Irreversible synthesis of an ultrastrong two-dimensional polymeric material.

Polymers that extend covalently in two dimensions have attracted recent attention1,2 as a means of combining the mechanical strength and in-plane energy conduction of conventional two-dimensional (2D) materials3,4 with the low densities, synthetic processability and organic composition of their one-dimensional counterparts. Efforts so far have proven successful in forms that do not allow full realization of these properties, such as polymerization at flat interfaces5,6 or fixation of monomers in immobilized lattices7-9. Another frequently employed synthetic approach is to introduce microscopic reversibility, at the cost of bond stability, to achieve 2D crystals after extensive error correction10,11. Here we demonstrate a homogenous 2D irreversible polycondensation that results in a covalently bonded 2D polymeric material that is chemically stable and highly processable. Further processing yields highly oriented, free-standing films that have a 2D elastic modulus and yield strength of 12.7 ± 3.8 gigapascals and 488 ± 57 megapascals, respectively. This synthetic route provides opportunities for 2D materials in applications ranging from composite structures to barrier coating materials.

Fluids and Electrolytes under Confinement in Single-Digit Nanopores.

Confined fluids and electrolyte solutions in nanopores exhibit rich and surprising physics and chemistry that impact the mass transport and energy efficiency in many important natural systems and industrial applications. Existing theories often fail to predict the exotic effects observed in the narrowest of such pores, called single-digit nanopores (SDNs), which have diameters or conduit widths of less than 10 nm, and have only recently become accessible for experimental measurements. What SDNs reveal has been surprising, including a rapidly increasing number of examples such as extraordinarily fast water transport, distorted fluid-phase boundaries, strong ion-correlation and quantum effects, and dielectric anomalies that are not observed in larger pores. Exploiting these effects presents myriad opportunities in both basic and applied research that stand to impact a host of new technologies at the water-energy nexus, from new membranes for precise separations and water purification to new gas permeable materials for water electrolyzers and energy-storage devices. SDNs also present unique opportunities to achieve ultrasensitive and selective chemical sensing at the single-ion and single-molecule limit. In this review article, we summarize the progress on nanofluidics of SDNs, with a focus on the confinement effects that arise in these extremely narrow nanopores. The recent development of precision model systems, transformative experimental tools, and multiscale theories that have played enabling roles in advancing this frontier are reviewed. We also identify new knowledge gaps in our understanding of nanofluidic transport and provide an outlook for the future challenges and opportunities at this rapidly advancing frontier.

Metal-Organic Framework Stability in Water and Harsh Environments from Data-Driven Models Trained on the Diverse WS24 Data Set.

Metal-organic frameworks (MOFs) are porous materials with applications in gas separations and catalysis, but a lack of water stability often limits their practical use given the ubiquity of water. Consequently, it is useful to predict whether a MOF is water-stable before investing time and resources into synthesis. Existing heuristics for designing water-stable MOFs lack generality and limit the diversity of explored chemistry due to narrowly defined criteria. Machine learning (ML) models offer the promise to improve the generality of predictions but require data. In an improvement on previous efforts, we enlarge the available training data for MOF water stability prediction by over 400%, adding 911 MOFs with water stability labels assigned through semiautomated manuscript analysis to curate the new data set WS24. The additional data are shown to improve ML model performance (test ROC-AUC > 0.8) over diverse chemistry for the prediction of both water stability and stability in harsher acidic conditions. We illustrate how the expanded data set and models can be used with a previously developed activation stability model in combination with genetic algorithms to quickly screen ∼10,000 MOFs from a space of hundreds of thousands for candidates with multivariate stability (upon activation, in water, and in acid). We uncover metal- and geometry-specific design rules for robust MOFs. The data set and ML models developed in this work, which we disseminate through an easy-to-use web interface, are expected to contribute toward the accelerated discovery of novel, water-stable MOFs for applications such as direct air gas capture and water treatment.

Caged AIEgens: Multicolor and White Emission Triggered by Mechanical Activation.

Aggregation-induced emission luminogens (AIEgens) that respond to mechanical force are increasingly used as force probes, memory devices, and advanced security systems. Most of the known mechanisms to modulate mechanoresponsive AIEgens have been based on changes in aggregation states, involving only physical alterations. Instances that employ covalent bond cleavage are still rare. We have developed a novel mechanochemical uncaging strategy to unveil AIEgens with diverse emission characteristics using engineered norborn-2-en-7-one (NEO) mechanophores. These NEO mechanophores were covalently integrated into polymer molecules and activated in both the solution and solid states. This activation resulted in highly tunable fluorescence upon immobilization through solidification or aggregation, producing blue, green, yellow, and orange-red emissions. By designing the caged and uncaged forms as donor-acceptor pairs for Förster resonance energy transfer (FRET), we achieved multicolor mechanofluorescence, effectively broadening the color spectrum to include white emission. Additionally, we computationally explored the electronic structures of activated NEOs, providing insights into the observed regiochemical effects of the substituents. This understanding, together with the novel luminogenic characteristics of the caged and activated species, provides a highly tunable reporter that traces progress with continuous color evolution. This advancement paves the way for future applications of mechanoresponsive materials in areas like damage detection and bioimaging.

Self-Amplified HF Release and Polymer Deconstruction Cascades Triggered by Mechanical Force.

Hydrogen fluoride (HF) is a versatile reagent for material transformation, with applications in self-immolative polymers, remodeled siloxanes, and degradable polymers. The responsive in situ generation of HF in materials therefore holds promise for new classes of adaptive material systems. Here, we report the mechanochemically coupled generation of HF from alkoxy-gem-difluorocyclopropane (gDFC) mechanophores derived from the addition of difluorocarbene to enol ethers. Production of HF involves an initial mechanochemically assisted rearrangement of gDFC mechanophore to α-fluoro allyl ether whose regiochemistry involves preferential migration of fluoride to the alkoxy-substituted carbon, and ab initio steered molecular dynamics simulations reproduce the observed selectivity and offer insights into the mechanism. When the alkoxy gDFC mechanophore is derived from poly(dihydrofuran), the α-fluoro allyl ether undergoes subsequent hydrolysis to generate 1 equiv of HF and cleave the polymer chain. The hydrolysis is accelerated via acid catalysis, leading to self-amplifying HF generation and concomitant polymer degradation. The mechanically generated HF can be used in combination with fluoride indicators to generate an optical response and to degrade polybutadiene with embedded HF-cleavable silyl ethers (11 mol %). The alkoxy-gDFC mechanophore thus provides a mechanically coupled mechanism of releasing HF for polymer remodeling pathways that complements previous thermally driven mechanisms.

A Thermally Stable SO2-Releasing Mechanophore: Facile Activation, Single-Event Spectroscopy, and Molecular Dynamic Simulations.

Polymers that release small molecules in response to mechanical force are promising candidates as next-generation on-demand delivery systems. Despite advancements in the development of mechanophores for releasing diverse payloads through careful molecular design, the availability of scaffolds capable of discharging biomedically significant cargos in substantial quantities remains scarce. In this report, we detail a nonscissile mechanophore built from an 8-thiabicyclo[3.2.1]octane 8,8-dioxide (TBO) motif that releases one equivalent of sulfur dioxide (SO2) from each repeat unit. The TBO mechanophore exhibits high thermal stability but is activated mechanochemically using solution ultrasonication in either organic solvent or aqueous media with up to 63% efficiency, equating to 206 molecules of SO2 released per 143.3 kDa chain. We quantified the mechanochemical reactivity of TBO by single-molecule force spectroscopy and resolved its single-event activation. The force-coupled rate constant for TBO opening reaches ∼9.0 s-1 at ∼1520 pN, and each reaction of a single TBO domain releases a stored length of ∼0.68 nm. We investigated the mechanism of TBO activation using ab initio steered molecular dynamic simulations and rationalized the observed stereoselectivity. These comprehensive studies of the TBO mechanophore provide a mechanically coupled mechanism of multi-SO2 release from one polymer chain, facilitating the translation of polymer mechanochemistry to potential biomedical applications.

Protein3D: Enabling analysis and extraction of metal-containing sites from the Protein Data Bank with molSimplify.

Metalloenzymes catalyze a wide range of chemical transformations, with the active site residues playing a key role in modulating chemical reactivity and selectivity. Unlike smaller synthetic catalysts, a metalloenzyme active site is embedded in a larger protein, which makes interrogation of electronic properties and geometric features with quantum mechanical calculations challenging. Here we implement the ability to fetch crystallographic structures from the Protein Data Bank and analyze the metal binding sites in the program molSimplify. We show the usefulness of the newly created protein3D class to extract the local environment around non-heme iron enzymes containing a two histidine motif and prepare 372 structures for quantum mechanical calculations. Our implementation of protein3D serves to expand the range of systems molSimplify can be used to analyze and will enable high-throughput study of metal-containing active sites in proteins.

Leveraging natural language processing to curate the tmCAT, tmPHOTO, tmBIO, and tmSCO datasets of functional transition metal complexes.

The breadth of transition metal chemical space covered by databases such as the Cambridge Structural Database and the derived computational database tmQM is not conducive to application-specific modeling and the development of structure-property relationships. Here, we employ both supervised and unsupervised natural language processing (NLP) techniques to link experimentally synthesized compounds in the tmQM database to their respective applications. Leveraging NLP models, we curate four distinct datasets: tmCAT for catalysis, tmPHOTO for photophysical activity, tmBIO for biological relevance, and tmSCO for magnetism. Analyzing the chemical substructures within each dataset reveals common chemical motifs in each of the designated applications. We then use these common chemical structures to augment our initial datasets for each application, yielding a total of 21 631 compounds in tmCAT, 4599 in tmPHOTO, 2782 in tmBIO, and 983 in tmSCO. These datasets are expected to accelerate the more targeted computational screening and development of refined structure-property relationships with machine learning.

How Do Differences in Electronic Structure Affect the Use of Vanadium Intermediates as Mimics in Nonheme Iron Hydroxylases?

We study active-site models of nonheme iron hydroxylases and their vanadium-based mimics using density functional theory to determine if vanadyl is a faithful structural mimic. We identify crucial structural and energetic differences between ferryl and vanadyl isomers owing to the differences in their ground electronic states, i.e., high spin (HS) for Fe and low spin (LS) for V. For the succinate cofactor bound to the ferryl intermediate, we predict facile interconversion between monodentate and bidentate coordination isomers for ferryl species but difficult rearrangement for vanadyl mimics. We study isomerization of the oxo intermediate between axial and equatorial positions and find the ferryl potential energy surface to be characterized by a large barrier of ca. 10 kcal/mol that is completely absent for the vanadyl mimic. This analysis reveals even starker contrasts between Fe and V in hydroxylases than those observed for this metal substitution in nonheme halogenases. Analysis of the relative bond strengths of coordinating carboxylate ligands for Fe and V reveals that all of the ligands show stronger binding to V than Fe owing to the LS ground state of V in contrast to the HS ground state of Fe, highlighting the limitations of vanadyl mimics of native nonheme iron hydroxylases.

Computational Screening of Putative Catalyst Transition Metal Complexes as Guests in a Ga4L612- Nanocage.

Metal-organic cages form well-defined microenvironments that can enhance the catalytic proficiency of encapsulated transition metal complexes (TMCs). We introduce a screening protocol to efficiently identify TMCs that are promising candidates for encapsulation in the Ga4L612- nanocage. We obtain TMCs from the Cambridge Structural Database with geometric and electronic characteristics amenable to encapsulation and mine the text of associated manuscripts to curate TMCs with documented catalytic functionality. By docking candidate TMCs inside the nanocage cavity and carrying out electronic structure calculations, we identify a subset of successfully optimized candidates (TMC-34) and observe that encapsulated guests occupy an average of 60% of the cavity volume, in line with previous observations. Notably, some guests occupy as much as 72% of the cavity as a result of linker rotation. Encapsulation has a universal effect on the electrostatic potential (ESP), systematically decreasing the ESP at the metal center of each TMC in the TMC-34 data set, while minimally altering TMC metal partial charges. Collectively these observations support geometry-based screening of potential guests and suggest that encapsulation in Ga4L612- cages could electrostatically stabilize diverse cationic or electropositive intermediates. We highlight candidate guests with associated known reactivity and solubility most amenable for encapsulation in experimental follow-up studies.

Improving Predictions of Spin-Crossover Complex Properties through DFT Calculations with a Local Hybrid Functional.

We conducted a study on the performance of the local hybrid exchange-correlation functional PBE0r for a set of 95 experimentally characterized iron spin-crossover (SCO) complexes [Vennelakanti, V.; J. Chem. Phys. 2023, 159, 024120]. The PBE0r functional is a variant of PBE0 where the exchange correction is restricted to on-site terms formulated on the basis of local orbitals. We determine the free parameters of the PBE0r functional against the experimental data and other hybrid functionals. With a Hartree-Fock (HF) exchange factor of 4%, the PBE0r functional accurately reproduces the electronic and free-energy trends predicted in prior DFT studies for these 95 complexes by using the B3LYP functional. Larger values of HF exchange stabilize high-spin states. The PBE0r-predicted bond lengths tend to exceed the experimental bond lengths, although bond lengths are less sensitive to HF exchange than in global hybrids. The predicted SCO transition temperatures T1/2 from PBE0r correlate moderately with the experimental transition temperatures, showing a slight improvement compared to the previous modB3LYP-predicted T1/2. This study suggests that the PBE0r functional is computationally cost-effective and offers the possibility of simulating larger complexes with accuracy comparable to global hybrid functionals, provided the HF-exchange parameter is carefully optimized.

Machine Learning Prediction of the Experimental Transition Temperature of Fe(II) Spin-Crossover Complexes.

Spin-crossover (SCO) complexes are materials that exhibit changes in the spin state in response to external stimuli, with potential applications in molecular electronics. It is challenging to know a priori how to design ligands to achieve the delicate balance of entropic and enthalpic contributions needed to tailor a transition temperature close to room temperature. We leverage the SCO complexes from the previously curated SCO-95 data set [Vennelakanti et al. J. Chem. Phys. 159, 024120 (2023)] to train three machine learning (ML) models for transition temperature (T1/2) prediction using graph-based revised autocorrelations as features. We perform feature selection using random forest-ranked recursive feature addition (RF-RFA) to identify the features essential to model transferability. Of the ML models considered, the full feature set RF and recursive feature addition RF models perform best, achieving moderate correlation to experimental T1/2 values. We then compare ML T1/2 predictions to those from three previously identified best-performing density functional approximations (DFAs) which accurately predict SCO behavior across SCO-95, finding that the ML models predict T1/2 more accurately than the best-performing DFAs. In addition, we study ML model predictions for a set of 18 SCO complexes for which only estimated T1/2 values are available. Upon excluding outliers from this set, the RF-RFA RF model shows a strong correlation to estimated T1/2 values with a Pearson's r of 0.82. In contrast, DFA-predicted T1/2 values have large errors and show no correlation to estimated T1/2 values over the same set of complexes. Overall, our study demonstrates slightly superior performance of ML models in comparison with some of the best-performing DFAs, and we expect ML models to improve further as larger data sets of SCO complexes are curated and become available for model training.

Improving gas adsorption modeling for MOFs by local calibration of Hubbard U parameters.

While computational screening with density functional theory (DFT) is frequently employed for the screening of metal-organic frameworks (MOFs) for gas separation and storage, commonly applied generalized gradient approximations (GGAs) exhibit self-interaction errors, which hinder the predictions of adsorption energies. We investigate the Hubbard U parameter to augment DFT calculations for full periodic MOFs, targeting a more precise modeling of gas molecule-MOF interactions, specifically for N2, CO2, and O2. We introduce a calibration scheme for the U parameter, which is tailored for each MOF, by leveraging higher-level calculations on the secondary building unit (SBU) of the MOF. When applied to the full periodic MOF, the U parameter calibrated against hybrid HSE06 calculations of SBUs successfully reproduces hybrid-quality calculations of the adsorption energy of the periodic MOF. The mean absolute deviation of adsorption energies reduces from 0.13 eV for a standard GGA treatment to 0.06 eV with the calibrated U, demonstrating the utility of the calibration procedure when applied to the full MOF structure. Furthermore, attempting to use coupled cluster singles and doubles with perturbative triples calculations of isolated SBUs for this calibration procedure shows varying degrees of success in predicting the experimental heat of adsorption. It improves accuracy for N2 adsorption for cases of overbinding, whereas its impact on CO2 is minimal, and ambiguities in spin state assignment hinder consistent improvements of O2 adsorption. Our findings emphasize the limitations of cluster models and advocate the use of full periodic MOF systems with a calibrated U parameter, providing a more comprehensive understanding of gas adsorption in MOFs.

A Semi-Automated, High-Throughput Approach for the Synthesis and Identification of Highly Photo-Cytotoxic Iridium Complexes.

The discovery of new compounds with pharmacological properties is usually a lengthy, laborious and expensive process. Thus, there is increasing interest in developing workflows that allow for the rapid synthesis and evaluation of libraries of compounds with the aim of identifying leads for further drug development. Herein, we apply combinatorial synthesis to build a library of 90 iridium(III) complexes (81 of which are new) over two synthesise-and-test cycles, with the aim of identifying potential agents for photodynamic therapy. We demonstrate the power of this approach by identifying highly active complexes that are well-tolerated in the dark but display very low nM phototoxicity against cancer cells. To build a detailed structure-activity relationship for this class of compounds we have used density functional theory (DFT) calculations to determine some key electronic parameters and study correlations with the experimental data. Finally, we present an optimised semi-automated synthesise-and-test protocol to obtain multiplex data within 72 hours.

Computational Discovery of Stable Metal-Organic Frameworks for Methane-to-Methanol Catalysis.

The challenge of direct partial oxidation of methane to methanol has motivated the targeted search of metal-organic frameworks (MOFs) as a promising class of materials for this transformation because of their site-isolated metals with tunable ligand environments. Thousands of MOFs have been synthesized, yet relatively few have been screened for their promise in methane conversion. We developed a high-throughput virtual screening workflow that identifies MOFs from a diverse space of experimental MOFs that have not been studied for catalysis, yet are thermally stable, synthesizable, and have promising unsaturated metal sites for C-H activation via a terminal metal-oxo species. We carried out density functional theory calculations of the radical rebound mechanism for methane-to-methanol conversion on models of the secondary building units (SBUs) from 87 selected MOFs. While we showed that oxo formation favorability decreases with increasing 3d filling, consistent with prior work, previously observed scaling relations between oxo formation and hydrogen atom transfer (HAT) are disrupted by the greater diversity in our MOF set. Accordingly, we focused on Mn MOFs, which favor oxo intermediates without disfavoring HAT or leading to high methanol release energies─a key feature for methane hydroxylation activity. We identified three Mn MOFs comprising unsaturated Mn centers bound to weak-field carboxylate ligands in planar or bent geometries with promising methane-to-methanol kinetics and thermodynamics. The energetic spans of these MOFs are indicative of promising turnover frequencies for methane to methanol that warrant further experimental catalytic studies.

Reversible O-O Bond Scission and O2 Evolution at MOF-Supported Tetramanganese Clusters.

The scission of the O-O bond in O2 during respiration and the formation of the O-O bond during photosynthesis are the engines of aerobic life. Likewise, the reduction of O2 and the oxidation of reduced oxygen species to form O2 are indispensable components for emerging renewable technologies, including energy storage and conversion, yet discrete molecule-like systems that promote these fundamental reactions are rare. Herein, we report a square-planar tetramanganese cluster formed by self-assembly within a metal-organic framework that reversibly reduces O2 by four electrons, facilitating the interconversion between molecular O2 and metal-oxo species. The tetranuclear cluster spontaneously cleaves the O-O bond of O2 at room temperature to generate a tetramanganese-bis(µ2-oxo) species, which, in turn, is competent for O-O bond reformation and O2 evolution at elevated temperatures, enabled by the head-to-head orientation of two oxo species. This study demonstrates the viability of four-electron interconversion between molecular O2 and metal-oxo species and highlights the importance of site isolation for achieving multi-electron chemistry at polynuclear metal clusters.

1D Hybrid Semiconductor Silver 2,6-Difluorophenylselenolate.

Organic-inorganic hybrid materials present new opportunities for creating low-dimensional structures with unique light-matter interaction. In this work, we report a chemically robust yellow emissive one-dimensional (1D) semiconductor, silver 2,6-difluorophenylselenolate─AgSePhF2(2,6), a new member of the broader class of hybrid low-dimensional semiconductors, metal-organic chalcogenolates. While silver phenylselenolate (AgSePh) crystallizes as a two-dimensional (2D) van der Waals semiconductor, introduction of fluorine atoms at the (2,6) position of the phenyl ring induces a structural transition from 2D sheets to 1D chains. Density functional theory calculations reveal that AgSePhF2 (2,6) has strongly dispersive conduction and valence bands along the 1D crystal axis. Visible photoluminescence centered around λp ≈ 570 nm at room temperature exhibits both prompt (110 ps) and delayed (36 ns) components. The absorption spectrum exhibits excitonic resonances characteristic of low-dimensional hybrid semiconductors, with an exciton binding energy of approximately 170 meV as determined by temperature-dependent photoluminescence. The discovery of an emissive 1D silver organoselenolate highlights the structural and compositional richness of the chalcogenolate material family and provides new insights for molecular engineering of low-dimensional hybrid organic-inorganic semiconductors.

Polymer Networks with Cubic, Mixed Pd(II) and Pt(II) M6L12 Metal-Organic Cage Junctions: Synthesis and Stress Relaxation Behavior.

Metal-organic cages/polyhedra (MOCs) are versatile building blocks for advanced polymer networks with properties that synergistically blend those of traditional polymers and crystalline frameworks. Nevertheless, constructing polyMOCs from very stable Pt(II)-based MOCs or mixtures of metal ions such as Pd(II) and Pt(II) has not, to our knowledge, been demonstrated, nor has exploration of how the dynamics of metal-ligand exchange at the MOC level may impact bulk polyMOC energy dissipation. Here, we introduce a new class of polymer metal-organic cage (polyMOC) gels featuring polyethylene glycol (PEG) strands of varied length cross-linked through bis-pyridyl-carbazole-based M6L12 cubes, where M is Pd(II), Pt(II), or mixtures thereof. We show that, while polyMOCs with varied Pd(II) content have similar network structures, their average stress-relaxation rates are tunable over 3 orders of magnitude due to differences in Pd(II)- and Pt(II)-ligand exchange rates at the M6L12 junction level. Moreover, mixed-metal polyMOCs display relaxation times indicative of intrajunction cooperative interactions, which stands in contrast to previous materials based on point metal junctions. Altogether, this work (1) introduces a novel MOC architecture for polyMOC design, (2) shows that polyMOCs can be prepared from mixtures of Pd(II)/Pt(II), and (3) demonstrates that polyMOCs display unique relaxation behavior due to their multivalent junctions, offering a strategy for controlling polyMOC properties independently of their polymer components.

Remolding and Deconstruction of Industrial Thermosets via Carboxylic Acid-Catalyzed Bifunctional Silyl Ether Exchange.

Convenient strategies for the deconstruction and reprocessing of thermosets could improve the circularity of these materials, but most approaches developed to date do not involve established, high-performance engineering materials. Here, we show that bifunctional silyl ether, i.e., R'O-SiR2-OR'', (BSE)-based comonomers generate covalent adaptable network analogues of the industrial thermoset polydicyclopentadiene (pDCPD) through a novel BSE exchange process facilitated by the low-cost food-safe catalyst octanoic acid. Experimental studies and density functional theory calculations suggest an exchange mechanism involving silyl ester intermediates with formation rates that strongly depend on the Si-R2 substituents. As a result, pDCPD thermosets manufactured with BSE comonomers display temperature- and time-dependent stress relaxation as a function of their substituents. Moreover, bulk remolding of pDCPD thermosets is enabled for the first time. Altogether, this work presents a new approach toward the installation of exchangeable bonds into commercial thermosets and establishes acid-catalyzed BSE exchange as a versatile addition to the toolbox of dynamic covalent chemistry.

Synthesis and Ring-Opening Metathesis Polymerization of a Strained trans-Silacycloheptene and Single-Molecule Mechanics of Its Polymer.

The cis- and trans-isomers of a silacycloheptene were selectively synthesized by the alkylation of a silyl dianion, a novel approach to strained cycloalkenes. The trans-silacycloheptene (trans-SiCH) was significantly more strained than the cis isomer, as predicted by quantum chemical calculations and confirmed by crystallographic signatures of a twisted alkene. Each isomer exhibited distinct reactivity toward ring-opening metathesis polymerization (ROMP), where only trans-SiCH afforded high-molar-mass polymer under enthalpy-driven ROMP. Hypothesizing that the introduction of silicon might result in increased molecular compliance at large extensions, we compared poly(trans-SiCH) to organic polymers by single-molecule force spectroscopy (SMFS). Force-extension curves from SMFS showed that poly(trans-SiCH) is more easily overstretched than two carbon-based analogues, polycyclooctene and polybutadiene, with stretching constants that agree well with the results of computational simulations.