In organic chemistry , MöBIUS AROMATICITY is a special type of
aromaticity believed to exist in a number of organic molecules .
In terms of molecular orbital theory these compounds have in common a
monocyclic array of molecular orbitals in which there is an odd number
of out-of-phase overlaps, the opposite pattern compared to the
aromatic character to Hückel systems . The spatial configuration of
the orbitals is reminiscent of a
CONTENTS * 1 Hückel-
HüCKEL-MöBIUS AROMATICITY 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 . 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
In 2005 the same P. v. R. Schleyer 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. They formed derivatives where osmium had 16 and 18
electrons and determined that Craig–
TRANSITION STATES Möbius systems are also found in transition states. The
determination of a transition state as Möbius or Hückel is involved
in deciding if a reaction with 4N or 4N+2 electrons is allowed or
forbidden. This uses the
ANSATZ WAVEFUNCTION "> L x {displaystyle L_{x}} and L z {displaystyle L_{z}} , we can see that general Mobius boundary conditions for the {displaystyle psi } wavefunction are: * ( x , 0 ) = ( x , L z ) {displaystyle psi (x,0)=psi (x,L_{z})} * ( 0 , z ) = ( L x , z ) {displaystyle psi (0,z)=psi (L_{x},-z)} or using the spherical azimuthal angle {displaystyle phi } : ( ) = ( + 2 ) {displaystyle psi (phi )=-psi (phi +2pi )} For an N {displaystyle N} -carbons, the proposed ansatz linear combination of atomic orbitals (LCAO) is: = j = 0 N 1 c j j = j = 0 N 1 e i j j = j = 0 N 1 e i 2 j / N j {displaystyle {psi _{lambda }}rangle =sum _{j=0}^{N-1}{{c_{j}^{lambda }}{varphi _{j}}}rangle =sum _{j=0}^{N-1}{e^{ilambda phi _{j}}{varphi _{j}}}rangle =sum _{j=0}^{N-1}{e^{ilambda 2pi j/N}{varphi _{j}}}rangle } where j = 2 j / N {displaystyle phi _{j}=2pi j/N} is the angle at each j {displaystyle j} -th carbon atom and j {displaystyle varphi _{j}} is the j {displaystyle j} -th AO. Thus, for circular carbon rings, the general Mobius boundary condition can be rewritten as: c j + N = c j {displaystyle c_{j+N}^{lambda }=-c_{j}^{lambda }} Using this equation and the Euler rule we can find the right {displaystyle lambda } value satisfying previous boundary conditions: e i 2 ( j + N ) / N = e i 2 j / N {displaystyle e^{ilambda 2pi (j+N)/N}=-e^{ilambda 2pi j/N}} e i 2 = 1 {displaystyle e^{ilambda 2pi }=-1} k = 2 k + 1 2 k = 0 , 1 , 2 , , ( N 1 ) {displaystyle lambda _{k}={frac {2k+1}{2}};;;k=0,1,2,ldots ,(N-1)} From the last equation we see that to fulfil the general boundary conditions, {displaystyle lambda } must be a half-integer number. The coefficients of the ansatz become: c j ( k ) = e i ( 2 k + 1 ) j / N {displaystyle c_{j}^{(k)}=e^{ipi (2k+1)j/N}} From figure above, it can also be seen that the overlap between two consecutive p z {displaystyle p_{z}} AOs is at a constant angle = / N {displaystyle omega =pi /N} , and for this reason resonance integral {displaystyle beta ^{prime }} it's considered as a constant into the Huckel matrix we will write later. It could be simply written as: = cos ( / N ) {displaystyle beta ^{prime }=beta cos(pi /N)} where {displaystyle beta } is the standard Huckel’s resonance integral value (the one with = 0 {displaystyle omega =0} ). Nevertheless, the presence of a C 2 {displaystyle C_{2}} axis as the only symmetry element brings to a full phase change at the end of the ring, e.i. between the first and the N {displaystyle N} -th carbon atoms. For this reason, in the Huckel matrix the resonance integral between carbon 1 {displaystyle 1} and N {displaystyle N} is {displaystyle -beta ^{prime }} . For the generic N {displaystyle N} carbons Mobius system, the Huckel matrix H {displaystyle mathbf {H} } is: H = ( 0 0 0 0 0 0 ) {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 width:31.648ex; height:17.509ex;" alt="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 & alphaend{pmatrix}" /> Eigenvalues equation can now be solved. 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 = E k {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 width:42.192ex; height:20.676ex;" alt=" 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" /> Hence we obtain a system of N {displaystyle N} equations, in which the first one ( k = 0 {displaystyle k=0} ) and the last one ( k = N 1 {displaystyle k=N-1} ) have a 1 {displaystyle -1} coefficient: { x 0 c 1 ( 0 ) + c 2 ( 0 ) c N ( 0 ) = 0 c j 1 ( k ) + x k c j ( k ) + c j + 1 ( k ) = 0 c N 1 ( N 1 ) + x N 1 c N ( N 1 ) c 1 ( N 1 ) = 0 {displaystyle {begin{cases}x_{0}c_{1}^{(0)}+c_{2}^{(0)}-c_{N}^{(0)}=0\vdots \c_{j-1}^{(k)}+x_{k}c_{j}^{(k)}+c_{j+1}^{(k)}=0\vdots \c_{N-1}^{(N-1)}+x_{N-1}c_{N}^{(N-1)}-c_{1}^{(N-1)}=0end{cases}}} All these equations can be easily solved using Euler's rule, leading to x k = 2 cos ( 2 k + 1 ) N {displaystyle x_{k}=-2cos {frac {(2k+1)pi }{N}}} hence E k = + 2 cos ( 2 k + 1 ) N = + 2 cos 2 k N {displaystyle E_{k}=alpha +2beta ^{prime }cos {frac {(2k+1)pi }{N}}=alpha +2beta ^{prime }cos {frac {2pi lambda _{k}}{N}}} SEE ALSO *
REFERENCES * ^ 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 *
RESONANT * 3c–2e * 4c–2e * Hypercoordination * 3c–4e * |