In organic chemistry,
Contents 1 Hückel-Möbius aromaticity 2 Transition states 3 Derivation of Hückel MO theory energy levels for Möbius topology 4 See also 5 References Hückel-Möbius aromaticity[edit] The Herges compound (6 in the image below) was synthesized in several photochemical cycloaddition reactions from tetradehydrodianthracene 1 and the ladderane syn-tricyclooctadiene 2 as a substitute for cyclooctatetraene.[5] Intermediate 5 was a mixture of 2 isomers and the final product 6 a
mixture of 5 isomers with different cis and trans configurations. One
of them was found to have a C2 molecular symmetry corresponding to a
Möbius aromatic and another Hückel isomer was found with Cs
symmetry. Despite having 16 electrons in its pi system (making it a 4n
antiaromatic compound) the Heilbronner prediction was borne out
because according to Herges the Möbius compound was found to have
aromatic properties. With bond lengths deduced from X-ray
crystallography a HOMA value was obtained of 0.50 (for the polyene
part alone) and 0.35 for the whole compound which qualifies it as a
moderate aromat.
It was pointed out by
The difference was demonstrated in a hypothetical pericyclic ring
opening reaction to cyclododecahexaene. The Hückel TS (left) involves
6 electrons (arrow pushing in red) with Cs molecular symmetry
conserved throughout the reaction. The ring opening is disrotatory and
suprafacial and both bond length alternation and NICS values indicate
that the 6 membered ring is aromatic. The Möbius TS with 8 electrons
on the other hand has lower computed activation energy and is
characterized by C2 symmetry, a conrotatory and antarafacial ring
opening and 8-membered ring aromaticity.
Another interesting system is the cyclononatetraenyl cation explored
for over 30 years by
Computed structure of trans-C9H9+, 2, illustrating the twisted nature of the ring, allowing incremental rotation of the orientation of p atomic orbitals around the ring: tracing the p orbitals all the way around the ring results in a phase inversion relative to the starting p orbital. The plane of the carbon skeleton (i.e., the nodal plane of the p orbitals) forms a Möbius strip. In 2005 the same P. v. R. Schleyer [10] questioned the 2003 Herges
claim: he analyzed the same crystallographic data and concluded that
there was indeed a large degree of bond length alternation resulting
in a HOMA value of -0.02, a computed NICS value of -3.4 ppm also did
not point towards aromaticity and (also inferred from a computer
model) steric strain would prevent effective pi-orbital overlap.
A Hückel-
The phenylene rings in this molecule are free to rotate forming a set
of conformers: one with a Möbius half-twist and another with a
Hückel double-twist (a figure-eight configuration) of roughly equal
energy.
In 2014, Zhu and Xia (with the help of Schleyer) synthesized a planar
Möbius system that consisted of two pentene rings connected with an
osmium atom.[13] They formed derivatives where osmium had 16 and 18
electrons and determined that Craig–
p z displaystyle p_ z AOs is attenuated by the incremental twisting between orbitals by cos ω displaystyle cos omega , where ω = π / N displaystyle omega =pi /N is the angle of twisting between consecutive orbitals, compared to the usual Hückel system. For this reason resonance integral β ′ displaystyle beta ^ prime is given by β ′ = β cos ( π / N ) displaystyle beta ^ prime =beta cos(pi /N) , where β displaystyle beta is the standard Hückel resonance integral value (with completely parallel orbitals). Nevertheless, after going all the way around, the Nth and 1st orbitals are almost completely out of phase. (If the twisting were to continue after the N displaystyle N th orbital, the ( N + 1 ) displaystyle (N+1) st orbital would be exactly phase-inverted compared to the 1st orbital). For this reason, in the Hückel matrix the resonance integral between carbon 1 displaystyle 1 and N displaystyle N is − β ′ displaystyle -beta ^ prime . For the generic N displaystyle N carbon Möbius system, the Hamiltonian matrix H displaystyle mathbf H is: H = ( α β ′ 0 ⋯ − β ′ β ′ α β ′ ⋯ 0 0 β ′ α ⋯ 0 ⋮ ⋮ ⋮ ⋱ ⋮ − β ′ 0 0 ⋯ α ) displaystyle mathbf H = begin pmatrix alpha &beta '&0&cdots &-beta '\beta '&alpha &beta '&cdots &0\0&beta '&alpha &cdots &0\vdots &vdots &vdots &ddots &vdots \-beta '&0&0&cdots &alpha end pmatrix . Eigenvalues for this matrix can now be found, which correspond to the energy levels of the Möbius system. Since H displaystyle mathbf H is a N × N displaystyle Ntimes N matrix, we will have N displaystyle N eigenvalues E k displaystyle E_ k and N displaystyle N MOs. Defining the variable x k = α − E k β ′ displaystyle x_ k = frac alpha -E_ k beta ' , we have: ( x k 1 0 ⋯ − 1 1 x k 1 ⋯ 0 0 1 x k ⋯ 0 ⋮ ⋮ ⋮ ⋱ ⋮ − 1 0 0 ⋯ x k ) ⋅ ( c 1 ( k ) c 2 ( k ) c 3 ( k ) ⋮ c N ( k ) ) = 0 displaystyle begin pmatrix x_ k &1&0&cdots &-1\1&x_ k &1&cdots &0\0&1&x_ k &cdots &0\vdots &vdots &vdots &ddots &vdots \-1&0&0&cdots &x_ k end pmatrix cdot begin pmatrix c_ 1 ^ (k) \c_ 2 ^ (k) \c_ 3 ^ (k) \vdots \c_ N ^ (k) \end pmatrix =0 . To find nontrivial solutions to this equation, we set the determinant of this matrix to zero to obtain x k = − 2 cos ( 2 k + 1 ) π N displaystyle x_ k =-2cos frac (2k+1)pi N . Hence, we find the energy levels for a cyclic system with Möbius topology, E k = α + 2 β ′ cos ( 2 k + 1 ) π N ( k = 0 , 1 , … , ⌈ N / 2 ⌉ − 1 ) displaystyle E_ k =alpha +2beta ^ prime cos frac (2k+1)pi N quad (k=0,1,ldots ,lceil N/2rceil -1) . In contrast, recall the energy levels for a cyclic system with Hückel topology, E k = α + 2 β cos 2 k π N ( k = 0 , 1 , … , ⌊ N / 2 ⌋ ) displaystyle E_ k =alpha +2beta cos frac 2kpi N quad (k=0,1,ldots ,lfloor N/2rfloor ) . See also[edit] Barrelene Baird's rule Bicycloaromaticity References[edit] ^ Möbius
v t e Chemical bonds Intramolecular (strong) Covalent By symmetry Sigma (σ) Pi (π) Delta (δ) Phi (φ) By multiplicity 1 (single) 2 (double) 3 (triple) 4 (quadruple) 5 (quintuple) 6 (sextuple) Miscellaneous Agostic Bent Coordinate (dipolar) Pi backbond Charge-shift Hapticity Conjugation Hyperconjugation Antibonding Resonant Electron deficiency 3c–2e 4c–2e Hypercoordination 3c–4e Aromaticity möbius super sigma homo bicyclo spiro σ-bishomo spherical Y- Metallic Metal aromaticity Ionic Intermolecular (weak) van der Waals forces London dispersion Hydrogen Low-barrier Resonance-assisted Symmetric Dihydrogen bonds C–H···O interaction Noncovalent other Mechanical Halogen Chalcogen Aurophilicity Intercalation Stacking Cation–pi Anion–pi Salt bridge Bond cleavage Het |

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