Category: Interesting chemistry

A little while back, I wrote about anomeric-like effects in the sulfur ring S₇.[cite]10.59350/rzepa.28407[/cite] I had started that exploration by retrieving the crystal structure from the ICSD (Inorganic crystal structure database) and then optimising these coordinates using a DFT method (MN15L/Def2-TZVPP to be precise). In demonstrating this effect to a student, I decided to create an initial guess for the molecule coordinates not from the crystal structure but by drawing and then minimising using a simple molecular mechanics force field – and only then subjecting it to DFT re-optimisation.[cite]10.14469/hpc/15924[/cite] It turns out the result was quite surprising in one respect and so here I tell the rest of the story.

I start by noting that there is one fundamental difference between a DFT optimisation of the geometry and a molecular mechanics procedure – the latter cannot respond to (stereo)electronic orbital interactions, such as those found in anomeric effects. Thus DFT optimisation (using a simple opt keyword) starting from the mechanics coordinates leads – surprisingly perhaps – to a transition state rather than an equilibrium species. Normally the opt keyword does not produce such results – although to be certain of course the opt(calcfc) should be used to guarantee that a minimum rather than a transition state is found. The DFT optimised geometry has a C₂ axis of symmetry, rather than the plane of symmetry expected for S₇, running through atom 7 and the mid point of atoms 1 and 3 (Figure 1). All the S-S bond lengths are almost equal (Figure 1) – there is little discrimination and no anomeric effects are reflected in this geometry.

Figure 1.

To find out what the transition state connects, an IRC (intrinsic reaction coordinate calculation) was performed (Figure 2). It is symmetrical about the transition state, and leads to the known conformation of S₇ in both directions, albeit with one strongly lengthened bond between S5-S7 on one side and between S6-S7 on the other side. As noted previously[cite]10.59350/rzepa.28407[/cite], this bond lengthening is a direct consequence of the anomeric orbital interactions. So the transition state is a low energy isomerisation, converting one anomeric isomer to the adjacent bond-lengthened one. To my knowledge, such a process has never been previously reported. It reminds one of mechanisms that exchange axial and equatorial positions in e.g. square planar or trigonal metal complexes.[cite]10.1021/ic0519988[/cite] (see also this link).

Figure 2.

The principle process occuring can be inferred by inspecting the dihedral angles S6-S7-S5-S4 and S2-S6-S7-S5 (Figures 3 and 4). The first changes from a dihedral close to 90° down to 0°, the second changes from 0° down to -90° and so directly relates to the orientation of a p-orbital on one sulfur and the adjacent S-S σ^*-bond. The anomeric effect shifts by one bond during this process.

Figure 3.

Figure 4.

The process can be animated as in Figure 5.

Figure 5.

The NBO7 orbital perturbation energies (kcal/mol) for transition state and equilibrium state[cite]10.59350/rzepa.28407[/cite] respectively are shown below. The former are all very close in value (note the absence of S7, through which the axis of symmetry passes) and hence induce no bond length discrimination, whereas the equilibrium state reveals the differences we have identified as an anomeric effect.

Transition state
LP S1 BD* S2-S6 5.81
LP S1 BD* S3-S4 7.17
LP S2 BD* S1-S3 6.82
LP S2 BD* S6-S7 7.33
LP S3 BD* S1-S2 7.17
LP S3 BD* S4-S5 5.81
LP S4 BD* S1-S3 6.82
LP S4 BD* S5-S7 7.33
LP S5 BD* S3-S4 5.53
LP S5 BD* S6-S7 5.27
LP S6 BD* S1-S2 5.53
LP S6 BD* S5-S7 5.27
Sum              75.9
Equilibrium geometry
LP S1 BD* S2-S6 5.03
LP S1 BD* S3-S4 7.08
LP S2 BD* S6-S7 12.34
LP S3 BD* S1-S2 7.06
LP S3 BD* S4-S5 7.04
LP S4 BD* S1-S3 7.08
LP S4 BD* S5-S7 5.03
LP S5 BD* S6-S7 12.34
LP S6 BD* S1-S2 10.09
LP S7 BD* S4-S5 10.09
Sum              83.2

This (accidentally discovered) transition state teaches us that the bond lengthening in S₇ is directly associated with orbital orientations. And never to ignore a strange result – learning what happened can teach us a great deal.

In 2008, the previously elusive hydroxycarbene, H-C-OH was finally reported[cite]10.1038/nature07010[/cite] as having been captured by matrix isolation, accompanied by the observation that “we unexpectedly find that H–C–OH rearranges to formaldehyde with a half-life of only 2h at 11K by pure hydrogen tunnelling through a large energy barrier in excess of 30 kcal mol^–1. A subsequent theoretical study of this tunnelling in 2017[cite]10.1016/j.proeng.2017.03.024[/cite] reported that “the half-life calculation after monodeuteration is 2.97 × 10¹⁶ hours, which is extremely longer than before monodeuteration that is only 2.5 hours using the same calculation methods“; in other words a kinetic isotope effect k^H/k^D of ~10¹⁶, which is by far the largest ever suggested.[cite]10.59350/qgwfn-rsc92[/cite] In 2011, the original study was extended to methylhydroxycarbene (X=Me)[cite]10.1126/science.1203761[/cite], again arguing for “Tunneling Control of a Chemical Reaction.” In this post,^† I explore an alternative mechanism for rearrangement of hydroxycarbene to formaldehyde using a “double hydrogen transfer” via a dimeric transition state (Figure 1).

Figure 1. Two mechanistic possibilities for hydrogen transfer in hydroxycarbene.

There is general agreement that the rearrangement via a [1,2]-hydrogen shift (a four electron Woodward-Hoffmann “forbidden” pericyclic process) occurs with a barrier of > 30 kcal/mol. I will start with a traditional DFT method, ωB97XD/Def2-TZVPP/SCRF=dichloromethane) to see if I can replicate this assertion,[cite]10.14469/hpc/15587[/cite],[cite]10.14469/hpc/15596[/cite]^‡ which yields 32.95 kcal/mol for the monomer free energy barrier. A CCSD(T)/Def2-TZVPP follow up gives 32.6 kcal/mol,[cite]10.14469/hpc/15861[/cite] so we may presume that ωB97XD is a reasonable DFT method. These values represent a very slow thermal reaction. Kinetic isotope effects (using KINISOT, DOI: 10.5281/zenodo.10403662) for this reaction are listed below.

KIE 1,2-H shift (ωB97XD)	KIE 1,2-H shift (CCSD(T) )	Temp, K
4.97	5.23	298.15
11.04	11.92	200.00
125.72	146.28	100.00
16,221.02	21,906.65	50.00
34,837,481,344.42	73,565,737,699.94	20.00

To locate a transition state for the dimer reaction, some subterfuge was used (for reasons that will become apparent). I needed a (computational) reaction that would generate two molecules of hydroxycarbene, which would then allow these two molecules to interact as they wished. Such a (hypothetical) reaction is shown in Figure 2.

Figure 2. Generation of two hydroxycarbene molecules from a precursor.

A transition state (X = Y = H, Figure 1) for this was located[cite]10.14469/hpc/15586[/cite] which is -4.6 kcal/mol lower than the free energy of two molecules of hydroxycarbene at 298K and -13.7 kcal/mol lower at 20K.[cite]10.14469/hpc/15860[/cite] At the CCSD(T)/Def2-TZVPP level @298K, the computed free energy of this TS[cite]10.14469/hpc/15874[/cite] is -2.3 kcal/mol lower than two isolated monomers.

The located transition states are shown in Figure 3, and it consists of two hydroxycarbene molecules with a hydrogen bond formed between the hydrogen of one hydroxyl group and the carbene lone pair of the other hydroxycarbene.

Figure 3. ωB97XD/Def2-TZVPP (red) and CCSD(T)/Def2-TZVPP (black) calculated TS for generation of two hydroxycarbene molecules. Click image for 3D model

There is support for such a hydrogen bond forming in the crystal structure database – see Figure 4.

Figure 4. Crystal structure of an N-heterocyclic carbene with methanol.[cite]10.1021/ol050773y[/cite],[cite]10.5517/cc971sb[/cite]

An IRC (Figure 5) is needed to make more sense of the transition state. At this point, we need not concern ourselves about the preceding reaction profile (IRC 8 to 0), which as I mentioned was a computational subterfuge to generate two hydroxy carbene monomers in close proximity.

Figure 5. Intrinsic reaction coordinate for the reaction shown in Figure 2.[cite]10.14469/hpc/15585[/cite]

It is what  happens next that is crucial, which the IRC animation (Figure 6) makes clear. This is shown pausing at the TS and you should focus on what happens next, which is a rotation followed by two successive (but not entirely synchronous) proton transfers. As appropriate for a TS, the energy past this point only goes down.

Figure 6. IRC animation ωB97XD/Def2-TZVPP for the reaction shown in Figure 2.[cite]10.14469/hpc/15585[/cite]

Further insight can be found by inspecting the gradient norm of the IRC (Figure 7).

Figure 7. The calculated gradient normals along the IRC.

1. From IRC 8 to 0, the generating reaction occurs (the “hidden intermediate” at IRC 1.5 is interesting but will not be discussed here).
2. From IRC 0 to -4, a rotation of the two fragments occurs, setting up the hydrogen transfers.
3. At IRC  -5.5 the first hydrogen transfers.
4. At IRC -6.1 the second hydrogen transfers.

The important observation is that at this stationary point (Figure 3), the computed free energy at 298K is -4.6 kcal/mol lower relative to two fully isolated hydroxycarbene molecules and it is even lower at 20K. We conclude from this analysis that when placed close to each other, two hydroxycarbenes react WITHOUT a barrier for exchanging hydrogens to form two molecules of formaldehyde. Hence the trick of generating the two hydroxycarbenes from a precursor to model this behaviour.

Kinetic isotope effects with deuterium substitution on both OH groups can be approximated using this new bimolecular transition state, via these outputs.[cite]10.14469/hpc/15856[/cite],[cite]10.14469/hpc/15586[/cite]

HCOH: KIE (no tunnelling)	KIE (Bell tunneling)	Temperature, K
3.366175	3.398699	298.15
5.622623	5.748042	200.00
25.480480	28.428388	100.00
504.392382	–	50.00
4,246,513.724875	–	20.00

I next look at a fluorinated version (X = Y = F).[cite]10.14469/hpc/15593[/cite],[cite]10.14469/hpc/15590[/cite]^♥ The transition state[cite]10.14469/hpc/15595[/cite] has C_2h symmetry (Figure 8).

Figure 8. Transition state for synchronous double hydrogen transfer for X=Y=F.

The IRC[cite]10.14469/hpc/15594[/cite] shows different behaviour (Figure 9, animation Figure 10). The dimer is a clear albeit very shallow intermediate now[cite]10.14469/hpc/15593[/cite] – rather than just a point on the reaction coordinate as for  X = Y = H, but the free energy of the TS (ωB97XD/Def2-TZVPP) is still lower by -4.4 kcal/mol compared to two isolated fluorohydroxycarbenes (ΔΔG -3.0 at the CCSD(T)/Def2-TZVPP level.[cite]10.14469/hpc/15929[/cite]).

Figure 9. IRC for synchronous double hydrogen transfer for X=Y=F (Figure 1).

Figure 10. IRC animation for synchronous double hydrogen transfer for X=Y=F. (Figure 1)

The hydroxycarbene dimer itself is shown below (click image to view model)

Figure 11. The structure of the FCOH H-bonded dimer.

The computed KIE are somewhat higher for the fluorinated molecule.[cite]10.14469/hpc/15865[/cite],[cite]10.14469/hpc/15860[/cite]

FCOH: KIE (no tunnelling)	KIE (Bell tunneling)	Temperature, K
6.587916	6.751828	298.15
14.724290	15.598102	200.00
187.689465	269.048707	100.00
31344.686809	32,350.352820	50.00
142,555,714,740.5	–	20.00

The conclusion is that whereas a unimolecular proton transfer to generate formaldehyde indeed passes (“tunnels”) through the significant barrier of a “forbidden” pericyclic reaction, an alternative bimolecular reaction is predicted to occur without a free energy barrier – the entropic penalty of combining two molecules is offset by the strong hydrogen bonds formed. Generating hydroxycarbene in a low temperature matrix suppresses the bimolecular mode, but when the matrix is warmed up, the two monomers can diffuse together to rapidly react. This speed can be achieved either through extreme tunnelling of one monomer, or by a barrierless concerted double hydrogen transfer via a dimer. Could it be that the fast disappearance of hydroxycarbene after formation might not be due to tunnelling control after all?

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^‡All the results are published as a FAIR data collection.[cite]10.14469/hpc/15584[/cite] ^†This post has DOI:10.59350/syhqn-7md47[cite]10.59350/syhqn-7md47[/cite] ^♥A half-fluorinated reaction of HCOH + FCOH shows a similar profile to the non-fluoro version[cite]10.14469/hpc/15598[/cite]

The recent report[cite]10.1126/science.aea3321[/cite] of what is termed a “half-Möbius” molecule is generating a lot of excitement. It has its origins in a project to make odd-numbered cyclocarbons on STM (scanning tunnelling microscope) surfaces. I had discussed even-numbered cyclocarbons in another post[cite]10.59350/g4309-gv109[/cite], where I also happened to include several odd-numbered examples, such as C₄₉ and C₅₁. In this study[cite]10.1126/science.aea3321[/cite] they were focussing on C₁₃ and a precursor to this was to be C₁₃Cl₂. As part of the microscopy, they noticed this latter species was asymmetric (chiral) and so started the story of a “half-Möbius” molecule (molecules with twists in their topology are of course chiral). I should at this stage say that the concept of a half-Möbius is quite new and thought provoking. Perhaps the simplest way of explaining why, is that a conventional Möbius molecule (as with the strip or ribbon) requires two full circuits of the edge of the ribbon to return to the start, whereas this half version requires a full four circuits to achieve the same. More about this later.

Since STM microscopy is not capable of yielding accurate molecular geometries and bond lengths, the authors proceeded to calculate these – a non trivial undertaking! Basically, because of orbital degeneracies, the wavefunction has important multi-reference character. I thought I would illustrate some of the outcomes here. I actually start with a single reference calculation, using the r²scan-3c method,[cite]10.1063/5.0040021[/cite] which I had previously shown[cite]10.59350/g4309-gv109[/cite],[cite]10.59350/k0kjg-hpc66[/cite] seemed to reliably reproduce geometries of even-numbered cyclocarbons such as C₄₈, and also predicted the onset of bond length alternation (BLA) for rings of about 58 carbon atoms or greater. Applied to C₁₃Cl₂ r²scan-3c yields a planar molecule (Figure 1) with symmetrically disposed bond lengths.[cite]10.14469/hpc/15786[/cite]

Figure 1. A r²SCAN-3c/Def2-mTZVPP optimised geometry for C₁₃Cl₂.[cite]10.14469/hpc/15786[/cite]

The reported[cite]10.1126/science.aea3321[/cite] calculation using CASPT2/cc-pVDZ[cite]10.5281/zenodo.15495263[/cite] shows different behaviour (Figure 2). The singlet state geometry is asymmetric and non-planar, with interesting BLA in both the C₆ and C₇ fragments, but also more “aromatic-looking” values of ~1.40Å at the chlorine-connected carbon.

Figure 2. A CASPT2/cc-pVDZ gas phase optimised geometry for A: singlet C₁₃Cl₂ and b: Triplet.

A pure CASSCF(12,12) calculation performed here (Figure 3) using the Def2-TZVPP basis set (a triple-ζ basis – the geometry above is for a smaller double-ζ basis) reproduces both the non-planarity and the BLA, but not the aromatic-like bond lengths, confirming that a higher level of theory which includes MP2-like electron correlation perturbation corrections is needed for these deceptively simple molecules.

Figure 3. A CASSCF(12,12)/Def2-TZVPP calculation for singlet C₁₃Cl_2.

But what about that twist?

Now to the next stage of the story. Using orbitals derived from the wavefunctions, the authors[cite]10.1126/science.aea3321[/cite] showed that only a 90° rotation (½π) occurred during a single trip around the edge of the molecular ribbon and hence 4*½π = 2π (360°) was required to achieve a return to the start. A full-Möbius molecule would achieve a 180° rotation or 1π for one circuit, and therefore requires only two circuits to achieve 2π. At this point, my thoughts turned to a well known topological theorem for these types of twisted systems[cite]10.1021/ja710438j[/cite], the Cãlugãreanu−White−Fuller theorem.[cite]10.21136/CMJ.1961.100486[/cite]. This defines a topological invariant known as the linking number (L_k) which itself is the sum two quantities, the sum of local twists T_w and a writhe W_r. The latter can be thought of as the extent to which coiling of the central curve of the object can relieve local twisting of the ribbon. It is stated as:

L_k = W_r + T_w (each of which can be expressed in units of π).^‡

One practical example is C₁₄H₁₄,[cite]10.1021/ja710438j[/cite], a molecule not entirely unrelated to C₁₃Cl₂. This has a “figure-eight”, lemniscular or “double-Möbius” topology (Figure 4).

Figure 4: A double-twist Möbius annulene, calculated at the B3LYP/6-31G level. For the (almost identical) geometry using the more modern r²scan-3c/Def2-mTZVPP, see the FAIR data archive.[cite]10.14469/hpc/15786[/cite]

When analysed using the expression above,[cite]10.1021/ja710438j[/cite] it shows values of L_k = 2π, W_r = 0.89π and T_w = 1.11π (B3LYP/6-31G* calculation).

Firstly, a bit of (unrecorded) history. When I discovered this little lemniscular lovely, I did what had been done above, namely I added the total rotation of the orbital basis (the p-π-orbitals on each carbon) and returned to my starting point in just one circuit. I obtained a rotational sum of ~180°, or ~π. I knew that returning to the starting point in one circuit (2π) meant it was not a conventional Möbius molecule but a “figure-eight” or double twist topological isomer (which Möbius, along with Listing, had identified!). This form had, not C₂ symmetry as per a single twist Möbius, but the higher D₂ chiral symmetry. The correct answer for this twist sum was therefore surely 2π, not 1π? I had lost ~1π worth of twist! And this is when I came across the above theorem, which put simply indeed allows any fraction of twist to be “lost” by its conversion into writhe, as can be seen from the values shown above.^†

So here is my question. Might it be possible that the same has happened to C₁₃Cl₂? A measured orbital rotation of ~90° or ~½π (and hence the term half-Möbius) would only be correct if the writhe for this molecule – the coiling of the central curve out of a plane – was zero. If instead the writhe also had a value of lets say ~½π, then

L_k = ~1π, comprising W_r = ~½π and T_w = ~½π

which would make it a conventional rather than half-Möbius molecule.

To conclude: the reported interpretation of C₁₃Cl₂ as a “half-Möbius” molecule is only correct if it does not “writhe” topologically to any significant extent. Watch this space for updates!

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^‡L_k can be both a positive or a negative integer, depending on which enantiomer is used, and hence acts as a chiral descriptor in the manner of the Cahn-Ingold-Prelog convention. ^†W_r and T_w do not have to have the same sign. Thus the value of T_w can be greater than that of L_k if W_r is opposite in sign. For an extreme example of these various effects see here.[cite]10.59350/j60gh-gzr35[/cite]. The proposed molecular trefoil knot has values of Lk 6π = Tw -0.8π + Wr +6.8π Not only are the twist and writhe of opposite sign, the knot is composed almost entirely of writhe and no twist!

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This post has DOI: 10.59350/5q3ka-2ag71

A few posts back, I contemplated the curly arrows appropriate for the formation of nitrosobenzene dimer from nitrosobenzene,[cite]10.59350/rzepa.28849[/cite] and commented on the odd nature of the N=N double bond formed in this process.[cite]10.59350/rzepa.29383[/cite]. Odd, because the valence bond representation of this dimer (1 below[cite]10.1021/ja00827a021[/cite]) has two formally positive adjacent nitrogen atoms. An energy decomposition analysis (NEDA[cite]10.1021/acs.organomet.5b00429[/cite]) of species 1 showed an unusually small negative interaction energy of -27.6 kcal/mol between the two nitrosobenzene fragments (typical ΔE values ~-130 to -180 kcal/mol[cite]10.59350/rzepa.29410[/cite]), commensurate with the facile equilibrium between two monomers and the dimer[cite]10.1002/mrc.1260251118[/cite] A little later I went on to speculate upon a similar theme for the more hypothetical nitric oxide dimer, a species 2 which again has two adjacent +ve charges[cite]10.59350/rzepa.29429[/cite] and even a smaller +ve NEDA for the triple bond! You can imagine discussing these results with organic chemists, who would normally shrink from placing two (formal) positive charges on adjacent atoms.

Browsing (as one does) the CSD crystal structure database, I came across a molecule shown as representation 5 above.[cite]10.1021/om980661t[/cite]. This rang a small alarm bell – why was the central nitrogen atom there shown as neutral? To balance the only +ve charge (on the pyridinium cation), the Ti had a single -ve charge. Representation 3 installs a second +ve charge on the second nitrogen, just as with 1 and 2. The ligand in question (PyN₂^-1) has an overall charge of -1, and together with the other three negatively charged ligands results in Ti^IV. The total formal count around the Ti is 6 (from Cp^-1) + 2×2 (2Cl^-1) + 6 (PyN₂^-1), making 16e, a fairly normal count for many Ti species and only two short of a filled valence shell of 18e. Alternative representation 4 shows only one +ve and one -ve charge in the molecule, but now the Ti formal valence shell has only 14e.

I decided firstly to find out if there was any supporting data for the N≡Ti triple bond as shown in 3 and 5. A search of the CSD database for species with a Ti-N bond (of any order or type) produces the following plot.

There is a distinct cluster in the region 1.7-1.8Å, which we may assume corresponds to the shortest TiN bonds, presumed to be triple. Two more diffuse clusters are in the region 1.9 – 2.0 (double bonds) and 2.0-2.3Å (single and other bonds). The crystal structure of HOPSUA shows as 1.735Å and hence appears to be in the triple (3) rather than the double bond (4) region. Moreover the measured TiNN angle is 165° whereas 4 might be expected to be more highly bent. The N-N bond length is 1.362Å.

MN15L/Def2-TZVPP and R²-SCAN-3c calculations give respectively TiN 1.784Å/1.806Å and NNTi 147°/152°.[cite]10.14469/hpc/15774[/cite] The observed NN distance of 1.362Å in HOPSUA and containing an N(+)-N(+) motif (calculated N-N 1.337Å/1.303Å) compares with the distribution below in the CSD, where the main features are ~1.3Å (double), 1.4Å (aromatic) and ~1.58Å (single). So the NN bond order in HOPSUA is rather less than double.

A NEDA (natural energy decomposition analysis)[cite]10.14469/hpc/15774[/cite] shows that the interaction energy of a neutral singlet pyridine with a singlet NTiCl₂Cp fragment (a metal nitrido complex[cite]10.5517/cc1jplh9[/cite]) is -89 kcal/mol. The charge transfer (CT) component is large (-677.4 kcal/mol) because combining two fragments, each deploying a lone pair of electrons to form less than a double bond requires transferring electrons out of this region. This overall interaction energy is larger than for nitrosobenzene, but it is still unusually small for a bond interaction and indeed perhaps a feature of systems which have two (formal) repulsive positive charges on adjacent atoms. For completeness, the interaction energy for the two fragments PyN₂ and TiCl₂Cp both as doublet states is -83.36 kcal/mol, with a much smaller charge transfer component of -307.4 kcal/mol compared to the NN bond. Despite the apparent disparity between the bond orders of the NN and NTi bonds, they appear to have similar interaction energies! The quartet-quartet decomposition is -184.5, whilst the ionic decompositions [singlet C5H5TiCl2(-) and C5H5N2(+)] vs [singlet C5H5TiCl2(+) and C5H5N2(-)] are respectively -210.6 and -244.6 kcal/mol.

Both 1 and 2 were previously mooted as systems where adjacent atoms bear formal +ve charges, and characterised by their unusually low interaction energies emerging from an energy decomposition analysis. We can now add the known system 3 to this class, albeit its interaction energy being somewhat higher than 1 or 2. There are probably many more of this type out there.

Sandwich compounds are the colloquial term used for molecules where a metal atom such as an iron dication is “sandwiched” between two carbon-based rings as ligands, most commonly cyclopentadienyl anion (the “bread”) as in e.g. Ferrocene – a molecule first discovered in 1951. An “inverted” sandwich is where the carbon ring is itself sandwiched between two metal ions and one such was reported this year [cite]10.1021/jacs.5c08414[/cite] containing benzene in the middle with scandium as the metal. The novelty of the subsequent four-electron reduction of the benzene “filler” and its ring opening to a linear hexadiene unit resulted in this being selected as one “molecule of the year” for 2025.

The first example of such an inverted sandwich was reported in 1983 (CAMZAP^‡)[cite]10.1021/ja00354a050[/cite] and this made me wonder how many examples have subsequently been discovered. A search of the CSD (crystal structure database) using the following query was undertaken.

This query places a centroid to the central benzene ring and measures its distances to the two metal ions, along with the angle subtended at the centroid. A C-C distance is also defined. The results (82 hits, increasing to 182 for the more general CR, R=C or H above) are shown below (Figure 1).

Figure 1a. Metal to centroid distances and angle at the benzene centroid.

Figure 1b. Benzene CC ring distances.

Figure 1c. Publication year for benzene as sandwich filler.

Figure 1d. Publication year for structures including substituted benzenes as sandwich filler.

An outlier in Figure 1c that dates from 1975 also corresponds to the outlier in Figure 1a seen as a blue dot and is actually an example of TWO metal Pd atoms sandwiched by benzene rings (inadvertently captured by the search definition above). There has been an explosive growth in the reports of crystal structures of such complexes since 2024, which suggests that this is currently an area of intense activity.

The molecule of the year referred to above goes by the CSD name BAFHUQ[cite]10.5517/ccdc.csd.cc2m4ffn[/cite] and its structure is shown below (Figure 2, M-centroid length 1.856Å). This molecule has a calculated[cite]10.14469/hpc/15647[/cite] (D4) dispersion stabilisation of 62.8 kcal/mol, deriving in large measure from interactions of the isopropyl groups on the P ligand.

Figure 2. BAFHUG

HOQNEK[cite]10.1021/acs.inorgchem.4c00149[/cite],[cite]10.5517/ccdc.csd.cc2j0r7k[/cite] with Ti as metal (Figure 3) is an example having the shortest M-benzene centroid length (1.69Å). The dispersion contribution in this case is 46.2 kcal/mol.

Figure 3. HOQNEK

CALNIP[cite]10.1039/C6DT03565E[/cite],[cite]10.5517/ccdc.csd.cc1l9wrj[/cite] has the shortest distance to the ring centroid of 1.56Å, with toluene as the sandwich filler.

Figure 4. CALNIP (Fe-centroid 1.56Å, K-centroid 2.96Å).

The trends above suggest that a new area of the reactivity of aromatic molecules such as benzene when sandwiched between two metal atoms may be emerging.

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^‡Actually a “triple decker” sandwich containing both ligand-metal-ligand and the inverted metal-ligand-metal motifs.

The annual “Molecules of the Year” selections are available for the year 2025. A theme was elemental allotropes and one such was carbon in the form of C₄₈ stabilised by formation of a catenane C₄₈.M3 (M = red ligand below)[cite]10.1126/science.ady6054[/cite] – it was not possible however to crystallise C₄₈.M3. When “unmasked” by removal of the M ligand, the true allotrope C₄₈ had a solution half-life of about 1 hour at 20°C. This follows the reports from 2019 onwards of a series of smaller cyclo[n]carbon allotropes, (n=6,10,12,13,14,16,18,20,26)[cite]10.1126/science.aay1914[/cite],[cite]10.59350/jdy16-7rv58[/cite] which were only characterised on a solid surface and not in solution.

Since I did not find 3D model coordinates for the 285 atom C₄₈.M3 in the article ESI, I generated them using the following procedures:

1. Using the Chemdraw CDXML file located in the article ESI, saving as a MDL molfile and opening in Avogadro2 (direct opening of the CDXML file fails) and running Extensions/Optimise geometry. This produces an approximate 3D model using a simple molecular mechanics force field.
2. These coordinates were then refined using the semi-empirical PM6 and PM7 QM methods implemented in Gaussian. The latter includes a dispersion attraction term whilst the former does not; the difference is clear to see.[cite]10.59350/x9m30-5aa79[/cite]
3. This system was also finally optimised using the r²-SCAN-3c[cite]10.1063/5.0040021[/cite],[cite]10.1063/5.0004608[/cite] “Swiss army knife” thrice corrected density functional. The initial geometry was based on PM6, using the tightopt keyword, followed by a frequency calculation. Interestingly, the final geometry is closer to PM6 than to PM7. Click on the graphic above to view this 3D model.

Table 1.
PM6 optimised
PM7 optimised
r2-SCAN-3c optimised. Click on image to view 3D model

The geometry of [n]-annulenes and [n]-cyclocarbons

Calculating the quantum mechanical geometry of both [n]-annulenes and by association [n]-cyclocarbons is non trivial.[cite]10.59350/nnctg-v6535[/cite],[cite]10.1021/acs.jpca.3c07797[/cite] Many DFT functionals for example tend to over-estimate the degree of C-C bond length alternation around the ring.[cite]10.3390/molecules25030711[/cite] Recognising this, the authors of this article[cite]10.1126/science.ady6054[/cite] calibrated their own adjustment to the veritable CAM-B3LYP functional against a redetermined crystal structure of [18]-annulene[cite]10.5517/ccdc.csd.cc2gzmz2[/cite],[cite]10.5517/cc1krr46[/cite],[cite]10.1021/acs.jpca.3c07797[/cite] calling the resulting functional OX B3LYP30[cite]10.1021/acsnano.4c14100[/cite] It was specifically optimized for extended conjugated systems by including 30% short-range exact HF exchange.[cite]10.1021/acs.jpca.3c07797[/cite] Significantly, [18]-annulene is an example of a 4n+2 (n = 4) cyclo-aromatic molecule for which significantly less bond alternation (if any) is expected, compared to so-called 4n-class antiaromatic molecules (e.g. cyclobutadiene (n=1)[cite]10.59350/7m9dm-an754[/cite]). The focus of this blog – C₄₈ – is doubly anti-aromatic (n=12), once in the σ- and then the π-frameworks and so its bonds would certainly be expected to alternate in length. This alternation directly results in the Raman activity observed in the C-C stretching regions (see Figure 4c in the article[cite]10.1126/science.ady6054[/cite] and reproduced in Figure 1a below).

Here I also explore the recent r²-SCAN-3c functional,[cite]10.1063/5.0040021[/cite],[cite]10.1063/5.0004608[/cite] which importantly has NOT been adjusted, re-parametrised or scaled to achieve a particular result for these molecules and which – unlike the OX B3LYP30 method – also includes dispersion corrections.^‡ Included in the table below are not only results for [18]-annulene and cyclo[48]carbon but two types of variation on the latter to test the scope of the functionals. The first was to include charged versions of C₄₈, which reduce the electron count by either 4 (changing both the π- and σ- electron manifolds to a 4n+2 count) or by ±2 (changing just one of the manifolds to 4n+2). Also included are C₁₄, C₁₈, C₄₆ (n=11; 4n+2 =46), C₅₀ (n=12; 4n+2 =50), C₅₈ (n=14, 4N+2=58), C₆₂ (n=15, 4N+2=62) and C₉₈ (n=24; 4n+2 = 98) which are all aromatic rings for which bond alternation should be much smaller, if it occurs at all. C₆₀ is included as a larger 4n anti-aromatic molecule. It also proved possible to model the geometry of the full C₄₈M3 catenane system as shown above using r²-SCAN-3c. FAIR data for the present calculations are published in a data repository.[cite]10.14469/hpc/15614[/cite].

Entry	System	Δra OX B3LYP30 [cite]10.14469/hpc/15617[/cite]	Δra r2-SCAN-3c [cite]10.14469/hpc/15615[/cite]	ΔG C2e
–				-75.872686
1	C14	0.00017 (2)[cite]10.14469/hpc/15640[/cite] 0.00001 (0)[cite]10.14469/hpc/15641[/cite]	0.00002 (0)	-76.124925 (+6.32)
2	C18[cite]10.1126/science.aay1914[/cite]	0.00000 (2)[cite]10.14469/hpc/15619[/cite] 0.06345 (0)[cite]10.14469/hpc/15620[/cite]	0.00000 (0)	-76.129778 (+3.28)
3	C18H18	0.0166 (0)[cite]10.14469/hpc/15625[/cite]	0.0178 (0) 0.0165[cite]10.5517/ccdc.csd.cc2gzmz2[/cite] 0.0192[cite]10.5517/cc1krr46[/cite]	–
4	C46	0.11003 (0)[cite]10.14469/hpc/15626[/cite]	0.00005 (0)	-76.135596 (-0.37)
5	C484+	0.06892[cite]10.14469/hpc/15627[/cite],[cite]10.14469/hpc/15628[/cite]	0.00005 (0)	-76.078196
6	C482+	0.07459 (0)[cite]10.14469/hpc/15629[/cite]	0.03059 (0)g	-76.112352
7	C48	0.11087b[cite]10.14469/hpc/15630[/cite]	0.05781c (0)	-76.135003 (0.0)
8	C48 (chloroform)	0.11087[cite]10.14469/hpc/15631[/cite],[cite]10.14469/hpc/15622[/cite]	0.05810 (0)	-76.135020
9	C48M3	n/a	0.05666 (in) 0.05676 (out)d	–
10	C482-	0.07077 (0)[cite]10.14469/hpc/15632[/cite]	0.02895 (0)g	-76.144764
11	C50	0.00006 (2)[cite]10.14469/hpc/15635[/cite] 0.00011 (1)[cite]10.14469/hpc/15633[/cite], 0.11021  (0) [cite]10.14469/hpc/15637[/cite]	0.00003 (0)	-76.135728 (-0.45)
12	C58	0.00001 (1),[cite]10.14469/hpc/15636[/cite] 0.11022 (0)[cite]10.14469/hpc/15634[/cite]	0.00011 (1, 178i),0.01159 (0)	-76.135900 (-0.56)
13	C60	0.11068 (0)[cite]10.14469/hpc/15624[/cite]	0.05300 (0)	-76.135547 (-0.34)
14	C62	0.11020 (0)[cite]10.14469/hpc/15618[/cite]	0.0001 (1, 373i) 0.02145 (0)	-76.135966 (-0.60)
15	C98	0.11035 (0) [cite]10.14469/hpc/15638[/cite]	0.04035 (0)	-76.135986 (-0.61)
16	C49	0.09494 (0) [cite]10.14469/hpc/15671[/cite]	0.010050 (0)	–
17	C51	0.10230 (0) [cite]10.14469/hpc/15670[/cite]	0.000110 (0)	–
18	C54	0.11032 (0) [cite]10.14469/hpc/15669[/cite]	0.00002 (0)	-76.135839

^aNumber of negative force constants at this geometry in parentheses. ^b Raman Activity 2088, 2192 (expt 1890 A_1g, 2012 E_2g cm^-1) ^cRaman Activity 1752, 1993 cm^-1. ^d Inside and outside an M ligand. Raman modes 1767, 2013 cm^-1. Calculating the intensity of these modes is still in progress, whilst program issues are resolved. ^eFree energy, Hartree (kcal/mol) normalised to C₂ vs C₂ itself at the r²-SCAN-3c level. ^gThis has one aromatic and one antiaromatic electron manifold.

Bond length alternation by table entry.

1. For cyclo[14]carbon, the two methods agree.
2. For cyclo[18]carbon, the two methods differ; the OX B3LYP30 model predicts significant bond alternation.
3. Both DFT methods however closely reproduce the bond length alternation in [18]annulene.[cite]10.59350/k0kjg-hpc66[/cite]
4. Cyclo[46]carbon is formally 4n+2 aromatic (n=11) in both σ and π manifolds and a clear difference in bond length alternation between the two DFT methods emerges, with OX B3LYP30 predicting strong alternation and r²-SCAN-3c none.
5. As with 2, entry 5 is also a 4n+2 aromatic (n=11) and again OX B3LYP30 predicts (weaker) alternation and again r²-SCAN-3c none.
6. The dication has a mixed 4n+2/4n system. The same is true of entry 10.
7. For the key entry 7, a 4n antiaromatic system, OX B3LYP30 predicts twice the bond alternation of r²-SCAN-3c. More surprisingly, the degree of alternation for OX B3LYP30 for this antiaromatic system is almost identical to that for eg entries 2, 5, 11 12, 14, 15, 18 all 4n+2 aromatic. So This functional is not responding to the 4n+2/4n rule in the normal bond length sense.
8. The original calculations were done for the gas phase. Including eg chloroform as a continuum solvent has no effect on the OX B3LYP30, and a very minor effect with r²-SCAN-3c in slightly increasing the alternation.
9. This is the full system, with three enclosing groups M. These have a small effect using r2-SCAN-3c, in very slightly decreasing the bond alternation depending on whether the carbon chain is enclosed by the ligand M or not.

11. This is the two-carbon homologue of C₄₈, and conforms to the 4n+2 rule. Again r²-SCAN-3c predicts no alternation (and indeed no Raman activity), whilst OX B3LYP30 again predicts a strangely invariant value of ~0.11Å.
12. This 4n+2 electron molecule has one particular point of interest. Whilst the OX B3LYP30 method sticks to its standard bond length variation of Δ_r ~0.11Å, r²-SCAN-3c starts to depart from its previous prediction of no alternation for 4n+2 systems. One -ve force constant corresponds to an imaginary (Kekule type[cite]10.59350/jdy16-7rv58[/cite]) mode of 177i cm^-1 and a geometric distortion along this mode now leads to a small bond alternation of 0.0116Å. This is particularly exciting since it has long been thought that given a large enough ring, bond alternation will start to manifest. Perhaps for r²-SCAN-3c the onset of such an effect is around 58 carbons? Previous estimates of this transition have been rather lower (~30).[cite]10.1063/1.476083[/cite]
13. This entry is included partially because of the fame of the C₆₀ fullerene allotrope (which has a very different structure). As a cyclocarbon it is again 4n antiaromatic. Both methods predict bond alternation, albeit that the value for OX B3LYP30 is twice that of r²-SCAN-3c.
14. This continues the trend first seen with entry 12; r²-SCAN-3c showing a doubling in the bond length alternation of a 4n+2 system.
15. With this size ring, the bond alternation of this 4n+2 system is beginning to approach the value for a 4n system, and hence the aromatic/antiaromatic distinction is beginning to vanish.

To summarise, the entries shown in red above correspond to systems for which OX B3LYP30 predicts an almost constant bond alternation of ~0.11Å whereas for the same system using r²-SCAN-3c, the bond alternation is essentially zero (green). The entries marked with orange or pink are 4n+2 aromatic (except entries 16 and 17 which have odd numbers of carbons) in one or both manifolds and for which the onset of intrinsic bond variation may have started.

Raman Activity.

For C₄₈M3, the measured activity as shown in figure 4c[cite]10.1126/science.ady6054[/cite] is reproduced below (Figure 1a). Two modes are active, the A_1g mode (the “kekule vibration“) and a less intense E_2g mode. The OX B3LYP30 functional appears to reproduce this, with the values quoted in this figure (A_1g 1959 and E_2g 2058 cm^-1 derived from scaling the OX B3LYP30 calculated values of 2088 and 2192 cm^-1 reported in the supporting information data file by 0.9386). The r²-SCAN-3c functional likewise predicts the A_1g mode (1752 cm^-1, unscaled) to be more intense than E_2g 1993 cm^-1 (Figure 1b). Before we conclude which method is achieving the better result, the effect induced by the surrounding M3 ligands should be taken into account. We see in the table that the predicted CC bond length variation in the unmasked ring (0.0578) is slightly decreased in C₄₈M3 to 0.0567, as are the bond lengths themselves (1.24839Å gas; 1.24699Å M3). The Raman modes in C₄₈M3 (1767, 2013 cm^-1) are calculated a little higher in wavenumber than C₄₈ itself, but final assignment will depend on calculation of the Raman intensity, which is ongoing (see comment below).

Figure 1a. Observed Raman activity for C₄₈

Figure 1b. Calculated r²-SCAN-3c Raman activity for C₄₈

Figure 1c. Calculated r²-SCAN-3c Raman activity for C₅₀

Figure 1d. Calculated r2-SCAN-3c Raman activity for C₅₈

Figure 1e. Calculated r²-SCAN-3c Raman activity for C₉₈

Conclusions

The synthesis of cyclo[48]carbon and its stabilized derivative C₄₈.M3 has provided a nice opportunity to investigate the strange phenomenon of bond alternation in cyclic carbon systems, coupled with experimental measurements of Raman activity and comparison with calculation. The recent constrained^♥ functional r²-SCAN-3c and perhaps similar ones such as the forthcoming COACH[cite]10.59350/rzepa.28931[/cite] might prove useful in modelling the properties of these unusual compounds of carbon. The r²-SCAN-3c method also suggests that the 4n+2 series cyclocarbons are slightly more stable in terms of free energy than the 4n series.

It is to be hoped that a 4n+2 series example can be synthesized. For one such, e.g. cyclo[58]carbon or especially cyclo[98]carbon, Raman activity is again predicted (Figure 1d,e), whilst for the smaller cyclo[50]carbon (Figure 1c), this activity is predicted absent. Providing a test of this behaviour might provide a motivation for the synthesis of these larger systems!

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^‡Thus the dispersion stabilisation for C₄₈ is -11.5 kcal/mol. ^♥Constraints are exact conditions that an ideal (DFT) functional should have. Though the exact density functional is not known, researchers have discovered analytical properties of such a functional.[cite]10.1109/SCW63240.2024.00027[/cite]. The functional SCAN[cite]10.1021/acs.jpclett.0c02405[/cite] satisfies the 17 derived constraints; many earlier functionals satisfy less than 6 or fewer.

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This post has DOI: 10.59350/g4309-gv109

In the previous post,[cite]10.59350/x5k75-t2m40[/cite] I was commenting that the transition state for borohydride reduction of a ketone contained some close contacts between the hydrogen of the borohydride and the hydrogen of water. A systematic search of the CSD reveals a modest number of such contacts have been observed in crystal structures (Table).  Since it is always good to have a reality check for constructed transition states, here I take a look at some of compounds showing the closest H…H contacts in the experimental database of structures.

The DFT procedures I used to calculate the geometries of the examples tabled below[cite]10.14469/hpc/15566[/cite] were

1. the classical B3LYP/Def2-TZVPP method, but enhanced with the D4 dispersion correction – the latter developed as a successor to the often used D3+BJ predecessor.
2. The DFT method named  r²scan-3c, a composite described by its developers as the “Swiss army knife of DFT methods” and  “r²SCAN-3c Works Well On Everything”.[cite]10.26434/chemrxiv.13333520.v2[/cite] rather grandly quotesThe specific features are described as  “The unaltered r2SCAN functional is combined with a tailor-made triple-ζ Gaussian atomic orbital basis set as well as with refitted D4 and geometrical counter-poise corrections for London-dispersion and basis set superposition error“.[cite]10.1063/5.0040021[/cite] The method scales efficiently to several hundred atoms. Results for both types of DFT calculation are  collected here.[cite]10.14469/hpc/15566[/cite]

Table. Calculated and observed BH…HO interactions
Molecule	rH…H using B3LYP+D4/ Def2-TZVPP	rH…H using r2SCAN-3c/ Def2-mTZVPP	Crystal structure, rH…H, Å	angle, BH…H, °	angle, OH…H, °
JATMUN	1.531	1.544	1.519 (1.513) [cite]10.1016/j.jorganchem.2005.02.037[/cite],[cite]10.5517/cc8gcz1[/cite]	95.1	158.6
OLEVIL	1.612	1.654	1.806 (1.67) [cite]10.1016/j.tetlet.2010.12.106[/cite],[cite]10.5517/ccvkd70[/cite]	107.7	171.4
OLEVEH	1.623	1.666	1.857 (1.757) [cite]10.1016/j.tetlet.2010.12.106[/cite],[cite]10.5517/ccvkd6z[/cite]	105.1	176.6
OWUKID	1.685	1.770	1.871 (1.778) [cite]10.1002/cplu.202000427[/cite],[cite]10.5517/ccdc.csd.cc252bp9[/cite]	131.7	157.8
SASVAS	1.757	1.872	1.901 (1.807) [cite]10.1002/chem.200902915[/cite],[cite]10.5517/cct86qz[/cite]	94.3	155.0
BOTFOJ	1.741	1.810	1.976 (1.856) [cite]10.5517/ccdc.csd.cc2k93ww[/cite]	129.6	171.0
MOPXOG	1.888	1.966	1.990 (1.872) [cite]10.1021/om5005989[/cite],[cite]10.5517/cc136fzq[/cite]	95.4	159.5
FOLREF	1.881	1.966	1.998 (1.908) [cite]10.1021/ol500970x[/cite],[cite]10.5517/cc12tj5l[/cite]	110.2	160.8

OLEVIL@r2SCAN-3c
OLEVIL @B3LYP+D4

In general, the B3LYP+D4 method predicts slightly shorter H…H contacts than does r²SCAN-3c. Comparison with experiment is tricky, since OH and BH distances obtained directly from a classical crystal structure refinement tend to emerge as ~0.1A too short (see eg [cite]10.59350/5dy8w-0zs92[/cite] for more detailed discussion). A simple correction for these values is shown in parentheses in the table above. However, given that the angle subtended at the hydrogen atom varies  enormously, this correction may too simplistic. Better would be if the original crystallographic data could be re-refined using the non-spherical atom model model described in [cite]10.59350/5dy8w-0zs92[/cite]  A provisional conclusion without such treatment might be that r²SCAN-3c is somewhat more accurately predicting the H…H distances.

OLEVIL is interesting because it contains two OH groups, with only one interacting with a proximate BH group. Calculating (r2scan-3c) ν_OH gives values of 3477 cm^-1 as perturbed by the close BH vs a BH unperturbed value of 3801 cm^-1  or Δν 324 cm^-1. The corresponding values using B3LYP+D4 are 3700 vs 3803, Δν 103 cm^-1. This large difference in perturbation predicted by these two DFT methods could be easily tested experimentally. Unfortunately, the experimental information reported for this compound[cite]10.1016/j.tetlet.2010.12.106[/cite] does not contain the OH stretching values, which might have been a good test of the claim that “r²SCAN-3c Works Well On Everything”.

This selection of compounds illustrates how one aspect of a transition state can be given a reality check by comparing a key interaction with experimentally determined crystal structures.