Research

An open machine-learning guide for non-specialists

Students in chemistry, physics, engineering, medicine and the arts increasingly need machine learning, but few of them take computer-science courses. For them we maintain Machine Learning for Humans, an open online guide whose notebooks were first written in 2018 for the machine-learning seminar at Florida State University. It is listed among our publications as the e-book Machine Learning and Quantum Computing Guide for Humans (2025). Short explanations are paired with executable Jupyter notebooks, and the only prerequisite is basic Python.

• Methods. Data preprocessing, decision trees, k-nearest neighbours, support-vector machines, self-organizing maps, genetic algorithms, naive Bayes, non-negative tensor factorization, Monte Carlo and Markov-chain methods and uncertainty quantification, followed by neural networks: convolutional and recurrent networks, generative adversarial networks, DeepDream and style transfer.
• Tools. Tutorials for scikit-learn, Keras, PyTorch and TensorFlow.
• Applications. Examples are grouped by field, from art and image manipulation to chemistry, engineering, medicine and physics, and include potential-energy surfaces and reaction networks.

The guide's quantum-computing chapter is described under Quantum computing. We use the guide in our courses at FSU and MSU.

Related topics: Machine learning, Deep learning, Scikit-learn, Project Jupyter, Open educational resources

Classification experiments as a guide to molecular descriptors

Choosing a molecular representation and a training set-up for a property-prediction model is often a matter of trial and error. With L. Alday-Toledo and co-workers, we proposed running quick classification experiments first, to gauge how hard a prediction task is and which descriptors carry the information it needs (arXiv:2212.01420, 2022).

• Set-up. For the 133,885 small organic molecules of the QM9 database, the enthalpy, Gibbs free energy, zero-point vibrational energy and heat capacity were each divided into five to seven classes of similar size, and neural networks with three hidden layers were trained to assign molecules to these classes.
• Descriptors. The sorted Coulomb matrix (435 numbers per molecule), its eigenvalues, a five-number atomic-composition vector that simply counts the atoms of each element, and combinations of these were compared, using the Adam and AdaBelief optimizers.
• Finding. With AdaBelief, atomic composition alone classified enthalpy and Gibbs free energy with more than 99% accuracy, about as well as the much larger Coulomb matrix. It was less accurate for zero-point energy and much less for heat capacity, where structural descriptors helped.

Such classifiers can quickly filter a database for molecules in a desired property range before costly electronic-structure calculations are run on them.

Related topics: Statistical classification, Molecular descriptor, Surrogate model, Enthalpy, Gibbs free energy

Do Hessians improve machine-learning potentials?

Machine-learning interatomic potentials are usually trained on energies and forces, and models trained mostly on stable geometries can fail elsewhere on the potential-energy surface. With J. S. Smith of NVIDIA, A. Rodriguez in our group tested whether adding Hessians, the second derivatives that describe the curvature of that surface, helps (J. Chem. Theory Comput. 21, 6698 (2025)). ANI-type neural-network potentials were trained on energies alone, on energies and forces, or on energies, forces and Hessians, using our OpenREACT-CHON-EFH dataset of small C, H, N, O molecules along reactions.

• Accuracy. Hessian-trained models gave lower energy errors for distorted geometries and along reaction paths, captured the curvature at transition states and reproduced most DFT vibrational frequencies.
• Simulations. They kept molecular dynamics stable above 500 K for most test molecules, and nudged-elastic-band calculations with them converged to a reaction barrier close to the DFT value, whereas models trained without Hessians did not converge.
• Cost. Hessians are expensive to compute, and training took about 25 times longer per epoch; in return, less training data was needed.

The study concluded that potentials for reaction simulations should include curvature information, a cost we later reduced with Projected Hessian Learning.

Related topics: Machine-learned interatomic potential, Potential energy surface, Hessian matrix, Transition state, Molecular vibration

OpenREACT-CHON-EFH: a reaction dataset with energies, forces and Hessians

To train and test curvature-aware potentials we released OpenREACT-CHON-EFH (Open REaction Dataset of Atomic ConfiguraTions comprising C, H, O, N with Energies, Forces, and Hessians) on Figshare in 2025 under a CC BY 4.0 licence (doi:10.6084/m9.figshare.29189858). It accompanies our study of Hessian training and builds on the reactions collected by Grambow, Pattanaik and Green (2020). Every geometry carries an energy, atomic forces and a full Hessian computed at the ωB97X-D/6-31G(d) level of density-functional theory.

• RTP. 35,087 optimized reactants, transition states and products from 11,961 elementary reactions: the core set for training and evaluation.
• IRC. 34,248 geometries along 600 intrinsic-reaction-coordinate paths, held out to test how well a model extrapolates along complete reaction pathways.
• NMS. 62,527 off-equilibrium geometries made by displacing path geometries along vibrational normal modes, to test robustness.

The dataset supports benchmarking of potentials that learn second derivatives and is also used in our Projected Hessian Learning work.

Related topics: Elementary reaction, Transition state, Reaction coordinate, Normal mode, Machine-learned interatomic potential

Projected Hessian Learning

Training on full Hessians makes machine-learning potentials more accurate, but a Hessian grows with the square of the number of atoms, so computing and storing it limits practical use. With S. Matin, N. Lubbers and K. Barros at Los Alamos National Laboratory and J. S. Smith at NVIDIA, we developed Projected Hessian Learning (PHL), which supervises curvature using only Hessian–vector products along random probe directions, at a cost close to that of force training (Mach. Learn.: Sci. Technol. 7, 055037 (2026)).

• Benchmark. Reactants, products, transition states, reaction paths and normal-mode-sampled geometries from OpenREACT-CHON-EFH. Training on energies and forces, on Hessian–vector products with one-hot or dense Gaussian probes, and on full Hessians was compared.
• Result. When the probe vectors are redrawn for every minibatch, both projected schemes match full-Hessian training in energy, force and Hessian accuracy, with epochs more than 24 times faster on the small molecules studied. With one fixed probe per molecule, dense Gaussian probes beat single-column probes, especially far from equilibrium.
• Code. A reference implementation for ANI models is at github.com/Austinrg14/PHL; Hessian and Hessian–vector-product training for HIP-NN models is available in the hippynn library.

The authors propose PHL as a scalable replacement for explicit Hessian supervision in larger molecular systems.

Related topics: Machine-learned interatomic potential, Hessian matrix, Automatic differentiation, Potential energy surface

CheMLFlow: reproducible machine-learning workflows for chemistry and materials

Machine-learning studies in chemistry often force researchers to assemble a whole pipeline, from data acquisition to the final report, even when they contribute to only one step. With B. Smith, N. Abdulrahiman and colleagues, we developed CheMLFlow, an open-source platform of modular components and ready-to-run reference pipelines for cheminformatics and materials informatics (arXiv:2608.04942, 2026; code at github.com/nijamudheen/CheMLFlow).

• Reproducible and auditable. Each run is defined by one version-controllable YAML or JSON file. Data splits are deterministic and saved, every step writes its outputs to the run directory, and runs record configuration and git hashes, so results, including failed runs, can be audited.
• Models. From random forests, support-vector machines, XGBoost and CatBoost to neural networks, Chemprop graph networks and the CheMeleon foundation model, with a design-of-experiments engine for large model and hyperparameter screens.
• Agent layer. Short, versioned “skills” let coding agents install the software, write configurations, launch runs and summarize results, while interpretation and final decisions stay with the researcher.
• Benchmarks. Quantum-mechanical (QM9), physicochemical and bioactivity (ChEMBL) prediction tasks reach performance comparable to published results, and a time-series example shows that the platform extends beyond molecular data.

Related topics: Cheminformatics, Materials informatics, Reproducibility, Workflow management system, Graph neural network

Recent advances in 2D materials: a multi-group review

Research on two-dimensional (2D) materials now spans theory, synthesis, characterization, devices and quantum physics. In 2023 a team of 48 authors from many groups, with J. A. Robinson (Penn State) as corresponding author and Y.-H. Lin, D. Maldonado-Lopez and J. L. Mendoza-Cortés among the contributors, surveyed recent progress across the field (ACS Nano 2023). The review covers:

• Theory and modelling. Modelling of defects and intercalants in 2D materials.
• Machine learning. Data-driven methods for synthesis and sensing.
• New materials. Growth of MXenes and of magnetic and low-symmetry 2D crystals.
• Optics and phonons. Optical and phonon properties of atomically thin layers.
• Devices. Mixed-dimensional heterostructures for logic and memory.
• Quantum transport. Transport in twisted homojunctions.

The review closes with perspectives on future directions for the field.

Related topics: Single-layer (2D) materials, MXenes, Twistronics, Heterojunction

Twistronics: engineering band gaps by twisting and stacking TMDC bilayers

Twistronics, the control of electronic properties through the twist angle between stacked two-dimensional layers, took off after superconductivity was found in magic-angle bilayer graphene in 2018. Semiconducting transition-metal dichalcogenides (TMDCs) add further knobs: the choice of metal and chalcogen, and the stacking of two different materials. In 2025 we reported a first-principles survey of bilayers built from six MX₂ compounds (M = Mo, W; X = S, Se, Te), using range-separated hybrid density functional theory on 30 homo- and heterobilayer combinations with different stackings and twist angles (Nanoscale Advances 2025).

• Twist angle. Twisted bilayers form moiré patterns. At critical angles (17.9°, 42.1°, 77.9° and 102.1° for WS₂ and WSe₂) the patterns are symmetric and the band gap becomes tunable. Under low strain, MoS₂, WS₂ and WSe₂ bilayers switch between direct and indirect gaps at specific angles, and MoS₂ can switch between semiconducting and metallic states.
• Stacking. Stacking and composition set the stability and the size of the gap; for example, the MoTe₂/WSe₂ heterostructure with a 60° shift keeps a direct band gap.
• Band flatness. The twist angle also tunes how flat the electronic bands are.

Combining heterostructure chemistry with controlled twist angles gives a toolkit for setting the size and character (direct or indirect) of band gaps in 2D semiconductors.

Related topics: Twistronics, Transition metal dichalcogenide monolayers, Moiré pattern, Direct and indirect band gaps, Hybrid functionals

Silicene nanoclusters with low reorganization energies

Good charge-transport materials change shape very little when they gain or lose an electron; in other words, they have a small internal reorganization energy (λ). In a 2018 study led by R. Pablo-Pedro with T. Van Voorhis, J. Kong and M. S. Dresselhaus (MIT) and S. Fomine (UNAM), with J. L. Mendoza-Cortés as co-author, density functional theory calculations predicted small reorganization energies for rectangular, hydrogen-passivated nanoclusters of silicene (H-SiNCs), the silicon analogue of graphene (Physical Review Applied 2018).

• Low reorganization energies. For several clusters both the electron and the hole reorganization energies are below 0.1 eV, with the lowest values between about 0.05 and 0.07 eV. These are comparable to pentacene for holes (0.097 eV) and fullerene for electrons (0.06 eV), two benchmark organic semiconductors.
• n-type character. All clusters studied have large electron affinities and strongly stabilized anions, which points to n-type behaviour.
• Design handle. Changing the cluster size along the zigzag and armchair directions tunes these properties.

Size-selected H-SiNCs are therefore proposed as n-type materials for silicon-compatible electronics and spintronics.

Related topics: Silicene, Marcus theory, Organic semiconductor, Electron affinity

Activating "inert" hexagonal boron nitride

Hexagonal boron nitride (hBN) is usually considered chemically inert because of its wide band gap and strong covalent bonds, which limits its use in catalysis and energy conversion. In 2021, in work co-led by M. Terrones (Penn State) and our group, cryomilling (mechanical milling in liquid nitrogen) was shown to turn hBN into a reactive support for single metal atoms (Materials Today 2021).

• Defects by cryomilling. Milling creates vacancies, with or without oxygen, giving defective BN (d-BN). Photoluminescence reveals mid-gap states and electron spin resonance detects localized radicals; density functional theory links both to the vacancies and their spin densities.
• Spontaneous metal loading. The radicals and the resulting shift of the Fermi level let d-BN reduce metal cations without an added reducing agent, anchoring single atoms (for example platinum) as well as mono- and bimetallic clusters.
• Catalysis. The anchored single atoms and nanostructures improve catalysis of the hydrogen-evolution reaction; the authors also point to uses in sensing and quantum information.

The same defective-BN platform was later used to anchor copper single atoms that inactivate E. coli within 30 minutes under sunlight (ACS Nano 2025).

Related topics: Boron nitride, Vacancy defect, Electron paramagnetic resonance, Heterogeneous catalysis

Arsenic-alloyed black phosphorus transistors

Black phosphorus is a layered semiconductor whose band gap and carrier mobility depend on the number of layers, which makes it attractive for optoelectronics and thin-film transistors. In 2019, in a collaboration led by the experimental groups of N. R. Pradhan and L. Balicas, with theory from H. Terrones and our group, few-layer black phosphorus with a quarter of its atoms replaced by arsenic (b-AsP, x = 0.25) was characterized (Nanoscale 2019).

• Raman spectroscopy. Density functional theory (DFT) explains the measured spectra: each Raman mode of black phosphorus splits into three, from P–P, P–As and As–As bonds.
• Transistors. Few-layer b-AsP field-effect transistors show an on/off current ratio of about 10³ at room temperature (about 10⁵ at 4 K) and an intrinsic field-effect mobility approaching 300 cm² V⁻¹ s⁻¹ at 300 K, rising to about 600 cm² V⁻¹ s⁻¹ at 100 K.
• Insulator–metal transition. At high gate voltages the resistivity decreases on cooling and saturates below about 100 K, a gate-induced insulator-to-metal transition.
• Anisotropy and doping. Transport is strongly anisotropic between the armchair and zigzag directions, as in pristine black phosphorus, and hybrid DFT places the Fermi level close to the top of the valence band, explaining the hole conduction.

With its tunable band gap and anisotropy, b-AsP is a candidate for optoelectronic applications.

Related topics: Phosphorene, Field-effect transistor, Raman spectroscopy, Anisotropy

Carbon-doped silicene: disorder, correlation and magnetism

Silicene, the silicon analogue of graphene, is compatible with silicon electronics but has no natural band gap. In 2020, with T. Van Voorhis (MIT), we used first-principles calculations on finite clusters and periodic sheets to understand how substituting carbon atoms into silicene changes its properties through lattice disorder and local electron correlation (Nano Letters 2020).

• Tunable band gap. Where the carbon atoms sit changes the electronic properties dramatically; the band gap is set by the amount of lattice disorder and by electron correlation.
• Mott–Anderson transition. Changing the dopant positions drives a Mott–Anderson transition, a change from extended to localized electronic states caused jointly by electron–electron repulsion and disorder.
• Magnetism. The doped sheets remain ferromagnetic even when disordered.

Carbon doping therefore offers a way to engineer the band gap, magnetism and correlation effects of silicene for silicon-based nanoelectronics such as field-effect transistors.

Related topics: Silicene, Anderson localization, Mott insulator, Ferromagnetism

Two-dimensional metal halides

Metal halides with MX and MX₂ formulas can be thinned from bulk crystals into two-dimensional sheets, but how this changes their electronic behaviour was poorly understood. In 2026 we built a database of 60 such compounds (X = F, Cl, Br, I) and computed their properties in bulk and slab form with the hybrid functional HSE06-D3 (Phys. Chem. Chem. Phys. 2026).

• Band gaps. Most compounds are semiconductors, with band gaps from 0 to 9 eV that depend on composition and on dimensionality.
• Indirect-to-direct switching. Nine compounds, including CdCl₂, CuI and VI₂, change from an indirect to a direct band gap when thinned, which could benefit energy-conversion applications.
• Magnetism. Many monolayers are magnetic; for example, CrI₂ and NiCl₂ change from antiferromagnetic in the bulk to ferromagnetic as monolayers, whereas MnCl₂ and FeCl₂ stay antiferromagnetic within the layer.
• Catalytic descriptors. Comparing band-edge positions with the water redox levels singles out CrI₂ and NiI₂ monolayers as candidates for photocatalytic water splitting.

The database is intended as a resource for choosing low-dimensional halides for electronic, optoelectronic, catalytic and spintronic uses.

Related topics: Metal halides, Single-layer (2D) materials, Direct and indirect band gaps, Photocatalytic water splitting

Rare isotopes in diamond for tests of fundamental symmetries

A permanent electric dipole moment of a nucleus would violate time-reversal symmetry and point to physics beyond the Standard Model. In “pear-shaped” nuclei such as protactinium-229, octupole deformation enhances the nuclear Schiff moment and hence the expected signal, but such isotopes can be produced only in tiny amounts, for example at MSU's Facility for Rare Isotope Beams. I. M. Morris, K. Klink, J. T. Singh, J. L. Mendoza-Cortés, S. S. Nicley and J. N. Becker proposed embedding them in synthetic diamond, where they would form colour centres that can be probed repeatedly (Phil. Trans. R. Soc. A 382, 20230169 (2024)).

• Why diamond. A defect in diamond can combine isolated, molecule-like electronic states with efficient capture of scarce atoms, repeated measurements on even a single atom and narrow optical linewidths.
• Calculations. Density-functional calculations (PBE+U geometries, HSE06 electronic structure) indicate that the most stable protactinium defect is the ion accompanied by two carbon vacancies, with defect states inside the diamond band gap.
• Non-radioactive proxy. A praseodymium defect with a similar electronic structure is also stable and could be used to prepare experiments without radioactive material.

The article appeared in the Theo Murphy meeting issue “Diamond for quantum applications”, where it was featured on the cover, and outlines a toolbox, inspired by quantum information processing, for studying rare isotopes.

Related topics: Electric dipole moment, T-symmetry, Crystallographic defects in diamond, Nitrogen-vacancy center, Facility for Rare Isotope Beams, Protactinium

Tuning photoredox catalysis in titanium metal–organic frameworks

Metal–organic frameworks (MOFs) that drive light-powered redox reactions are attractive photocatalysts because their pore size and electronic structure can be tuned by changing their building blocks. In a 2023 collaboration led by F. J. Uribe-Romo (University of Central Florida) with several groups, including ours, a library of eleven isoreticular titanium MOFs with the formula Ti₆O₉(linker)₃ was used to separate these two effects (J. Am. Chem. Soc. 2023).

• Structure. The linkers are linear oligo(p-arylene) dicarboxylates of different lengths, some carrying electron-donating amine groups. Powder X-ray diffraction and total scattering show parallel one-dimensional titanium-oxo "nanowires" connected by the linkers in a rod-packed hexagonal net.
• Steric versus electronic effects. Longer linkers enlarge the pores, while donor groups change the HOMO–LUMO gap. In the photocatalytic oxidation of benzyl alcohol, substrate uptake correlates with the reaction rate, and both longer linkers and more donor groups increase activity.
• Performance. The best frameworks react almost 20 times faster than the benchmark MOF MIL-125(Ti).

The results show that pore size and electronic structure must be considered together when designing MOF photocatalysts.

Related topics: Metal–organic framework, Photoredox catalysis, Photocatalysis, Benzyl alcohol

Platinum-free catalysts for hydrogen evolution

Platinum is the benchmark catalyst for the hydrogen-evolution reaction (HER) in water electrolysis, but it is scarce and expensive. In two collaborations, synthesis and electrochemistry were combined with our density functional theory (DFT) calculations to explain the activity of cheaper catalysts.

• Alloyed dichalcogenides on graphene (2017). With M. Terrones (Penn State), R. Lv and co-workers, centimetre-scale films alternating graphene with W_xMo_1−xS₂ alloy layers were made by wet chemistry and annealing at only 300 °C. The best composition, W_0.4Mo_0.6S₂, reaches 10 mA cm⁻² at 96 mV with a Tafel slope of 38.7 mV dec⁻¹, is at least twice as efficient as WS₂ or MoS₂ and keeps its activity over 1000 cycles. DFT shows that alloying activates the otherwise inert basal plane, with the lowest hydrogen-adsorption barrier at x ≈ 0.4 (ACS Nano 2017).
• Sulfur-doped molybdenum phosphide (2019). With Y. Yang (University of Central Florida), a self-organized, binder-free S-doped MoP nanoporous layer was made by anodization followed by two-step chemical vapour deposition. It needs an overpotential of 86 mV to reach 10 mA cm⁻² in acid (Tafel slope 34 mV dec⁻¹) and is also active in basic and neutral electrolytes. DFT indicates that H₂ forms through the Volmer–Heyrovsky route and that sulfur doping makes this easier than on MoS₂ (ACS Catalysis 2019).

Related topics: Hydrogen evolution reaction, Electrolysis of water, Molybdenum disulfide, Tafel equation

How a cobalt complex selectively reduces CO₂ to CO

Electrochemical reduction of carbon dioxide to carbon monoxide is a step toward recycling carbon, but molecular catalysts often waste electrons by reducing protons to hydrogen. The cobalt tetraaza complex [Co^IIN₄H]²⁺ makes CO with 45% Faradaic efficiency (CO:H₂ = 3:2). In 2018, with M. Head-Gordon and A. T. Bell (UC Berkeley), we used density functional theory to identify its most likely mechanism (Phys. Chem. Chem. Phys. 2018), building on our earlier infrared and computational study of CO₂ activation by the same complex (J. Phys. Chem. C 2015).

• Selectivity. Comparing the catalyst's affinity for the species present (acetonitrile, water, CO₂ and bicarbonate) explains why CO₂ reduction wins over proton reduction: after a two-electron reduction, formation of [Co^IN₄H]⁺–CO₂⁻ is strongly favoured.
• Catalytic cycle. Protonation gives [Co^IN₄H]⁺–CO₂H, which reacts with a second CO₂ molecule to give [Co^IIN₄H]²⁺–CO and bicarbonate; CO₂ acts as a Lewis acid that assists C–O bond cleavage.
• No formic acid. High activation barriers block the formic acid pathway, consistent with the observed products, and the calculated reduction potentials and turnover frequency agree with published experiments.
• Testable prediction. The rate of CO formation should increase linearly with the CO₂ partial pressure.

Related topics: Electrochemical reduction of carbon dioxide, Homogeneous catalysis, Faraday efficiency, Lewis acids and bases

Graphene electrodes that are impermeable yet electronically transparent

Graphene blocks molecules yet is only one atom thick, which raises the question of whether a metal buried beneath it can still influence reactions on its outer surface. In 2018, with J. Rodríguez-López's group (University of Illinois Urbana-Champaign), we built graphene heterostructures that combine a buried underlayer with a molecular-catalyst adlayer and used them to tune the oxygen-reduction reaction (ORR) (ACS Nano 2018).

• Approach. Systematic voltammetry and imaging of patterned electrodes probed how each modification changes the ORR mechanism and overpotential.
• Adlayers. Porphyrin catalyst adlayers showed higher activity for the two-electron pathway than the bare graphene basal plane.
• Underlayers. The buried layer lowered the ORR activation potential in the order Pt > Au > SiO_x, even though it is sealed off from the solution.
• Mechanism and stability. Calculations attribute the enhancement to metal d-electrons permeating through the graphene. The electrodes tolerated cyanide poisoning and long-term cycling, showing that graphene can protect noble metals while still allowing electrochemical interaction.

The results inform the design of two-dimensional and core–shell electrocatalysts.

Related topics: Graphene, Oxygen reduction reaction, Porphyrin, Heterojunction

Batteries

Stabilizing ether electrolytes in high-voltage lithium batteries

Ether-based liquid and polymer electrolytes are promising for high-energy batteries with lithium or sodium metal anodes, but they fail at both electrodes: at the low potential of the anode they undergo uncontrolled anionic polymerization, and at the cathode they oxidize at the working potentials of the most attractive cathode chemistries. In 2019, a joint experimental and computational study with L. A. Archer's group at Cornell University addressed both problems (Nature Communications 2019).

• Anode side. A cationic chain-transfer agent, tris(hexafluoroisopropyl) phosphate, stops the runaway polymer growth that degrades the electrolyte at the lithium anode.
• Cathode side. Cathode–electrolyte interphases made from preformed anionic polymers, or from oligomers derived from the additive lithium bis(oxalato)borate (LiBOB), protect the ether. Our density functional theory calculations examined how these oligomers form, how stable they are and how they oxidize.
• Outcome. The strategies raise the oxidative stability of ether electrolytes by about 0.3 V, beyond conventionally accepted limits. Lithium-metal cells with nickel–manganese–cobalt oxide (NCM) cathodes and LiBOB retained more than 80% of their capacity for at least 200 cycles between 3 and 4.2 V.

Related topics: Lithium metal battery, Solid-state battery, Diglyme

Potassium and sodium in few-layer graphene electrodes

Graphite, the standard lithium-ion anode, works poorly for more abundant alkali metals: potassium tends to plate on its surface instead of intercalating, and sodium barely intercalates at all. With the electrochemistry group of J. Rodríguez-López (University of Illinois Urbana-Champaign), we use ultrathin few-layer graphene as a model electrode and combine electrochemistry and spectroscopy with density functional theory (DFT).

• Fast potassium (2018). Conditioning the electrode in a Li⁺ electrolyte forms a solid-electrolyte interphase (SEI) through which K⁺ intercalates efficiently, with distinct staging peaks at scan rates up to 100 mV s⁻¹, charge–discharge at up to about 360C (a full discharge in 10 s) and stable cycling for 1000 cycles at 10C (J. Am. Chem. Soc. 2018).
• Two ions at once (2021). Li⁺ intercalation follows the Nernst equation (54–58 mV per decade). In mixed Li⁺/K⁺ electrolytes the slope falls to about 30 mV per decade once lithium is abundant, a signature of co-intercalation; to our knowledge this was the first study of two alkali ions intercalating together. DFT shows that separate Li-only and K-only domains are always stable, whereas mixed layers are stable only when lithium is dilute (Chemical Science 2021).
• Reversible sodium (2022). As our calculations had predicted, fluorinating the graphene surface allows Na⁺ to intercalate, reaching about NaC_14–18 compared with NaC₁₈₆ reported for graphite, but only when a Li-based SEI is formed first (J. Electrochem. Soc. 2022).

Related topics: Potassium-ion battery, Sodium-ion battery, Graphite intercalation compound, Nernst equation

Designing surface-modified graphene for beyond-lithium batteries

Sodium, potassium, magnesium and calcium are more abundant than lithium, but graphite cannot store most of them efficiently. Our calculations ask whether modifying only the outer surface of few-layer graphene can change the energetics of ion insertion in the layers below.

• Surface modifiers (2020). First-principles calculations compared doping the top layer with heteroatoms and covering the electrode with a single-side fluorinated graphene or a Janus hydrofluorographene monolayer. With a suitable modifier, Li⁺, Na⁺ and K⁺ intercalation all become thermodynamically favourable, making reversible sodium storage with a large capacity possible; charge transfer between the modifier and the graphene explains the effect (ACS Appl. Mater. Interfaces 2020).
• Six ions, three stages (2024). Hybrid DFT on four- and five-layer graphene topped with a half-fluorinated graphene layer followed Li, Na, K, Be, Mg and Ca from stage 3 to stage 1. For all six ions the most favourable site lies directly beneath the modifier; the modifier reverses the unfavourable intercalation of Na, Be, Mg and Ca in pristine graphene, and four layers bind the ions more strongly than five (J. Appl. Phys. 2024).

The 2020 predictions motivated the sodium experiments described above.

Related topics: Sodium-ion battery, Multivalent battery, Fluorographene, Intercalation (chemistry)

Solid electrolytes for lithium batteries

Solid electrolytes could replace the flammable liquids in lithium batteries, but few combine high room-temperature conductivity with stability over a wide voltage range.

• Helical covalent polymers (2021). In work led by W. Zhang's group (University of Colorado Boulder), single-crystal X-ray diffraction showed that an anionic covalent polymer folds into densely packed double helices whose bundles form one-directional, negatively charged channels for lithium ions. Without added lithium salt it conducts 1.2 × 10⁻³ S cm⁻¹ at room temperature, with a lithium transference number of 0.84, an activation energy of 0.14 eV and a stability window of 0.2–5 V; a nonflammable ionic liquid acts as a solvating medium that raises the conductivity more than 1000-fold. With this ionic liquid, an all-solid-state cell with a high-voltage NMC811 cathode cycled reversibly (CCS Chemistry 2021).
• Antiperovskites under strain (2026 preprint). Lithium chalcohalide antiperovskites are candidate solid electrolytes. Our calculations of their energetics and lattice dynamics show that placing the smaller anion inside the octahedron favours stability: Li₃OCl, Li₃OBr, Li₃OI, Li₃FTe and Li₃FPo are stable cubic phases, Li₃FS and Li₃FSe become dynamically stable under compression, and a few percent strain tunes the band gap within the electrochemical stability window (arXiv 2026).

Related topics: Solid-state battery, Fast-ion conductor, Ionic liquid

Air electrodes for rechargeable zinc–air batteries

Rechargeable zinc–air batteries need low-cost air electrodes that catalyse both the oxygen-reduction reaction (ORR) during discharge and the oxygen-evolution reaction (OER) during charging, without platinum-group metals. Arrays of nitrogen-doped carbon nanotubes (NCNTs) are good ORR catalysts, but their hydrophobic surface and base-growth mode limit their performance in practical cells. In 2018, with Y. Yang's group (University of Central Florida), we reported an "apically dominant" design (Adv. Energy Mater. 2018).

• Design. CoNi nanoparticles are encapsulated in the tips (apical domain) of NCNTs grown on nickel foam (CoNi@NCNT/NF).
• Performance. The electrode is more active for both ORR and OER than conventional base-grown NCNT arrays. As the air electrode of a zinc–air coin cell it gave a peak power density of 127 mW cm⁻², an energy density of 845 Wh kg_Zn⁻¹ and rechargeable operation for more than 90 h, outperforming platinum-group-metal catalysts.
• Mechanism. Density functional theory attributes the ORR activity mainly to synergy between the nanotubes and active sites near the embedded CoNi particles.

Related topics: Zinc–air battery, Oxygen reduction reaction, Oxygen evolution, Carbon nanotube

Sodium insertion in layered birnessite

Layered manganese oxides such as birnessite can host ions between their sheets, but how intercalation reshapes their structure and electronic properties is not well understood. In 2026, with M. Videa and J. L. Cholula-Díaz, we followed the insertion of Na⁺ into potassium birnessite up to saturation using hybrid density functional theory (J. Chem. Phys. 2026).

• What was computed. Formation energies, simulated Raman spectra and X-ray diffraction patterns, diffusion barriers for Na⁺ and K⁺, binding energies and spin-polarized densities of states.
• Structural changes. Intercalation changes the oxidation states of manganese, distorts the lattice and alters its symmetry; Na⁺ ions become more loosely bound as the layers approach saturation.
• Electronic and magnetic changes. Intercalation tunes the band gap and the magnetism, and some intercalated structures behave as bipolar magnetic semiconductors, of interest for spintronics.

The results connect ion uptake in a layered oxide to measurable structural signatures and to its electronic and magnetic behaviour.

Related topics: Birnessite, Manganese dioxide, Sodium-ion battery, Intercalation (chemistry)

Hydrogen and methane storage

Where it began: designing frameworks for methane and hydrogen, 2008 to 2012

With first-principles force fields in hand (see Interatomic potentials (force fields) and atomistic simulation), we designed new frameworks on the computer before anyone made them. For methane-fuelled vehicles, simulations of gas delivery and molecular-dynamics tests of stability gave 14 new covalent organic frameworks, two of which exceed the Department of Energy target at 35 bar (J. Phys. Chem. A 2011). For hydrogen, which binds too weakly to most materials at room temperature, we added metals.

• Metalated linkers. Quantum calculations on 48 metal-containing versions of five common COF linkers led to COF-301-PdCl₂, predicted to store 60 g of hydrogen per liter at 298 K and 100 bar, 1.5 times the Department of Energy 2015 target (J. Phys. Chem. Lett. 2012).
• Alkali doping. Lithium, sodium and potassium doping of simple, low-cost frameworks brought eleven compounds to the 2010 Department of Energy weight target at room temperature (J. Phys. Chem. A 2012).
• Thesis. These designs are the core of the Ph.D. thesis (Caltech 2012); the projects above continue them with chelating, multi-binding and ferrocene sites.

Related topics: Hydrogen storage, Adsorbed natural gas, Molecular dynamics

Quantum-level design rules for hydrogen-storage materials

Storing hydrogen by physisorption in porous solids is attractive because uptake and release are fast and reversible, but H₂ usually binds too weakly for practical storage at room temperature; an adsorption enthalpy of roughly 7–15 kJ mol⁻¹ is considered ideal for charging and discharging at ambient temperature. In 2020 we used dispersion-corrected density functional theory (B3LYP-D3) to map how H₂ binds to 240 model complexes that mimic the metal sites of metal–organic and covalent organic frameworks (J. Phys. Chem. C 2020).

• Chemical space. Eleven transition metals (Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Pd and Pt) chelated by five organic linkers, in different oxidation states, spin states and coordination geometries, from square-planar and tetrahedral to octahedral.
• Key findings. Dispersion and electrostatics dominate the interaction. Many sites bind H₂ with at least 10 kJ mol⁻¹, a key requirement for practical storage, and the average binding of the metal complexes is close to the ideal range; Pt(II) complexes bind most strongly.
• Design implications. The results give quantitative guidance for choosing metals and linkers so that framework pores present many sites with suitable binding strength.

This work underpins our framework designs for room-temperature hydrogen storage described below.

Related topics: Hydrogen storage, Physisorption, Dihydrogen complex, Metal–organic framework

Metal-chelating covalent organic frameworks for room-temperature hydrogen storage

Most porous materials store less than 10 g L⁻¹ of hydrogen at room temperature because they bind H₂ too weakly. Our strategy is to add linkers that chelate (bind) transition metals to lightweight covalent organic frameworks (COFs), so that the pores offer stronger binding sites.

• Design principles (2016). Quantum-mechanical calculations on five linkers, each chelated with 11 transition metals, showed that abundant first-row metals (Sc to Cu) bind H₂ as strongly as precious metals, or more strongly. We then built 30 new COFs and simulated their uptake from 0 to 700 bar at 298 K with quantum-derived force fields and grand canonical Monte Carlo. Co-, Ni- and Fe-based frameworks gave the highest uptake and delivery, up to about 40 g L⁻¹ at 700 bar (J. Am. Chem. Soc. 2016; highlighted in MRS Bulletin).
• Multi-binding-site COFs (2024). The MSUCOF family combines three linkers with seven metals (Co, Cu, Fe, Mn, Ni, Pd and Pt) in 24 designs. In simulations at 298 K and up to 700 bar, Co-, Ni-, Mn-, Fe-, Pd- and Pt-based MSUCOFs reach the U.S. Department of Energy (DOE) 2025 targets, and the cheaper first-row metals often outperform palladium and platinum (Energy & Fuels 2024, front cover).

These capacities are computational predictions for the materials alone, whereas the DOE targets (5.5 wt% and 40 g L⁻¹ for 2025; 6.5 wt% and 50 g L⁻¹ ultimately) apply to complete storage systems, including the tank.

Related topics: Hydrogen storage, Covalent organic framework, Chelation, Monte Carlo method

A ferrocene-functionalized framework beyond the ultimate hydrogen-storage targets

Extending the multi-binding-site strategy, in 2026 we designed MSUCOF-4-FeCp by attaching ferrocene groups (an iron atom sandwiched between two cyclopentadienyl rings) to the covalent organic framework IRCOF-102, and evaluated it with first-principles multiscale simulations (npj Energy Materials 2026).

• Binding sites. The ferrocene units provide multiple binding sites with binding energies of 15–20 kJ mol⁻¹, strong enough for high capacity yet weak enough to deliver the hydrogen under practical conditions.
• Capacity. At 298 K and 700 bar the framework is predicted to store 18.0 wt% and 72.6 g L⁻¹ of hydrogen, above the DOE ultimate targets of 6.5 wt% and 50 g L⁻¹ (which apply to complete storage systems).
• Cost. Ferrocene offers a lower-cost alternative to incorporating precious metals in COFs.

The design builds directly on the metal-chelating frameworks described above, replacing chelated metals with ferrocene groups as the source of binding sites.

Related topics: Ferrocene, Metallocene, Hydrogen storage, Covalent organic framework

Layered metal oxides for solar and electrochemical water oxidation

Splitting water into hydrogen and oxygen requires efficient, earth-abundant catalysts for the demanding oxygen-evolution half-reaction. Layered manganese, cobalt and nickel oxides are natural candidates; birnessite, a layered manganese oxide found in nature, resembles the manganese cluster that oxidizes water in Photosystem II.

• Tuning birnessite to capture sunlight (2015). First-principles calculations on birnessite with 13 different intercalated cations, from Li⁺ to Y³⁺, showed how Jahn–Teller distortions and the ordering of Mn³⁺ centres tune the band gap, from 2.63 to about 2.20 eV (indirect gap) and from 3.09 to about 2.50 eV (direct gap). Band alignment points to Sr, Ca, B and Al birnessites as candidates for water splitting, and separating a single oxide layer turns the gap from indirect to direct (J. Phys. Chem. C 2015).
• Ordering metal ions layer by layer (2025). With M. J. Zdilla's group (Temple University), few-layer potassium cobalt and nickel oxides with uniform, randomly mixed and alternating CoO₂/NiO₂ layers were compared. The alternating stack performs best, lowering the oxygen-evolution overpotential by about 200–400 mV, and our calculations relate the gain to the density of states and orbital localization of the layer-segregated material (ACS Catalysis 2025).

Both studies show that which ions sit in or between the layers, and how they are arranged, matters as much as the overall composition.

Related topics: Birnessite, Oxygen evolution, Photocatalytic water splitting, Jahn–Teller effect, Photosystem II

An ultrastable platinum diselenide catalyst for fuel cells

The oxygen-reduction reaction (ORR) at the cathode limits the efficiency and cost of hydrogen fuel cells, which rely on platinum catalysts. Platinum dichalcogenides had been overlooked as ORR catalysts because they are semiconducting and their stacked layers expose little surface. In October 2024, with B. E. Koel's group at Princeton University, we showed that electrochemical cycling restructures defective platinum diselenide (PtSe₂) into an ultrastable ORR catalyst (Nature Materials 2024).

• Restructuring. Defective PtSe₂ was cycled electrochemically in an oxygen-saturated electrolyte; the restructured surface carries platinum nanoparticles and apical active sites.
• Activity and durability. After 42,000 cycles the catalyst showed 1.3 times the specific activity and 2.6 times the mass activity of a commercial Pt/C catalyst, and after 126,000 cycles it still outperformed Pt/C with minimal decay.
• Mechanism. Hybrid density functional theory attributes the performance to synergy between the platinum nanoparticles and the apical active sites on the PtSe₂ surface.

The study shows that defect engineering combined with reaction-driven restructuring can turn a semiconducting layered material into a durable electrocatalyst.

Related topics: Oxygen reduction reaction, Proton-exchange membrane fuel cell, Platinum diselenide, Electrocatalyst

An organic crystal that converts heat into mechanical work

Solids that abruptly expand or contract at a phase transition could serve as actuators, but in organic crystals this behaviour is rare: few such crystals are reversible, most degrade after about a dozen cycles, and others take minutes or hours to reset. In 2021, with J. J. Gassensmith's group, we reported a fully reversible organic crystalline switch that works near ambient temperature (J. Am. Chem. Soc. 2021).

• Actuation. When heated above 45 °C the crystals contract almost instantly to 90% of their length, and they return to their original length when cooled below 35 °C.
• Durability. The crystals actuate for more than 200 cycles and remain stable at room temperature for years.
• Application. The fast, predictable motion was used to build a thermal fuse that switches a circuit on and off in real time.

The work shows that an organic crystal can convert heat into mechanical work repeatedly and predictably near room temperature.

Related topics: Actuator, Phase transition, Shape-memory polymer, Fuse (electrical)

Semiconductors with crystalline porosity

Covalent organic frameworks (COFs) are crystalline, permanently porous solids built from light elements joined by strong covalent bonds, but they are usually electrical insulators, which limits their use in electronics. Using first-principles calculations, we showed that placing transition-metal atoms between the layers of boroxine- and triazine-linked COFs can turn them into porous semiconductors with tunable electronic structure.

• Iron (2017). Seven new frameworks with iron atoms between the organic layers were designed with dispersion-corrected hybrid DFT. The parent framework has a band gap of about 2.6 eV; most iron-intercalated structures are semiconductors with gaps of about 1.1–2.0 eV, and the gap depends on which rings the iron atoms sit over (J. Phys. Chem. C 121, 21160 (2017)).
• First-row transition metals (2019). Thirty-one materials, COF-TM-x, incorporate each metal from scandium to zinc. The choice and concentration of the metal tune the band gap and density of states, with the metal's d electrons playing the main role (Phys. Chem. Chem. Phys. 21, 8785 (2019)).
• Bilayers (2023). In 63 metal-intercalated COF bilayers, the concentration and position of the metals strongly changed electronic, magnetic and spintronic properties, giving candidate water-splitting catalysts, visible-light semiconductors, half-metals and bipolar magnetic semiconductors (Phys. Scr. 98, 105926 (2023)).

Because the porous framework is preserved, such materials could combine high surface area with tailored electronic properties in nanoporous electronic devices.

Related topics: Covalent organic framework, Intercalation (chemistry), Electronic band structure, Transition metal, Semiconductor, Density functional theory

Restricted multilayer theory of gas adsorption

The classic models of gas uptake in porous solids make opposite assumptions: the Langmuir model allows a single adsorbed layer, and the Brunauer–Emmett–Teller (BET) model an unlimited number. Neither fits crystalline adsorbents such as metal–organic frameworks (MOFs), whose pores can hold only a few layers. We derived a restricted multilayer theory in which the pore geometry limits the number of layers (ACS Appl. Mater. Interfaces 10, 13244 (2018)).

• Key idea. The maximum number of adsorbed layers, n_max, follows from the geometry of the framework. The isotherm reduces to the Langmuir model for n_max = 1 and to BET for n_max → ∞.
• Inputs and validation. Only geometric properties of the framework, such as pore size, pore volume and surface area, and the interaction energy between gas and framework are needed. The theory was validated against hydrogen-uptake data for seven MOF materials.
• Design rules. It yields the optimal binding strength for hydrogen storage and a universal formula for the linker length, and hence pore size, that maximizes hydrogen delivery.

Although tested for hydrogen in MOFs, the theory is expected to apply to other light gases and other crystalline porous materials.

Related topics: Adsorption, Langmuir adsorption model, BET theory, Metal–organic framework, Hydrogen storage

Building units and nets of metal–organic frameworks

Metal–organic frameworks are built by linking metal-containing clusters with organic molecules, and the shape of these clusters, called secondary building units (SBUs), decides which networks can form. With Omar Yaghi's and Michael O'Keeffe's groups, we described the geometry, connectivity and composition of 131 metal carboxylate SBUs collected from the Cambridge Structural Database, and listed the frameworks in which each one appears (Chem. Soc. Rev. 2009).

• Compendium. Drawings, connectivity and composition of 279 such units were compiled for the review (2008; also listed under Code and data).
• Use. The catalogue became a reference for designing new frameworks: a chemist chooses the building unit whose shape gives the desired network, then the organic linker that connects it.

Related topics: Metal–organic framework, Cambridge Structural Database

Toward magnetically mediated superconductivity in metal-intercalated bilayer graphene

Graphite becomes superconducting when alkali or alkaline-earth metals such as calcium are intercalated between its layers. In a 2019 preprint with J. T. Haraldsen (University of North Florida) and A. V. Balatsky (Nordita), we asked whether transition metals could play a similar role in bilayer graphene (arXiv 2019). Using spin-polarized density functional theory, we placed each first-row transition metal, from scandium to zinc, in a stable honeycomb arrangement between two graphene sheets and calculated the electronic structure, the induced magnetic moments and the exchange coupling between the metal atoms.

• Electronic structure. The carbon p-bands become degenerate and hybridize with the metal d-bands, forming an electron pocket below the Fermi level at the Γ point. We compare this to the carbon p-band behaviour that underlies superconductivity in intercalated graphite.
• Magnetism. Comparing ferromagnetic and antiferromagnetic arrangements shows that only V and Ni favour ferromagnetic order, that Sc, Ti, Cr, Mn, Fe and Co favour antiferromagnetic order, and that Cu and Zn carry no magnetic moment; Cr, Mn and Co have the largest exchange interactions.
• Outlook. Spin coupling of the hybridized bands could provide bosonic modes while preserving graphene's linear dispersion. Superconductivity is proposed as a possibility for experiments to test, not demonstrated.

The work outlines a design route toward tunable magneto-electric Dirac materials based on transition-metal intercalation.

Related topics: Bilayer graphene, Graphite intercalation compound, Unconventional superconductor, Exchange interaction

Terahertz control of surface topology

Using light to switch topological states could give local control over the electronic properties of quantum materials. In 2025, the terahertz scanning-tunnelling-microscopy group of T. L. Cocker and our group, both at MSU, reported a light-driven topological phase transition at the surface of tungsten ditelluride (WTe₂), a layered material that is a likely Weyl semimetal (Nature Photonics 2025).

• Method. Terahertz pulses are concentrated at the apex of an atomically sharp tip. The enhanced field couples resonantly to an interlayer shear mode through a ferroelectric dipole at the interface and drives the surface into a metastable structure in which the top atomic layer has shifted.
• Subatomic imaging. Differential imaging, interpreted with our hybrid density functional theory calculations, resolved a shift of 7 ± 3 pm in the top atomic plane.
• Electronic consequences. Tunnelling spectroscopy links the changes to the electron and hole pockets of the band structure, and the data suggest a reversible, light-induced annihilation of the topologically protected Fermi-arc surface states in that single layer.

The experiment shows that a topological surface state can be switched locally by moving a single atomic layer, and that the change can be read out with subatomic precision.

Related topics: Terahertz radiation, Weyl semimetal, Tungsten ditelluride, Scanning tunneling microscope, Fermi arc

Spintronic properties of metal-intercalated COF bilayers

Two-dimensional covalent organic frameworks (COFs) are porous, crystalline layered materials whose structure is easy to tune, but weak π-electron delocalization limits their use in electronics. In 2023 we used hybrid density functional theory to study a COF built from benzene, boroxine (B₃O₃) and triazine (C₃N₃) rings, reduced to just two crystalline layers, with first-row transition metals inserted between them (Physica Scripta 2023).

• Structures. We compared a pristine bilayer, a trilayer and 63 bilayers with metal atoms at different concentrations and positions.
• Key findings. The concentration and position of the metals change the electronic, magnetic and spintronic properties drastically. Depending on the configuration, the bilayers are direct semiconductors with gaps in the visible range, half-metals, half-semiconductors or bipolar magnetic semiconductors. In the last three classes the electronic states at the band edges are spin-polarized, a property that spintronic devices such as spin filters exploit.
• Catalysis. Some structures are proposed as candidate water-splitting catalysts.

The study extends our earlier work on metal intercalation in COFs (Phys. Chem. Chem. Phys. 2019) and suggests that bilayer COFs could serve as tunable platforms for organic spintronics.

Related topics: Covalent organic framework, Spintronics, Half-metal, Intercalation (chemistry)

Switching and moving the Dirac cone of bilayer graphene

Monolayer graphene owes its fast, massless electrons to the Dirac cone, where its valence and conduction bands meet linearly. Bilayer graphene lacks this cone and behaves as a semiconductor whose band gap can be tuned by doping or an applied voltage. In two 2018 studies we used dispersion-corrected density functional theory to ask whether metal atoms intercalated between the sheets could restore or reshape the cone.

• Recreating the cone with V, Nb and Ta. One vanadium atom at the centre of the honeycomb (BLG-1V) produces a Dirac cone at the K point. Adding more vanadium destroys it, and an antiferromagnetic arrangement of two vanadium atoms opens a small band gap (about 0.1 eV), while niobium and tantalum give metals whose Dirac point lies 1.0–1.4 eV above the Fermi level. Symmetry, spin and metal concentration are the controlling parameters (J. Chem. Phys. 2018).
• Moving the cone with Sc to Zn. A second study designed 20 materials, bilayer and bulk (graphite-like) versions of graphene intercalated with each first-row transition metal. In both forms, the position of the Dirac point depends on which metal is inserted (J. Phys. Chem. C 2018).
• Mechanism. In both studies the states near the Fermi level come from carbon 2p_z orbitals mixing with metal 3d_yz orbitals, so out-of-plane metal atoms control graphene's in-plane electrons.

The V, Nb and Ta compounds had not yet been made, but are predicted to be stable; the studies suggest intercalation as a chemical dial for carbon electronics.

Related topics: Dirac cone, Bilayer graphene, Intercalation (chemistry), Graphene

Spin currents in the quantum spin Hall phase

Quantum spin Hall insulators, the two-dimensional topological insulators, are hard to model with standard density-functional theory (DFT), whose electron–electron potential depends on the spin density but not on the spin currents that spin–orbit coupling sets up. In spin-current DFT (SCDFT) it does. With J. K. Desmarais and A. Erba, we applied SCDFT, in a developmental version of the CRYSTAL program, to a bismuth (001) bilayer that changes from an ordinary to a topological insulator under tensile strain (Phys. Rev. B 106, L201109 (2022)).

• Band inversion. Tensile strains of 5 to 20% were applied; the transition, driven by an s+p_z ↔ p_x+ip_y band inversion, sets in at about 15%.
• Dirac cone. At the onset of the transition, a Dirac cone appears at the Γ point of the valence band only when spin currents are included, and the spin–orbit gap at Γ is larger in SCDFT (0.40 eV) than in DFT (0.28 eV). A first-order k·p model rationalizes the valence-band structure.
• Takeaway. Including spin currents improves the computed electronic structure even qualitatively, which matters for materials whose properties are governed by spin–orbit coupling.

Related topics: Quantum spin Hall effect, Topological insulator, Density functional theory, Spin–orbit interaction, Dirac cone

First-principles force fields for gases in porous materials

To predict how much gas a porous crystal can hold, a simulation needs an accurate description of the weak dispersion forces between the gas molecules and the framework. With William A. Goddard III at Caltech we computed these interactions with second-order Møller–Plesset perturbation theory (MP2) and other first-principles methods, turned them into van der Waals force fields for methane and hydrogen in covalent and metal–organic frameworks, and used the force fields in grand canonical Monte Carlo simulations of gas uptake.

• Methane in COFs. Simulations with an MP2-based force field matched measured methane uptake in covalent organic frameworks within 2%, revealed multilayer adsorption coexisting with pore filling, and identified COF-1 as exceeding the U.S. Department of Energy volumetric target (J. Phys. Chem. A 2010). The approach began as an undergraduate thesis (2008).
• Hydrogen. A new first-principles van der Waals force field for hydrogen, validated against experiments, guided the design of frameworks that reach the Department of Energy's 2015 energy-density target (Ph.D. thesis, 2012). These designs are described under Energy storage.
• Review. How quantum calculations, Monte Carlo and molecular dynamics are used to study and improve hydrogen storage in frameworks (Chem. Soc. Rev. 2009).

Related topics: Force field (chemistry), Møller–Plesset perturbation theory, London dispersion force, Grand canonical ensemble

Interatomic potentials for large deformations of 2D materials

Simulations of how two-dimensional crystals stretch and break depend on interatomic potentials, which are hard to parameterize far from equilibrium. X. Zhang, H. D. Espinosa (Northwestern University) and co-workers, including our group, developed a fitting framework modelled on machine-learning practice (npj Comput. Mater. 7, 113 (2021)).

• Method. A multi-objective genetic algorithm (NSGA-III) fits the parameters to many density-functional targets at once: equilibrium properties such as cohesive energy, elastic constants and surface energies, and non-equilibrium ones such as bond-dissociation curves, vacancy formation energies, uniaxial stress–strain curves and a phase-transition pathway. Properties are split into training and screening sets, and correlation and principal-component analyses identify redundant targets and guide each new iteration.
• MoSe₂ case study. Buckingham, Stillinger–Weber, Tersoff and modified REBO potentials were re-parameterized for monolayer MoSe₂. The Tersoff form, the most flexible of the four, performed best, and adding acoustic phonon modes as targets improved its phonon properties. All fitted potentials still wrongly predicted the 1T phase to be the most stable.

The authors expect the framework to help parameterize potentials for many other 2D and bulk materials.

Related topics: Interatomic potential, Molybdenum diselenide, Multi-objective optimization, Genetic algorithm, Principal component analysis

Rotaxanes, hydrogels and metal–organic complex arrays

The Ph.D. thesis (Caltech, 2012) argued that each problem needs a method matched to its length scale, and applied quantum mechanics, first-principles force fields, molecular dynamics and Monte Carlo simulations to very different systems.

• Metal–organic complex arrays. Their structures were predicted and characterized with quantum mechanics and force-field molecular mechanics, borrowing methods used to model enzymes.
• Hydrogels. Transition-state theory gave the rates of the reactions that build polymer hydrogel networks, which pointed to good concentrations for polymerization and supplied parameters for coarse-grained force fields, in which beads stand for groups of atoms so that simulations reach much longer times.
• Rotaxanes. Free energies from quantum mechanics explained the template-directed synthesis of rotaxane dumbbells of the Stoddart type, including the role of the counter-anion. We returned to rotaxanes in 2023 with first-principles molecular dynamics, described under Molecular machines.

Related topics: Rotaxane, Hydrogel, Coarse-grained modeling, Transition state theory

From force fields to machine-learned potentials

Machine learning offers a way to keep the accuracy of quantum mechanics at the cost of a force field. At Florida State University, undergraduates worked in 2018 on a force-field generator for molecules and materials (project description, PDF), and from 2019 to 2023 the U.S. Army Research Office supported our work on machine-learning techniques for force fields (see Funding). Today we train machine-learned interatomic potentials on energies, forces and second derivatives (Hessians) with collaborators at NVIDIA and Los Alamos National Laboratory.

• Curvature helps. Potentials trained with Hessians predict structures away from equilibrium, reaction paths, transition states and vibrational spectra better, and need less training data, although training with Hessians costs more (J. Chem. Theory Comput. 2025).
• Faster curvature training. Projected Hessian Learning supervises curvature with Hessian–vector products and matches full-Hessian accuracy while training more than 24 times faster per epoch (Mach. Learn.: Sci. Technol. 2026).

Both projects are described in detail under Machine learning and AI; reaction force fields that know the curvature of the energy landscape are one of the directions on which we are heading.

Related topics: Machine-learned interatomic potential, Hessian matrix, Potential energy surface

High-throughput search for semiconductors for artificial photosynthesis

Artificial photosynthesis, using sunlight to make fuels by splitting water, needs semiconductors that absorb visible light, drive hydrogen or oxygen evolution and survive in water, ideally made of earth-abundant elements. To look for them we built SALSA (Substitution, Approximation, evoLutionary Search and Ab initio calculations), a high-throughput workflow (J. Appl. Phys. 134, 235706 (2023)).

• Composition generation. An ionic translation model trained on the Inorganic Crystal Structure Database proposes more than 30,000 undiscovered semiconductor compositions, ternary and quaternary hybrids of known binary compounds.
• Redox stability. Candidates are kept only if their estimated reduction and oxidation potentials let them survive hydrogen- or oxygen-evolution conditions in water: a reduction potential below 0 V and an oxidation potential above 1.23 V against the normal hydrogen electrode, within a tolerance.
• Structures and band gaps. Crystal structures of the remaining candidates are predicted with the evolutionary code USPEX, and their band gaps are recalculated with the HSE06 hybrid functional.

The workflow identified dozens of promising candidates; the five that passed every step, such as PbCuSeCl and Ag₄Cl₂S, all contain two kinds of anion. The SALSA code is archived on Zenodo (doi:10.5281/zenodo.23045693). The search began at Florida State University, where undergraduates worked in 2018 on materials for capturing sunlight (project description, PDF).

Related topics: Artificial photosynthesis, Photocatalytic water splitting, Solar fuel, High-throughput screening, Crystal structure prediction, Hybrid functional

Posets, Eulerian numbers and zeta values

Partially ordered sets (posets) connect combinatorics and number theory with our work on the mathematical foundations of machine learning. In a 2022 preprint, E. Dolores-Cuenca and J. L. Mendoza-Cortés used them to revisit results of Srinivasa Ramanujan (arXiv:2205.05208).

• Operads and order polytopes. Finite posets form an operad, an algebraic structure that describes how operations compose, and the paper studies this operad and its algebras. The order polytope of each poset is then used to study combinatorial properties of values of the Riemann zeta function, ζ(k).
• Zeta-value identities. Reading the factor kⁿ in one of Ramanujan's zeta-value identities as the order polynomial of a set of n unrelated points, the authors generalize the identity to other posets.
• Eulerian numbers. Some of Ramanujan's results on Eulerian numbers are re-proved by viewing his series as carrying the algebraic structure of disjoint unions of points.
• Linear independence. The conjecture that 1, ζ(2), ζ(3), … are linearly independent over the rationals is recast as an equivalent statement about maps built from posets; the conjecture itself remains open.

Related topics: Partially ordered set, Operad, Order polytope, Eulerian number, Riemann zeta function, Srinivasa Ramanujan

Polychrony and chinampas

Polychronization, introduced by E. M. Izhikevich in computational neuroscience, describes networks whose nodes fire in precisely timed patterns. With J. A. Arciniega-Nevárez and colleagues, we turned this idea into a graph-theoretic framework (Algorithms 16, 193 (2023)).

• Model. A node fires either when it is stimulated externally or when two signals reach it at the same time: each signal has unit strength and the firing threshold is two. Groups of nodes that fire in such time-locked patterns are called polychrony groups.
• Cascades and profit. A cascade is a polychrony group in which more nodes are activated by other nodes than by external stimuli; the difference is its profit.
• Chinampas. The activation diagram of a cascade is called a chinampa, after the farm plots of pre-Hispanic Mexico. Chinampas of profit zero and one are counted exactly, and counting one family of chinampas turns out to be equivalent to computing Stanley's order polynomials of associated posets.
• Algorithms. One algorithm reconstructs a cascade from its externally stimulated nodes, and another predicts whether a given node will fire at a given time; both scale as O(n) in the best case and O(n²) in the worst.

The framework links timing-based models of neural computation with combinatorics.

Related topics: Spiking neural network, Chinampa, Graph theory, Partially ordered set, Order polynomial

Order theory in machine learning

Neural networks with integer weights and ReLU-type activations correspond to tropical rational functions, whose geometry is captured by polytopes. In a 2024 preprint (revised in 2025), E. Dolores-Cuenca, A. Guzmán-Sáenz, S. Kim, S. López-Moreno and J. L. Mendoza-Cortés studied the networks whose associated polytope is the order polytope of a partially ordered set, or poset (arXiv:2412.06097).

• Poset filters. Posets with four elements give network modules that act like 2×2 convolutional filters with fixed weights. They can be added to any network, for example as a pooling layer, without extra trainable parameters.
• Training behaviour. Like maxout units, poset pooling updates the network weights more precisely during backpropagation than average, max or mixed pooling. In tests on CIFAR-10, CIFAR-100, Fashion-MNIST and an ImageNet subset, it improved accuracy in some networks (on CIFAR-10, from 76.8% with max pooling to 78.9% in a quaternion network) and was on par in others.
• Algebraic structure. Poset networks and tropical polynomials form algebras over the operad of posets, and composing networks corresponds to generalized Minkowski sums and convex envelopes of their Newton polytopes.

Order theory may thus offer interpretable, geometry-aware building blocks for deep learning. Code: Poset-filters.

Related topics: Order theory, Tropical geometry, Convolutional neural network, Pooling layer, Newton polytope

Quantum computing for non-specialists

Our open guide Machine Learning for Humans, listed among our publications as the e-book Machine Learning and Quantum Computing Guide for Humans, includes a chapter of quantum-computing notebooks. It grew out of the courses on machine learning and quantum computing that Prof. Mendoza-Cortés began teaching at Florida State University in 2018 and, like the rest of the guide, assumes only basic Python.

• Foundations. Qubits, Pauli matrices, superposition and the Bloch sphere, and the physical systems used to make qubits, from spins and cold atoms to trapped ions and photons.
• Gate-based computing. Notebooks use Qiskit and the IBM Q Experience to build small circuits and run them on simulators and on IBM hardware; worked examples include Grover's search and a quantum half-adder.
• Quantum annealing. A D-Wave notebook introduces annealing-based quantum computing and quantum-assisted machine learning.

Because the circuits run on simulators or through IBM's cloud service, the notebooks can be used in classes without access to quantum hardware. The Quantum Computing Live Book (2026) builds on these courses with a full two-semester sequence.

Related topics: Quantum computing, Qiskit, Bloch sphere, Grover's algorithm, Quantum annealing, D-Wave Systems

Quantum Computing Live Book

The Quantum Computing Live Book (2026) is our open, executable textbook on quantum computing, written by F. Velasquez (an MSU undergraduate), W. Comaskey and E. Dolores-Cuenca (former group members) and J. L. Mendoza-Cortés. It is archived on Zenodo under a CC BY 4.0 licence (doi:10.5281/zenodo.21825989). Its approach is to read the theory, then run it: each topic comes with code, Jupyter notebooks, animations and exercises, and the book is updated continuously rather than fixed as a static text.

• Semester 1: foundations and algorithms. Postulates and Dirac notation, single qubits and the Bloch sphere, entanglement and teleportation, circuits and universality, software stacks, density operators and noise, qubit hardware, the quantum Fourier transform, phase estimation and Shor's algorithm, Grover search, Hamiltonian simulation with the Jordan–Wigner mapping, and variational methods for molecules and materials.
• Semester 2: information, error correction and materials. Quantum error correction from stabilizer codes to a first look at the surface code, entanglement and entropy, quantum Shannon theory, quantum key distribution, error mitigation, and end-to-end workflows for materials problems.
• Hands-on. Notebooks run on simulators and real hardware, including Qiskit with IBM Quantum and D-Wave annealers; cheat sheets and extra notebooks accompany the chapters.

The book grew out of our courses on machine learning and quantum computing.

Related topics: Quantum computing, Quantum algorithm, Quantum error correction, Quantum key distribution, Qiskit

Cubane clusters in enzymes and synthetic chemistry

Cube-shaped clusters of metal and oxygen or sulfur atoms, called cubanes or heterocubanes, sit at the active sites of key enzymes: the Mn₄CaO₅ oxygen-evolving complex that splits water in photosynthesis, the iron–molybdenum cofactor of nitrogenase that fixes nitrogen, and the Fe₄S₄ clusters of aconitase in the citric acid cycle and of many electron-transfer proteins. In a review with the group of M. J. Zdilla, we connected these natural clusters with the large body of synthetic cubane chemistry (Coord. Chem. Rev. 450, 214168 (2022)).

• Synthetic cubanes. The review surveys synthetic cubanes by coordination environment and cluster geometry, across different core metals and bridging ligands.
• Linking the two fields. It compares these synthetic systems with their natural counterparts, relating structure and coordination environment to function in reactions such as water oxidation, nitrogen fixation and single-electron transfer.

By bringing biological and synthetic examples together, the review presents heterocubanes as a unifying scaffold for understanding, and eventually imitating, some of nature's most demanding redox reactions. The theme continues in our work on synthetic heterocubane mimics of Photosystem II.

Related topics: Oxygen-evolving complex, Nitrogenase, Iron–sulfur protein, Aconitase, Bioinorganic chemistry

Models of the oxygen-evolving complex

Plants split water with a cluster of four manganese atoms, a calcium atom and five oxygen atoms in Photosystem II. Synthetic clusters that resemble it, structurally and electronically, help explain how the natural catalyst forms the oxygen–oxygen bond and suggest new catalysts for oxygen evolution. With the groups of Theodor Agapie and William A. Goddard III at Caltech, we combined synthesis, isotopic labeling and quantum-mechanical calculations on cube-shaped manganese–oxygen complexes that transfer oxygen atoms or take up oxygen from water. Oxygen transfer turned out to be controlled by how easily the surrounding ligands detach, and labeling showed oxygen atoms migrating within the cluster, supporting recent proposals for the mechanism of the natural catalyst (J. Am. Chem. Soc. 2013).

• Oxidation potentials. The Ph.D. thesis computed the oxidation potentials of tetranuclear CaMn₃O₄ and Mn₄O₄ clusters and proposed new molecules that resemble the oxygen-evolving complex (Caltech 2012).
• Review. A book chapter with the same groups reviewed what quantum calculations and synthetic model clusters reveal about the structure of the complex and how it forms the oxygen–oxygen bond (CRC Press 2015).
• Today. With NSF support from 2026 to 2029 we study synthetic heterocubane mimics of the complex (see the cubane topic above).

Related topics: Oxygen-evolving complex, Photosystem II, Artificial photosynthesis

Real-time monitoring of glucose use by liver cells

Liver cells help keep blood glucose in balance in response to hormones such as insulin, but how the changing timing of insulin release affects liver metabolism is not well understood. A. G. Adams, M. G. Roper and co-workers, including our group, built a modular microfluidic system that follows glucose consumption by human liver-derived HepG2 cells in real time (Anal. Chem. 91, 5184 (2019)).

• Bioreactor. A microfluidic bioreactor holds about 10⁶ HepG2 cells, which can be cultured in it for up to 10 days.
• Assay. For measurements, the bioreactor outflow is mixed with the reagents of an enzymatic glucose assay and split into droplets whose fluorescence reports the glucose concentration, over 0–12 mM and with a time resolution of about 96 s.
• Result. The system detected rapid, insulin-induced changes in glucose consumption.

The same format could be extended to other liver metabolites, such as lactate, pyruvate or ketone bodies, to relate the dynamics of pancreatic hormones to liver metabolism, a question relevant to diabetes research.

Related topics: Microfluidics, Droplet-based microfluidics, Hep G2, Insulin, Carbohydrate metabolism

Predicting how an energetic material packs

The power of an energetic material grows with its density, and density depends on how the molecules pack in the crystal. Earlier calculations had suggested that di-tetrazine-tetroxide (DTTO), a molecule not yet synthesized, could be unusually dense and release a great deal of energy. With William A. Goddard III's group at Caltech we predicted how it would crystallize (J. Comput. Chem. 2015).

• Isomers. Density functional theory compared 20 possible isomers of DTTO and singled out the two most stable.
• Packing search. For each of the two, Monte Carlo sampling with the Universal and Dreiding force fields searched the ten space groups most common for organic crystals, and the best packings were then optimized with DFT.
• Result. One isomer is predicted to crystallize in space group P2₁2₁2₁ with a density of 1.96 g/cm³, the other in Pbca with 1.98 g/cm³, among the highest of current energetic materials (RDX: 1.81 g/cm³; CL-20: 2.01 g/cm³). Across the predicted polymorphs, stability increases with density while the planarity of the molecules decreases.

Related topics: Crystal structure prediction, Crystal polymorphism, Explosive, Space group

Moving parts in rigid crystals: amphidynamic covalent organic frameworks

Amphidynamic crystals combine a rigid, ordered lattice with parts that move rapidly, a combination that is hard to design because flexible groups usually disrupt crystallization. In a 2018 collaboration with the groups of F. J. Uribe-Romo and J. K. Harper (University of Central Florida), covalent organic frameworks (COFs) were decorated with oligo(ethylene oxide) side chains of different lengths (Chem. Commun. 2018).

• Preserved crystallinity. Because the frameworks are open, the side chains do not interfere with crystallization, and the materials adopt the predicted crystal structure.
• Measuring motion. By varying the side-chain length and using solid-state ¹³C NMR relaxation, the team distinguished the dynamics of the framework from those of the side chains.
• Atomistic picture. Our calculations helped explain the motion of the individual atoms.

The work shows that complex, designed internal motion can be built into crystalline organic frameworks.

Related topics: Covalent organic framework, Molecular machine, Solid-state nuclear magnetic resonance, Polyethylene glycol

Polymorphism in YbCuBi: how the shape of a layer controls heat and elasticity

Zintl compounds with honeycomb layers, such as YbCuBi, tolerate vacancies and conduct heat poorly, properties that matter for thermoelectric materials, but how their structure controls heat transport was unclear. Below about 410 K, YbCuBi changes from flat to corrugated copper–bismuth layers, so the same compound offers two structures to compare. With the groups of Alexandra Zevalkink (MSU) and Raphaël Hermann (Oak Ridge National Laboratory), inelastic neutron scattering, resonant ultrasound spectroscopy and thermal-conductivity measurements were combined with our first-principles calculations (Chem. Mater. 2026).

The study ties the corrugation of the honeycomb layers directly to the elastic and thermal-transport properties, a guide for designing other Zintl materials with tailored thermal behavior.

Related topics: Zintl phase, Crystal polymorphism, Neutron scattering, Thermal conductivity and resistivity

Can strain stabilize an unstable crystal? Lithium antiperovskites

Lithium chalcohalide antiperovskites, Li₃BA with a chalcogen B and a halogen A, are candidate solid electrolytes for lithium batteries, but some of the cubic structures are unstable. With MSU physicist Phillip Duxbury, we computed the energetics, lattice dynamics and electronic structure of these compounds and of their anion-interchanged variants, Li₃AB (arXiv 2026).

• The smaller anion goes inside. Placing the smaller anion inside the octahedron favors stability; Li₃FTe, Li₃FPo, Li₃OCl, Li₃OBr and Li₃OI are both energetically and dynamically stable.
• Strain. Li₃FS and Li₃FSe are energetically stable but dynamically unstable at equilibrium; compressing them in all three directions makes them stable.
• Band gaps. Interchanging the anions changes the band gap substantially, and a few percent of strain tunes it within the electrochemical stability window that matters for solid-state batteries.

Related topics: Antiperovskite, Solid-state electrolyte, Phonon, Strain engineering

False positives in photocatalytic nitrogen fixation

Light-driven conversion of nitrogen gas into ammonia (photocatalytic nitrogen fixation) is widely studied on metal-oxide nanoparticles, but many synthesis recipes use nitrogen-containing ligands such as oleylamine. In 2024, with the groups of K. E. Knowles and M. C. Hatzell, we showed that such ligands can masquerade as catalytic activity (Adv. Funct. Mater. 2024).

• The artefact. Spinel ferrite nanocrystals (CoFe₂O₄, NiFe₂O₄ and ZnFe₂O₄) made with oleylamine produced 1–2 ppm of ammonia within 4 h, mostly before illumination, and the contamination survived standard washing. Such contamination can exceed genuine photocatalytic activity by more than two orders of magnitude.
• Controls. Comparing nitrogen-free nanocrystals under N₂ and argon, only NiFe₂O₄ gave a small, statistically significant light-induced signal.
• Theory. Hybrid density functional theory places the conduction-band edges of CoFe₂O₄ and ZnFe₂O₄ surfaces below the level needed to reduce N₂, ruling them out, whereas one spin channel of NiFe₂O₄ lies close to it, consistent with its small signal.
• Recommendations. Use nitrogen-free syntheses, report synthesis and washing steps in full, and run catalyst-free, dark and argon controls; the same caution applies to related reactions such as photocatalytic urea synthesis.

Related topics: Nitrogen fixation, Photocatalysis, Oleylamine, Spinel group

Catalysts for negative-emissions science

Negative-emissions technologies, which remove CO₂ from the atmosphere, are considered critical for limiting global warming, and electrocatalytic or photocatalytic CO₂ reduction is expected to play an important role in them. Their wider use requires efficient, selective catalysts, yet rational design is difficult: the reactions are complex, many structural variables affect behaviour at heterogeneous interfaces, and the range of candidate materials is vast.

In a 2022 perspective written with C. M. Hill (University of Wyoming), J. M. Velázquez (University of California, Davis) and L. Whittaker-Brooks (University of Utah), we examined the disconnect between the catalytic systems that can be made and probed experimentally and those that can be modelled with appropriate rigour (iScience 2022). The article discusses how materials design and synthesis, characterization and theory can be brought together to close this gap and speed up the rational design of catalysts for CO₂ conversion and other applications of negative-emissions science.

Related topics: Carbon dioxide removal, Electrochemical reduction of carbon dioxide, Photoelectrochemical reduction of carbon dioxide, Heterogeneous catalysis

Aluminium catalysts for epoxide polymerization

Ring-opening polymerization of epoxides gives polyethers used in adhesives, elastomers and biomedical devices. Aluminium catalysts are attractive because aluminium is abundant and inexpensive, but how they control the reaction was unclear. With the groups of N. A. Lynd and R. C. Ferrier Jr., we have combined experiments with density functional theory (DFT) to establish their mechanisms.

• The Vandenberg catalyst (2018). This classical aluminium catalyst makes isotactic (stereoregular) polyethers, but the origin of its selectivity was unclear. The most likely pathway starts from a rigid bis(μ-oxo)dialuminium (BOD) resting state and passes through a mono(μ-oxo)dialuminium (MOD) transition state. Because the BOD binds both the last and the second-to-last oxygen atoms of the growing chain, inserting an epoxide of the opposite configuration loses a favourable Al–O interaction, an energy difference of about 2 kcal mol⁻¹ that explains the isoselectivity. Spectroscopy and kinetics on model BOD and MOD complexes support this picture, a synthetic BOD analogue gave isotactically enriched poly(allyl glycidyl ether), and in situ ¹H NMR gave an activation enthalpy of 21 kcal mol⁻¹ (Macromolecules 2018).
• Nitrogen–aluminium adducts (2023). Combining experiments and DFT, we determined the complete mechanism by which N–Al adduct catalysts polymerize functional epoxides, establishing these adducts as a customizable platform for controlled epoxide polymerization (Polymer Chemistry 2023).

These mechanisms guide the design of aluminium-based catalysts for polyethers with controlled structure.

Related topics: Ring-opening polymerization, Epoxide, Tacticity, Aluminium compounds

Thermal stability and optical response of a rotaxane

Rotaxanes are mechanically interlocked molecules, a ring threaded onto a dumbbell-shaped axle, and are building blocks for molecular machines. In 2023 we combined force-field simulations with ab initio molecular dynamics (atom-centred density matrix propagation) to study a prototypical rotaxane, 2R-D-2PF₆ (J. Phys. Chem. A 2023).

• Simulations. Trajectories of 1.2 ps were run at 300, 500, 700, 900, 2000 and 2500 K, and a thermal fluctuation index computed along each trajectory was used to assess stability at room temperature and above.
• Non-covalent interactions. Reduced-density-gradient analyses and Hirshfeld surfaces map the non-covalent contacts that hold the ring and axle together.
• Optical response. We computed the dipole moment, static and frequency-dependent polarizabilities and first and second hyperpolarizabilities; the linear and nonlinear optical responses are promising for nonlinear-optical materials. Time-dependent density functional theory predicts a broad ultraviolet–visible absorption band.

The broad absorption suggests that such molecules could also be used to harvest solar energy.

Related topics: Rotaxane, Molecular machine, Molecular dynamics, Nonlinear optics

f-Orbital bonding in the later actinides

How much the 5f orbitals of the later actinides take part in chemical bonding is debated, partly because few well-defined molecular examples exist. S. S. Galley, S. C. Bart and co-workers, in a collaboration that included our group, prepared a family of isostructural complexes M(DOPO^q)₃, in which DOPO^q is a redox-active dioxophenoxazine ligand, for the lanthanides Ce, Nd, Sm and Gd and the actinides Am, Bk and Cf (J. Am. Chem. Soc. 141, 2356 (2019)). The authors describe it as the first nonaqueous isostructural series that includes the later actinides.

• Experiments. All complexes were characterized by X-ray crystallography and electronic absorption spectroscopy, and the lanthanide members also by ¹H NMR spectroscopy and SQUID magnetometry.
• Bonding. Multireference CASSCF calculations show that the metal–ligand bonding is largely ionic, in contrast with earlier studies that assigned a small covalent contribution to berkelium and californium.
• Method check. CASSCF analyses of the experimental structures and of structures optimized with relativistic unrestricted DFT (UDFT-ZORA) gave the same bonding picture.

UDFT-ZORA geometries combined with CASSCF therefore offer a practical first route to the structures and electronic properties of f-elements that are hard to synthesize or characterize.

Related topics: Actinide, Lanthanide, Covalent bond, Multi-configurational self-consistent field, Berkelium, Californium

Relativistic effects on NMR shielding in heavy-metal compounds

Interpreting nuclear magnetic resonance (NMR) spectra of molecules that contain heavy atoms requires relativistic theory, because relativistic effects make large contributions to the shielding constants (σ). A. D. Zapata-Escobar, A. F. Maldonado and G. A. Aucar (Universidad Nacional del Nordeste and CONICET, Argentina), together with our group, analysed these effects in platinum, cadmium and mercury compounds (Magnetochemistry 9, 165 (2023)).

• Method. LRESC-Loc, a recently introduced form of the linear response with elimination of small components (LRESC) method, expresses each relativistic correction in localized molecular orbitals, that is, in terms of core shells, lone pairs and bonds. This separates corrections that depend on the ligands from those set by the metal's core.
• Systems. PtX₄²⁻ and PtX₆²⁻ (X = F, Cl, Br, I), and XCl₂Te₂Y₂H₆ (X = Cd, Hg; Y = N, P).
• Platinum. In the four-coordinate halides the one-body spin–orbit term is the main relativistic mechanism, although mass-velocity and Darwin terms also matter; in the six-coordinate halides the mass-velocity and Darwin terms outweigh it.
• Cadmium and mercury. Replacing nitrogen by phosphorus leaves the ligand-dependent corrections of similar size, except the one-body spin–orbit term, which changes sign; the orbital analysis traces this to the metal–tellurium bond orbitals.

Related topics: Nuclear magnetic resonance spectroscopy, Chemical shift, Relativistic quantum chemistry, Spin–orbit interaction, Localized molecular orbitals

Four-component relativistic calculations on actinide complexes

Dealing with spent nuclear fuel and legacy waste requires knowing how actinides bind to organic ligands, but for these heavy elements relativistic effects must be included, and the most rigorous methods are costly. With the group of G. A. Aucar, we compared relativistic methods for an isostructural series of plutonium, americium, berkelium and californium complexes, each with three ligands of the redox-active molecule DOPO (2,4,6,8-tetra-tert-butyl-1-oxo-1H-phenoxazin-9-olate) (Phys. Chem. Chem. Phys. 25, 5592 (2023)).

• Methods. All-electron, four-component Dirac–Hartree–Fock calculations with the DIRAC program, on molecules of up to 67 atoms and 421 electrons (6,131 basis functions), were compared with scalar and spin–orbit ZORA calculations using Hartree–Fock and hybrid density-functional theory in ADF.
• Structures. Where crystal structures were available for comparison (Pu, Am and Bk), the four-component geometries agreed most closely with experiment, while ZORA-DFT gave a reasonable first estimate at much lower cost.
• Bonding. Relativistic effects control how the 5f orbitals take part in metal–ligand bonding; for americium, spin–orbit coupling was the largest relativistic contribution to the computed bond energy.

The results support lower-cost ZORA-DFT as a first approximation for actinide complexes that are hard to study experimentally, with four-component methods reserved for refinement.

Related topics: Actinide, Relativistic quantum chemistry, Dirac equation, Radioactive waste, Berkelium, Californium

Designing chelating agents to separate actinides

Radioactive waste from Cold War-era nuclear-weapons production still contaminates sixteen sites in the United States, and cleaning it up requires separating actinides from complex mixtures. In a 2018 preprint led by A. Gannon, we combined a high-throughput structure search with relativistic density-functional theory (DFT) to design chelating ligands, molecules that grip a metal ion at several points, for thorium(IV), protactinium(IV) and uranium(IV) (arXiv:1806.09733).

• Screening. The HostDesigner program joined two catecholamide ligands with linkers drawn from a library of 8,266 candidates and ranked the resulting structures with molecular mechanics. The twelve best structures for each actinide were then optimized with relativistic DFT (the ZORA approximation, with a dispersion-corrected B3LYP functional).
• Secondary coordination sphere. The central hypothesis was that tuning interactions beyond the atoms bound to the metal can improve selectivity. Covalent linkers joining the two ligands made the complexes more stable than non-covalent ligand–ligand interactions alone, and the non-covalent interactions improved selectivity.
• Differences between actinides. Binding energies of the candidates spread over less than 11 kcal mol⁻¹ for thorium but about 45 kcal mol⁻¹ for protactinium, and one linker bound uranium almost 47 kcal mol⁻¹ more strongly than the rest.

Several of the linkers identified could also be used with other ligands.

Related topics: Actinide, Chelation, Nuclear reprocessing, Radioactive waste, Protactinium

Reversible chemistry: from molecular triangles to the first 3D covalent organic frameworks

Dynamic covalent chemistry uses reactions that can run backwards, so that a mixture of building blocks can correct its mistakes and settle into the most stable product. At the University of Illinois, with Jeffrey Moore's group, this idea gave a one-step, template-free synthesis of a rigid molecular triangle by alkyne metathesis, in 86% yield, because the product precipitates as it forms (J. Org. Chem. 2005). At UCLA, with Omar Yaghi's group, reversible boronic-acid condensation linked triangular and tetrahedral building blocks into the first three-dimensional covalent organic frameworks: crystals held together only by strong covalent bonds, stable to 400–500 °C, with very high surface areas and extremely low densities (Science 2007).

• Selectivity without a template. Under thermodynamic control the triangle forms selectively, and the same dynamic chemistry also gave a new ring made of four units, a cyclic tetramer.
• Targeted networks. The 3D frameworks were designed by combining triangular and tetrahedral building blocks so as to target two specific three-dimensional networks.
• Legacy. These frameworks became the starting point of our work on porous materials, and the paper received the AAAS Newcomb Cleveland Prize, which honors an outstanding paper published in Science.

Related topics: Dynamic covalent chemistry, Alkyne metathesis, Covalent organic framework, Newcomb Cleveland Prize