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The active set contains four oxygen pi orbitals and two carbon pi orbitals, with eight electrons in six orbitals. The noninteracting levels, using the parameters inferred below, areEach bonding oxygen orbital has equal-phase coefficients on its two O atoms; its antibonding partner reverses one coefficient. The two pi directions are z in the prospective ring plane and y perpendicular to it. carbon pi and pi-star use z-directed p orbitals with equal and opposite coefficient signs. In this separated-fragment comparison the occupations are those of neutral triplet O${}_2$ plus neutral ethene. Do not refill the combined list solely by energy, which would move an electron between infinitely separated fragments and describe a charge-transfer state instead.

Use negative bonding Hückel resonance integrals, ${\beta }_{OO}=-2.0$ and ${\beta }_{CC}=-2.5\mathrm{eV}$. The oxygen HOMO is pi-antibonding, whereas the ethene HOMO is pi-bonding. The frozen-orbital Koopmans estimate therefore givesConsequently ${\alpha }_O\simeq -14.1\mathrm{eV}$ and ${\alpha }_C\simeq -8.0\mathrm{eV}$. These Hückel Coulomb parameters are effective parameters of this spectroscopically calibrated model. Real ionization energies include relaxation and correlation, so they are not exact matrix elements of an isolated-atom Hamiltonian.

Let ${\sigma }_L,{\sigma }_R$ be localized bonding orbitals of the two new C–O bonds, and let ${\sigma }_L^{\ast },{\sigma }_R^{\ast }$ be their antibonding partners. Their symmetry combinations areThe two remaining active orbitals are out-of-ring oxygen lone-pair combinations, $n_{\pm }=(p_{y,L}\pm p_{y,R})/\sqrt{2}$. The opposite-phase combination $n_{-}$ is the HOMO in the supplied ordering and $n_{+}$ is HOMO−1. Both are filled, as are the two C–O bonding combinations, givingThe original active-orbital sketches show these phase relations, while the correlation diagram gives a qualitative energy ordering. Precise splittings between unrelated symmetry blocks are not determined by the fragment ionization energies. The retained O–O sigma bond, the C–C sigma bond and the other framework orbitals lie outside this six-orbital active comparison; the eight-electron count does not describe all valence electrons of the product.

In the electronic state correlation diagram, the singly occupied reactant orbitals for triplet oxygen have symmetries $B_1$ and $A_2$, so their spatial product is $B_2$. The reactant state is therefore a triplet ${}^3{B}_2$, whereas the closed-shell product is ${}^1{A}_1$. A strictly spin-conserving path cannot correlate triplet reactants directly to the ground-state product. It correlates to an excited triplet product configuration or requires intersystem crossing; the symmetric direct reaction consequently has a substantial electronic obstacle.

Simply changing the oxygen spin does not repair the occupied carbon-orbital correlation. The lowest oxygen singlet manifold has spatial components of $A_1$ and $B_2$ symmetry in this subgroup. The $A_1$ singlet can have an avoided crossing with another $A_1$ configuration and reach the product singlet; the $B_2$ singlet cannot mix directly with the product ${}^1{A}_1$ while $C_{2v}$ is maintained. Following either closed-shell oxygen singlet component in the orbital diagram still leaves the carbon-pi electron pair in the rising ${\sigma }_{+}^{\ast }$ orbital. Its redistribution into the final occupied orbitals requires configuration mixing at an avoided crossing. A barrier is therefore expected even for the symmetry-constrained concerted singlet addition; singlet does not imply barrierless. The state curves in the original diagram are schematic, illustrating this avoided-crossing argument rather than computed energies.^[ref]

The real singlet reaction can lower this obstacle through asymmetric or stepwise pathways that are absent from the stipulated direct approach. A primary multireference study explicitly distinguishes singlet and triplet addition through biradical intermediates, supporting that limitation of a one-coordinate symmetry argument.^[ref] Neither correlation alone nor a spin label supplies a numerical activation energy.

The two substituents are $-\mathrm{C}{\mathrm{H}}_2-\mathrm{O}-\mathrm{C}(\mathrm{O})-\mathrm{CH}=\mathrm{CH}-\mathrm{Ph}$, with the alkene E, and $-\mathrm{C}(\mathrm{OH})(\mathrm{C}{\mathrm{H}}_3{)}_2$. The free bond at the left of each drawing attaches to the cyclohexene.

The A-D COSY correlation and $16.2\mathrm{Hz}$ coupling identify the trans cinnamate double bond. Five aromatic protons and four phenyl-carbon environments identify its monosubstituted phenyl ring. F is a two-proton OCH${}_2$ signal; its HMBC correlation to carbonyl 1 joins that methylene through oxygen to the ester. The two three-proton singlets N/O, their correlations to oxygenated quaternary carbon 10, and the missing OH resonance give the tertiary-alcohol substituent.

For one specified channel, the photolysis rate coefficient isIt depends on spectral actinic flux, absorption cross section, and channel quantum yield. Solar zenith angle, season, overhead absorbers, clouds and aerosol scattering affect the photon field; branching and collisional quenching affect the chemical yield. Absorption alone does not guarantee the desired dissociation.

Curve A is oxygen photodissociation; curve B is ozone photodissociation. Molecular oxygen needs relatively short-wavelength ultraviolet photons, which are strongly screened at lower altitudes, so its rate increases steeply with height. Ozone can dissociate using longer wavelengths that penetrate much further, producing the much larger, less height-sensitive rate shown by B. The plotted quantity is a first-order coefficient with units ${\mathrm{s}}^{-1}$, rather than a bimolecular rate constant.

The dominant OH-to-HO${}_2$ pseudo-first-order rate is $k_5[{\mathrm{O}}_3]=90{\mathrm{s}}^{-1}$, corresponding to $0.0111\mathrm{s}$. Recycling HO${}_2$ through atomic oxygen and NO has rate $k_6[\mathrm{O}]+k_8[\mathrm{NO}]=0.0218{\mathrm{s}}^{-1}$, corresponding to $45.9\mathrm{s}$ for an individual HO${}_2$ radical. A fixed-total two-state exchange relaxes at the sum of forward and reverse rates, giving about $0.0111\mathrm{s}$ for the partition perturbation. Its fast partition is thereforeThis supports a steady-state approximation in chemical kinetics for rapidly recycled intermediates if the parent concentrations vary slowly. It does not by itself establish stationarity of the entire HOx family. The later source–termination estimate gives a family turnover time of about 16 hours, far longer than partition relaxation; a stationary absolute abundance additionally assumes sufficiently persistent radiation and background conditions. This is the distinction between fast family partition and slow family balance.

With $[{\mathrm{H}}_2]=5\times 10^{-6}[\mathrm{M}]$, a reaction of an electronically excited oxygen atom with hydrogen gives one OH and, after rapid H conversion, one HO${}_2$. The radical source becomesThis is about 4.75 times the original source. Holding water, oxygen, NO and radiation fixed keeps the fast radical partition approximately unchanged, while bimolecular termination makes total radical abundance scale as $\sqrt{S}$. Thusabout 2.18 times the original catalytic loss. This predicts a stronger ozone-loss tendency within the specified stationary box model, not a numerically determined percentage change in the ozone layer. The latter requires coupled odd-oxygen production, changing ozone concentrations, transport and the longer-term water budget; those have been held fixed here.

At night, nitrate radical addition and ozone attack are relevant oxidations of isoprene; direct daytime photochemistry is absent, though small OH sources can still exist. The nitrate-containing products identify the NO${}_3$-addition route. Add NO${}_3$ to the terminal carbon next to the methyl-substituted alkene carbon, producing a nitrooxy substituent and an allylic radical. Resonance moves the radical to the far terminal carbon while shifting the double bond. Oxygen trapping gives a peroxy radical with the carbon skeletonA nitrate-forming termination branch, for example the organic-nitrate branch of reaction with NO, replaces the terminal peroxyl group by a nitrate ester, giving ${\mathrm{O}}_2\mathrm{NOC}{\mathrm{H}}_2-\mathrm{C}(\mathrm{C}{\mathrm{H}}_3)=\mathrm{CH}-\mathrm{C}{\mathrm{H}}_2\mathrm{ON}{\mathrm{O}}_2$. An alternative branch forms an alkoxy radical, such as peroxy-radical reaction with NO${}_3$ producing RO${}^{\bullet }$, NO${}_2$ and O${}_2$. Reaction of the terminal CH${}_2$O${}^{\bullet }$ group with O${}_2$ then gives an aldehyde plus HO${}_2$, producing ${\mathrm{O}}_2\mathrm{NOC}{\mathrm{H}}_2-\mathrm{C}(\mathrm{C}{\mathrm{H}}_3)=\mathrm{CH}-\mathrm{CHO}$. These are covalent O–NO${}_2$ ester groups, not C–NO${}_2$ nitro groups. The delocalized radical can be trapped after conformational change, so this mechanism does not require one universal alkene geometry for both product channels. Product structures alone do not fix a unique branching ratio.

Base first gives phenoxide. Use its para-carbanion/quinoid resonance contributor: the para carbon bearing the tether attacks the terminal chloromethylene carbon in an intramolecular substitution, ejecting chloride. This forms a five-membered ring at that carbon and produces a para-spirocyclohexadienone, with the phenoxide oxygen becoming a carbonyl oxygen. The first stage is a dearomatizing spirocyclization of a phenoxide.

On acidification, carbonyl protonation makes the dienone electrophilic. A tether C–C bond attached to the spiro centre migrates to the adjacent dienone carbon, with the accompanying pi-bond shifts; loss of a proton restores aromaticity. This dienone-phenol rearrangement converts the spiro five-membered ring into the fused six-membered ring while regenerating the phenolic OH. The apparent bond formation meta to OH is thus explained by para spirocyclization followed by skeletal migration, rather than requiring the final fused connectivity to form directly in the base step.

Activate the oxime hydroxyl oxygen with thionyl chloride to give a good N–O leaving group. In the Beckmann rearrangement branch, the C–C bond anti to that leaving group migrates onto nitrogen as the N–O bond breaks. Here the migrating carbon is the spiro centre. Inserting N between that centre and the former oxime carbon expands the five-membered oxime ring to a six-membered nitrilium ion ring. Water attacks the electrophilic nitrilium carbon, followed by proton transfers and tautomerization to the observed spiro lactam. The lactam carbonyl oxygen is introduced in the aqueous work-up.

The competing Beckmann fragmentation cleaves the same anti C–C bond instead of completing migration onto N. Formation of the C≡N bond gives an open-chain nitrile and a tertiary carbocation at the former spiro centre. Loss of a proton from the adjacent chain carbon then gives the exocyclic alkene attached to the retained cyclopentane. The key distinction is N insertion and ring expansion for the lactam, versus C–C cleavage followed by elimination for the unsaturated nitrile. Stabilization of the resulting tertiary carbocation makes fragmentation plausible.

B contains both $-\mathrm{C}{\mathrm{H}}_2-\mathrm{CH}(\mathrm{Cl})-$ and $-\mathrm{C}{\mathrm{H}}_2-\mathrm{CH}(\mathrm{OCOC}{\mathrm{H}}_3)-$ monomeric units distributed along the same carbon backbone: a vinyl chloride-vinyl acetate copolymer. One possible local segment isThe ester is attached through oxygen to the backbone carbon. A random copolymer should not be drawn as two long homopolymer blocks. The stated $75:25$ monomer feed is not enough to prove an incorporated $75:25$ unit ratio: monomer reactivity ratios and conversion affect composition and sequence statistics. Under an explicitly ideal equal-incorporation assumption that ratio could be used as a schematic average, not a periodic repeat.

Saponification cleaves the acetate ester side groups while the intended carbon backbone remains intact:The ideal target C is therefore a vinyl chloride-vinyl alcohol copolymer, containing $-\mathrm{C}{\mathrm{H}}_2-\mathrm{CH}(\mathrm{Cl})-$ and $-\mathrm{C}{\mathrm{H}}_2-\mathrm{CH}(\mathrm{OH})-$ units in the original sequence positions. The carbon–oxygen bond to the backbone remains; hydrolysis removes the acetyl group from oxygen. Incomplete hydrolysis would leave a three-component population of chloride, acetate and alcohol units. New hydroxyl groups can strengthen adhesion and coating interactions through hydrogen bonds.

The first equivalent of titanocene methylidene methylenates the ester: the carbonyl oxygen and titanium form an oxatitanacyclobutane, whose fragmentation gives $\mathrm{PhC}(\mathrm{OMe})=\mathrm{C}{\mathrm{H}}_2$ and a titanium–oxygen product. The second equivalent adds across this electron-rich double bond. The initial adduct is a titanacyclobutane with cyclic connectivity $\mathrm{C}{\mathrm{p}}_2\mathrm{Ti}-\mathrm{C}{\mathrm{H}}_2-\mathrm{C}(\mathrm{Ph})(\mathrm{OMe})-\mathrm{C}{\mathrm{H}}_2-$, closing from the last methylene to titanium. The ester's phenyl-substituted carbon lies opposite Ti; the two new methylenes originate from the two methylidene equivalents.

Beta-methoxy elimination opens that metallacycle to B, $\mathrm{C}{\mathrm{p}}_2\mathrm{Ti}(\mathrm{OMe})[\mathrm{C}{\mathrm{H}}_2\mathrm{C}(\mathrm{Ph})=\mathrm{C}{\mathrm{H}}_2]$, the allyltitanium reagent. The C–OMe bond breaks, OMe transfers to Ti, one Ti–C bond becomes the new C=C bond, and the other Ti–C bond remains as the allylic nucleophile. The cyclic adduct is therefore a precursor to the opened reagent B; treating B as merely the one-equivalent enol ether loses the second reagent carbon.

The aldehyde oxygen coordinates to titanium. Allylic carbon transfer to its carbonyl carbon, accompanied by movement of the C=O pi electrons to oxygen, gives the titanium alkoxide $\mathrm{PhC}{\mathrm{H}}_2\mathrm{C}{\mathrm{H}}_2\mathrm{CH}(\mathrm{O}-\mathrm{Ti})\mathrm{C}{\mathrm{H}}_2\mathrm{C}(\mathrm{Ph})=\mathrm{C}{\mathrm{H}}_2$. Acidic aqueous workup cleaves O–Ti and protonates oxygen, giving $\mathrm{PhC}{\mathrm{H}}_2\mathrm{C}{\mathrm{H}}_2\mathrm{CH}(\mathrm{OH})\mathrm{C}{\mathrm{H}}_2\mathrm{C}(\mathrm{Ph})=\mathrm{C}{\mathrm{H}}_2$. A six-membered allyl-transfer arrangement is a convenient mechanistic model. No enantioselectivity is imposed by these achiral reagents.

Schwartz reagent adds syn across the terminal alkene, placing zirconium at its terminal carbon and hydrogen at the substituted carbon. The organozirconium intermediate is $\mathrm{PhCH}(\mathrm{OMe})\mathrm{CH}(\mathrm{Me})\mathrm{C}{\mathrm{H}}_2\mathrm{ZrC}{\mathrm{p}}_2\mathrm{Cl}$. The two faces of the achiral starting alkene give enantiomeric organometallic intermediates.

Boron trifluoride coordinates the ether oxygen, making the benzylic C–OMe bond susceptible to cleavage. The C–Zr bond then transfers its carbon terminus intramolecularly to the benzylic carbon as the activated methoxy group departs, closing a cyclopropane. This can be represented as a concerted, strongly assisted displacement rather than requiring a freely equilibrating long-lived benzylic carbocation. hydrozirconation fixes the relative disposition of the new C–H and C–Zr bonds, and the preferred closure conformation places the bulky phenyl and methyl groups on opposite faces. The result is racemic trans-1-methyl-2-phenylcyclopropane, with much less steric crowding than the cis alternative. The ring contains the original two alkene carbons and the original benzylic carbon; no carbene carbon is added.

Let $\mathrm{Ar}=2,6$-dimethylphenyl and $\mathrm{A}{\mathrm{r}}^{\prime }=2$-methoxyphenyl. After conversion of the Pd(OAc)${}_2$ precatalyst into an active phosphine–Pd(0) species, oxidative addition of $\mathrm{A}{\mathrm{r}}^{\prime }\mathrm{Cl}$ gives ${\mathrm{L}}_{\mathrm{n}}\mathrm{P}{\mathrm{d}}^{\mathrm{II}}(\mathrm{A}{\mathrm{r}}^{\prime })\mathrm{Cl}$. Carbonate-assisted replacement of chloride by the alcohol's alkoxide produces ${\mathrm{L}}_{\mathrm{n}}\mathrm{P}{\mathrm{d}}^{\mathrm{II}}(\mathrm{A}{\mathrm{r}}^{\prime })[\mathrm{OC}(\mathrm{Me}{)}_2\mathrm{Ar}]$.

In the key beta-carbon elimination, the alkoxide oxygen forms a C=O bond while the bond between its carbinol carbon and $\mathrm{Ar}$ cleaves, transferring $\mathrm{Ar}$ to palladium. This gives acetone and a diarylpalladium(II) complex, ${\mathrm{L}}_{\mathrm{n}}\mathrm{P}{\mathrm{d}}^{\mathrm{II}}(\mathrm{A}{\mathrm{r}}^{\prime })(\mathrm{Ar})$. reductive elimination creates the new biaryl bond and regenerates Pd(0). Thus the tertiary alcohol functions as an aryl donor by carbon–carbon bond cleavage, rather than being incorporated as a carbinol substituent. The overall organic products are the cross-coupled biaryl and acetone. The carbinol carbon has no hydrogen, so ordinary beta-hydride elimination from this alkoxide cannot explain the observed aryl transfer. The bulky electron-rich phosphine helps access the aryl-chloride oxidative addition and the sterically congested biaryl-forming elimination.

Both free primary allylic alcohols undergo Sharpless asymmetric epoxidation. With slightly more than two equivalents of hydroperoxide, the expected fully oxidized product is the diepoxide, with both original C=C bonds converted to epoxides and both primary alcohols retained. The two central pre-existing stereocentres and their ether substituents are retained.

The stereochemical construction is unambiguous by applying the L-(+)-tartrate mnemonic separately at each end: orient that alkene horizontally with its CH${}_2$OH group at lower right, and deliver oxygen from the back face. In the source's original in-plane orientation this gives the left epoxide oxygen behind the plane and the right epoxide oxygen in front. The methyl and hydroxymethyl groups retain their original relative arrangement about each former double bond. One cannot simply draw both oxygens on the same face of the original extended chain: the two allylic-alcohol ends have opposite drawing orientations. titanium binding to each free hydroxyl directs oxygen transfer; the pre-existing stereocentres can create matched/mismatched rate differences, but the stated reagent control predicts the stereochemistry of both epoxidations. Molecular sieves maintain a dry active titanium–tartrate system.^[ref]

acetone first forms a proline enamine by reversible iminium formation and alpha-deprotonation. The enamine attacks the aromatic aldehyde in a hydrogen-bonded arrangement involving the catalyst's carboxylic acid. hydrolysis of the resulting iminium releases the beta-hydroxyketone and regenerates proline. The usual L-proline arrangement gives the $R$ secondary alcohol.

The sterically crowded aryl–amide axis is selected at the same bond-forming step: rapid rotation in the aldehyde permits the catalyst to react preferentially with the conformer that puts the smaller carbonyl-oxygen side of the amide towards the approaching aldol transition structure and the bulky diisopropylamino side away. A chair-like enamine–aldehyde model minimizes the interaction of the amide with the catalyst-side axial hydrogen; reversing the amide orientation creates the less favourable diastereomeric transition state. Replacing planar CHO by tetrahedral CH(OH)CH${}_2$COMe then raises the rotation barrier and traps the axis. This is a dynamic kinetic resolution, not resolution of two configurationally stable starting atropisomers.

To specify the predicted major axis without relying on a flat bond-line formula, look from the aromatic ipso carbon towards the amide carbonyl carbon. Put the higher-priority aryl path, the trimethylsilyl-bearing ortho side, at nine o'clock. The preferred model puts the carbonyl oxygen at approximately six o'clock and the diisopropylamino group at twelve o'clock. This is $S_a$ using the near-to-far high-priority convention; oxygen outranks nitrogen at the far end. Coupled with the alcohol assignment, the model predicts $(R,S_a)$. The drawing defines the configuration even if an alternative viewing direction is used. The dynamic-resolution mechanism is established by the primary aldol study; the product-axis assignment here is a stereochemical-model prediction for the printed substrate.^[ref]

The conventional lattice-defect description is vacancy-mediated diffusion: an ion moves into a neighbouring vacancy and leaves a vacancy behind. sodium chloride is another familiar example. In a real alpha-alumina specimen the dominant species and transport path depend on oxygen chemical potential, dopants and grain boundaries, so this label should not be read as a universal microscopic assignment for every aluminium and oxygen diffusion experiment. A vacancy mechanism is different from an interstitial ion moving between otherwise empty interstitial sites.

The free allylic alcohol directs epoxidation of the adjacent alkene rather than the remote double bond. Applying the D-(−)-tartrate mnemonic gives the epoxide oxygen on the back face in the source orientation, with both new C–O bonds dashed if the original six-membered ring is kept in the plane. The CH${}_2$OH substituent at the lower epoxide carbon consequently projects towards the front in an explicit tetrahedral product drawing. The remote double bond is retained. The opposite tartrate would reverse both newly created epoxide stereocentres; it would not alter the location of the epoxidized alkene.