CHM 481 Presentation Report Bryant

8/8/2019 CHM 481 Presentation Report Bryant

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On the Topologyand ElectronicStructure of

Mbius AnnulenesCHM 481

Lehigh University | Spring 2009 | Professor Flowers

2009

James Scott BryantSensata Technologies

3/29/2009


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ABSTRACT

Recent advances in the understanding of - and -electron molecular framework conjugation

have increased the theoretical design and successful synthesis of molecules exhibiting chiral aromaticity.

Mbius molecules, which are intrinsically chiral, violate the standard rules governing the formation of

Hckel-type aromatics, but have been successfully synthesized and are stable. Topology and stability

studies of simple Mbius annulenes (e.g. the cyclo C9H9+ cation) with various twist (Tw) and writhe (Wr)

chiral indices indicate that odd linking number (Lk) molecules sustain 4n -electron aromaticity, while

analogous molecules with an even Lk obey the 4n+2 -electron rule. New research indicates that

focusing on Tw rather than Lk, as was previously suggested, in the molecular orbital treatment of

annulenes will result in greater delocalized stability.

AN INTRODUCTION TO MBIUS TOPOLOGY AND ELECTRONIC STRUCTURE

Imagine a strip of paper formed into a loop so that the two ends meet each other. The three-

dimensional figure might be described as a flattened ring with both an inside and an outside surface.

Now imagine temporarily disconnecting the ends, twisting one end 180, and reconnecting them.

Amazingly this simple operation has transformed the flattened ring into a three-dimensional shape with

only one side. The shape is called a Mbius band.

In 1865, mathematician August Ferdinand Mbius published a treatise entitled ber die

Bestimmung des Inhaltes eines Polyders1 which was to become the seminal work on one-sided

surfaces. Despite Johann Benedict Listing having published an analogous work four years previous,2

Mbius would eventually get the credit and namesake for the aforementioned structure. Mbius strips

with a higher number of 180 twists are called paradromic, or sometimes Listing, rings.

The Mbius band is formally described3 as a non-orientable (one-sided) surface obtained by

cutting a closed band or cylinder into a single strip, giving one of the two ends thus produced a one-half

(180) twist, and then reattaching the two ends.4 It is important to note that the twist may be spread out

through the structure and does not necessarily need be located at any one discrete point. While it is true

that the twist in a Mbius band does reduce the structures sidedness to one, it is more useful to state that

the twist changes the structure from one that is orientable to one that is not, as in Figure 1.5 Sidedness,

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mathematically defined as having a normal vector on the surface of the object transferable to any other

point on the object without crossing a boundary,5 is an extrinsic property, whereas orientability is intrinsic.

A surface is called orientable if it is not possible to move a shape on the surface in such a way that it is

transformed into its mirror image5

Figure 1 Examples of orientable and non-orientable molecules and their 2D vectors diagrams.5

It was first reported in 1964 that the possibility to synthesize Mbius molecules existed.6 Until

recently, the difficulties associated with product stabilization thwarted attempts to create such molecules.

The successful synthesis of the first small Mbius molecule7 was reported in 2003. When created

chemically, Mbius bands are severely strained. The electronic structure contains both sp2-hybridized

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atoms within the familiar trigonal planar portion and pyramidalized sp2 atoms within the remainder, as in

Figure 2.7 The pyramidalized portion contributes the most to the overall molecular strain. The twist of the

molecule may be defined as referenced to either two edges of a band or to a -electron system. Closed

molecular ribbons may exhibit twist by the coordination of their atomic framework or by their electronic

topology. Twists related to electronic topology are typically introduced by the alternation of E and Z

double bonds. The first band-type Mbius strip was synthesized in 1982.8,9

Figure 2 Stabilized Mbius bands contain both a planar and a pyramidalized portion.7

A Mbius strip may be formed by twisting one end of the band either clockwise or

counterclockwise, and the two resulting structures are enantiomeric. If the strip itself were static, it would

belong to the C2 symmetry group, as in Figure 3,4 but under experimental conditions, the twist is less than

theoretically delocalized, tending to stick at double bonds. Given a ladder-like molecule such as one

that has three ethylene rungs and two polyether ropes, the localized twist will result in six possible

conformations. When time-averaged, the molecule belongs to the D6 symmetry group.5 The chirality of a

Mbius strip is inherent in its topology. Topological isomerism10 (switching between one- and two-sided

isomer) might be applicable to the design of quantum computers.11 Interestingly, recent experimentation

has resulted in molecules that switch topologies depending upon the solvent polarity.12 As will be

elucidated further, this also switches on and off its aromaticity.

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Figure 3 C2 symmetry of an idealized Mbius molecule.4

If a Mbius molecular band is separated down the rungs of its ladder, unlike an untwisted band,

it will form a single ring with four half-twists, and twice the size of the original band. This has been

experimentally observed using an ozonolysis reaction, as illustrated in Figure 4.5 Figure 55 shows more

complicated topological alterations that have not yet been performed.5

Figure 4 Ozonolysis of a Mbius molecule.5

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Figure 5 Complicated Mbius operations and their theorized resultant molecules.5

The topology of Mbius strips is mathematically described using two independent chiral indices,

twist (Tw) and writhe (Wr). Although theoretically decentralized, in reality Tw is a local phenomenon and is

calculated by integrating rotation angles () around the molecule. Wr is truly decentralized. It is

measured using a double integration and represents the degree to which the band is coiled about itself.

Mathematicians Clugreanu, White, and Fuller, developed a relationship between Tw and Wr after White

observed that there is no topological difference between a band given two half-twists and a band that has

been wrapped around an object twice with no twists before connecting the ends.13,14 The expression, Lk =

Tw + Wr, introduces a third term, the linking number (Lk) which is always an integer quantity. Lk is such

that any non-zero value for a cyclic compound indicates a chiral molecule. As shown in Figure 6,15 a zero

or even Lk precludes any 2p atomic orbital (AO) phase shift around a molecule, whereas an odd Lk results

in at least one 2p AO shift.

Figure 6 Exhibition of the phase shift associated with odd Lk Mbius molecules.15

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Aromaticity may generally be described as the delocalization of -electrons leading to a

significant increase in stability. Hckels Rule states that any cyclic ring molecule will be aromatic if it has

4n+2 -electrons, where n is any positive integer or zero. In 1964, Heilbronner proposed that Mbius

molecules with an odd number of half-twists should not theoretically obey the 4n+2 rule.6 Specifically,

any cyclic ring molecule with an even number or zero half-twists will obey the 4n+2 rule for aromaticity

and be antiaromatic with 4n -electrons, whereas the reverse is true regarding cyclic ring molecules with

an odd number of half-twists. Attempts at synthesizing Mbius molecules were largely unsuccessful due

to the high amount of ring strain placed upon the molecule during twisting and reduced p-p resonance

overlap. In larger molecules, where ring strain is less pronounced but flexibility is higher, potential Mbius

molecules transition into their more stable Hckel configuration.16,17 In the past decade, several Mbius

molecules have been characterized, including transition-state species,18 an unstable intermediate,19 a

non-conjugated cyclic structure,9 and a metallacyclic species.20

A derivation of Heilbronners discovery may be expressed thus: If the molecular orbitals (MOs)

derived from a cyclic conjugated AO basis set array containing one phase shift are populated with 4n

electrons (n being a positive integer), the resulting system will exhibit closed-shell stability.15 Mbius

molecules of higher order with an even Lk will not exhibit a phase shift.

The determination and quantification of aromaticity is not so straightforward, however. It is

inaccurate to judge the aromaticity of a molecule based upon geometric patterns alone since it is a

multidimensional phenomenon.21 In 1959, Winstein22,23 introduced the term homoaromaticity to describe

systems in which either the -electron or -electron conjugation is interrupted by the addition of bridging

groups. As will be discussed, the homoaromaticity of Mbius molecules is currently being researched.

METHOD

Rappaport and Rzepa have examined a series of Mbius (CH)n systems containing between two

and seven half-twists of the 2p AO basis to evaluate the differences between theoretical and empirical

molecular topology and stability. The B3LYP/6-31G(d) density functional approach24 was used to

construct molecules with closed-shells.15 The two independent chiral indices (Tw, Wr) and their

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summation (Lk) were each theoretically calculated and experimentally measured. It is important to note

that the linking number does not change with elastic deformation of the molecule which leads directly to

the ability of molecules to transform strain imposed by Tw into a less destabilizing Wr, as in Figure 7.15 Lk

is calculated using Equation 1, where s is arc length, r1(s) is the first link, t1(s) is a unit vector orthogonal

to the first link, and is a factor corresponding to the distance between two links.

Equation 1 Lk calculation.15

Appropriate mathematical modifications were made to account for molecular structures consisting

of a number of atom-to-atom segments and not a topologically smooth ideal Mbius form. Rappaport has

made the computer software used to calculate the numbers reported available upon request.15

Figure 7 An illustration of the conversion from Tw to Wr for a given Lk.15

RESULTS AND DISCUSSION

A series of 4n+2 annulenes (Figure 815) were imparted with between two and six half-twists using

trans-alkene motifs. A trend emerged in the calculation of the resultant Tw and Wr suggesting that any

simple annulene series contains a minimum value of twist (and a maximum writhe). When Tw is

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minimized it is also maximally delocalized, both in non-alternating bond lengths and similar values of the

local rotation angle . The lack of variation in could be regarded as another expression/metric of

aromaticity.4 Annulene 1 represents the minimum Tw within the illustrated series in relation to ring size.

For those annulenes with larger rings, bond lengths varied more substantially, and twist favored

localization at the longest bond.

Figure 8 4n+2 annulenes utilized for topological and electronic examination.15

Figure 915 illustrates the effects of imparting between two and seven half-twists on the 2p AO

basis exhibiting D2-D7 symmetry respectively. Interestingly, Rappaport discovered that the NICS(0) index

values25 are equal to or less than -10 ppm, characteristic of diamagnetic ring currents stronger than that of

benzene and hence stronger aromaticity (in one sense). It was Wr that varied with increasing Lk while Tw

remained relatively constant. The molecules successfully relieved twist strain by transforming it into a

more delocalized writhe. The implications of this transformation are obvious and important: Since there is

not significantly more strain in a molecule with multiple half-twists as one with fewer half-twists, the

synthesis of higher order Mbius molecules is theoretically plausible.

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Figure 9 Molecules exhibiting D2 (a) through D7 (f) symmetry.15

An analysis of a [16]annulene with a single half-twist previously theorized by Castro26 was used in

comparison to a similar [16]annulene experimentally synthesized by Herges.4 The theorized molecule

was chosen for its low twist value (Tw = 0.24) and its nearly symmetrical bond lengths. The synthesized

annulene was created using a benzo-group backbone for necessary stabilization. Both molecules are

illustrated in Figure 10.15 While both molecules have Lk = 1, the twist value of the synthesized molecule

was markedly higher (Tw = 0.58) due to the rigidity imposed by the backbone structure, limiting the

molecules ability to transform as much twist strain into torsion (writhe) and thus dampening its overall

stability via interference with extended conjugation.15

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Figure 10 Backbone stabilization used in the synthesis of Mbius molecules.15

Lk, Tw, and Wr values were also calculated for a [28]hexaphyrin and a [26]hexaphyrin, both double

twist Mbius molecules, as illustrated in Figure 11.15 Phyrins were chosen for their ability to assume a

lemniscular (figure-eight) topology.27 The [28]hexaphyrin, a crystalline solid, exhibited 4n -electron

antiaromaticity (Tw = 1.19, Wr = 0.81), whereas the [26]hexaphyrin exhibited 4n+2 -electron aromaticity

(Tw = 1.13, Wr = 0.87), both further supporting the Heilbronner refinement of the original Hckel

aromaticity rule.

Figure 11 [28]hexaphyrin and [26]hexaphyrin.15

Of the thirteen annulenes with Lk = 2 examined by Rappaport, ten were determined to be

transition states or a higher-order saddle point, and the remaining three were those for which the

B3LYP/6-31G(d) method predicts the symmetric Dn (n = 2 7) geometry to be a minimum in the potential

surface.15 Rappaport also observed that with increasing ring size with Tw held relatively constant, the

mean value of the local rotation angles decreases despite the standard deviation of increasing. In

other words, there is increased bond angle variation as ring size increases caused by less efficient -

electron conjugation. Variations in local rotation angles are more favorable than overlapping 2p

orbitals as ring size increases.

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CONCLUSIONS

Research into the structure and synthesis of simple and higher-order, intrinsically chiral Mbius

molecules (otherwise known as paradromic or Listing rings) has recently accelerated after roughly four

decades of incorrectly assuming that the difficulty inherent in designing single half-twist compounds would

be amplified when designing those with more twists. The emerging characterization of Mbius

homoaromatic molecules broadens the field and deepens the understanding of aromaticity.

Linking number (Lk) is defined as the summation of the independent chiral indices twist (Tw) and

writhe (Wr). In a refinement of the Hckel rule, cyclic conjugated annulenes with a zero or even Lk will

exhibit aromatic properties with 4n+2 electrons, and those with an odd Lk require 4n electrons to achieve

a closed-shell ground state. Mbius molecules are maximally stable when Tw has been minimized. Any

non-zero Lk implies molecular chirality.

The Mbius molecules with the greatest potential for study due to their idealized nature include a

[9]annulene cation (C9H9+), a perfluorinated annulene anion (C9F9-),28 and a [12]annulene transition state.29

Possessing the ability to quantitatively determine those molecules with the greatest stability will assist in

choosing the isomer with the greatest chance of a successful synthesis. While the torsion strain

transformation from the -electron framework (Tw) to the bending strain in the -electron framework (Wr)

has a direct analogy in nature (seen in the structure of supercoiled DNA30), there are no as yet known

naturally occurring Mbius molecules. Unexplored areas of research include the development of MO

theory accounting for variations in Tw and Wr, whether delocalized Mbius systems are capable of

sustaining characteristic patterns of reactivity with selection rules for electrophilic or nucleophilic

substitution,4 and the potential synthesis of doubly aromatic molecules by combining Mbius aromatic

character with -electron planar orbital overlap.

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REFERENCES

(1) Mbius, A. F., Berichte ber die Verhandlungen der Kniglich Schsischen Gesellschaft derWissenschaften zu Leipzig, 1865, 17, 31-68

(2) Listing, J. B.,Abhandlungen der Mathematischen Classe der Kniglichen Gesellschaft derWissenschaften zu Gttingen, 1861, 10, 97-182

(3) Derbyshire, J., Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem inMathematics, 2004, Penguin, New York, 381

(4) Rzepa, H. S., Chem. Rev., 2005, 105(10), 3697-3715

(5) Herges, R., Chem. Rev., 2006, 106(12), 4820-4842

(6) Heilbronner, E., Tetrahedron Lett., 1964, 1923-1928

(7) Ajami, D., et. al., Nature, 2003, 426, 819-821

(8) Walba, D. M., et. al., J. Am. Chem. Soc., 1982, 104, 3219

(9) Walba, D. M., et. al., New J. Chem., 1993, 17, 661

(10) Frisch, H. L., et. al., J. Am. Chem. Soc., 1961, 83, 3789

(11) Ioffe, L. B., et. al., Nature, 2002, 415, 503

(12) Stpie, M., et. al.,Angew. Chem. Int. Edn., 2007, 46, 7869-7873

(13) White, J. H.,Am. J. Math., 1969, 91, 693-728

(14) Tanada, S., et. al., Nature, 2002, 417, 397-398

(15) Rappaport, S. M., et. al., J. Am. Chem. Soc., 2008, 130(24), 7613-7619

(16) Garrat, P. J.,Aromaticity, 1986, Wiley and Sons, New York

(17) Oth, J. F. M., et. al., Tetrahedron Lett., 1968, 60, 6259-6264

(18) Zimmerman, H. E., J. Am. Chem. Soc., 1966, 88, 1564-1567

(19) Mauksch, M., et. al.,Angew. Chem. Int. Edn., 1998, 37, 2395-2398

(20) Tunyogi, T., et. al., Inorg. Chem., 2008, 47, 2049-2055

(21) Katritzky, A. R., et. al., Chem. Rev., 2001, 101, 1421-1449

(22) Winstein, S. J., J. Am. Chem. Soc., 1959, 81, 6524-6525

(23) Allan, C. S. M., et. al., J. Chem. Theory Comput., 2008, 4 (11), 1841-1848

(24) Frisch, M. J., Gaussian 03, 2004, Gaussian, Inc., Wallingford, CT

(25) Schleyer, P. v. R., et. al., J. Am. Chem. Soc., 1996, 118, 6317-6318

(26) Castro, C., et. al., Org. Lett., 2002, 4, 3431-3434

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(27) Rzepa, H. S., Org. Lett., 2008, 10, 949-952

(28) Kastrup, C. J., et. al., Chem. Commun., 2002, 642-643

(29) Castro, C., et. al., J. Am. Chem. Soc., 2005, 127, 9704-9805

(30) Bouchiat, C., et. al., Euro. Phys. J. E.: Soft Matter, 2000, 2, 377-402

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