Use the D3h molecular point group for phosphorus pentafluoride, with $z$ along the two axial P–F bonds. For each symmetry operation, only atoms left in place contribute to the character of the Cartesian displacement representation; each contributes the trace of the corresponding $3\times 3$ spatial matrix. In the class order $E,2C_3,3C_2^{\prime },{\sigma }_h,2S_3,3{\sigma }_v$ this givesFor example a $C_2^{\prime }$ fixes P and one equatorial F, and each fixed atom has Cartesian trace $-1$, giving $-2$. reflection in the equatorial plane fixes four atoms, each with trace $1$, giving $4$. Reduction with the character table givesSubtract translations $E^{\prime }+A_2^{\prime \prime }$ and rotations $A_2^{\prime }+E^{\prime \prime }$ to obtainIts dimension is $2+6+2+2=12=3(6)-6$, as required for a nonlinear six-atom molecule. Thus there are eight symmetry-distinct fundamental frequencies in the harmonic picture, with each E mode doubly degenerate. This analysis concerns vibrations about one trigonal-bipyramidal minimum; it does not treat large-amplitude rearrangements between minima.

Using mass $7\mathrm{u}$ for the dominant lithium isotope gives a transition wavenumber ${\tilde{\nu }}_0=14925\mathrm{c}{\mathrm{m}}^{-1}$. Inserting the thermal speed into the stated Doppler broadening expression givesThe supplied formula is the Gaussian half width at half maximum; the corresponding full width is about $0.256\mathrm{c}{\mathrm{m}}^{-1}$. Meanwhile exponential population decay with lifetime $\tau$ produces a natural linewidth expressed as a full widthHere $c$ must be in $\mathrm{cm}{\mathrm{s}}^{-1}$ in the second expression. Equivalently the fine-structure separation is about $9.0$ GHz, the Doppler half width about $3.83$ GHz, and the natural linewidth expressed as a full width about $5.89$ MHz. The natural linewidth is therefore negligible compared with thermal broadening; low density also makes collision broadening less important.

The doublet is resolvable in principle, but only marginally against the Doppler width. The separation exceeds one Doppler full width, so two components can survive, although their profiles overlap strongly. The $J=3/2$ and $J=1/2$ upper levels have different statistical weights, so the weaker member need not make an equally conspicuous maximum. A spectral resolving power $R\gtrsim {\tilde{\nu }}_0/0.3\simeq 5\times 10^4$ is the basic instrumental requirement; a width well below $0.1\mathrm{c}{\mathrm{m}}^{-1}$ and adequate signal would make the test more convincing. An arbitrarily fine spectrometer cannot remove the physical Doppler overlap. This conclusion uses the stipulated thermal, optically thin picture; additional photospheric velocity fields or optical-depth effects would require more information.

Neither formula requires a bridging carbonyl to repair an electron deficiency: the terminal-CO donor–acceptor structures in (d) and (e) already give an electron-counting rationale for the metal–metal interaction. A predominantly terminal-carbonyl model is a reasonable starting proposal for A; bridging or semibridging CO is more geometrically plausible for the shorter Fe–Fe contact in B. Larger metal separation makes a carbonyl simultaneously engaging both Mo atoms less favourable, while carbonyl redistribution or tilting can be easier around a compact Fe pair.

This is a qualified structural prediction, not a determination from molecular formula. A bridging CO can change the relative hapticity and local electron allocation, and the two metal centres are inequivalent; a large frequency shift by itself need not prove a symmetric bridge. Infrared carbonyl frequencies combined with diffraction would test terminal versus bridging or semibridging assignments. The given low-temperature preparation can trap a coordination pattern before complete carbonyl redistribution.

The wording places both dicarbollide cages on the same iron, leaving the second iron outside the cages. The corresponding proposal is $[L_2\mathrm{Fe}\to \mathrm{Fe}(\mathrm{CO}{)}_4{]}^{2-}$, with the first iron sandwiched between the two open ${\eta }^5$ faces and the second carrying four carbonyls. It is not the symmetric arrangement with one cage on each iron, which would contradict the specified single cage-interacting metal site.

The donor fragment $[\mathrm{F}{\mathrm{e}}^{\mathrm{II}}{L}_2{]}^{2-}$ has $d^6+2(6)=18$ electrons; the external neutral $\mathrm{Fe}(\mathrm{CO}{)}_4$ fragment has $8+4(2)=16$. Metal-to-metal donation therefore provides a natural 18-electron acceptor adduct. In covalent-single-bond oxidation-state bookkeeping, the analogous assignments Fe(III)/Fe(−I) give $d^5+12+1=18$ and $d^9+8+1=18$. In either account the whole species has two negative charges. The outside iron can have an approximately trigonal-bipyramidal five-coordinate arrangement when the Fe–Fe direction and its four CO ligands are counted.

The important structural distinction is a bis(dicarbollide) iron donor joined to an exo tetracarbonyl iron acceptor. Formula and electron counting alone do not uniquely fix all carbonyl angles or prove a particular crystalline conformation.

Write $L=[{\mathrm{C}}_2{\mathrm{B}}_9{\mathrm{H}}_{11}{]}^{2-}$ for the open-face dicarbollide ligand. The structure consistent with the stated attachment is $[L\mathrm{Mo}(\mathrm{CO}{)}_3\to \mathrm{Mo}(\mathrm{CO}{)}_5{]}^{2-}$, with a direct Mo–Mo connection, three carbonyls on the cage-capping Mo and five on the outside Mo. The first Mo has an ${\eta }^5$ interaction with the five-vertex face and completes a 12-vertex metallacarborane cage; the second is exo to that cage. The two tetramethylammonium ions are counterions.

The unsaturated $\mathrm{Mo}(\mathrm{CO}{)}_5$ fragment is a 16-electron Lewis acceptor; $[L\mathrm{Mo}(\mathrm{CO}{)}_3{]}^{2-}$ is an 18-electron metal-base fragment in ionic counting, $d^6+6e(L)+6e(3CO)$. Donation from a filled Mo-based orbital supplies the external fragment with a pair. This donor–acceptor representation avoids the mistaken claim that both initially saturated 18-electron fragments simply add a new conventional bond without changing anything.

An alternative formal oxidation-state allocation for a single covalent Mo–Mo bond is cage Mo(I) and external Mo(−I): the local counts are $d^5+6+6+1=18$ and $d^7+10+1=18$. These are electron-bookkeeping alternatives to the metal-base/metal-acid picture, not a measurement of integral site charges. With terminal carbonyls, the external $\mathrm{Mo}(\mathrm{CO}{)}_5$ group can be approximately octahedral when the Mo–Mo direction occupies its sixth coordination direction. The cage-containing unit must accommodate an exo metal interaction in addition to its facial cage/CO environment; its geometry need not be a perfectly undistorted isolated half-sandwich.

For $[{\mathrm{B}}_{12}{\mathrm{H}}_{12}{]}^{2-}$, twelve BH vertices supply $12(2)$ skeletal electrons and the charge adds two:For the cobaltacarborane, nine BH vertices supply 18, two CH vertices supply six and CpCo supplies two by (a):Both are 12-vertex closo icosahedra with 13 skeletal electron pairs. In the cobalt compound, the Cp ring is external to the cage. cobalt caps the five-membered open face of the 11-vertex dicarbollide fragment and is a vertex of the resulting icosahedron, giving the usual adjacent-carbon cobaltacarborane framework. Counting all five Cp carbons as additional polyhedral vertices would be incorrect.

React pentaborane with an alkyne under relatively mild conditions to obtain a nido ${\mathrm{C}}_2{\mathrm{B}}_4{\mathrm{H}}_8$ framework, conventionally nido-2,3-dicarbahexaborane(8) for acetylene. The two entering carbon atoms initially retain their adjacency. A specific mild example uses 2-butyne with triethylamine-assisted ${\mathrm{B}}_5{\mathrm{H}}_9$ at or below room temperature to give nido-$\mathrm{M}{\mathrm{e}}_2{\mathrm{C}}_2{\mathrm{B}}_4{\mathrm{H}}_6$. This is a practical route to an explicitly nido carborane, rather than an assumption that an alkyne supplies a closo cage under every condition.^[ref]

More energetic gas-phase reaction or pyrolysis promotes hydrogen loss, boron-fragment loss and cage rearrangement, giving closo carboranes, for example $1,5$-${\mathrm{C}}_2{\mathrm{B}}_3{\mathrm{H}}_5$ and $1,6$-${\mathrm{C}}_2{\mathrm{B}}_4{\mathrm{H}}_6$, with other cage sizes possible in the mixture. Thermolysis of a previously formed nido carborane is an alternative route to this stage. The point is not a clean same-composition nido-to-closo isomerization: the change of hydrogen content and sometimes cage size must be retained in the formulas.

Milder conditions favour kinetically trapped adjacent-carbon products; higher temperatures permit migration to carbon-separated arrangements, often increasing favourable B–C contacts relative to adjacent C–C/B–B contacts. Conditions and Lewis bases also influence which separated-carbon cage is isolated; temperature alone does not prescribe a single quantitative isomer distribution. Primary thermolysis studies and the small-carborane literature document both cage rearrangement and reagent-dependent product mixtures.^[ref]

An isolobal analogy compares fragment frontier orbitals, not atomic sizes or total electron counts. The neutral BH fragment has four valence electrons; two make the external B–H bond, leaving two skeletal electrons in three frontier orbitals directed towards possible cluster neighbours. For the neutral $\mathrm{CpCo}$ fragment, $\mathrm{Cp}={\mathrm{C}}_5{\mathrm{H}}_5$ contributes five electrons and cobalt nine, giving 14. Removing the 12 electrons assigned to filled, predominantly non-skeletal fragment orbitals again leaves two electrons in three cluster-directed frontier orbitals.

The corresponding frontier sets have comparable numbers, spatial symmetries and occupations, allowing replacement of a BH vertex by CpCo in a suitable cluster-electron count. This is why the metal fragment can cap an open carborane face without adding an arbitrary new set of skeletal electrons. Exact orbital energies, ligand preferences and bond lengths need not be identical.^[ref]

Radiation changes the solution's chemistry by radiolysis; it does not turn Am(VI) nuclei into Am(V) nuclei. An emitted alpha particle deposits energy into many water and solute molecules, generating ionized/excited species, hydrated electrons, hydrogen atoms and molecular products including hydrogen peroxide. Reducing products can supply the electron needed forFor example, hydrogen peroxide can be oxidized while reducing two americyl(VI) ions:The $0.3$ mM loss of Am(VI) equals the $0.3$ mM appearance of Am(V), preserving the measured dissolved americium total while transferring $0.3\mathrm{mmol}{\mathrm{L}}^{-1}$ of electron equivalents. Oxidizing radiolysis products are also present, and the net reduction depends on acid, dissolved gases and scavengers; not every radiolytic event is reducing. In nitric acid, primary kinetic work identifies hydrogen peroxide and radiolytically produced nitrous acid as important reductants, with a rate controlled by total americium activity rather than necessarily by the instantaneous Am(VI) concentration.^[ref]

An alpha decay decreases mass number by four and atomic number by two; a beta-minus decay keeps the mass number fixed and increases atomic number by one. ThereforeThe beta-minus step converts a neutron to a proton with emission of an electron and an electron antineutrino. It is not capture of an orbital electron, which would change the atomic number in the opposite direction.

The left ring system is spiro, and the right-hand substituent is $\mathrm{CH}(\mathrm{OMe})\mathrm{Me}$, not OMe directly attached to the cyclohexene ring. Let $Q$ denote the right cyclohexenyl group bearing that methoxyethyl substituent cis to its linker. The connection to the left alkene carbon is $-(\mathrm{C}{\mathrm{H}}_2{)}_3-\mathrm{Q}$. These observations suggest two Diels–Alder constructions. A left-hand dienehas the original cyclopentane carbon as one terminal diene atom; cycloaddition with an ethylene equivalent gives the spiro six-membered ring. Its residual double bond is between the two internal diene atoms, putting the linker on the required alkene carbon.

Build $Q$ and its four-carbon alcohol appendage first. Protect the OH group of the indicated 4-Bromobutan-1-ol, for example as a THP ether. Convert the primary bromide to its grignard reagent and formylate with DMF, giving $\mathrm{THPOC}{\mathrm{H}}_2(\mathrm{C}{\mathrm{H}}_2{)}_3\mathrm{CHO}$. olefination with an allylidenetriphenylphosphorane, $\mathrm{P}{\mathrm{h}}_3\mathrm{P}=\mathrm{CHCH}=\mathrm{C}{\mathrm{H}}_2$, supplies a conjugated diene $\mathrm{THPOC}{\mathrm{H}}_2(\mathrm{C}{\mathrm{H}}_2{)}_3\mathrm{CH}=\mathrm{CHCH}=\mathrm{C}{\mathrm{H}}_2$; select the required diene geometry rather than assume a single E/Z outcome.

Cycloaddition with methyl vinyl ketone produces a cyclohexene with a protected four-carbon alcohol chain at one saturated ring carbon and an acetyl group at its adjacent ring carbon. Choose and isolate the adjacent-substituent cis regioisomer, matching the two dashed substituent bonds in the target. Endo preference and diene geometry can aid the selection, but an unsymmetrical diene is not guaranteed to give a unique regioisomer. Reduce the acetyl ketone to $\mathrm{CH}(\mathrm{OH})\mathrm{Me}$, then methylate that alcohol with sodium hydride/methyl iodide while the chain-end OH is protected. Remove the THP group under mild acid conditions to reveal $\mathrm{Q}(\mathrm{C}{\mathrm{H}}_2{)}_4\mathrm{OH}$, preserving the secondary methyl ether and the ring alkene.

Oxidize this primary alcohol to $\mathrm{Q}(\mathrm{C}{\mathrm{H}}_2{)}_3\mathrm{CHO}$. Add vinylmagnesium bromide to give $\mathrm{Q}(\mathrm{C}{\mathrm{H}}_2{)}_3\mathrm{CH}(\mathrm{OH})\mathrm{CH}=\mathrm{C}{\mathrm{H}}_2$, then oxidize the allylic alcohol to the enone $\mathrm{Q}(\mathrm{C}{\mathrm{H}}_2{)}_3\mathrm{C}(=\mathrm{O})\mathrm{CH}=\mathrm{C}{\mathrm{H}}_2$. This carbonyl carbon is the original alcohol-end carbon of 4-bromobutanol; after olefination it will be the linker-bearing carbon of the left ring alkene.

In the other branch, reduce the specified cyclopentanone to cyclopentanol, convert to bromocyclopentane, and react with triphenylphosphine to obtain the cyclopentylphosphonium salt. Deprotonation gives the cyclopentylidene phosphorane, whose ylidic carbon is part of the five-membered ring. Wittig olefination of the just-prepared enone gives the substituted diene displayed above. This carbonyl–ylide pairing keeps the ring carbon terminal in the diene and avoids accidentally constructing a fused ring instead of a spiro ring.

Finally, cycloadd an ethylene equivalent. Ethylene itself requires suitably forcing/high-pressure conditions; a more reactive alternative is phenyl vinyl sulfone, followed by reductive removal of the sulfone substituent using a method compatible with the methyl ether and retained C=C bonds. The latter installs the same two otherwise unsubstituted saturated carbons. Control or separate any additional cycloaddition isomers, and retain the previously selected cis configuration on $Q$.

The carbon count provides a strong check: cyclopentanone supplies five, bromobutanol four, DMF formylation one, the first allylidene ylide three, methyl vinyl ketone four, O-methylation one, vinylmagnesium two, and the final ethylene equivalent two, totalling 22 target carbons. Three original bromobutanol carbons form the linker; its fourth becomes the substituted carbon of the spiro ring's alkene. The proposal is a multistep constitutional/relative-stereochemical route with explicit regioisomer and diastereomer selection, not an unsupported claim of a single quantitative cycloaddition product.

Use the nitro group as a masked carbonyl and a carbon nucleophile. The most informative disconnection reverses the six-membered aldol annulation. Before that closure, the cyclopentanone beta carbon bearsThe carbon carrying $\mathrm{N}{\mathrm{O}}_2$ will become the lower carbonyl of the final six-membered ring by a Nef reaction. It comes from the specified nitromethane. The remote methyl ketone supplies the upper methyl-substituted alkene carbon after aldol condensation.

First, use a mild base to generate a nitromethane-derived nitronate and add it conjugatively to the indicated cyclopent-2-en-1-one. Protonation gives 3-(nitromethyl)cyclopentanone. Deprotonate its nitromethyl group and perform a second Michael addition to methyl vinyl ketone, giving $\mathrm{cyclopentanone}-!C3-\mathrm{CH}(\mathrm{N}{\mathrm{O}}_2)\mathrm{C}{\mathrm{H}}_2\mathrm{C}{\mathrm{H}}_2\mathrm{COC}{\mathrm{H}}_3$. The nitro-stabilized carbon is appreciably more acidic than a simple ketone alpha carbon, permitting selective nitronate formation; control the acceptor stoichiometry to avoid a further Michael alkylation of the remaining nitro-bearing H.

Form the enolate/enamine at the cyclopentanone carbon between its carbonyl and the side-chain-bearing carbon. Intramolecular attack on the remote methyl ketone closes a six-membered ring; the two shared atoms are the original enone alpha and beta carbons. Hydrolysis if required and dehydration create the alkene between the original alpha carbon and the newly formed tertiary carbinol carbon. That carbinol carbon retains its methyl substituent, placing the methyl exactly as required. The original cyclopentanone carbonyl remains the carbonyl of the five-membered ring.

Finally, form the secondary nitronate and carry out controlled acidic Nef hydrolysis to replace its $\mathrm{CH}(\mathrm{N}{\mathrm{O}}_2)$ unit by $\mathrm{C}=\mathrm{O}$. The resulting fused product contains the two required ketones. Carrying out the annulation before exposing this second ketone reduces competing enolate/aldol pathways that would be available in the corresponding triketone. The nitro intermediate's competing deprotonation means that the aldol step needs conditions designed to access the ring carbonyl enolate/enamine; simply treating every carbonyl position as equally reactive would not establish the route.

The carbon ledger is $5$ from cyclopentenone, $1$ from nitromethane and $4$ from methyl vinyl ketone, giving the nine-carbon fused skeleton plus its methyl substituent. A primary-nitro Nef reaction performed before the second carbon alkylation would instead give an aldehyde and would lose this useful protected-carbonyl construction.

Both side chains have the order ring–$\mathrm{C}{\mathrm{H}}_2\mathrm{CH}=\mathrm{CHMe}$, and both printed double bonds have the Z geometry. The two carbon side chains are cis to one another; equivalently the labelled ring H and ring Me are cis on the opposite face. A useful common late precursor therefore has two cis ring–$\mathrm{C}{\mathrm{H}}_2\mathrm{CHO}$ substituents, which can undergo two Z-selective ethylidene Wittig reactions.

Use the starting 2-methylcyclohex-2-en-1-one in a reductive alpha-alkylation. Conjugate reduction under dissolving-metal/enolate-trapping conditions, followed by a primary electrophile $\mathrm{BrC}{\mathrm{H}}_2\mathrm{C}{\mathrm{H}}_2\mathrm{OTBS}$, installs a protected two-carbon alcohol side chain at the original methyl-bearing alpha carbon. This gives 2-methyl-2-(2-siloxyethyl)cyclohexanone. The carbon at that position is now quaternary, with its original methyl retained. Alternatively, reduce the enone to 2-methylcyclohexanone and deliberately form/select its more substituted enolate before alkylation; its opposite kinetic enolate would give the wrong regioisomer.

Convert the remaining ring ketone to an exocyclic enoate by Horner–Wadsworth–Emmons olefination with a phosphonoacetate:Hydrogenate this sole alkene to give ring–$\mathrm{C}{\mathrm{H}}_2\mathrm{C}{\mathrm{O}}_2\mathrm{Et}$ at that carbon. Select the diastereomer with this substituent cis to the pre-existing protected hydroxyethyl side chain, so the newly installed ring H is cis to ring Me. Less hindered-face hydrogenation can assist this relationship, but the route should include separation/verification of the desired cis product rather than assert complete stereoselectivity without data.

Reduce the ester with lithium aluminium hydride to ring–$\mathrm{C}{\mathrm{H}}_2\mathrm{C}{\mathrm{H}}_2\mathrm{OH}$ and remove the TBS group from the other side chain. Both appendages are now two-carbon primary alcohols. Oxidize them to aldehydes under non-overoxidizing conditions, for example Dess–Martin oxidation, obtaining the cis dialdehyde with two ring–$\mathrm{C}{\mathrm{H}}_2\mathrm{CHO}$ groups. Treat with excess salt-controlled, nonstabilized ethylidenetriphenylphosphorane under conditions favouring kinetic Z olefination; isolate the Z,Z product. Each aldehyde carbon becomes the inner vinylic carbon of its crotyl group, while the ylide supplies $\mathrm{CHMe}$.

This order postpones both stereodefined alkenes until after hydrogenation, so they are not accidentally saturated. It also avoids assuming that a crotyl grignard reagent adds without allylic transposition or that direct alcohol deoxygenation fixes the required ring stereochemistry. An achiral synthesis gives a racemic mixture pair with the specified relative configuration.

There are three largely independent constructions: a substituted Robinson-annulation core, a protected vicinal diol, and the ester acyl fragment. Disconnect the ester first to an allylic alcohol on the bicyclic ring and the acid $\mathrm{H}{\mathrm{O}}_2\mathrm{CCH}(\mathrm{Et})\mathrm{CH}(\mathrm{Me}{)}_2$. Disconnect the benzylidene acetal to a side chain $-\mathrm{C}{\mathrm{H}}_2\mathrm{CH}(\mathrm{OH})\mathrm{C}{\mathrm{H}}_2\mathrm{OH}$ and benzaldehyde. Reversing 1,2-reduction of the allylic alcohol gives the corresponding bicyclic enone; its six-membered enone ring is a Robinson disconnection back to a suitably alpha-substituted cyclohexanone and methyl vinyl ketone.

An efficient way to protect the side chain before annulation is to prepare 4-(bromomethyl)-2-phenyl-1,3-dioxolane from $\mathrm{BrC}{\mathrm{H}}_2\mathrm{CH}(\mathrm{OH})\mathrm{C}{\mathrm{H}}_2\mathrm{OH}$ and benzaldehyde with mild acid and water removal. The desired five-membered acetal protects the adjacent hydroxyl groups, while its external primary bromide remains an alkylating site. Alpha alkylation of the indicated cyclohexanone with this bromide gives a ring bearing $-\mathrm{C}{\mathrm{H}}_2\mathrm{CH}(\mathrm{O}-)\mathrm{C}{\mathrm{H}}_2\mathrm{O}-$ in the benzylidene-protected form. There is no need to expose the eventual ring enone to a nonselective late dihydroxylation.

Generate the enolate at the other, less substituted alpha position of the cyclohexanone and add methyl vinyl ketone. Hydrolysis of an enamine-mediated Michael variant or direct enolate trapping gives the appropriate 1,5-dicarbonyl. Intramolecular aldol condensation closes a six-membered ring onto the original cyclohexanone carbonyl. Dehydration gives the fused bicyclic enone, with the protected three-carbon side chain on the adjacent carbon of the original ring. Regioselecting the initial Michael addition is essential: reaction at the side-chain-bearing alpha site would place that substituent at the wrong fusion position.

Reduce the new enone carbonyl by 1,2-addition, for example sodium borohydride with cerium(III) chloride under Luche-type conditions, to retain the displayed ring C=C bond and produce the allylic alcohol. The benzylidene acetal should be maintained through these neutral/basic operations. The target has no wedge/dash assignments, so a constitutional solution does not require a claim of a unique fused-ring/alcohol diastereomer; isolate the required constitutional annulation product.

For the acyl branch, form the enolate of the indicated methyl butanoate, add acetone, then dehydrate the beta-hydroxy ester:Hydrogenate this isolated alkene before coupling it to the bicyclic substrate, giving $\mathrm{Me}{\mathrm{O}}_2\mathrm{CCH}(\mathrm{Et})\mathrm{CHM}{\mathrm{e}}_2$. saponification gives $\mathrm{H}{\mathrm{O}}_2\mathrm{CCH}(\mathrm{Et})\mathrm{CHM}{\mathrm{e}}_2$; activate it as its acid chloride or use a mild ester-coupling reagent, then acylate the bicyclic allylic alcohol. Performing the hydrogenation on the separate acyl branch avoids reducing the target ring alkene. Strong acid should be avoided during the final esterification because the benzylidene group is acid-labile.

The carbon audit is particularly useful: the fused rings contain six carbons from cyclohexanone plus four from methyl vinyl ketone; the side chain has three carbons from the protected bromodiol; the benzylidene unit supplies its acetal carbon and phenyl group. In the acyl fragment, the four butanoate acyl carbons plus three acetone carbons give seven; the starting ester methoxy carbon is lost during hydrolysis.

C, the neutral arachno ${\mathrm{B}}_6{\mathrm{H}}_{12}$, is the most plausible proton donor among A, B and C. Loss of an open-framework bridging proton produces the experimentally accessible, charge-delocalized conjugate base $[{\mathrm{B}}_6{\mathrm{H}}_{11}{]}^{-}$. In contrast, deprotonating A or B would build a dianion and incurs additional electrostatic cost. The neutral, open hydrogen-rich cluster therefore has a more favourable route to a monoanionic conjugate base. This is a qualitative acidity prediction for the listed species, not a numerical solvent-independent $p{K}_a$ or a claim that all B–H hydrogens are equally acidic.