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644f1f7c0d87b493e3910993 | 0 | Hypoxia, generally considered as an oxygen concentration of <2%, is found in various diseases such as ischemia, 2,3 chronic kidney disease and cancer. Indeed, oxygen concentrations as low as around 1% may occur in ischemia and cancer, and selective in vivo visualization of such severe hypoxia is especially important for monitoring and understanding these diseases. To analyze hypoxia in cellulo and in vivo, various fluorescence probes have been developed. However, existing nearinfrared (NIR) small molecular probes that can be effectively used in vivo are also activated at oxygen concentrations around 5%. Therefore, there is a need for the development of NIR fluorescence probes able to selectively detect severe hypoxia as in vivo analytical and diagnostic tools. |
644f1f7c0d87b493e3910993 | 1 | Our research group has focused on the reductive cleavage of azo groups under hypoxia and developed NIR probes based on fluorescence resonance energy transfer (FRET) between dicarbocyanines and Black Hole Quencher (BHQ). Although these probes are selectively activated in severe hypoxia, they are not optimal for in cellulo and in vivo applications due to their large molecular size and susceptibility to photobleaching. As another strategy, we have recently developed fluorescence probes that detect hypoxia by incorporating an azo group into an O rhodamine or Si rhodamine (SiR) scaffold. For example, we have developed green-emitting MAR, red-emitting MASR and NIR-emitting azoSiR640 (2,6-diMe azoSiR640), and confirmed that they can detect hypoxia in cultured cells and mouse models. These fluorescence probes initially have no fluorescence due to fluorescence quenching by the azobenzene moiety incorporated in the fluorophore. Upon cleavage of the azo group by reductase-catalyzed reactions that are accelerated in hypoxia, corresponding fluorescent rhodamines are produced, enabling selective detection of hypoxia. However, the oxygen concentration threshold of these rhodamine-based hypoxia probes varies depending on the fluorescent scaffold. For example, MASR and 2,6-diMe azoSiR640 switched on at around 0.1% and 5% oxygen concentration, respectively. These characteristics may be due to differences in the rate of reverse oxidation of the azo moiety depending on the fluorescent scaffold (Figure ). Thus, we considered that the development of NIR probes to selectively detect severe hypoxia of around 1% oxygen concentration might be achieved by using different fluorophores. |
644f1f7c0d87b493e3910993 | 2 | The mono-julolidine-fused analog of SiR (JSiR) is widely used as a NIR fluorophore for biological imaging. In this study, we therefore examined JSiR as the scaffold fluorophore of a NIR fluorescence probe for hypoxia. We also evaluated the responsiveness of the synthesized probe, T-azoJSiR640, to oxygen concentration and confirmed its utility to selectively detect severe hypoxia in vivo. |
644f1f7c0d87b493e3910993 | 3 | Building on our previously reported MASR and 2,6-diMe azoSiR640, we designed and developed a novel fluorescence probe for hypoxia by introducing an azobenzene moiety at a free amino group of JSiR (Figure ). Since bulky substitutions at the 2 nd or 6 th positions of the benzene ring in SiR block nucleophilic attack at the 9 th position of the xanthene ring, for example, by water and cysteine containing molecules, we firstly designed a probe possessing carboxylic acid at the 2 nd position of the benzene ring of JSiR. However, this proved synthetically difficult, requiring tedious synthetic and purification steps. So, we next designed a probe possessing a thiophene ring instead of the benzene ring. We synthesized this probe, Tc-azoJSiR640, as shown in Scheme 1, but it proved ineffective due to its poor cellular membrane permeability (Figure ). Since the reason for this was considered to be the presence of the anionic form of the carboxylic acid moiety, we converted it to a methoxycarbonyl group and synthesized T-azoJSiR640. Compared to Tc-azoJSiR640, the membrane permeability of T-azoJSiR640 was apparently improved, as described later. Therefore, we focused on T-azoJSiR640 and evaluated its optical properties and susceptibility to hypoxia. |
644f1f7c0d87b493e3910993 | 4 | showed a broad absorbance spectrum with no fluorescence in sodium phosphate buffer (100 mM; pH 7.4) (Figures )). The observed complete quenching of fluorescence was considered to be due to ultrafast cis/trans conformational change of the azo bond of T-azoJSiR640 in the excited state, as in the case of our previously reported probes. On the other hand, T-JSiR640 showed marked fluorescence with a peak at 662 nm in the NIR range (Figure )), and its fluorescence quantum yield was 0.12 (Table ). We also confirmed that T-JSiR640 showed essentially the same fluorescence spectra over the pH range from 5.0 to 9.0, indicating that fluorescence signal of T-JSiR640 is not affected by pH (Figure ). These results suggested that T-azoJSiR640 is able to generate NIR fluorescence upon cleavage of the azo group under hypoxia. |
644f1f7c0d87b493e3910993 | 5 | To determine whether T-azoJSiR640 could detect hypoxia, we first performed an in vitro assay using rat liver microsomes, which contain various reductases such as NADPH-cytochrome P450 reductase. The probe T-azoJSiR640 showed a fluorescence increase only in the presence of rat liver microsomes and NADPH under hypoxia, indicating that the probe could detect hypoxia under reducing conditions catalyzed by microsomal reductases (Figure ). |
644f1f7c0d87b493e3910993 | 6 | To evaluate whether T-azoJSiR640 could visualize hypoxia in living cells, we used it to perform live-cell fluorescence imaging of A549 lung adenocarcinoma cells under various oxygen concentrations. T-azoJSiR640 showed almost no fluorescence under normoxia (Figures )), while it showed significant increase of NIR fluorescence at oxygen concentration around 1％. These results suggested that T-azoJSiR640 was able to selectively detect severe hypoxia. In contrast with the above results, our previously reported 2,6-diMe azoSiR640 was also activated at around 5% oxygen concentration (Figures )). To confirm that the observed fluorescence signals were due to reduction by flavoproteins, we measured fluorescence images in the presence of a NADPH oxidase inhibitor, diphenyleneiodonium chloride (DPI). The fluorescence increase was dramatically suppressed in the presence of DPI, suggesting that T-azoJSiR640 is reduced by flavoproteins such as NADPH-cytochrome P450 reductase. |
644f1f7c0d87b493e3910993 | 7 | We assumed that this selective activation of T-azoJSiR640 around 1% oxygen concentration is due to the high rate of reverse oxidation of the azo anion radical produced in the first reduction step (Figure ). This would explain why the fluorescence signal of T-azoJSiR640 is suppressed around 20-3% oxygen concentration, but is activated in extremely low oxygen concentration (around 1%), thereby allowing the selective detection of severe hypoxia. |
644f1f7c0d87b493e3910993 | 8 | In vivo fluorescence imaging of liver ischemia in mouse models. Finally, we performed fluorescence imaging of liver ischemia of mouse to determine whether T-azoJSiR640 could detect severe hypoxia in vivo. The probe T-azoJSiR640 was administered to mice by intravenous injection, and after 30 min, the portal vein was ligated to induce ischemia. Before ligation of the portal vein, no fluorescence increase was observed in the liver, while a rapid and significant fluorescence increase was observed over the entire liver after the induction of ischemia (Figures )). To confirm that T-JSiR640 was produced, we examined the production of T-JSiR640 by LC-MS/MS analysis (Figures General Procedures and Materials. Reagents and solvents were of the best grade available, purchased from Tokyo Chemical Industries, Wako Pure Chemical, Aldrich Chemical Co., Dojindo, and Invitrogen, and were used without further purification. Reactions were monitored by means of TLC, ESI mass spectrometry and HPLC. All compounds were purified by silica gel chromatography or preparative HPLC. |
644f1f7c0d87b493e3910993 | 9 | T-azoJSiR640 was weighed, dissolved in methanol and aliquoted to 1.5 mL Microtube Black for Shading (WATSON ® BIO LAB). The solvent was evaporated and the tubes were stored at -25℃. The samples were reinstated in an appropriate solvent before use. The probe solution was prepared individually for each experiment and not reused. T-JSiR640 was solved in DMSO and stored in Safe-Lock Tubes 1.5 mL at -25°C. The concentrations of stock solutions were calculated from the by ε values. |
644f1f7c0d87b493e3910993 | 10 | Preparation of rat liver microsomes. All animal experiments were performed according to institutional guidelines. Rats (Wistar, boar, 6-7 weeks old at the beginning of the experiment) were purchased from CLEA Japan. They were treated with 60 mg/5 mL/kg sodium phenobarbital intraperitoneally once daily for 3 days, fasted overnight, and sacrificed by exsanguination from the abdominal aorta. The livers containing 0.15 M KCl at pH 7.4 were in 3 equal volumes of the same buffer and centrifuged (8,500 rpm, 20 min, 4°C) twice. Then the supernatant was collected and centrifuged (34,000 rpm, 80 min, 4°C) to collect rat liver microsomal fractions. Rat liver microsomes contained 71.2 mg protein/mL and 0.479 nmol P450/mg protein. The microsome fraction was diluted in 100 mM sodium phosphate buffer at pH 7.4 for assay. |
644f1f7c0d87b493e3910993 | 11 | The oxygen concentrations was controlled with a multi gas incubator MCO-5MUV (Sanyo) by N₂ substitution. Fluorescence confocal microscopy images were acquired using a Leica Application Suite Advanced Fluorescence (LAS-AF) instrument equipped with a TCS SP5 and 40× or 10× objective lens. Gain and pinhole values were set at 150% and 95.0 m (Figures ) or 68.0 m (Figure ). |
644f1f7c0d87b493e3910993 | 12 | In vivo fluorescence imaging of liver ischemia in mouse models. All procedures were approved by the Animal Care and Use Committee of the University of Tokyo. Female Jcl: ICR mice (7 weeks) were used. T-azoJSiR640 (200 M) in PBS (150 L) containing 4% ethanol was administered by intravenous injection. After probe administration, the mice were anesthetized with a combination of domitor, butorphanol and midazolam. Fluorescence images were captured at 10 min, 20 min and 30 min after administration of the probe, and then the portal vein was ligated with suture thread at 30 min. Further fluorescence images were captured at 40 min, 50 min and 60 min. All fluorescence images were captured with the CRi Maestro imaging system (CRi Inc., Woburn, MA). |
649d997c6e1c4c986b8b5616 | 0 | 0.1 kcal mol -1 , (ii) benzene has a calculated homolytic bond dissociation energy of 147.0 kcal mol -1 . The increased thermodynamically stability of C-C bonds in aromatic rings is unsurprising given to their character and electron delocalisation across the ring. C-C bonds are also sterically protected. They are often buried within the molecular framework and the orbitals involved in bonding are kinetically inaccessible. As such chemoselectivity becomes a key issue, with surrounding C-H bonds often the first sites to react with reagents and catalysts that would otherwise be capable of breaking C-C bonds. C-C bond activation has been achieved on the surface of heterogenous catalysts, within the active sites of enzymes, and under homogenous conditions using metal complexes. The systems which are best understood are arguably those that contained well-defined transition metal sites (Co-Ir, Ni-Pt), where partially occupied valence d-orbitals facilitate C-C bond breaking. Often model substrates that contain weakened C-C bonds and/or extensive ring strain are studied. For example, hydrocarbons with smaller ring sizes are routinely investigated as the C-C bond strength decreases across the series cyclohexane > cyclopentane > cyclobutane > cyclopropane. Reactivity tends to follow established mechanisms (Scheme 1). |
649d997c6e1c4c986b8b5616 | 1 | • A) -alkyl elimination, wherein a metal bound alkyl ligand is fragmented into the corresponding metal alkyl and alkene units. • B) oxidative addition, wherein the C-C bond is cleaved by addition to a low oxidation state metal complex, increasing the metal oxidation state by two and creating two new M-C bonds. • C) A cycloaddition reaction between a hydrocarbon and metal reagent, creating a strained metallocycle which can then undergo C-C bond activation (e.g. through -elimination or an electrocyclic reaction). Scheme 1. Carbon-carbon bond activation mechanisms, via a) -alkyl elimination or migration; b) oxidative addition; c) cycloaddition reaction and subsequent rearrangement. |
649d997c6e1c4c986b8b5616 | 2 | Though important progress is being made toward transition metal mediated C-C bond activation, there is an increasing drive away from late transition metal-based systems. Late transition metals are commonly expensive and toxic, with further issues regarding the sustainability and ethics of the mining practices used to obtain the requisite minerals for refining. Main-group metals (e.g. Mg, Al, Zn, Sn) are promising alternatives to their transition metal counterparts for applications in synthesis and catalysis. These elements are commonly earth-abundant, inexpensive, and more widely distributed in the Earth's crust compared to the late transition metals. Except for Sn, they are non-toxic and accordingly safer to handle. |
649d997c6e1c4c986b8b5616 | 3 | For some (e.g. Al) there are even established networks and processes for recycling, auguring well for a future circular economy. A limited number of examples of metal free C-C bond activation have been reported for systems using Frustrated Lewis Pairs, boron-, silicon-, phosphorous-, and organic-compounds. In this review, we summarise the current examples of main-group metal mediated C-C bond activation. |
649d997c6e1c4c986b8b5616 | 4 | The discussion is split into three distinct approaches, namely -alkyl migration, oxidative addition, and those initiated by cycloaddition reactions. Much of the early mechanistic work on -alkyl migration was performed with group 4 metallocene complexes as part of understanding polymerization catalysis . Through discussion of mechanism, we aim to highlight the divergent chemistry shown by main-group metal complexes compared to their transition metal counterparts and touch on the potential implications in synthesis The reversible stoichiometric formation of iso-butene gas and trimethylaluminium was observed via sp 3 C-C -bond activation, presumed to occur through a -methyl migration reaction (Scheme 2). The release of three equivalents of iso-butene gas provides an entropic driving force for the forward reaction. |
649d997c6e1c4c986b8b5616 | 5 | In 1999, Dakternieks and co-workers reported a related reaction at a Sn complex, observed during fragmentation in a mass spectrometer. Application of a high cone voltage (> 60 V) to a acetonitrile solution of the tris(neo-pentyl) stannyl cation [Sn{CH2C(Me)3}3] + 2 showed formation of methyl tin cations and release of isobutene gas. Reaction of the deuterium labelled analog [Sn{CD2C(Me)3}3] + showed the formation of isobutene gas with the alkene protons D-labelled, consistent with a -alkyl elimination process. An alternate Sn-C bond homolysis and radical pathway was not ruled out and cannot be discounted under fragmentation conditions in the mass spectrometer. |
649d997c6e1c4c986b8b5616 | 6 | In 2020, we reported C-C -bond cleavage of strained alkylidene cyclopropanes using magnesium reagents (Scheme 3, top). This work was extended to include reaction of the related magnesium(II) hydride complex 6 [Mg{(-H){CH{C(CH3)NDipp}2}]2 (Dipp = 2,6-diisopropylphenyl) with the same set of substrates (Scheme 3, bottom). Stoichiometric reaction of 6 with methylidene cyclopropane (7a) and methylidene cyclobutane (7b) yielded the ring-opened alkenyl magnesium complexes 9a and 9b in good yields. Though no intermediates were observed spectroscopically, DFT calculations and related literature supported a hydromagensiated intermediate 8a-b as a prerequisite to -alkyl elimination and thus C-C bond activation. Additional evidence for the hydromagensiated intermediates was gathered through reaction of unstrained methylidene cyclopentane (7c) and methylidene cyclohexane (7d) with 6, which formed the hydromagnesiated products 8c-d in high yields. Calculated activation barriers for -bond C-C bond cleavage at 8c-d were unfeasible under the reaction conditions (G ‡ 298K > 40 kcal mol -1 ), indicating that the release of ring strain in the three-and four-membered systems is an important driving force for the reaction. Activation strain analysis was used to explain the differences in reactivity between the analogous zinc and magnesium hydride complexes. Namely, that magnesium was observed to ring open cyclobutane rings, whereas zinc was not. The more electropositive metal (Mg) was shown to be better able to stabilize the hydrocarbon fragment at the C-C activation transition state, lowering the kinetic barrier relative to Zn. 6 and 10 have an identical ligand coordination, at such it can be concluded that chemoselectivity in these reactions can be controlled through choice of the main-group metal. |
649d997c6e1c4c986b8b5616 | 7 | Reaction of the magnesium alkenyl complex 9a with an excess of phenyl silane (PhSiH3) led to formation of the linear and cyclic silane compounds 13-14 and reformation of magnesium hydride 6. A catalytic protocol for the hydrosilylation of strained -C-C bonds was developed from these findings. Reaction of 4a-b and 7a-b with excess phenyl silane and 10 mol% of 6 showed high conversion to the respective cyclic silanes 13-18. In the case of 4b, catalytic hydrosilylation yields a mixture of E and Z stereoisomers of the product 18, with a ratio of E:Z of 1:1.1. Scheme 5 shows the proposed catalytic cycle, each step of which is supported by experimental and computational data. The stepwise process follows: |
649d997c6e1c4c986b8b5616 | 8 | i) hydromagnesiation of the alkene through a 1,2-insertion reaction of the magnesium hydride to the alkene ii) -alkyl migration; iii) -bond metathesis to regenerate magnesium hydride catalyst 6 and the linear silane. The thermodynamic products 13-16 are likely formed from an intramolecular hydrosilylation also catalysed by the magnesium complex 6. The related zinc complex 10 was unable to catalyse the hydrosilylation of the -C-C bond of 7a, again highlighting the divergent reactivity of different main-group metals. |
649d997c6e1c4c986b8b5616 | 9 | Oxidative addition, and its microscopic reverse, reductive elimination, are some of the most fundamental kcal mol -1 . Both the anti-aromatic character and strain of the four-membered ring contribute to the weakening of this C-C -bond. In contrast, a C-C bond within the six-membered ring system has been estimated as 114.4 kcal mol -1 . Addition of transition metals to biphenylene results exclusively in -C-C bond activation via oxidative addition at the central C4 ring (M = Fe, Co, Ni, Ru, Rh, Pd, Os, Ir, Pt, Au). Very recently, selective cleavage of the C-C bonds in biphenylene during potassium reduction of rareearth metal complexes (Sc, Lu) was reported. In 2020, Kinjo and co-workers reported an aluminyl anion stabilised by a cyclic (alkyl)(amino) ligand, prepared by potassium graphite reduction of the corresponding aluminium dimer in the presence of 12crown-4-ether. |
649d997c6e1c4c986b8b5616 | 10 | Multimetallic transition metal complexes can facilitate unique C-C bond cleavage reactions. For example, the first C-C bond cleavage of benzene by any metal was achieved by a trimetallic titanium hydride cluster. A multimetallic approach is of interest as a potentially more accurate mimic of heterogenous catalysts employed industrially, which likely involve cooperative behaviour of multiple metal sites during C-C cleavage. Investigation of a reaction of a series of aluminium complexes with biphenylene has led to some remarkable examples of control over chemoselectivity in C-C bond activation, with bimetallic mechanisms often being invoke to explain the origin of selectivity. |
649d997c6e1c4c986b8b5616 | 11 | In 2021, our group reported the reaction of 25 with biphenylene. This was the first report of Both pathways are highly exergonic, consistent with the nonreversible formation of the products observed experimentally. A direct oxidative addition of the central C 1 -C 7 -bond of biphenylene to 25 was calculated to occur by a high activation barrier (G ‡ 298K = 42.0 kcal mol -1 ), likely to be inaccessible under the reaction conditions. Further calculations using activation strain analysis suggest that the inaccessible energy barriers for oxidative addition are likely a result of the strain required to achieve orbital overlap between the aluminium complex's lone pair and C 1 -C 7 *-orbital in biphenylene. In 2022, Braunschweig and co-workers reported an in situ generated base-stabilised aryl aluminylene complex capable of deconstructing benzene and toluene via C-C bond scission (Scheme 12). The In this regard, it is notable that many of the emerging applications of main group metals in C-C bond activation rely on low-oxidation state complexes with coordinated ligands (e.g. NHCs). The coordination event not only lowers the HOMO-LUMO gap it also changes the geometry at the main group metal centre. |
649d997c6e1c4c986b8b5616 | 12 | Both may be important in reaching accessible transition states for C-C bond activation that might not otherwise be possible with the ligand-free counterparts. Similarly, cooperative effects between two or more main group metals offer alternative pathways to break C-C bonds with reagents that are constrained to a certain set of orbital interactions. Bimetallic pathways have been invoked in several of the systems known to date, with two main group metals acting in concert; binding, distorting, and destabilising hydrocarbon frameworks to achieve C-C bond activation. |
649d997c6e1c4c986b8b5616 | 13 | In the immediate future, it is likely that new and interesting examples of C-C bond activation with main group metals will be discovered. Investigation of low-valent aluminium reagents appears to be a There is a clear need for development of main group catalysts for C-C bond functionalisation. The ability to alter hydrocarbon scaffolds of complex organic molecules and to valourise simple hydrocarbons or aromatics through catalysis are particularly attractive approaches that have long been associated with late transition metals. In the longer term, such catalytic transformations could underpin sustainable chemical manufacturing practices including the valourisation of molecules from biomass or the recycling of hydrocarbon-based polymers. The efforts described above show that main group systems have the potential to make important contributions in these areas that complement transition metal systems while also addressing key aspects of element scarcity, supply chain risk, and sustainability. |
673db9b27be152b1d0d75696 | 0 | The structure and properties of methylaluminoxane (MAO) continue to attract interest in the context of olefin polymerization using metallocene or other single-site catalysts. Recent attention has been focused on the synthesis and structure of modified MAO and solid MAO, which are typically prepared using non-hydrolytic methods. The identity of the reactive components of hydrolytic MAO, prepared via the controlled hydrolysis of Me3Al, remained largely unknown. This mystery, however, was partly solved by the recent isolation and characterization of a reactive sheet structure, (MeAlO)26(Me3Al)9 (hereinafter "26,9"), by Luo et al. Theory suggests the bulk of hydrolytic MAO contains relatively unreactive large cages, or other structures, with tetrahedral O3AlMe and trigonal O groups in six-membered or larger (MeAlO)n rings. Reactive structures consist of strained cages, especially those with vicinal (MeAlO)2 rings, extended cages or nanotubes with reactive groups formed via the addition of structural Me3Al to the vertices and ends of these structures, and two-dimensional sheets analogous to 26,9 with reactive sites incorporating structural Me3Al along their edges. |
673db9b27be152b1d0d75696 | 1 | Fairly recently, reactive MAO species were detected during the hydrolysis of Me3Al in polar solvent using the technique of ESI-MS. This sensitive analytical technique relies on ion-pair formation between reactive MAO species and a Lewis base donor such as (Me3SiO)2SiMe2 (OMTS). In sufficiently low amounts, OMTS reacts reversibly with MAO according to Scheme 1. During the hydrolysis process, anions with the composition [(MeAlO)n( )mMe] -(hereinafter abbreviated [n,m] -) were detected with variable intensity. Those formed most rapidly had n ≥ 7 with m = 4, of which the most intense were initially [7,4] -or [8,4] -depending on experimental conditions. Experiments in odifluorobenzene using stoichiometric Me3Al and H2O featured [7,4] -as the most intense anion. However, in fluorobenzene, at the same stoichiometry [8,4] -was the major anion detected (Figure , bottom), unless excess Me3Al was present, in which case [7,4] -was the major anion detected at short reaction times (Figure , top). At longer time scales, all mixtures evolved to form higher m/z anions of which the [16,6] -anion was most intense as is seen with commercial hydrolytic MAO (h-MAO). The preponderance of even-numbered anions (i.e. n = 14, 16, 18) at longer reaction times suggested an oligomerization mechanism involving sheet structures formed from linear MAO species. Anions with m/z ratios < 709 were present in only trace amounts during these experiments, despite the use of continuous reaction monitoring techniques. Species that were reproducibly detected include [5,3] -(M + = 521 Da) and [6,4] -(M + = 651 Da) and its hydrolyzed analogue [6,4-OH] -with m/z = M + +2 = 653 Da. In these spectra, the hydrolyzed anion was more intense than its parent ion (Supporting Information Figure S-1). The lowest molecular weight (MW) anion detected in these experiments had m/z = 465 and nominal formula [(MeAlO)4(Me3Al)3OH] -, i.e. [4,3-OH] -. Since none of these anions are detected in commercial h-MAO some may correspond to intermediates in the hydrolysis of Me3Al. Previous theoretical studies identified chains, rings, and sheets as the most stable neutral MAO structures for n ≤ 8 and m ≤ 5. Building on this foundation, we undertook a comprehensive investigation of structure-stability-reactivity relationships using a hierarchy of computational methods (DFT, MP2, and CCSD(T)) to explain the distribution of species observed in these monitoring experiments. This multi-level theoretical approach not only provided insights into the experimental observations but also revealed significant limitations in DFT's ability to accurately describe the relative stabilities of small MAO species. |
673db9b27be152b1d0d75696 | 2 | While specific MAO aluminum sites have been discussed in previous studies, a systematic classification scheme has been lacking. We developed a comprehensive classification system based on the local coordination environment of aluminum centers, with particular focus on their reactivity. The classification follows a hierarchical organization: 1) Decreasing number of Al-C bonds, with priority given to terminal over bridging methyl groups, 2) increasing number of Al-O bonds, and 3) increasing coordination number of the bound oxygen atoms. The complete classification scheme is provided in Supporting Information (Figure S-2), while Figure illustrates the most relevant site types for this study. |
673db9b27be152b1d0d75696 | 3 | High MW MAOs, such as the recently isolated sheet 26,9, primarily contain unreactive type L, featuring four-coordinate (4-C) aluminum centers bound to three oxygen atoms and one terminal methyl group. Reactive aluminum sites in such structures are predominantly located at sheet edges, with their reactivity patterns influenced by the overall MAO morphology. In contrast, our study focuses on lower MW MAO species (n = 1-8), where the high edge-to-bulk ratio results in a predominance of potentially reactive aluminum sites. |
673db9b27be152b1d0d75696 | 4 | To systematically investigate Al site reactivity patterns in MAO, we analyzed a series of structures (Figure ) chosen based on previous computational work, with refinements guided by highlevel CCSD(T) calculations. For each composition, we included both the global minimum structure and energetically competitive isomers, as summarized in Table . The complete set of analyzed structures is provided in Supporting Information, Figure . |
673db9b27be152b1d0d75696 | 5 | Table includes G-qh-tr/n values for the Me3Al hydrolysis reaction ½(n+m) Al2Me6 + n H2O → n,m + 2n CH4 calculated in fluorobenzene medium. This improves upon previous condensedphase thermodynamic stability calculations for the same reaction by incorporating explicit consideration of solvent free volume to better account for reduced entropy in solutions, and quasiharmonic treatment of low-energy vibrational modes. Monomeric MAO (n = 1) can incorporate up to three Me3Al molecules, with 1,3 being the thermodynamic preferred product. Two chain-like isomers of 1,3 exist, distinguished by their oxygen coordination environments (O3-and O4-chains). These exist in Me3Al association/dissociation equilibrium with the 1,2 O3-chain. A similar pattern emerges for the dimer (n = 2), where the thermodynamically favored product is the chain-like 2,4 structure. This species also forms both O3-and O4-chain isomers and exists in equilibrium with its Me3Al dissociation product, the 2,3 O3-chain. |
673db9b27be152b1d0d75696 | 6 | As molecular size increases, the structural motif evolves from chains to rings (n = 3,4) and ultimately to sheets, which dominate in the size range studied here. The prevalence of sheet structures extends well beyond our studied range, with previous computational work predicting their stability up to 18,6. The recent isolation and structural characterization of a larger 26,9 sheet demonstrates that this motif exists in higher MW MAO. The predominance of sheets allows us to focus exclusively on this structural motif, as alternative morphologies such as cages and tubes, while potentially relevant for large species, represent higher energy configurations in the low MW regime (n ≤ 8) examined here. |
673db9b27be152b1d0d75696 | 7 | To understand anion formation mechanisms, we studied the reactions of neutral MAO species with OMTS, which generate [Me2Al(OMTS)][n,m] ion-pairs in fluorobenzene medium during ESI-MS experiments. We initially considered the two reaction pathways shown in Scheme 1, yielding separated, non-interacting ions in the gas phase. Using M06-2X, we computed gas-phase electronic energy changes for every unique aluminum site (Figure ) in each MAO structure listed in Table . This comprehensive analysis required over 300 individual calculations, as each site was evaluated for both possible ionization mechanisms. The complete results, including neutrals and their corresponding anions shown in consistent orientations, are provided in Supporting Information Table S-2, along with their respective reaction energies. |
673db9b27be152b1d0d75696 | 8 | Statistical analysis of the computational results (Table ) reveals the frequency of reactive sites follows the order B > I > G > C > J > F > N > E > A. While gas-phase ion generation energies are universally high and positive, as expected, comparison of ionization mechanisms shows Me -abstraction generally requires less energy than Me2Al + cleavage across all site types except A. However, the large standard deviations in these energies prevent drawing statistically significant conclusions about the preferred ionization mechanism. |
673db9b27be152b1d0d75696 | 9 | Examination of the computed structures (Table -2) reveals that regardless of the initial ionization site or mechanism (Me2Al + cleavage or Me -abstraction), the resulting anions often converge to identical geometries. This structural adaptability explains the large standard deviations in the gas phase ionization energies, as the reorganization depends strongly on the local environment of the initial site -a feature not captured in our site classification scheme. Among the different site types, type C shows the lowest average energy for Me2Al + cleavage, while type N sites are most favorable for Me -abstraction. |
673db9b27be152b1d0d75696 | 10 | Analysis of individual sites reveals that type C exhibits the lowest energy for Me2Al + cleavage, while type E is most favorable for Me -abstraction. The high reactivity of type C sites towards Me2Al + cleavage can be rationalized structurally: the initial Me2Al + group is bound to both a 4-C oxygen and a bridging methyl group, and its removal restores two characteristic features of bulk MAO: 3-C oxygen atoms and tetrahedral aluminum centers with terminal methyl groups. Me2Al + cleavage from type C sites typically generates a terminal OAlMe3 -moiety, which can form chelating interactions with adjacent AlMe2 groups when sterically feasible. This same OAlMe3 -end group can also form via Me -abstraction from type E sites (3-C OAlMe2 groups). While type E sites are energetically most favorable for Me -abstraction, they are relatively uncommon as they tend to convert to less reactive type B sites. |
673db9b27be152b1d0d75696 | 11 | Moving to more experimentally relevant conditions, we examined these reactions in fluorobenzene medium to match the ESI-MS monitoring experiments. For the most favorable ionization sites, we calculated both electronic (Ei) and quasiharmonic Gibbs energy changes (ΔGi-qh-tr) with results compiled in Table S-3. In contrast to the uniformly high, positive gas-phase energies, these solution-phase energies approach zero or become negative. These calculations consider fully separated ions; the effects of ion-pairing are addressed separately below. |
673db9b27be152b1d0d75696 | 12 | The stabilities (ΔΔG-qh-tr/n) relative to our sheet model for 16,6 are also provided in Table -3 (Supporting Information), which presents the most stable anions for each oligomerization degree. Table presents a focused subset of these results, including only those species with negative formation energies or those observed experimentally. The data is organized by ionization mechanism, with entries 1-8 corresponding to Me2Al + cleavage and the remaining entries to Me -abstraction. Anions predicted by calculations but not detected in ESI-MS experiments are highlighted in light blue. |
673db9b27be152b1d0d75696 | 13 | ΔGi-qh-tr G-qh-tr/n Entry [n,m] - Neutral Site M06-2X MP2 CCSD(T) M06-2X MP2 CCSD(T) M06-2X MP2 [a] When the same anion is obtained from two (or more) different neutrals (Table -2, Supporting Information), the thermodynamically preferred neutral is used to compute Ei and ΔGi-qh-tr. [b] All sites of the most stable neutral yielding the same anion (Table -2) are indicated. [c] For the most stable anions of each oligomerization degree. |
673db9b27be152b1d0d75696 | 14 | The computational results align well with experimental observations of anion intensities in ESI-MS spectra. Of the anions formed via Me2Al + cleavage, [4,3-OH] -and [5,3] -are invariably the weakest in intensity at any stage of the hydrolysis reaction. In contrast, the strongest at the earlier stages can be either [7,4] -or [8,4] -(Figure ). The calculated ΔGi-qh-tr values correctly predict for each value of n, which [n,m] -anions should predominate in these mixtures. For example, for n = 6, four anions including [6,4-OH] -(vide infra) could be formed from various precursors (Table 3, entries 3, 4, 11 and 12). Among these, only [6,4-OH] -and [6,4] - have negative ΔGi-qh-tr values, and these are indeed the species observed in the ESI-MS spectra, with their relative experimental intensities (Table ) showing qualitative agreement with theoretical predictions. |
673db9b27be152b1d0d75696 | 15 | Of the trace anions observed, [4,3-OH] -and [6,4-OH] -possibly contain OH groups. These anions are consistently more intense than the corresponding anions without OH groups, for instance, [6,4-OH] -is always more intense than [6,4] -(Table ). These hydroxylated anions are not seen in commercial h-MAO (nor h-MAO synthesized on a preparative scale ). The persistence of these anions during in situ monitoring experiments suggests that their precursors are also persistent. This would only be possible when there is a deficit of Me3Al in the solution. The bimolecular reaction between MAO OH groups and monomeric Me3Al is diffusion-controlled with a low barrier. Earlier, we suggested that these hydrolyses proceed in poorly dispersed aqueous suspensions, wherein much Me3Al is incorporated into an insoluble gel, regardless of the initial stoichiometry. |
673db9b27be152b1d0d75696 | 16 | Possible precursors to these anions are related to each other, and possibly to other MAO species in this size range. Figure illustrates these relationships. For example, 4,4-OH has a 4-C sheet structure and serves as a precursor to the [4,3-OH] -anion through cleavage of the Me2Al + group (highlighted in blue, Figure ). The 4,4-OH sheet is related to the 4-C 5,4 sheet through the reaction of the OH group with Me3Al and the rearrangement of the highlighted Me2Al group. This 5,4 sheet can also serve as a precursor to the [5,3] -anion, though it is not the thermodynamically most favorable neutral structure (Table ). The 6,4 4-C sheet is a possible precursor to the [6,4] -anion. It is related to the 5,4 4-C sheet through the net addition of MeAlO (Al highlighted in green, Figure ). However, for this composition, the ring structure is thermodynamically preferred over the sheet. |
673db9b27be152b1d0d75696 | 17 | These trace anions are present with relative intensities that do not always reflect ΔGi-qh-tr in Table . For example, the intensity of [4,3-OH] -and [6,4-OH] -differ by about a factor of 440 (Table ) corresponding to an energy difference of 15 kJ mol -1 . While ΔGi-qh-tr values differ by ca. 30 kJ mol -1 , ΔΔG-qh-tr/n values at MP2 level are actually quite close to experiment. We suspect that the intensities of some observed anions, especially those with OH groups, also correlate with their importance as intermediates in the primary hydrolysis of Me3Al. |
673db9b27be152b1d0d75696 | 18 | We have studied the formation of [Me2Al(OMTS)] [16,6] in fluorobenzene medium and have explored transition structures for both Me -abstraction and Me2Al + cleavage from the stable 16,6 4-C sheet or a higher energy i-16,7 precursor, respectively in gas phase. Those studies indicate that the lowest energy pathway involves Me -abstraction from Me3Al-OMTS by sheet 16,6 to form this outer-sphere ion pair. Here, we compare those results with smaller MAO sheets, though we note that only 6,4 seems to operate by the same mechanism. We also study 26,9 and the formation of [Me2Al(OMTS)] [26,8] as well as 16,6 and [Me2Al(OMTS)] [16,6] not only at M06-2X/TZVP level but also single point RI-MP2 calculations at the M06-2X/TZVP geometry. The results are summarized in Table . |
673db9b27be152b1d0d75696 | 19 | Analysis of the gas-phase energetics reveals a systematic overestimation of ion-pair formation energies at the M06-2X/TZVP level compared to MP2 calculations, Method dependence extends further up the hierarchy of methods, with MP2 also overestimating energies relative to CCSD(T), though the magnitude of this overestimation is smaller. Specifically, for MAO sheets with n = 5-8, the average difference between MP2 and CCSD(T) energies is 6.9 kJ mol -1 , while M06-2X deviates from MP2 by an average of 16.1 kJ mol -1 . However, the substantial standard deviation in the M06-2X/MP2 energy differences (10.8 kJ mol -1 ) is comparable to the mean itself, precluding the identification of reliable size-dependent trends in ion-pair formation energetics. |
673db9b27be152b1d0d75696 | 20 | Comparing ion-paired ΔEi values in fluorobenzene (Table ) with non-ion-paired values (Table ) reveals significant ion-pairing effects. For sheets with n = 6-8, the ion-pairing contribution ranges from 54 to 58 kJ mol -1 , yielding and average value of 55.7± 1.9 kJ mol -1 . However, this consistency does not extend across the full range of MAO sizes. |
673db9b27be152b1d0d75696 | 21 | The smallest studied sheet (n = 5) exhibits a notably larger ionpairing energy (ca. 75 kJ mol -1 ) while the recently characterized large sheet (n = 26) shows an intermediate value of (60.8 kJ mol - 1 ) more comparable to other smaller sheets, including 16,6. Neither ΔEi or ΔGi-qh-tr values alone correlate well with the relative intensities of anions observed in ESI-MS monitoring experiments (Table ). This discrepancy is particularly evident for [n,4] -ion-pairs with (n = 6-8), where similar calculated ΔGi-qh-tr predict comparable stabilities, contrary to experimental observations under certain conditions. A more complete understanding emerges when incorporating neutral precursor stability into the analysis. The composite parameter, ΔΔG-qh-tr/n, which combines both ionization energetics and neutral stability, shows improved correlation with experimental anion intensities. This finding highlights the importance of the neutral precursor stability in determining the observed distribution of MAO anions. |
673db9b27be152b1d0d75696 | 22 | The analysis of the 26,9 sheet structure reveals a remarkable finding. The calculated ΔGi-qh-tr value (-55.9 kJ mol -1 ) suggests this species should be approximatively four orders of magnitude more reactive towards OMTS than other MAO structures studied. This predicted exceptional reactivity presents an apparent paradox, given the recent successful isolation and structural characterization of 26,9 (following the addition of nearly 1 equiv. of OMTS to MAO). While the incorporation of neutral precursor stability through ΔΔG-qh-tr/n values moderates the differences, [Me2Al(OMTS)] [26,8] is still predicted to be the most stable ionpair amongst all those studied. |
673db9b27be152b1d0d75696 | 23 | There is a striking disconnect between our theoretical predictions for the 26,9 and experimental ESI-MS observations. While calculations suggest [26,8] -should dominate the mass spectra, the ESI-MS spectra of commercial MAO with OMTS as an additive (Figure S-4) and those recorded during hydrolysis of Me3Al (Figure ) paint a markedly different picture. |
673db9b27be152b1d0d75696 | 24 | The [26,8] -anion is never intense during the monitoring experiments nor is it that noticeable in commercial material using OMTS as an additive. This discrepancy suggests that 26,9 exists only as a minor component in commercial MAO, or at least not present in the amounts dictated by thermodynamics, otherwise, it would be expected to overwhelm other species in the ESI-MS spectrum. |
673db9b27be152b1d0d75696 | 25 | As shown in Figure , the formation of [Me2Al(OMTS)][26,8] ion-pair involves a low energy intermediate which is only 27 kJ mol -1 higher in G-qh-tr than the separated reactants and is electronically more stable. This intermediate has an interesting structure, in which a Me3Al adduct of OMTS is involved in AlMe bridging to a Lewis acidic 26,8 neutral. We have been unable to locate transition structures connecting this intermediate with either starting materials or products. This suggests that the barriers for Me2Al + cleavage are likely low and predominantly entropic in nature for this specific sheet. |
673db9b27be152b1d0d75696 | 26 | Interestingly, in aged MAO, an anion with the formula [26,7] - is reasonably prominent (Figure S-4), while [26,8] -is weak but detectable in both aged and unaged samples. This observation of larger MAO anions in aged samples raises intriguing questions about the formation and stability of these species over time. The recent isolation of the 26,9 structure, which crystallized after several months of treatment with a near-stoichiometric amount of OMTS (relative to the available activator), may provide a clue. We hypothesize that this crystallization might result from a distinct aging process of the neutral MAO components that remained in the clathrate phase. |
673db9b27be152b1d0d75696 | 27 | Our results demonstrate that the accurate modeling of MAO structures requires theoretical work at the MP2 level or higher to reliably identify global minima. The widely used M06-2X DFT method artificially stabilizes structures with 4-coordinate oxygen atoms, leading to incorrect predictions of 5-coordinate aluminum centers. This methodological artifact has significant implications for understanding MAO structure and reactivity, particularly for small oligomers. |
673db9b27be152b1d0d75696 | 28 | The observed reactivity patterns of small MAO sheets towards anionization explain the relative intensities of anions detected during early-stage ESI-MS monitoring of Me3Al hydrolysis. This correlation becomes particularly clear when considering both ionization energies and neutral precursor stabilities in polar media. However, a quantitative prediction of intensity differences remains challenging, likely due to complex factors. Perhaps the most important factor is the actual mechanism for competitive hydrolysis vs. aggregation of the neutral MAO molecules formed. |
673db9b27be152b1d0d75696 | 29 | Our analysis of larger sheets, particularly 26,9 and 16,6, reveals an unexpected discrepancy: while calculations predict that 26,9 is much more reactive towards ion-pair formation than 16,6, ESI-MS observations show opposite relative intensities. Further experimental and theoretical studies are needed to fully understand the nature and distribution of active species in MAO solutions. |
673db9b27be152b1d0d75696 | 30 | In addition, DLPNO-MP2 /def2-TZVP optimizations in the gas phase were conducted for small neutral MAO structures and their anions, while electronic energies of the optimized structures were calculated at RI-MP2 /def2-TZVP level of theory. Frozen-core DLPNO-CCSD(T) /def2-TZVPD single point calculations were also carried out at the corresponding DLPNO-MP2/def2-TZVP optimized geometries. For ion-pairing studies reported in Table , the smaller def2-TZVP basis set was used with the DLPNO-CCSD(T) method. DLPNO-MP2, RI-MP2, and DLPNO-CCSD(T) calculations were carried out using Orca 5.0.3, with tight DLPNO thresholds for energy calculations and default thresholds for geometry optimizations. All other calculations were carried out using Gaussian 16. Quasi-harmonic corrections (qh) to the entropy and enthalpy were employed using the cut-off frequency of 100 cm -1 and corrections to reduced translational entropy (tr) in solution were calculated by the method described by Whitesides and coworkers. All the corrections were employed using the Goodvibes script, modified to include molarities and molecular volumes of fluorobenzene, which were required to calculate free volume. |
673db9b27be152b1d0d75696 | 31 | Theoretical study of small methylaluminoxane sheets (MeAlO)n(Me3Al)m indicates that DFT electronically stabilizes those featuring 5-coordinate Al and 4-coordinate O compared with higher level MP2 and CCSD(T) calculations which favor 4coordinate Al. The anions formed from these precursors all feature 4-coordinate Al and their relative stabilities explain the appearance of ESI-MS spectra recorded during the hydrolysis of Me3Al. |
67252d617be152b1d0b2c1ef | 0 | The discovery of new materials is needed for a wide range of applications from batteries, to catalysts, to solar cells, and more. Any new material is only of practical interest if it is stable, and therefore, the formation energy is an important property to predict for any new proposed material. Density functional theory (DFT) has long been used to study a wide variety of material properties, including formation energy. High throughput DFT calculations have been used to screen for stability and in some cases have yielded large databases of crystal structures and their formation energies. These databases have been used by many researchers for training machine learning (ML) systems to predict the stability of materials and attempt to discover new, stable materials. |
67252d617be152b1d0b2c1ef | 1 | Many material properties are calculated using density functional theory (DFT), but there always exists some difference between the DFT and experimental values. Modeling this error can help calibrate DFT results and make them more accurate. For example, the formation energy of crystal structures calculated using DFT are imperfect, and they are often corrected to more closely match experimental values. There is a set of about 1500 calorimetry experiments, published by Kingsbury et. al., for which both the experimental formation energy and the crystal structure are believed to be known. Figure clearly shows that there is a difference between theory and experiment, and applying a calibration, or "correction," would be a sensible way to improve the accuracy of DFT-calculated formation energies. In this work, we will discuss two classes of errors, how each can be modeled, and the benefits and drawbacks of accounting for each type of error. |
67252d617be152b1d0b2c1ef | 2 | Two classes of errors that may account for the theory-experiment gap are additive errors and proportional errors. An additive error does not scale with the property being measured; it is just a constant shift. For example, an uncalibrated scale with a container on it would always overestimate the weight of the object in the container by the same amount, regardless of the weight of the object. An equation of the form shown in equation 1 would be used to correct for this type of error, where P corrected is the corrected property value, P measured is the measured property value, and b additive is a constant correction factor, which would generally be fit to some calibration data where both P measured and a higher fidelity measurement of property P are known. |
67252d617be152b1d0b2c1ef | 3 | A proportional error scales with the property being measured, so larger prediction values come with larger errors. For example, a car's speedometer, which measures wheel rotation rate and converts that to a speed by assuming a tire radius, will overestimate the car's speed by the ratio of the real tire radius to the assumed tire radius. The error will be a constant multiple of the measured speed. So, this imperfect speedometer has zero error when the speed is zero, but as the speed increases, so does the magnitude of the error. An equation of the form shown in equation 2 would be used to correct for this type of error. |
67252d617be152b1d0b2c1ef | 4 | Previously published methods for correcting DFT formation energies assume that only additive errors are present. Specifically, they fit a constant energy shift to some or all of the reference state energies. Additive error correction implicitly assumes that the slope of the DFT formation energies versus the true energies must be equal to one. |
67252d617be152b1d0b2c1ef | 5 | The standard correction amounts to an additive correction to the reference state for each element (listed in the methods section), and the correction is only applied if the element is an anion in the chemical structure of interest. This method corrects for the clearly non-1 slope shown in figure by fitting larger correction values for more reactive elements, like fluorine and oxygen, and smaller correction values for less reactive elements, like antimony and hydrogen. |
67252d617be152b1d0b2c1ef | 6 | The standard correction method gives unphysical energy predictions for any composition that is near a reference state for which an additive correction is applied. If we start with any material and alter its composition and structure so that it becomes more and more similar to a reference state material, the formation energy must ultimately converge to zero; however, this does not hold true when applying the standard correction. |
67252d617be152b1d0b2c1ef | 7 | It is clear from equation 3 that as the system of interest approaches the reference state, and therefore the energy of the system of interest, E DFT , approaches the reference state energy, E DFT_reference , the corrected formation energy, ∆E formation , approaches the energy correction value, E correction (composition), not zero. |
67252d617be152b1d0b2c1ef | 8 | For example, if we start with a material that is 70% Br, 30% Cu, and we increase the fraction of bromine until 100% bromine is approached, the uncorrected DFT formation energy, E DFT -E DFT_reference , will approach zero regardless of the calculation details. However, if we apply a formation energy correction for bromine, as is done in the standard correction method, then the "corrected" formation energy will approach bromine's energy correction value, not zero. |
67252d617be152b1d0b2c1ef | 9 | This problem is avoided if E correction (composition) smoothly approaches zero as the reference state is approached. The standard correction method aimed to achieve this by only correcting anions. The number of anions in the structure should decrease smoothly to zero as the reference state is approached; however, in order to correct only the anions, one needs to be able to guess the oxidation state of the atoms in the structure, which is non-trivial and unreliable. This problem is described in the next section, "problem 2." |
67252d617be152b1d0b2c1ef | 10 | If that problem is not resolved, and E correction (composition) does not smoothly approach zero as the reference state is approached, then the very mathematical structure of the standard correction or any additive correction guarantees physically impossible predictions near the corner of the phase diagram. Concerningly, one would also expect this problem to extend at least partway into the rest of the phase diagram to a lesser but still potentially significant degree. It is worth noting that many of the correction values used in the standard correction method are on the order of -0.5 eV/atom, and energy differences on the order of 0.05 eV/atom are significant when considering the stability of a bulk material. If a given correction energy is -0.5 eV/atom, then size of this unphysical energy problem is exactly -0.5 eV at the cor-ner of the phase diagram. As one moves away from the corner of the phase diagram, for how long does a significant problem persist? The answer is not clear and use-case-dependent. The impact of the correction method on phase diagrams is shown later in this work, but even before any results are presented, this is a concerning idea, especially because all of the halogens have rather large energy corrections in the standard correction method, and transition metal halides are frequently 2/3 or 3/4 halogen by moles, so there are many commonly occurring compositions that are relatively close to the problematic corner of the phase diagram. |
67252d617be152b1d0b2c1ef | 11 | The problem is exacerbated by active learning because many active learning algorithms are designed to seek low formation energies, so structures with surprisingly low formation energies are sampled at a better-than-chance rate. If the correction value is negative, as is the case with many species in the standard correction method, structures near the corner of the phase diagram for a corrected species are predicted to be stable. Therefore, active learning algorithms trained on data corrected with the standard method will oversample the faulty region near the corner of the phase diagram. For examples of active learning exhibiting this behavior, refer to the phase diagrams in the supplemental information. In turn, this leads to false "discoveries" of new stable materials and encourages further exploration of this portion of the search space, with both DFT calculations and physical experiments, for an unfounded reason. |
67252d617be152b1d0b2c1ef | 12 | The standard correction method is only applied to when the corrected elements appear in a negative oxidation state, but this requires guessing the oxidation states of each species in the compound in order to compute the correction to the formation energy. Oxidation states are not trivial to guess for many structures, especially ones with semi-metals and non-metals that can exist in polyatomic anions, like chlorate. The standard correction method uses pymatgen's bond valance analyzer, then pymatgen's oxi_state_guesses method, then a maximum electronegativity method to guess the oxidation states. The methods are attempted in that order, so oxi_state_guesses is only used if BV-Analyzer fails, and maximum electronegativity is only used if both BVAnalyzer and oxi_state_guesses fail. |
67252d617be152b1d0b2c1ef | 13 | The first two methods are prone to failure for more exotic structures, and the fallback method, maxiumum electronegativity, is very simplistic and can yield incorrect predictions. For example, the most electronegative element in any chlorate salt is chlorine, but chlorine is not in a negative oxidation state. Additionally, for any structure that is mostly a non-metal with a very small amount of metal, this method would assume that all of the nonmetal atoms are negatively charged, when in fact only some of them are. |
67252d617be152b1d0b2c1ef | 14 | Furthermore, this series of oxidation state guesses is in principle subject to updates as the pymatgen code base is improved over time, and therefore, using the standard correction method, the predicted stability of a material and the very shape of the con-vex hull is subject to change based on the version of pymatgen. This is clearly an undesirable property of a correction method. |
67252d617be152b1d0b2c1ef | 15 | The assumption made by the standard correction method, that the DFT reference states are off by a shift, is a valid hypothesis, but if it were the correct hypothesis, the standard method would correct the non-1 slope. This is not what is seen when we consider oxides, where the standard correction has a large effect: In figure ; the non-1 slope in 1D is not corrected in 1E. This presents evidence that the hypothesis that only additive errors are present in DFT is either incorrect or insufficient. |
67252d617be152b1d0b2c1ef | 16 | In this work, we explore the possibility that proportional errors are present in DFT and some reasons to accept this possibility. First, a visual inspection of the uncorrected formation energy data shown in figure suggests that a non-1 slope may be a valid way to model the error that is present. Similar evidence appears when plotting atomization energies calculated using DFT versus the experimental values. |
67252d617be152b1d0b2c1ef | 17 | Furthermore, a very similar trend is seen with DFT-calculated band gaps. Work previously published by Wolverton et. al. shows a slope of about 0.9 on a plot of DFT-calculated band gap energies versus experimental band gap energies. Not only is the slope non-1, but also it is very similar to the slope observed in the formation energy case in figure . Furthermore, this is a raw DFT output, not a difference of DFT outputs, where error cancellation could cause additive errors to appear as proportional errors. |
67252d617be152b1d0b2c1ef | 18 | Finally, in the specific case of formation energy calculation, error cancellation may cause additive errors to appear as proportional errors. Assume, for instance, that a particular reference state is off by an additive error. One would expect the same additive error to appear in a system very similar to the reference state. Therefore, when the formation energy is calculated by subtracting the reference energy from the energy of this near-reference-state system, the additive error will mostly cancel, leaving behind a small additive error. The near-reference system will also have a small-magnitude formation energy. In summary, small-magnitude formation energy systems would likely exhibit small additive errors due to a large degree of error cancellation. However, if we consider a material that has a large-magnitude formation energy, it must be dissimilar from the reference state in a meaningful way. Therefore, when we calculate the formation energy, we would expect less error cancellation between the additive error of the system and that of the reference state. In summary, large-magnitude formation energy systems would be expected to exhibit larger additive errors due to a lesser degree of error cancellation. Fur-thermore, we would expect DFT's additive errors cause either an overestimation or an underestimation of the formation energies, not just a random increase in the spread of the formation energies we move away from the reference state. In mathematical terms, this means we would expect a proportional error, not a proportional absolute error. Altogether, we would expect this error cancellation effect to cause additive errors to appear as proportional errors in the case of formation energies. |
67252d617be152b1d0b2c1ef | 19 | Of course, K proportional can be factored out, and we will cancel out b additive , which yields equation 6 and leads directly into the correction method proposed in this work. It is possible that the additive error for the system of interest is not the same as that of the reference state, and therefore the two b additive values in equation 5 may not be exactly equivalent; however due to the error cancellation effect described in the previous section, the portion of the additive error that does not cancel out would likely appear as another proportional error. Therefore, to a first order approximation, such errors can also be modeled by K proportional once it is factored out, as shown next. |
67252d617be152b1d0b2c1ef | 20 | In this work, we propose an extremely simple, one-parameter, linear correction method that assumes DFT energies are indeed subject to proportional error. We call it the 110% PBE correction method because it multiplies energies given by DFT with the widely used Perdew-Burke-Ernzerhof (PBE) exchangecorrelation functional by a factor of about 1.1. The functional form is shown below in equation 6. |
67252d617be152b1d0b2c1ef | 21 | This correction method simply fits a zero-intercept line to the data shown in figure , resulting in the fit shown in figure 1B. Since it does not include per-element, additive reference state corrections, "problem 1" described in the previous section is eliminated. More specifically, it is clear from equation 6, that as the corner of the phase diagram is approached and E DFT_system approaches E DFT_reference , the corrected formation energy, ∆E formation , approaches zero regardless of the value of the correction constant, which is the only physically valid behavior. |
67252d617be152b1d0b2c1ef | 22 | Finally, figure shows that our correction performs well on the oxides, where the standard method does not correct the proportional errorr. So, "problem 3" is also resolved, and this provides evidence in support of the hypothesis that proportional error is present, and the 110% PBE correction method is really addressing the underlying problem. |
67252d617be152b1d0b2c1ef | 23 | Figure summarizes the fits for each model. In the column on the left, the formation energies are normalized per atom, and the column on the right, they are normalized per oxygen atom. Comparing figure to figure 1C suggests that the one-parameter 110% PBE correction method presented in this work fits the data better than the standard correction method. Figure shows that the 110% PBE correction method exhibits a mean absolute error (MAE) of 0.10 eV/atom, and figure shows that the standard method exhibits a validation error of 0.12 eV/atom, which is halfway between the MAE of the 110% PBE correction method and the MAE of the raw, uncorrected data. Furthermore, figure shows that the 110% PBE correction method continues to perform well when we consider the oxides alone while the standard correction does not. The 110% PBE model gives half the MAE of the standard model. |
67252d617be152b1d0b2c1ef | 24 | When we compute the MAE values for the plots in figure 1D-F per atom instead of per oxygen atom, we arrive at 0.25 eV/atom for plot D, 0.15 eV/atom for plot E, and 0.07 eV/atom for plot F. So, for this important subset of the data, the 110% PBE correction method performs better than on the broader dataset, whereas the standard method performs worse. Specifically, the MAE per atom for the 110% PBE method is reduced from 0.10 to 0.07 eV/atom (from figure to figure ), wherease the MAE per atom for the standard method is increased from 0.12 to 0.15 eV/atom (from figure to figure ). Further comparisons are plotted in the supplemental information. It is important to emphasize that the benefit of using the 110% PBE correction method instead of the standard method is the fact that it addresses the three key problems laid out in the introduction, not that it exhibits a meaningfully better mean absolute error. Addressing these problems is so important that the 110% PBE correction method would be preferable even if its mean absolute error were slightly higher; however, since it performs as well as the standard method, that tradeoff need not be analyzed. |
67252d617be152b1d0b2c1ef | 25 | First, switching from the standard to the 110% PBE correction method, several structures that were previously proposed as possible discoveries of new, stable materials now appear to be unstable. Figures and show binary phase diagrams for copperbromine and copper-sulfur, respectively. The blue points are from the Materials Project database, and the purple points are from the GNoME database. Any purple point below the blue line would constitute a new stable material, discovered in the GNoME project and not previously known by the Materials Project database. In figure , there are three purple points on the right side that represent new discoveries by the GNoME project, which used the standard correction method. These are predicted to be stable by 0.5 eV/atom below the previously known convex hull, which is a very large value. However, in figure , those same three points are shown using the 110% PBE correction method, and the structures appear to be slightly unstable or very close to the previously known convex hull. |
67252d617be152b1d0b2c1ef | 26 | Our result is far more likely to be correct because the original result would require copper-bromine bonds that are stronger than the strongest known bonds in chemistry. The point on the far right is CuBr 35 and has a corrected formation energy of -0.526 eV/atom under the standard method, which is, not coincidentally, very similar to the standard method's bromine energy correction: -0.534 eV/atom. Since there are 36 atoms in the unit cell, this means the formation energy per unit cell is -18.94 eV. This would imply that if one had pure bromine and added 1/36th, or 3% copper, the formation of the bonds made by those copper atoms would release 18.94 eV per copper atom. For reference, the N-N triple bond has a bond energy of about 9.8 eV. So, the bonds formed by each of those copper atoms would have to be more than twice as energetically favorable as combining atomic nitrogen radicals into N 2 , which is an unphysical result. |
67252d617be152b1d0b2c1ef | 27 | Figure shows the copper-sulfur binary phase diagram, and the shape of the convex hull and the energies of the points on it change depending on the correction method. The convex hulls shown in figure , and 3B are visually different. Additionally, figure 3A (standard correction) shows that CuS 2 is stable, whereas figure 3B (our correction) shows that it is not. CuS 2 would require copper to be in an unusual +4 oxidation state, which indicates that our correction again provides results that are more likely to be physically accurate. |
67252d617be152b1d0b2c1ef | 28 | Finally, the stability of the most stable materials on the binary phase diagrams presented in this work are quite different depending on which correction method is applied. In figure (standard In the left column, all data is plotted where both DFT and experimental energies are known, and the energies are normalized per atom. In the right column, the energies are normalized per oxygen atom, and only structures that contain oxygen are plotted. In the first row (A and C), no correction is applied. In the second row (B and E), the standard correction is applied. In the third row (C and F), the 110% PBE correction is applied. All DFT+U calculations and structures are excluded here. Note that the mean absolute error (MAE) is higher in the column on the right because it is normalized by oxygen atom. The corresponding MAE values, normalized per atom are 0.25 eV/atom for plot D, 0.15 eV/atom for plot E, and 0.07 eV/atom for plot F. correction), the most stable CuS structure has a formation energy of about -0.45 eV/atom, and in figure 3B (our correction), the same structure has a formation energy of only -0.2 eV/atom. Both of these formation energies constitute stable compounds, but this is a large difference in formation energy that may have implications on how easy or hard the material likely is to synthesize. |
67252d617be152b1d0b2c1ef | 29 | Looking at formation energies published by the National Institute of Standards and Technology (NIST) indicates that our correction provides more accurate results in some cases where the two correction methods give very different corrected formation energies. For example, according to NIST, the formation energy of zirconium (IV) chloride is -1.6 eV/atom. According to our correction, the formation energy of the most stable zirconium (IV) chloride structure in the materials project or GNoME databases is -1.6 eV/atom, and with the standard correction, the value is -1.9 eV/atom. This provides some further evidence that our correction gives more accurate results. These findings are not isolated to the particular binary phase diagrams shown in figures 2 and 3. Figure shows that, for every binary composition for which there is data, switching from the standard correction method to the 110% PBE correction method always results in a decrease in the number of discovered materials. |
67252d617be152b1d0b2c1ef | 30 | the 110% PBE correction method is also more robust than the standard correction method when the training data is randomly subsampled. Figure shows how the corrected formation energy changes as the training data is randomly subsampled. For each correction method, the structure whose energy changes the most as the training data is re-subsampled, is shown in figure . In this worst-case scenario, using the 110% PBE correction method causes a 0.05 eV/atom change in the corrected formation energy (95% confidence interval). However, using the standard correction method, this figure increases to 0.22 eV/atom. |
67252d617be152b1d0b2c1ef | 31 | We compared the standard correction and the 110% PBE correction to results from the r2SCAN exchange-correlation functional. r2SCAN is believed to be more accurate than PBE, but it is five times as computationally expensive. We fit our linear method with a correction for each of the Hubbard U elements (discussed in more detail in the supplemental information, called the 110% PBE method) to the experimental data. Then, we applied both the 110% PBE method and the standard correction method to the r2SCAN data. Figure shows that the 110% PBE model outperforms the standard correction method by about a factor of two with respect to the mean absolute difference (MAD). Furthermore, the 110% PBE method compares to experiment only marginally worse than the much more expensive r2SCAN. The MAD between r2SCAN and experiment is 0.082 eV/atom, and the MAD between the 110% PBE model and experiment is 0.096 eV/atom. So, for use cases where computational expense is a factor, applying our correction to PBE may be preferable to using r2SCAN. It seems possible that the biggest benefit of SCAN is that the slope is closer to one, which avoids the need for additive corrections. Work by Kingsbury et. al. discusses the value of being able to mix results from PBE and r2SCAN . We suggest that applying the 110% PBE correction method to the PBE data is a good first step toward this end goal. |
67252d617be152b1d0b2c1ef | 32 | In this work, we propose one new correction method, but in the future, others may be published as well. Therefore, whenever publishing data, it is imperative that the uncorrected energies are published. This way, the users of the data can select the correction method that is most relevant to their use case or develop new correction methods. |
67252d617be152b1d0b2c1ef | 33 | Additionally, these large corrections on anions can do more harm than good in certain circumstances. For example, when analyzing the convex hull and which materials are on it, it is more accurate to use the uncorrected energies than the standardcorrected energies. If the absolute value of the formation energy is important, then using the 110% PBE method is likely to help. In the supplemental information, we plot different subsets of the data, comparing to both experiment and r2SCAN, to show that the 110% PBE correction method outperforms the standard correction by method more and more as the standard correction method has a greater effect. This suggests that the key benefit of our simple, lightweight correction is to avoid the harm that can be caused by the standard correction method. |
67252d617be152b1d0b2c1ef | 34 | The fundamental idea that underlies this work is that DFT may indeed be subject to proportional error, and this idea has implications wherever DFT data is fit to experiment. For example, in DFT+U, the Hubbard U values are generally fit to experimental band gaps and formation energies. Pseudopotentials also often include a fit to experimental energies. In both cases, proportional errors are usually ignored in the fit. If proportional errors, not just additive errors, were modeled while fitting Hubbard U values or pseudopotentials, one would arrive at a different and perhaps better fit. More broadly, for anyone using experimental data to calibrate DFT data, it may be useful to consider the presence of proportional errors. This idea leaves room for a wide variety of future work. |
67252d617be152b1d0b2c1ef | 35 | In this work, we suggest that DFT is subject to proportional error. We show how the standard formation energy correction method, which considers only additive errors, causes unphysical formation energy predictions near the corners of phase diagrams and how that can lead to incorrect conclusions and false discoveries. We propose a one-parameter correction method, which considers proportional error, and show that it does not suffer from the same problems. We also show that it is more robust than the standard correction, and that the 110% PBE method outperforms the standard correction method with respect to matching both experiment and higher level theory: the r2SCAN functional. Overall, we stress the importance of publishing uncorrected results and considering proportional error in all cases where DFT data is fit to experiment. This plot demonstrates the improved robustness of our correction compared to the standard correction with respect to randomly subsampling the training data. 30% of the data was held out for validation, and the remaining 70% of the data was used for training but subsampled randomly 40 times. Each draw selected a random half of the training data. Then, for each draw, both the standard correction method and our method were fit on the training data and used to correct the holdout data. The standard deviation of the predictions on all points in the holdout dataset were calculated, and the one that varied the most is plotted here. In other words, the structure whose corrected formation energy was most sensitive to randomly subsampling the training data is shown here. Specifically, the residual (actual -predicted) formation energy per atom is plotted using correction models fit to each random subset of the training data. The 95% confidence interval is also plotted. and frequent conversations with SKS and JSB. BAR wrote the paper, and JSB, and SKS provided edits and comments. |
678edcb0fa469535b9e18efc | 0 | Analytic representations of coupled potential energy surfaces (PESs) of polyatomic molecules are valuable resources because one can perform detailed dynamics simulation of the corresponding system with appropriately large ensembles, ultralong simulation times, accurate electronic structure (which would be unaffordable without fitting), and accurate dynamics algorithms. The development of PES-learning methodologies has enabled highly accurate fitting of coupled global PESs for complex systems and systems with dense manifolds of states. Critical aspects of an accurate learning of a set of global coupled PESs include: (1) The fit should maintain -exactly or to a practically useful approximation -the invariance properties of potential energy surfaces, namely the invariance with respect to the translation, rotation, and permutation of identical nuclei. (2) The PESs should become flat in asymptotic regions, i.e., when a subsystem is dissociated from the rest of the system. (3) For coupled PESs in the regions where adiabatic PESs cross, i.e. conical intersections, the learned surfaces should provide the correct topology of the conical intersections without the potential error of double crossing. |
678edcb0fa469535b9e18efc | 1 | Over the years, various methods have been developed to attain these desirable features. For example, the potential energy invariance with respect to translation and rotation and invariance with respect to permutation of identical nuclei is often achieved by coordinate transformation, that is, one represents the molecular geometry by appropriate internal coordinates instead of atomic Cartesian coordinates, and by using permutationally invariant polynomials of internuclear distances, or constraining permutation invariance in equivariant message-passing graph neural networks. Proper asymptotic behavior can be challenging because one often has insufficient data for training because the asymptotic space is infinite, but it can achieved by constraining the form of the fit, for example by using a parametrically managed activation function. Correct topology of the PESs in the vicinity of conical intersections can be enforced by transforming the learning target from adiabatic PESs and nonadiabatic couplings into diabatic potential energy matrices (DPEMs). We have developed a series of methods to achieve semiautomatic learning of coupled adiabatic PESs and their couplings. The first generation of these semiautomatic methods is called diabatization by deep neural network (DDNN); it automatically discovers the DPEMs on-the-fly during the learning of adiabatic PESs, and therefore it greatly reduces the human effort by eliminating the need to construct DPEMs as a pre-fitting step, for example by finding diabatic molecular orbitals or diabatic configuration state functions. Next we presented permutationally restrained DDNN (PR-DDNN) in which we added approximate permutation invariance by adding a permutational-invariance term in the cost function. Our next version, called parametrically managed DDNN (PM-DDNN), enforces physical asymptotic behavior of PESs by introducing a parametrically managed activation function in the architecture of PR-DDNN. With these developments, the coupled PESs learned by PM-DDNN include all three of the desirable features mentioned above. |
678edcb0fa469535b9e18efc | 2 | Our recently published coupled PESs of the O3 system in the dense manifolds of 3 A′ and 5 A′ states have demonstrated the power of PM-DDNN. Construction of useful analytic representations for such dense manifolds of states would have been almost impossible with traditional diabatization methods. This is especially notable because our coupled PESs are global, i.e., valid for any nuclear geometry, whereas many surface fittings in the literature are focused only on the potential along a single reaction coordinate or in a semiglobal domain. |
678edcb0fa469535b9e18efc | 3 | We have been especially interested in global coupled PESs of small atmospheric molecules due to their significant role in determining thermal energy content and heat flux in the flows around hypersonic vehicles. For example, the exothermic recombination of atomic oxygen on the surface of vehicles contributes to heat shield erosion. Among the open questions are if and when electronically excited species of O2 and O may be generated and to what extent these high-energy excited species affect the species concentrations, state populations, chemical reactions, energy relaxation, and heat balance of the ultrahightemperature environment in the strong shock waves. Electronically nonadiabatic dynamics simulations (i.e., simulations in which the electronic state changes) of O + O 2 collisions can be valuable in addressing this question. The starting point for such simulations is a set of analytic representations of the dense manifolds of global coupled PESs of O3 in the various electronic symmetries. |
678edcb0fa469535b9e18efc | 4 | In this work, we report an improvement of our previously published 14 global coupled PESs of 3 A′ states of O3. The improvement includes three aspects: (1) We used an improved functional form for the parametrically managed activation function; (2) We obtained smoother PESs by introducing a higher weight for low-lying states; (3) We improved the asymptotic behavior of the coupled PESs by using a better low-dimensional potential in the parametric management step. These improvements are important for improving simulations of O + O2 collisions, and they are of broader interest as well because they are generally applicable to PES fitting of other systems. |
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