Ionic Covalent Organic Framework as a Dual Functional Sensor for Temperature and Humidity.

Visual sensing of humidity and temperature by solids plays an important role in the everyday life and in industrial processes. Due to their hydrophobic nature, most covalent organic framework (COF) sensors often exhibit poor optical response when exposed to moisture. To overcome this challenge, the optical response is set out to improve, to moisture by incorporating H-bonding ionic functionalities into the COF network. A highly sensitive COF, consisting of guanidinium and diformylpyridine linkers (TG-DFP), capable of detecting changes in temperature and moisture content is fabricated. The hydrophilic nature of the framework enables enhanced water uptake, allowing the trapped water molecules to form a large number of hydrogen bonds. Despite the presence of non-emissive building blocks, the H-bonds restrict internal bond rotation within the COF, leading to reversible fluorescence and solid-state optical hydrochromism in response to relative humidity and temperature.

Reactions of N2O and CO on neutral Rh10On clusters: a density functional study.

Density functional theory calculations were performed to identify product, reactant and intermediate dissociative/associative structures for the oxygen abstraction and addition reactions: Rh10On + CO → Rh10On-1 + CO2, n = 1-5 and Rh10On + N2O → Rh10On+1 + N2, n = 0-4 reactions. In the case of the oxygen abstraction reactions, the energetics of the reaction path were very similar in energy regardless of the number of oxygen atoms on the Rh10On cluster, whereas for the addition of oxygen to the Rh10On cluster, the reaction was found to become significantly less exothermic with each successive addition of oxygen.

Time-resolved keto-enol tautomerization of the medicinal pigment curcumin.

Curcumin is a medicinal agent that exhibits anti-cancer and anti-Alzheimer's disease properties. It has a keto-enol moiety that gives rise to many of its chemical properties including metal complexation and acid-base equilibria. A previous study has shown that keto-enol tautomerization at this moiety is implicated in the anti-Alzheimer's disease effect of curcumin, highlighting the importance of this process. In this study, tautomerization of curcumin in methanol, acetone and acetonitrile was investigated using time-resolved 1H nuclear magnetic resonance spectroscopy. Curcumin undergoes hydrogen-deuterium exchange with the solvents and the proton resonance peak corresponding to the hydrogen at the α-carbon position (Cα) decays as a function of time, signifying deuteration at this position. Because tautomerization is the rate limiting step in the deuteration of curcumin at the Cα position, the rate of tautomerization is inferred from the rate of deuteration. The rate constant of tautomerization of curcumin shows a temperature dependence and analysis using the Arrhenius equation revealed activation energies (Ea) of tautomerization of (80.1 ± 5.9), (64.1 ± 1.0) and (68.3 ± 5.5) kJ mol-1 in methanol, D2O/acetone and D2O/acetonitrile, respectively. Insight into the role of water in tautomerization of curcumin was further offered by density functional theory studies. The transition state of tautomerization was optimized in the presence of water molecules. The results show a hydrogen-bonded solvent bridge between the diketo moiety and Cα of curcumin. The Ea of tautomerization of curcumin shows a strong dependence on the number of water molecules in the solvent bridge, indicating the critical role played by the solvent bridge in catalyzing tautomerization of curcumin.

Enhancing the Crystallinity of Keto-enamine-Linked Covalent Organic Frameworks through an in situ Protection-Deprotection Strategy.

ß-Keto-enamine-linked 2D covalent organic frameworks (COFs) have emerged as highly robust materials, showing significant potential for practical applications. However, the exclusive reliance on 1,3,5-triformylphloroglucinol (Tp aldehyde) in the design of such COFs often results in the production of non-porous amorphous polymers when combined with certain amine building blocks. Attempts to adjust the crystallinity and porosity by a modulator approach are inefficient because Tp aldehyde readily forms stable ß-keto-enamine-linked monomers/oligomers with various aromatic amines through an irreversible keto-enol tautomerization process. Our research employed a unique protection-deprotection strategy to enhance the crystallinity and porosity of ß-keto-enamine-linked squaramide-based 2D COFs. Advanced solid-state NMR studies, including 1D 13 C CPMAS, 1 H fast MAS, 15 N CPMAS, 2D 13 C-1 H correlation, 1 H-1 H DQ-SQ, and 14 N-1 H HMQC NMR were used to establish the atomic-level connectivity within the resultant COFs. The TpOMe -Sqm COFs synthesized utilizing this strategy have a surface area of 487 m2 g-1 , significantly higher than similar COFs synthesized using Tp aldehyde. Furthermore, detailed time-dependent PXRD, solid-state 13 C CPMAS NMR, and theoretical DFT studies shed more light on the crystallization and linkage conversion processes in these 2D COFs. Ultimately, we applied this protection-deprotection method to construct novel keto-enamine-linked highly porous organic polymers with a surface area of 1018 m2 g-1 .

Photoresponsive Covalent Organic Frameworks: Visible-Light Controlled Conversion of Porous Structures and Its Impacts.

Covalent organic frameworks are a novel class of crystalline porous polymers that enable molecular design of extended polygonal skeletons to attain well-defined porous structures. However, construction of a framework that allows remote control of pores remains a challenge. Here we report a strategy that merges covalent, noncovalent, and photo chemistries to design photoresponsive frameworks with reversibly and remotely controllable pores. We developed a topology-guided multicomponent polycondensation system that integrates protruded tetrafluoroazobenzene units as photoresponsive sites on pore walls at predesigned densities, so that a series of crystalline porous frameworks with the same backbone can be constructed to develop a broad spectrum of pores ranging from mesopores to micropores. Distinct from conventional azobenzene-based systems, the tetrafluoroazobenzene frameworks are highly sensitive to visible lights to undergo high-rate isomerization. The photoisomerization exerts profound effects on pore size, shape, number, and environment, as well as molecular uptake and release, rendering the system able to convert and switch pores reversibly and remotely with visible lights. Our results open a way to a novel class of smart porous materials with pore structures and functions that are convertible and manageable with visible lights.

Saturated Linkers in Two-Dimensional Covalent Organic Frameworks Boost Their Luminescence.

The development of highly luminescent two-dimensional covalent organic frameworks (COFs) for sensing applications remains challenging. To suppress commonly observed photoluminescence quenching of COFs, we propose a strategy involving interrupting the intralayer conjugation and interlayer interactions using cyclohexane as the linker unit. By variation of the building block structures, imine-bonded COFs with various topologies and porosities are obtained. Experimental and theoretical analyses of these COFs disclose high crystallinity and large interlayer distances, demonstrating enhanced emission with record-high photoluminescence quantum yields of up to 57% in the solid state. The resulting cyclohexane-linked COF also exhibits excellent sensing performance for the trace recognition of Fe3+ ions, explosive and toxic picric acid, and phenyl glyoxylic acid as metabolites. These findings inspire a facile and general strategy to develop highly emissive imine-bonded COFs for detecting various molecules.

Enhancing Dynamic Spectral Diffusion in Metal-Organic Frameworks through Defect Engineering.

The crystal packing of organic chromophores has a profound impact on their photophysical properties. Molecular crystal engineering is generally incapable of producing precisely spaced arrays of molecules for use in photovoltaics, light-emitting diodes, and sensors. A promising alternative strategy is the incorporation of chromophores into crystalline metal-organic frameworks (MOFs), leading to matrix coordination-induced emission (MCIE) upon confinement. However, it remains unclear how the precise arrangement of chromophores and defects dictates photophysical properties in these systems, limiting the rational design of well-defined photoluminescent materials. Herein, we report new, robust Zr-based MOFs constructed from the linker tetrakis(4-carboxyphenyl)ethylene (TCPE4-) that exhibit an unexpected structural transition in combination with a prominent shift from green to blue photoluminescence (PL) as a function of the amount of acid modulator (benzoic, formic, or acetic acid) used during synthesis. Time-resolved PL (TRPL) measurements provide full spectral information and reveal that the observed hypsochromic shift arises due to a higher concentration of linker substitution defects at higher modulator concentrations, leading to broader excitation transfer-induced spectral diffusion. Spectral diffusion of this type has not been reported in a MOF to date, and its observation provides structural information that is otherwise unobtainable using traditional crystallographic techniques. Our findings suggest that defects have a profound impact on the photophysical properties of MOFs and that their presence can be readily tuned to modify energy transfer processes within these materials.

Covalent Organic Framework as a Metal-Free Photocatalyst for Dye Degradation and Radioactive Iodine Adsorption.

Exploring a covalent organic framework (COF) material as an efficient metal-free photocatalyst and as an adsorbent for the removal of pollutants from contaminated water is very challenging in the context of sustainable chemistry. Herein, we report a new porous crystalline COF, C6-TRZ-TPA COF, via segregation of donor-acceptor moieties through the extended Schiff base condensation between tris(4-formylphenyl)amine and 4,4',4″-(1,3,5-triazine-2,4,6-triyl)trianiline. This COF displayed a Brunauer-Emmett-Teller (BET) surface area of 1058 m2 g-1 with a pore volume of 0.73 cc g-1. Again, extended π-conjugation, the presence of heteroatoms throughout the framework, and a narrow band gap of 2.2 eV, all these features collectively work for the environmental remediation in two different perspectives: it could harness solar energy for environmental clean-up, where the COF has been explored as a robust metal-free photocatalyst for wastewater treatment and as an adsorbent for iodine capture. In our endeavor of wastewater treatment, we have conducted the photodegradation of rose bengal (RB) and methylene blue (MB) as model pollutants since these are extremely toxic, are health hazard, and bioaccumulative in nature. The catalyst C6-TRZ-TPA COF showed a very high catalytic efficiency of 99% towards the degradation of 250 parts per million (ppm) of RB solution in 80 min under visible light irradiation with the rate constant of 0.05 min-1. Further, C6-TRZ-TPA COF is found to be an excellent adsorbent as it efficiently adsorbed radioactive iodine from its solution as well as from the vapor phase. The material exhibits a very rapid iodine capturing tendency with an outstanding iodine vapor uptake capacity of 4832 mg g-1.

Emissive Substoichiometric Covalent Organic Frameworks for Water Sensing and Harvesting.

Emissive covalent organic frameworks (COFs) have recently emerged as next-generation porous materials with attractive properties such as tunable topology, porosity, and inherent photoluminescence. Among the different types of COFs, substoichiometric frameworks (so-called Type III COFs) are especially attractive due to the possibility of not only generating unusual topology and complex pore architectures but also facilitating the introduction of well-defined functional groups at precise locations for desired functions. Herein, the first example of a highly emissive (PLQY 6.8%) substoichiometric 2D-COF (COF-SMU-1) featuring free uncondensed aldehyde groups is reported. In particular, COF-SMU-1 features a dual-pore architecture with an overall bex net topology, tunable emission in various organic solvents, and distinct colorimetric changes in the presence of water. To gain further insights into its photoluminescence properties, the charge transfer, excimer emission, and excited state exciton dynamics of COF-SMU-1 are investigated using femtosecond transient absorption spectroscopy in different organic solvents. Additionally, highly enhanced atmospheric water-harvesting properties of COF-SMU-1 are revealed using FT-IR and water sorption studies.The findings will not only lead to in-depth understanding of structure-property relationships in emissive COFs but also open new opportunities for designing COFs for potential applications in solid-state lighting and water harvesting.

Chemically Gradient Hydrogen-Bonded Organic Framework Crystal Film.

Hydrogen-bonded organic frameworks (HOFs) are ordered supramolecular solid structures, however, nothing much explored as centimetre-scale self-standing films. The fabrication of such crystals comprising self-supported films is challenging due to the limited flexibility and interaction of the crystals, and therefore studies on two-dimensional macrostructures of HOFs are limited to external supports. Herein, we introduce a novel chemical gradient strategy to fabricate a crystal-deposited HOF film on an in situ-formed covalent organic polymer film (Tam-Bdca-CGHOF). The fabricated film showed versatility in chemical bonding along its thickness from covalent to hydrogen-bonded network. The kinetic-controlled Tam-Bdca-CGHOF showed enhanced proton conductivity (8.3×10-5  S cm-1 ) compared to its rapid kinetic analogue, Tam-Bdca-COP (2.1×10-5  S cm-1 ), which signifies the advantage of bonding-engineering in the same system.

Dual Metalation in a Two-Dimensional Covalent Organic Framework for Photocatalytic C-N Cross-Coupling Reactions.

Covalent organic frameworks (COFs) are promising hosts in heterogeneous catalysis. Herein, we report a dual metalation strategy in a single two-dimensional-COF TpBpy for performing a variety of C-N cross-coupling reactions. [Ir(ppy)2(CH3CN)2]PF6 [ppy = 2-phenylpyridine], containing two labile CH3CN groups, and NiCl2 are used as iridium and nickel-metal precursors, respectively, for postsynthetic decoration of the TpBpy COF. Moving from the traditional approach, we focus on the COF-backbone host for visible-light-mediated nickel-catalyzed C-N coupling reactions. The controlled metalation and recyclability without deactivation of both catalytic centers are unique with respect to previously reported coupling strategies. We performed various photoluminescence, electrochemical, kinetic, and Hammett correlation studies to understand the salient features of the catalyst and reaction mechanism. Furthermore, theoretical calculations delineated the feasibility of electron transfer from the Ir center to the Ni center inside the confined pore of the TpBpy COF. The dual metal anchoring within the COF backbone prevented nickel-black formation. The developed protocol enables selective and reproducible coupling of a diverse range of amines (aryl, heteroaryl, and alkyl), carbamides, and sulfonamides with electron-rich, neutral, and poor (hetero) aryl iodides up to 94% isolated yield. The reaction can also be performed on a gram scale. Furthermore, to establish the practical implementation of this approach, we have applied the synthetic strategy for the late-stage diversification of the derivatives of ibuprofen, naproxen, gemfibrozil, helional, and amino acids. The methodology could also be applied to synthesize pharmacophore N,5-diphenyloxazol-2-amine and Food and Drug Administration-approved drugs, including flufenamic acid, flibanserin, and tripelennamine.

A Covalent Organic Framework for Cooperative Water Oxidation.

The future of water-derived hydrogen as the "sustainable energy source" straightaway bets on the success of the sluggish oxygen-generating half-reaction. The endeavor to emulate the natural photosystem II for efficient water oxidation has been extended across the spectrum of organic and inorganic combinations. However, the achievement has so far been restricted to homogeneous catalysts rather than their pristine heterogeneous forms. The poor structural understanding and control over the mechanistic pathway often impede the overall development. Herein, we have synthesized a highly crystalline covalent organic framework (COF) for chemical and photochemical water oxidation. The interpenetrated structure assures the catalyst stability, as the catalyst's performance remains unaltered after several cycles. This COF exhibits the highest ever accomplished catalytic activity for such an organometallic crystalline solid-state material where the rate of oxygen evolution is as high as ∼26,000 µmol L-1 s-1 (second-order rate constant k ≈ 1650 µmol L s-1 g-2). The catalyst also proves its exceptional activity (k ≈ 1600 µmol L s-1 g-2) during light-driven water oxidation under very dilute conditions. The cooperative interaction between metal centers in the crystalline network offers 20-30-fold superior activity during chemical as well as photocatalytic water oxidation as compared to its amorphous polymeric counterpart.

Outstanding Charge Mobility by Band Transport in Two-Dimensional Semiconducting Covalent Organic Frameworks.

Two-dimensional covalent organic frameworks (2D COFs) represent a family of crystalline porous polymers with a long-range order and well-defined open nanochannels that hold great promise for electronics, catalysis, sensing, and energy storage. To date, the development of highly conductive 2D COFs has remained challenging due to the finite π-conjugation along the 2D lattice and charge localization at grain boundaries. Furthermore, the charge transport mechanism within the crystalline framework remains elusive. Here, time- and frequency-resolved terahertz spectroscopy reveals intrinsically Drude-type band transport of charge carriers in semiconducting 2D COF thin films condensed by 1,3,5-tris(4-aminophenyl)benzene (TPB) and 1,3,5-triformylbenzene (TFB). The TPB-TFB COF thin films demonstrate high photoconductivity with a long charge scattering time exceeding 70 fs at room temperature which resembles crystalline inorganic materials. This corresponds to a record charge carrier mobility of 165 ± 10 cm2 V-1 s-1, vastly outperforming that of the state-of-the-art conductive COFs. These results reveal TPB-TFB COF thin films as promising candidates for organic electronics and catalysis and provide insights into the rational design of highly crystalline porous materials for efficient and long-range charge transport.

Control of Crystallinity of Vinylene-Linked Two-Dimensional Conjugated Polymers by Rational Monomer Design.

The interest in two-dimensional conjugated polymers (2D CPs) has increased significantly in recent years. In particular, vinylene-linked 2D CPs with fully in-plane sp2 -carbon-conjugated structures, high thermal and chemical stability, have become the focus of attention. Although the Horner-Wadsworth-Emmons (HWE) reaction has been recently demonstrated in synthesizing vinylene-linked 2D CPs, it remains largely unexplored due to the challenge in synthesis. In this work, we reveal the control of crystallinity of 2D CPs during the solvothermal synthesis of 2D-poly(phenylene-quinoxaline-vinylene)s (2D-PPQVs) and 2D-poly(phenylene-vinylene)s through the HWE polycondensation. The employment of fluorinated phosphonates and rigid aldehyde building blocks is demonstrated as crucial factors in enhancing the crystallinity of the obtained 2D CPs. Density functional theory (DFT) calculations reveal the critical role of the fluorinated phosphonate in enhancing the reversibility of the (semi)reversible C-C single bond formation.

Performance of GFN1-xTB for periodic optimization of metal organic frameworks.

Tight-binding approaches bridge the gap between force field methods and Density Functional Theory (DFT). Density Functional Tight Binding (DFTB) has been employed for a wide range of systems including proteins, clays and 2D and 3D materials. DFTB is 2-3 orders of magnitude faster than DFT, allowing calculations containing up to ca. 5000 atoms. The efficiency of DFTB comes via pre-computed integrals, which are parameterized for each pair of atoms, and the requirement for this parameterization has previously prevented widespread use of DFTB for Metal-Organic Frameworks. The GFN-xTB (Geometries, Frequencies, and Non-covalent interactions Tight Binding) method provides parameters for elements up to Z ≤ 86. We have therefore employed GFN-xTB to periodic optimizations of the Computation Ready Experimental (CoRE) database of MOF structures. We find that 75% of all cell parameters remain within 5% of the reference (experimental) value and that bonds containing metal atoms are typically well conserved with a mean average deviation of 0.187 Å. Therefore GFN-xTB provides the ability to calculate MOF structures more accurately than force fields, and ca. 2 orders of magnitude faster than DFT. We therefore propose that GFN-xTB is a suitable method for screening of hypothetical MOFs (Z ≤ 86), with the advantage of accurate binding energies for adsorption applications.

Effect of unwanted guest molecules on the stacking configuration of covalent organic frameworks: a periodic energy decomposition analysis.

Elucidating the precise stacking configuration of a covalent organic framework, COF, is critical to fully understand their various applications. Unfortunately, most COFs form powder crystals whose atomic characterisations are possible only through powder X-ray diffraction (PXRD) analysis. However, this analysis has to be coupled with computational simulations, wherein computed PXRD patterns for different stacking configurations are compared with experimental patterns to predict the precise stacking configuration. This task is often computationally challenging firstly because, computation of these systems mostly rely on the use of semi-empirical methods that need to be adequately parametrised for the system being studied and secondly because some of these compounds possess guest molecules, which are not often taken into account during computation. COF-1 is an extreme case in which the presence of the guest molecule plays a critical role in predicting the precise stacking configuration. Using this as a case study, we mapped out a full PES for the stacking configuration in the guest free and guest containing system using the GFN-xTB semi-empirical method followed by a periodic energy decomposition analysis using first-principles Density Functional Theory (DFT). Our results showed that the presence of the guest molecule leads to multiple low energy stacking configurations with significantly different lateral offsets. Also, the semi-empirical method does not precisely predict DFT low energy configurations, however, it accurately accounts for dispersion. Finally, our quantum-mechanical analysis demonstrates that electrostatic-dispersion model suggested Hunter and Sanders accurately describes the stacking in 2D COFs as opposed to the newly suggested Pauli-dispersion model.

Combination of Knoevenagel Polycondensation and Water-Assisted Dynamic Michael-Addition-Elimination for the Synthesis of Vinylene-Linked 2D Covalent Organic Frameworks.

Vinylene-linked two-dimensional conjugated covalent organic frameworks (V-2D-COFs), belonging to the class of two-dimensional conjugated polymers, have attracted increasing attention due to their extended π-conjugation over the 2D backbones associated with high chemical stability. The Knoevenagel polycondensation has been demonstrated as a robust synthetic method to provide cyano (CN)-substituted V-2D-COFs with unique optoelectronic, magnetic, and redox properties. Despite the successful synthesis, it remains elusive for the relevant polymerization mechanism, which leads to relatively low crystallinity and poor reproducibility. In this work, we demonstrate the novel synthesis of CN-substituted V-2D-COFs via the combination of Knoevenagel polycondensation and water-assisted dynamic Michael-addition-elimination, abbreviated as KMAE polymerization. The existence of C=C bond exchange between two diphenylacrylonitriles (M1 and M6) is firstly confirmed via in situ high-temperature NMR spectroscopy study of model reactions. Notably, the intermediate M4 synthesized via Michael-addition can proceed the Michael-elimination quantitatively, leading to an efficient C=C bond exchange, unambiguously confirming the dynamic nature of Michael-addition-elimination. Furthermore, the addition of water can significantly promote the reaction rate of Michael-addition-elimination for highly efficient C=C bond exchange within 5 mins. As a result, the KMAE polymerization provides a highly efficient strategy for the synthesis of CN-substituted V-2D-COFs with high crystallinity, as demonstrated by four examples of V-2D-COF-TFPB-PDAN, V-2D-COF-TFPT-PDAN, V-2D-COF-TFPB-BDAN, and V-2D-COF-HATN-BDAN, based on the simulated and experimental powder X-ray diffraction (PXRD) patterns as well as N2 -adsorption-desorption measurements. Moreover, high-resolution transmission electron microscopy (HR-TEM) analysis shows crystalline domain sizes ranging from 20 to 100 nm for the newly synthesized V-2D-COFs.

Solvent-Influenced Fragmentations in Free-Standing Three-Dimensional Covalent Organic Framework Membranes for Hydrophobicity Switching.

The ordered open organic frameworks membranes are attractive candidates for flow-assisted molecular separations. The physicochemical properties of such membranes mostly depend on their selectively chosen functional building blocks. In this work, we have introduced a novel concept of functional switchability of three-dimensional covalent organic framework (3D-COF) membranes through a simple solvent-influenced fragmentation method. This room-temperature interfacial synthesis provides free-standing 3D-COF membranes with distinct physicochemical properties from the same building monomers. Notably, the change of solvent from chloroform to ethyl acetate switches the membrane property from hydrophilic (water contact angle 60°) to hydrophobic (water contact angle 142°) nature. The hydrophobic 3D-COF membrane selectively passes oil molecules from an oil-water emulsion with a gravitational flux of 1536 L m-2 h-1 .

A Nanographene-Based Two-Dimensional Covalent Organic Framework as a Stable and Efficient Photocatalyst.

Synthesis of covalent organic frameworks (COFs) with desirable organic units furnishes advanced materials with unique functionalities. As an emerging class of two-dimensional (2D) COFs, sp2 -carbon-conjugated COFs provide a facile platform to build highly stable and crystalline porous polymers. Herein, a 2D olefin-linked COF was prepared by employing nanographene, namely, dibenzo[hi,st]ovalene (DBOV), as a building block. The DBOV-COF exhibits unique ABC-stacked lattices, enhanced stability, and charge-carrier mobility of ≈0.6 cm2 V-1 s-1 inferred from ultrafast terahertz photoconductivity measurements. The ABC-stacking structure was revealed by the high-resolution transmission electron microscopy and powder X-ray diffraction. DBOV-COF demonstrated remarkable photocatalytic activity in hydroxylation, which was attributed to the exposure of narrow-energy-gap DBOV cores in the COF pores, in conjunction with efficient charge transport following light absorption.

Module-Patterned Polymerization towards Crystalline 2D sp2 -Carbon Covalent Organic Framework Semiconductors.

Despite rapid progress over the past decade, most polycondensation systems even upon a small structural variation of the building units eventually result in amorphous polymers other than the desired crystalline covalent organic frameworks. This synthetic dilemma is a central and challenging issue of the field. Here we report a novel approach based on module-patterned polymerization to enable efficient and designed synthesis of crystalline porous polymeric frameworks. This strategy features a wide applicability to allow the use of various knots of different structures, enables polycondensation with diverse linkers, and develops a diversity of novel crystalline 2D polymers and frameworks, as demonstrated by using the C=C bond-formation polycondensation reaction. The new sp2 -carbon frameworks are highly emissive and enable up-conversion luminescence, offer low band gap semiconductors with tunable band structures, and achieve ultrahigh charge mobilities close to theoretically predicted maxima.