmonic potentials and ignores dynamic system-environment coupling. The second approach obtains excitation energies for a series of time-correlated snapshots. This dynamic approach relies on the accurate truncation of the cumulant expansion but treats the dynamics of the chromophore and the environment on equal footing. Both approaches show significant potential for making strides toward more accurate optical spectroscopy simulations of complex condensed phase systems.

ParaMol: A Package for Automatic Parameterization of Molecular Mechanics Force Fields

The ensemble of structures generated by molecular mechanics (MM) simulations is determined by the functional form of the force field employed and its parameterization. For a given functional form, the quality of the parameterization is crucial and will determine how accurately we can compute observable properties from simulations. While accurate force field parameterizations are available for biomolecules, such as proteins or DNA, the parameterization of new molecules, such as drug candidates, is particularly challenging as these may involve functional groups and interactions for which accurate parameters may not be available. Here, in an effort to address this problem, we present ParaMol, a Python package that has a special focus on the parameterization of bonded and nonbonded terms of druglike molecules by fitting to ab initio data. We demonstrate the software by deriving bonded terms' parameters of three widely known drug molecules, viz. aspirin, caffeine, and a norfloxacin analogue, for which we show that, within the constraints of the functional form, the methodologies implemented in ParaMol are able to derive near-ideal parameters. Additionally, we illustrate the best practices to follow when employing specific parameterization routes. We also determine the sensitivity of different fitting data sets, such as relaxed dihedral scans and configurational ensembles, to the parameterization procedure, and discuss the features of the various weighting methods available to weight configurations. Owing to ParaMol's capabilities, we propose that this software can be introduced as a routine step in the protocol normally employed to parameterize druglike molecules for MM simulations.

Population analysis with Wannier orbitals

We formulate Wannier orbital overlap population and Wannier orbital Hamilton population to describe the contribution of different orbitals to electron distribution and their interactions. These methods, which are analogous to the well-known crystal orbital overlap population and crystal orbital Hamilton population, provide insight into the distribution of electrons at various atom centers and their contributions to bonding. We apply this formalism in the context of a plane-wave density functional theory calculation. This method provides a means to connect the non-local plane-wave basis to a localized basis by projecting the wave functions from a plane-wave density functional theory calculation to a localized Wannier orbital basis. The main advantage of this formulation is that the spilling factor is strictly zero for insulators and can systematically be made small for metals. We use our proposed method to study and obtain bonding and electron localization insights in five different materials.