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At least 73 records · Page 4

Combining Reactive Quantum-Mechanical Molecular-Dynamics Simulations with Mutagenesis, Crystallography, and Enzyme Kinetics to Reveal Plausible Steps of Isocyanide Hydratase Catalysis

A complete understanding of enzyme mechanisms requires atomistic details of chemical reactions. Quantum-based molecular dynamics simulations (QMD) are a potential source of this information, but trade-offs between accuracy and computational cost have limited their use. We previously developed extended Lagrangian Born–Oppenheimer molecular dynamics (XL-BOMD) methods that leverage a negligible compromise in accuracy to substantially decrease the cost of QMD simulations. Here, we develop a reactive QMD approach using the latest XL-BOMD formulation, which enables efficient simulations of highly reactive systems, and use it to investigate mechanisms of intermediate formation in isocyanide hydratase (ICH) catalysis. In QMD simulations, molecular analogs of ICH active site residues reacted with para-nitrophenyl isocyanide, forming a thioimidate. Analysis of simulated atomic configurational and charge dynamics revealed a pathway where protonation of the isocyanide carbon occurs prior to thioimidate formation and suggested a possible role of Asp17 as a proton donor in the early phase of ICH catalysis. To test whether the pathway seen using the reactive QMD approach might be relevant to ICH catalysis, we performed X-ray crystallography and pre-steady-state enzyme kinetics studies of wild-type and D17N mutant ICH. Both the structure and kinetics are sensitive to the D17N mutation in a manner that is consistent with the order of the reaction steps seen in the simulations. Mobile protons play essential roles in many enzymes, yet they are difficult to observe experimentally, making the ordering of proton-dependent steps ambiguous in many enzyme mechanisms. The ability to directly simulate model reactions for the design of experiments that provide information about enzyme mechanisms involving mobile protons demonstrates the significance of our reactive QMD approach and motivates further biological applications.

36 MATERIALS SCIENCE↗

Model-based iterative reconstruction with adaptive regularization for artifact reduction in electron tomography

Obtaining high-quality 3D reconstructions from electron tomography of crystalline particles embedded in lighter support elements is crucial for various material systems such as catalysts for fuel cell applications. However, significant challenges arise due to the limited tilt range, sparse and low signal-to-noise ratio of the measurements. In addition, small metal particles can cause strong streaking and shading artifacts in the 3D reconstructions when using conventional reconstruction algorithms due to the presence of Bragg diffraction and the large scattering cross-section difference between the materials of the particles and the background support regions. These artifacts lead to errors in the downstream characterization affecting extraction of critical features such as the size of the metal particles, their distribution and the volume of the lighter support regions. In this paper, we present a two-stage algorithm based on metal artifact reduction, utilizing model-based iterative reconstruction methods with adaptive adjustment of regularization parameters. Our approach yields high-quality 3D reconstructions compared to traditional algorithms, accurately capturing both the metal particles as well as the background support. We demonstrate the effectiveness of our algorithm through simulated and experimental bright-field electron tomography data, showing significant improvements in reconstruction quality compared to traditional methods.

97 MATHEMATICS AND COMPUTING↗

Catalytic enhancement in the performance of the microscopic two-stroke heat engine

Here we consider a model of heat engine operating in the microscopic regime: the two-stroke engine. It produces work and exchanges heat in two discrete strokes that are separated in time. The working body of the engine consists of two d-level systems initialized in thermal states at two distinct temperatures. Additionally, an auxiliary nonequilibrium system called catalyst may be incorporated with the working body of the engine, provided the state of the catalyst remains unchanged after the completion of a thermodynamic cycle. This ensures that the work produced by the engine arises solely from the temperature difference. Upon establishing the rigorous thermodynamic framework, we characterize twofold improvement stemming from the inclusion of a catalyst. Firstly, we prove that in the noncatalytic scenario, the optimal efficiency of the two-stroke heat engine with a working body composed of two-level systems is given by the Otto efficiency, which can be surpassed by incorporating a catalyst with the working body. Secondly, we show that incorporating a catalyst allows the engine to operate in frequency and temperature regimes that are not accessible for noncatalytic two-stroke engines. We conclude with a general conjecture about the advantage brought by a catalyst: including the catalyst with the working body always allows to improve efficiency over the noncatalytic scenario for any microscopic two-stroke heat engines. We prove this conjecture for two-stroke engines where the working body is composed of two d-level systems initialized in thermal states at two distinct temperatures, as long as the final joint state leading to optimal efficiency in the noncatalytic scenario is not a product state, or at least one of the d-level system is not thermal.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Spectroscopic and Computational Studies of Spin-Orbit Coupling of Lanthanide Oxides

Spin-orbit (SO) coupling makes it possible for spin forbidden transitions or reactions within non-relativistic quantum theory feasible. Thus, quantification of such interactions has important implications in photophysics and chemical catalysis. This work aims to quantify SO interactions using lanthanide (Ln) oxides as target molecular systems. Through these systems, we examine the impact on the SO coupling by electron configurations and 4f orbital occupancies of Ln elements and sizes of the metal oxides. By examining these factors, we explore how SO coupling is affected by the number of Ln 4f electrons for a given size of molecules and if the Ln 4f orbitals remain atomic in nature in these small clusters. Ln oxides are produced in laser ablation molecular beams, identified with time-of-fight mass spectrometry, and characterized with laser spectroscopy and relativistic quantum chemical computations. Spectroscopic measurements include mass-analyzed threshold ionization (MATI), zero electron kinetic energy, and slow electron velocity-map imaging spectroscopies. Relativistic computations treat scalar relativistic corrections, electron correlations, and SO interactions. The main results are SO terms and energies of the neutral molecules and singly charged cations, ionization energies of the neutral species, metal-metal and metal-oxygen vibrational frequencies of the ions and, in some cases, neutrals as well, and charge effects on the bonding and structures.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

ReactionMechanismSimulator.jl: A modern approach to chemical kinetic mechanism simulation and analysis

Abstract We present ReactionMechanismSimulator.jl (RMS), a modern differentiable software for the simulation and analysis of chemical kinetic mechanisms, including multiphase systems. RMS has already been applied to problems in combustion, pyrolysis, polymers, pharmaceuticals, catalysis, and electrocatalysis. RMS is written in Julia, making it easy to develop and allowing it to take advantage of Julia's extensive numerical computing ecosystem. In addition to its extensive library of optimized analytic Jacobians, RMS can generate and use Jacobians computed using automatic differentiation and symbolically generated analytic Jacobians. RMS is demonstrated to be faster than Cantera and Chemkin in several benchmarks. RMS also implements an extensive set of features for analyzing chemical mechanisms, including a library of easy‐to‐call plotting functions, molecular structure resolved flux diagram generation, crash analysis, traditional sensitivity analysis, transitory sensitivity analysis, and an automatic mechanism analysis toolkit. RMS implements efficient adjoint and parallel forward sensitivity analyses. We also demonstrate the ease of adding new features to RMS.

Johnson, Matthew S.↗

A Perspective on Multiscale Modeling of Explicit Solvation-Enabled Simulations of Catalysis at Liquid–Solid Interfaces

Catalysis at liquid-solid interfaces is profoundly influenced by the interfacial solvent structure, which affects catalytic activity, selectivity, and reaction pathways. This perspective discusses state-of-the-art multiscale modeling methods that integrate quantum mechanics and molecular mechanics approaches to apply explicit solvent molecules to capture these interfacial phenomena. Specifically, the construction of multiscale models, the importance of capturing the interfacial solvent structure, and the computational strategies used to achieve this are explored, and the challenges in balancing chemical accuracy with computational expense are highlighted. Additionally, this perspective addresses the limitations of current methods. Opportunities for integrating machine learning are proposed. Here, by advancing the efficiency and user friendliness of multiscale modeling, it is argued that deeper insights into heterogeneous catalysis in liquid phases can be provided, which will ultimately contribute to the development of more efficient catalytic processes.

Ab initio molecular dynamics↗

Continuity of reaction kinetics across the pressure and materials gaps in CO oxidation on FeO–Pt interfaces

Translating atomic-scale insights from surface science studies of model catalysts to practical powder catalysts remains a persistent challenge in heterogeneous catalysis. Here, in this study, we demonstrate mechanistic continuity across the pressure and materials gaps during CO oxidation at the FeO-Pt interface using in situ microscopy, spectroscopy and computational modelling. Under reaction conditions, coordinatively unsaturated Fe (Fe cus ) sites at the interface enable selective O 2 activation on CO-saturated surfaces, circumventing the CO-poisoning limitation of platinum-group metals. We identify parallel reaction pathways involving the *O 2 -*CO intermediate. Remarkably, activation energies remain consistent at 12-15 kJ mol −1 (0.12-0.16 eV) from ultrahigh vacuum to atmospheric pressures and from FeO/Pt(111) model catalysts to FeO/Pt powder catalysts, validating mechanistic insights derived from surface science studies. Our findings show an example of bridging the long-standing divide between model and practical catalyst systems, establishing an effective approach to capture catalytic behaviours under operational conditions and advancing mechanism-driven catalyst design.

03 NATURAL GAS↗

Density Functional Tight-Binding Models for Band Structures of Transition-Metal Alloys and Surfaces across the d -Block

First-principles electronic structure simulations are an invaluable tool for understanding chemical bonding and reactions. While machine-learning models such as interatomic potentials significantly accelerate the exploration of potential energy surfaces, electronic structure information is generally lost. Particularly in the field of heterogeneous catalysis, simulated electron band structures provide fundamental insights into catalytic reactivity. This ab initio knowledge is preserved in semiempirical methods such as density functional tight binding (DFTB), which extend the accessible computational length and time scales beyond first-principles approaches. In this paper here we present Shell-Optimized Atomic Confinement (SOAC) DFTB electronic-part-only parametrizations for bulk and surface band structures of all d-block transition metals that enable efficient predictions of electronic descriptors for large structures or high-throughput studies on complex systems outside the computational reach of density functional theory.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Atomic resolution coherent x-ray imaging with physics-based phase retrieval

Coherent x-ray imaging and scattering from accelerator based sources such as synchrotrons continue to impact biology, medicine, technology, and materials science. Many synchrotrons around the world are currently undergoing major upgrades to increase their available coherent x-ray flux by approximately two orders of magnitude. The improvement of synchrotrons may enable imaging of materials in operando at the atomic scale which may revolutionize battery and catalysis technologies. Current algorithms used for phase retrieval in coherent x-ray imaging are based on the projection onto sets method. These traditional iterative phase retrieval methods will become more computationally expensive as they push towards atomic resolution and may struggle to converge. Additionally, these methods do not incorporate physical information that may additionally constrain the solution. In this work, we present an algorithm which incorporates molecular dynamics into Bragg coherent diffraction imaging (BCDI). This algorithm, which we call PRAMMol (Phase Retrieval with Atomic Modeling and Molecular Dynamics) combines statistical techniques with molecular dynamics to solve the phase retrieval problem. We present several examples where our algorithm is applied to simulated coherent diffraction from 3D crystals and show convergence to the correct solution at the atomic scale.

47 OTHER INSTRUMENTATION↗

Institute for Catalysis in Energy Processes (ICEP) (Final Report)

The Institute for Catalysis in Energy Processes (ICEP) was a multi-PI program located at the Northwestern University Center for Catalysis and Surface Science from 2006-2024. Over several renewal cycles and several organizing themes, ICEP addressed fundamental questions in catalysis science. In turn, the scientific questions addressed were directly relevant to the efficient use of the nation’s resources to produce fuels and commodity chemicals, harness alternate energy sources for chemical reactions such as light or electricity, reduce emissions and waste, and minimize the impact of our use of plastics. Selective oxidation, (oxidative) dehydrogenation, deNOx, CO 2 reduction, hydrogen release, polymer decomposition / depolymerization, and many other reactions of intense interest to the US Department of Energy were studied in the center. Major ICEP strengths were in catalyst synthesis, in measurements that elucidated catalyst properties, in reaction mechanisms and kinetics, and in predictive theory and modeling. ICEP researchers were often the inventors or developers of materials and methods that were brought to bear on the Institute’s catalytic systems. Key characterization tools advanced through this project included resonance Raman and related techniques, sum frequency generation, and X-ray standing wave spectroscopy. Center members relied extensively on computational tools such as density functional theory and microkinetic modeling to fully investigate catalyst structures and catalytic mechanisms. ICEP researchers were early developers of atomic layer deposition (ALD) for catalyst synthesis and were early pioneers of metal organic frameworks for catalysis. ICEP innovations are found in areas of intense research such as single-atom catalysts, catalytic deconstruction of plastics, and catalytic MOFs, to name a few. Research areas initiated by ICEP indirectly led to DOE EFRCs, startup companies, and other achievements. Over its 18 years of existence, ICEP involved two different PIs (Stair and Notestein) and 24 senior investigators across several academic disciplines at Northwestern University. There were tight collaborations with Argonne National Laboratory, including specialized equipment and experiments permanently located there. The center supported ~400 person-years of effort by graduate students and postdocs, either directly through DOE support or indirectly by leveraging independent support provided to the trainees, e.g. through the NSF GRFP or internal fellowships. The project resulted in 328 manuscripts and numerous conference presentations.

36 MATERIALS SCIENCE↗

Illuminating the Material World: Autonomous Microscopy to Understand Order, Disorder, and Everything In Between

Artificial intelligence (AI) holds immense promise for revolutionizing microscopy, yet its widespread adoption has been hindered by challenges ranging from user inexperience to limited model transferability and difficulties in operationalizing machine learning. This presentation showcases our approach to developing practical autonomy for materials discovery, aiming to accelerate the integration of AI into everyday microscopy workflows. As shown in Fig. 1, I will focus on three key areas: understanding order-disorder transitions, quantifying point defects, and achieving truly device-scale microscopy. First, I will demonstrate the power of multi-modal knowledge graphs for integrating diverse microscopy data. By combining imaging, spectroscopy, and diffraction data, these graphs provide a holistic view of material behavior, capturing the intricate relationships between different modalities [1,2]. I will present a case study on how these models illuminate the structural and chemical changes associated with irradiation in oxide thin films, revealing critical insights for designing materials for extreme environments like spaceflight and nuclear energy. Specifically, I will show how multi-modal analysis clarifies the evolution of order-disorder transitions under irradiation, a key factor influencing material performance in these applications. Next, I will address the challenge of quantifying point defects in 2D materials. We demonstrate the application of computer vision and transfer learning to accurately identify and classify various defect types, such as vacancies and substitutional atoms, and to quantify their concentrations. This information is crucial for understanding and tailoring the properties of 2D materials for applications in electronics, optoelectronics, and catalysis. For example, I will show how our models can characterize the topological distribution of point defects in MXene transition metal carbides, providing valuable insights for optimizing their performance in energy storage and separation science. Finally, I will discuss our progress toward autonomous device-scale microscopy [3,4]. We are fundamentally redesigning electron microscopes around the principles of machine reasoning, enabling automation beyond basic tasks like sample navigation and data acquisition to include sophisticated experimental design. This approach paves the way for truly reproducible and massively scaled analysis campaigns. I will emphasize the importance of autonomous microscopy platforms for high-throughput materials discovery and characterization, facilitating the rapid screening of materials for a broad range of applications and accelerating the development of next-generation technologies.

36 MATERIALS SCIENCE↗

Steric Mapping, Ligand Dynamics, and Cycloisomerization Catalysis with Redox Robust Mn I/0/-I Dicarbenes

Manganese is perhaps the most electronically versatile element, yet the redox properties, reactivity, and catalytic applications of low-valent Mn 0 /Mn −I complexes remain underexplored due to the propensity for Mn 0 to dimerize, quenching highenergy metalloradicals. We report a series of redox-active monometallic Mn I , Mn 0 and Mn −I complexes containing a BH 2 - bridged dicarbene, characterized using a suite of experimental and cutting-edge computational (DFT) methods. Slow electron transfer kinetics at Mn I/0 are observed, with computations and electrochemical simulations in excellent agreement with experimental values. Despite the lack of steric bulk at the BH 2 -bridged Mn 0 , the t Bu groups at the dicarbene provide adequate steric protection to prevent dimerization, with percent buried volume (% V bur ) serving as a valuable steric ranking tool. We also show that a %V bur > 83% prevents dimerization for a diverse array of Mn 0 complexes from the literature. Ligand sterics of BPh 2 -and BH 2 -bridged complexes dictate reaction outcomes when Mn I and Mn −I are exposed to nucleophiles and electrophiles, respectively, while Mn0 facilitates the radical cycloisomerization catalysis of 6-iodo-1- hexene at room temperature. Furthermore, this work underscores the importance of ligand sterics in rationalizing reactivity patterns at Mn and provides valuable insights for designing chelating ligands that can selectively leverage Mn I/0/‑I states in redox-mediated catalytic reactions.

Ligands↗

Actuating Liquid Crystals Rapidly and Reversibly by Using Chemical Catalysis

Abstract Microtubules and catalytic motor proteins underlie the microscale actuation of living materials, and they have been used in reconstituted systems to harness chemical energy to drive new states of organization of soft matter (e.g., liquid crystals (LCs)). Such materials, however, are fragile and challenging to translate to technological contexts. Rapid (sub‐second) and reversible changes in the orientations of LCs at room temperature using reactions between gaseous hydrogen and oxygen that are catalyzed by Pd/Au surfaces are reported. Surface chemical analysis and computational chemistry studies confirm that dissociative adsorption of H 2 on the Pd/Au films reduces preadsorbed O and generates 1 ML of adsorbed H, driving nitrile‐containing LCs from a perpendicular to a planar orientation. Subsequent exposure to O 2 leads to oxidation of the adsorbed H, reformation of adsorbed O on the Pd/Au surface, and a return of the LC to its initial orientation. The roles of surface composition and reaction kinetics in determining the LC dynamics are described along with a proof‐of‐concept demonstration of microactuation of beads. These results provide fresh ideas for utilizing chemical energy and catalysis to reversibly actuate functional LCs on the microscale.

Chemistry↗

Exploration of the Electronic and Catalytic Properties of [Co 5 MS 8 (PEt 3 ) 5 ] 1+ Nanoclusters: A Computational Study

Recent studies have demonstrated the relative stability of undercoordinated hexanuclear cobalt sulfide nanoclusters (NCs) with different charge states. Considering that these small metal NCs have atomically precise structures and high reactivity due to the open shell of the transition metals, and provide selectivity toward ligand loss, they are a vital model for catalysis. In this paper, the electronic structures of these NCs are investigated. These NCs are then used as the reference state to analyze the catalytic properties with respect to hydrogen evolution reaction (HER) and CO 2 reduction (CO 2 R). Further, to understand the effect of heteroatom incorporation, the geometry and reactivity of ten different metal dopants are analyzed. This work shows that the type of metal incorporation greatly affects the electronic structure and formation energies for ligand binding and catalysis. Particularly, the d-orbital occupancy in the cobalt atoms remains largely unchanged, while the heteroatom greatly influences the reactivity of the undercoordinated NCs. Most notably, this work highlights that transition metals in [Co 5 MS 8 (PEt 3 ) 5 ] 1+ NCs would competitively prefer electrochemical adsorption of H over COOH, while the main group metals prefer COOH adsorption.

catalysis↗

Degrees of rate control and AutoDiff-driven direct sensitivity analysis in heterogeneous catalysis

Despite the wide application and benefits of the degree of rate control (DRC) analysis, several details remain argued, particularly about the conservation of DRCs at transient (TR) and steady-state (SS) conditions, especially for complex reaction networks. This work argues that previous proofs about the conservation properties of DRCs have been incomplete, and we provide new mathematical proofs at TR and SS conditions. In addition, we use both analytical (automatic differentiation) and numerical (finite difference) approaches to compute DRCs for the case study of ethane hydrogenolysis (EH) over Pt(111). This work confirms that at both TR and SS conditions, the sum of all DRCs, i.e., sum of the degrees of kinetic (DKRC) and thermodynamic rate control (DTRC), is conserved at zero. At SS conditions, the sum of DKRC is conserved at 1 while the sum of DTRC is conserved at −1. In corroboration of previous works, we show that the DTRC for any adsorbate at SS is equal to the product of the species coverage and a constant. In contrast, at TR conditions, the individual sums of both DTRC and DKRC are not conserved and can be any real number, with potential implications for the novel field of dynamic catalysis. Finally, we show that the conventional finite difference (FD) approach, only useful at SS, is prone to inaccuracy and very sensitive to the value of the differential change applied. The optimal differential value also varies significantly with system and rate definition. Consequently, we describe and illustrate in this work the application of the automatic differentiation (AD) approach for the more accurate determination of DRCs at both TR and SS conditions.

Automatic differentiation↗

Prediction of hydration energies of adsorbates at Pt(111) and liquid water interfaces using machine learning

Aqueous phase heterogeneous catalysis is important to various industrial processes, including biomass conversion, Fischer–Tropsch synthesis, and electrocatalysis. Accurate calculation of solvation thermodynamic properties is essential for modeling the performance of catalysts for these processes. Explicit solvation methods employing multiscale modeling, e.g., involving density functional theory and molecular dynamics have emerged for this purpose. Although accurate, these methods are computationally intensive. This study introduces machine learning (ML) models to predict solvation thermodynamics for adsorbates on a Pt(111) surface, aiming to enhance computational efficiency without compromising accuracy. In particular, ML models are developed using a combination of molecular descriptors and fingerprints and trained on previously published water–adsorbate interaction energies, energies of solvation, and free energies of solvation of adsorbates bound to Pt(111). These models achieve root mean square error values of 0.09 eV for interaction energies, 0.04 eV for energies of solvation, and 0.06 eV for free energies of solvation, demonstrating accuracy within the standard error of multiscale modeling. Feature importance analysis reveals that hydrogen bonding, van der Waals interactions, and solvent density, together with the properties of the adsorbate, are critical factors influencing solvation thermodynamics. Furthermore, these findings suggest that ML models can provide rapid and reliable predictions of solvation properties. This approach not only reduces computational costs but also offers insights into the solvation characteristics of adsorbates at Pt(111)–water interfaces.

Adsorption↗

Metalloborophenes: Structural Diversity and Emerging Properties of Metal–Boron Two‐Dimensional Frameworks

Metalloborophenes, an emerging subclass of 2D materials, have attracted growing attention owing to their exceptional structural diversity and highly tunable electronic and magnetic properties. Constructed from vacancy‐rich borophene frameworks stabilized by electron donation from incorporated metal atoms, metalloborophenes merge the chemical versatility of boron with the functional richness of metal dopants. The first experimental realization of Cu–borophene nanoribbons in 2024 marked a pivotal advance, confirming long‐standing theoretical predictions and revitalizing interest in this new frontier of boron‐based 2D chemistry. Despite this progress, most studies to date remain conceptual and theoretical, constrained by challenges in scalable synthesis, dopant precision, and substrate control. Computational investigations have revealed a broad landscape of stable metalloborophene structures, exhibiting metallic, semiconducting, and magnetic behavior across diverse dopant families, including alkali, alkaline‐earth, transition, and lanthanide elements. These tunable characteristics open promising avenues for applications in spintronics, catalysis, and hydrogen storage. This review provides a comprehensive overview of metalloborophenes, emphasizing the interplay between structure, stability, and functionality, and outlining future directions toward bridging predictive modeling with experimental realization of this rapidly evolving class of 2D materials.

2D materials↗

Polynomial Chaos Surrogate Construction for Random Fields with Parametric Uncertainty

Engineering and applied science rely on computational experiments to rigorously study physical systems. The mathematical models used to probe these systems are highly complex, and sampling-intensive studies often require prohibitively many simulations for acceptable accuracy. Surrogate models provide a means of circumventing the high computational expense of sampling such complex models. In particular, polynomial chaos expansions (PCEs) have been successfully used for uncertainty quantification studies of deterministic models where the dominant source of uncertainty is parametric. We discuss an extension to conventional PCE surrogate modeling to enable surrogate construction for stochastic computational models that have intrinsic noise in addition to parametric uncertainty. We develop a PCE surrogate on a joint space of intrinsic and parametric uncertainty, enabled by Rosenblatt transformations, which are evaluated via kernel density estimation of the associated conditional cumulative distributions. Furthermore, we extend the construction to random field data via the Karhunen–Loève expansion. We then take advantage of closed-form solutions for computing PCE Sobol indices to perform a global sensitivity analysis of the model which quantifies the intrinsic noise contribution to the overall model output variance. Additionally, the resulting joint PCE is generative in the sense that it allows generating random realizations at any input parameter setting that are statistically approximately equivalent to realizations from the underlying stochastic model. The method is demonstrated on a chemical catalysis example model and a synthetic example controlled by a parameter that enables a switch from unimodal to bimodal response distributions.

97 MATHEMATICS AND COMPUTING↗