A Quantitative Metric for Conjugation in Polyene Hydrocarbons Having

Article pubs.acs.org/JPCA

A Quantitative Metric for Conjugation in Polyene Hydrocarbons Having a Single Classical Structure Jerry Ray Dias* Department of Chemistry University of Missouri Kansas City, Missouri 64110-2499, United States

ABSTRACT: A quantitative measure of the extent of conjugation is introduced. The number of Dewar resonance structures (DS) in a conjugated hydrocarbon quantitatively determines its extent of conjugation interaction. The greater the degree of conjugation the more stable is the conjugated hydrocarbon. The number of single bonds joining the exo-double bonds of a purely cross-conjugated hydrocarbon is equal to its number of Dewar resonance structures (DS). Polyene cross-conjugated hydrocarbons are minimally conjugated and possess only one classical structure with a succession of fixed single and exo-double bonds. There are three increasing levels of conjugation for systems with a single classical structure according to whether they are cross-conjugated, linear extended two-way conjugated, and ring-enhanced conjugated. Numerous analytical expressions have been derived. Our valence-bond (VB) approach involves studying the interrelationship of several well-known series of conjugated hydrocarbons having only a single classical structure, i.e., the number of Kekulé resonance structures is one (K = 1). This work provides a totally new perspective to the concept of conjugation.

■

INTRODUCTION Ever pervasive concepts used in chemical research and teaching are aromatic stabilization energy (ASE), conjugation, hyperconjugation, inductive effects, resonance energy, and strain energy and their relative contributions to a molecule’s properties. Separate reviews on the role of aromaticity,1 pπdelocalization,2 inductive effects,3 and hyperconjugation4 in chemistry have recently appeared in the literature. Another review by Wu and Schleyer5 emphasized the interplay of twoway hyperconjugation in determining conformation between planar and perpendicular forms of 1,3-butadiene and related systems; loss of conjugation in both perpendicular 1,3butadiene and tub-shaped cycloocta-1,3,5,7-tetraene is substantially compensated by enhanced two-way hyperconjugation.5 Herein, we will propose extended two-way conjugation to explain the increased energy stabilization of acyclic linear conjugated over cross-conjugated polyenes. Hyperconjugative effects are absent in planar conjugated hydrocarbons, like most unsubstituted polycyclic aromatic hydrocarbons (PAHs). A review on conjugation per se is currently lacking and is the partial goal of this paper. It is a necessary component that needs to be sorted out when evaluating the exclusive magnitude of ASE. It is well-known that the more Kekulé resonance structures that a PAH has the more stable it is and numerous © 2014 American Chemical Society

equations have been proposed relating energy based on the number of Kekulé resonance structures (K > 1).6 A qualitative and intuitive phrase frequently iterated is “the more conjugated, the more stable”. Undergraduate organic textbook treatments of conjugation of 1,3-butadiene shows resonance between the principal structure CH2CH−CHCH2 and the less important structures •CH2CHCH−CH2•, ⊖CH2 CHCH−CH2⊕, and ⊕CH2CHCH−CH2⊖, where the latter three structures have either a lone electron or charge on a remote unstarred and starred position [for example, page 521 in ref 7]; alternant hydrocarbon systems have no odd size rings and can have their conjugated carbons partitioned into sets of starred (*) and unstarred (0) conjugated sites such that a site in one set has only nearest neighbors solely belonging to the other set. We will represent these latter three 1,3-butadiene resonance structures by a single Dewar resonance structure with a long bond denoting communication between these two selected positions (cf. with graphical abstract); this communication occurs electronically through space and through a succession of connecting bonds, i.e., the Dewar long bonds Received: August 19, 2014 Revised: October 10, 2014 Published: October 13, 2014 10822

dx.doi.org/10.1021/jp5083906 | J. Phys. Chem. A 2014, 118, 10822−10836

The Journal of Physical Chemistry A

Article

Figure 1. Sachs graphs for the determination of the aN−2 coefficient of the characteristic polynomial and aN−2ac coefficient of the matching polynomial for 5,6-dimethylenecyclohexa-1,3-diene; note there are two arrangements for 3 K2 components in the hexagonal ring above (third and fourth Sachs graphs). For determination of aN−2ac, the Sachs cycle graphs are omitted.

identify long-range spin-pairing. This same textbook [page 577, ref 7] also shows three Dewar resonance structures (DS = 3) for benzene as long bonds between para positions. Pauling referred to Dewar resonance structures as first excited structures and showed that they were required for calculation of bond orders for essentially fixed single bonds like those joining two naphthalene moieties in perylene.8 Herein, we strive to place this well-known phrase into a more quantitative framework for conjugated systems having only one classical structure (i.e., K = 1). Restricting our study to conjugated hydrocarbons with one classical structure avoids any complication that might arise from overlapping Kekulé delocalization.

■

The characteristic polynomial contains both conjugation and cyclic delocalization contributions, whereas the matching polynomial only contains conjugation contributions. The important point here is that the number of Dewar resonance structures is a function of conjugation only since it is derived from the matching polynomial, which is devoid of cyclic Sachs graph contributions. The use of Sachs graphs for constructing characteristic polynomials can be easily found in a number of accessible literature.12,13 To review, consider Sachs formula, which sums up all possible combinations of disjoint bonds (K2 components) and cycles (Cm components), which determines the coefficients of the characteristic polynomial aN =

∑ (−1)c(s)2r(s) s∈S

THEORY

where 0 ≤ n ≤ N, s is a Sachs graph, Sn is the set of Sachs graphs with exactly n vertices, c(s) is the total number of K2 and Cm components, and r(s) is the total number of rings (cycles) in s. All combination of lK2 and kCm such that 2l + km = n are enumerated. By definition a0 = 0 and a2 = −q is the number of edges (molecular graph σ -bonds). The number of Sachs graphs for calculation of a6 and a6ac for 5,6-dimethylenecyclohexa-1,3diene (benzene-quinododimethide) is illustrated in Figure 1. Bond Resonance Energy (BRE). Let the HMO standard resonance integral (β) between two adjacent carbon pπ orbitals and a hypothetical π−system in which a given Cp−Cq π−bond blocks cyclic conjugation through the selected π-bond be designed by assuming that βpq = iβ and βqp = −iβ where i is the square root of −1. In this hypothetical π-system no circulation of π-electrons is expected along the circuits sharing the Cp−Cq bond in common.14 The BRE (or t-BRE where t stands for topological) for the Cp−Cq bond is then defined as the destabilization energy of this hypothetical π-system relative to

Dewar Resonance Structures. The characteristic polynomial and matching (acyclic) polynomial are well-known HMO polynomials that are used to determine topological resonance energies (TRE).6,9 Programs for the characteristic and matching polynomials are available.10 The sum of the positive roots to the first polynomial give the total pπ-electronic energy (Eπ) of a PAH and the sum of the positive roots the latter polynomial gives the pπ-electronic energy of to the hypothetical reference to original PAH because the matching polynomial excludes cyclic Sachs graph contributions. The difference in these two pπ-electronic energies gives the TRE of the PAH. One can obtain the number of Dewar resonance structures for conjugated hydrocarbons from the last two coefficients [aN−2ac and aNac, i.e., the (N − 2nd) and Nth terms] of the matching polynomials by using the following equation:11 DS = |aN − 2 ac + (N /2)aN ac| = |aN − 2 ac| − (N /2)K 10823



Article

Figure 2. Linear versus cross-conjugated series and their analytical expression for their number of Dewar resonance structures (DS).

and Gutman hypothetical reference system G − ci is determined as follows. Here G − ci has the same topology as G but is assumed to have no ith circuit, i.e., the contributions the ith circuit to the coefficients of P(G − ci ;X) are ignored when using the Sachs theorem. Thus, the characteristic polynomial of G − ci is determined by

the actual one. For peripheral bonds, we use the following graph-theoretical method to determine the characteristic polynomial [P(Gref ;X) ] of the hypothetical reference πsystem (Gref) associated with a given actual system (G) which circumvents using a determinant as done by Aihara:14 P(Gref ; X) = P(G − e ; X) − P(G − (e); X)

P(G − ci ; X) = P(G ; X) + 2P(G − ri ; X) = P(Gref ; X)

where G − e is the molecular graph of G with edge Cp−Cq deleted, and G − (e) is the molecular graph of G with edge Cp− Cq and its corresponding p,q vertices deleted. As an example, consider the molecular graph of benzene C6:

where G − ri is the molecular graph of G in which the ith ring (ri) is deleted.15 For benzene molecular graph, this gives P(G − ci ;X) = P(C6; X) + 2P(ϕ; X) = (X6 − 6X4 + 9X2 − 4) + 2 = X6 − 6X4 + 9X2 − 2, where the empty graph ϕ = 1. For all monocyclic systems, the two different measures give the same precise value for P(Gref ; X) and so do all the members of the ppolyphenyls and p-polyphenyl-α,ω-quinododimethide series. In general, the Bosanac and Gutman CCE is given by

P(C6ref ; X) = P(L6 ; X) − P(L4 ; X)

where L6 and L4 are the linear molecular graphs of 1,3,5hexatriene and 1,3-butadiene, respectively. For benzene P(C6ref ;X) = X6 − 6X4 + 9X2 − 2, which gives the pπ-electronic energy for the hypothetical reference as E(Cref)π = 7.72741β. Per Aihara, BRE is given by

CCE = E(G)π − E(Gref )π

which is different from that for BRE in that the hypothetical reference Gref is different. Aihara has shown that that the Bosanac and Gutman’s CCE is closely related to his CRE.16 CCE is also called energy effect, ef(G,Z), which is the effect of cycle Z on the total π-energy of a conjugated molecule whose molecular graph is G and was used to show that the empty rings of total resonant sextet (TRS) benzenoids had a lower energy effect [smaller CCE = ef(G,Z) values] compared to the sextet rings on the total π-energy of the TRS benzenoids.17 For the graph theoretical procedure for determining BRE and CCE, the reader should consult our prior paper.18

BRE = E(G)π − E(Gref )π

where E(G)π and E(Gref)π are the pπ electronic energies of the actual and hypothetical reference molecular graphs, respectively.14 For benzene BRE = 8β − 7.72741β = 0.273β. Cyclic Conjugation Energy (CCE). In 1977, Bosanac and Gutman15 proposed a measure of a kind of circuit resonance energy (CRE) for polycyclic π-conjugated systems that is closely related to Aihara’s BRE. The difference between the two different measures is the hypothetical reference system used. The characteristic polynomial [P(G − ci ;X)] of the Bosanac 10824



Article

Figure 3. Dewar resonance structures for the infinite linear and comb cross-conjugated polyenes. The infinite linear polyene is conjugated in both directions without limit with and infinite number of Dewar resonance structures, whereas the cross-conjugated comb polyene is only locally conjugated with only three Dewar resonance structures per unit.

■

RESULTS AND DISCUSSION

estimates of double lengthening and single bond shortening using the ab initio structure for 90° twisted 1,3-butadiene are 0.005 and 0.028 Å, respectively;20 note that this model is not perfect because 90°-twisted s-trans-1,3-butadiene engages in two-way hyperconjugation, which is absent in the planar conformers.5 Using Pauling’s valence-bond equation for pπelectron delocalization by long bond Dewar resonance structures (DS = 1 for butadiene), we get an increase of the formal bond length of the double bonds in 1,3-butadiene by

Acyclic Linear versus Cross-Conjugation. The prototype for conjugation is 1,3-butadiene. Evidence for pπ-electron delocalization in 1,3-butadiene was presented by showing that its structural consequence was to increase the formal bond length of its double bonds by 0.007 Å (compared to their ethylene bond length of 1.3305 Å) and to decrease of the length of its formal single bond by 0.016 Å.19 More recent 10825



Article

Figure 4. Isodesmic bond separation energies calculated from experimental heats of formation using protobranching (2.8 kcal/mol) and hypeconjugation (5.5 kcal/mol) corrections recommended in ref 24.

0.006 Å (compared to 1.3305 Å) and to decrease of the length of its formal single bond by 0.009 Å. We will show that from a valence-bond point of view, the number of Dewar resonances structures (DS) is responsible for π-electron delocalization in conjugated polyenes and is a strong measure of conjugation. Dewar resonance structures in planar fully conjugated systems contain only electronic information and are devoid of variation in carbon hybridization, hyperconjugation, and cis/ trans effects. In Figure 2 it is evident that for cross-conjugated polyenes, known as dendralenes (called comb graphs by Hosoya)21 and radialenes, the DS values coincide with the number of single bonds joining the exo-double bonds. These are minimum degree conjugation systems. The DS values for the acyclic cross-conjugated polyenes increase linearly as the number of carbons (Nc) and the DS values for the acyclic linear polyenes increase as the square of the number of carbons; note here that we do not distinguish between cis and trans linear polyenes. If we plot the HMO total pπ-electronic energy of the cross-conjugated polyenes against their DS values, we obtained a linear regression, but for the linear polyenes, we obtain a parabolic-like curve; note that both cross-conjugated and linear polyenes have the same end-groups. Thus, linear acyclic

polyenes with a single topological path energetically gain something more than do cross-conjugated systems. We attribute this extra stabilization in the linear acyclic polyenes to extended two-way conjugation. To illustrate this extended two-way conjugation, consider the infinite linear polyene (polyacetylene) and the cross-conjugated comb polyene in Figure 3. The curved double headed arrows are Dewar long bond resonance structures. The σ-bond marked in bold for the linear polyene becomes a double bond in all the Dewar resonance structures as we extend the long bond successively to the right as shown and to the left (not shown), i.e., the infinite linear polyene is conjugated in both directions throughout with an infinite number of Dewar resonance structures. This picture of two-way conjugation suggests that, for the infinite linear polyene bond length, equalization occurs. Of course real-world polyacetylene has a finite length where bond length equalization is approached only in its center segment. More will be discussed about the convergence bond alternation to bond equalization under the Polyacetylene section. The infinite crossconjugated comb polyene in Figure 3 is quite different because the σ-bond marked in bold becomes a double bond in only one 10826



Article

Figure 5. Comparison of the bond lengths in Å determined by ab initio MP2/ccpVTZ (upper numbers) and by valence-bond using the number Dewar resonance structures (lower numbers) for docosa-1,3,5,7,9,11,13,15,17,19,21-undecaene.

Figure 6. Valence-bond calculation of the double bond lengths of docosaundecaene and the infinite linear polyene using the number of Dewar resonance structures.

polyacetylene, has a zero band gap. 22 Carotenes are tetraterpene natural products with 11 linearly conjugated double bonds but no similar natural product representatives of cross-conjugated molecular systems are known. The members of linear polyenes with corresponding sizes to the

of three Dewar resonance structures as shown, i.e., conjugation is locally confined to its nearest neighbors. It is known that cross-conjugated polyenes are less stable than linear polyenes, but the infinite member has a band gap whereas the infinite linear polyene, known as all trans10827



Article

Figure 7. Valence-bond calculation of the single bond lengths of docosaundecaene and the infinite linear polyene using the number of Dewar resonance structures.

cross-conjugated polyenes have greater DS values giving them greater stability but smaller band gaps because of their greater biradical character (BRC). For the definition of BRC, the reader should refer to our prior work.23 Two-way hyperconjugation5 is balanced between the corresponding possible nonplanar members of the acyclic linear polyene and the crossconjugated polyene series. Enthalpy data for [2]-dendralene (1,3-butadiene; Δf Ho = 26.0 kcal/mol) and [4]-dendralene (Δf Ho = 61.9 kcal/mol) can be found at http://webbook.nist.gov/. Figure 4 summarizes the energy of stabilization due to conjugation using experimental heats of formation. These isodesmic results incorporate corrections recommended in the reference for protobranching (2.8 kcal/mol) and hypeconjugation (5.5 kcal/mol).24 All these enthalpy equations should result in Δf Ho = 0 since the bonds made equals the bonds broken, i.e., involve equal bonding balanced equations. Any difference arises from the extra stabilization due to conjugation. For 1,3-butadiene (upper equation), we see that if there was no additional stabilization due to conjugation, the standard enthalpy of formation for butadiene should be 40.73 kcal/mol (i.e., 26 + 14.73); this value agrees with the recommended value of 14.8 kcal/mol.24

As shown by the middle isodesmic bond separation equation, adding a second conjugation mode results in an even greater stabilization energy increment, i.e., 25.46 kcal/mol or almost twice the 14.73 kcal/mol value for the first conjugation. Therefore, the total conjugation stabilization energy for 1,3,5hexatriene is 40.19 kcal/mol. This value may be compared the conjugation stabilization energy for the cross-conjugated system, 2,3-divinyl-1,3-butadiene (lower isodesmic equation), which has a value of 36.41 kcal/mol. Since the number of Dewar resonance structures (DS) is a direct measure of conjugation and DS = 1 for butadiene and DS = 3 for both 1,3,5-hexatriene and 2,3-divinyl-1,3-butadiene, this gives a conjugation stabilization energy of around 12.8 kcal/mol for each Dewar resonance structure. Since benzene has DS = 3 and ASE = 28.8 kcal/mol, combining these results give conjugation energy + ASE = RE = 3 × 12.8 + 28.8 = 67.2 kcal/mol, which compares favorably with the value of 65.1 kcal/mol given in Scheme 2 and Table 7 of ref 24. The bond lengths for [3]- and [4]-dendralene have been determined for various conformations.25 Dewar resonance structures contain only electronic information and are devoid of steric, hyperconjugation, and conformational consequences. A 10828



Article

Figure 8. Three series of of conjugated ring systems with only one classical Kekulé structure (i.e., K = 1) but increasing number of Dewar resonance structures (DS).

series of polyenes have been studied in regard to their end group effect at the Hückel MO level with recursion equations.27 We can directly compare the cross-conjugated polyene with the linear polyenes in Figure 2 because both have the same endmost like double bonds, which are absent in the radialenes. Example valence-bond calculations (VB-DS) of the bond lengths are given in Figures 6 and7. This VB-DS model predicts that in the middle of the infinite linear polyene chain that both the single and double bond lengths of converge to 1.3939 Å, close to the bond lengths of benzene, which we believe is intuitively reasonable. However, higher level calculations that include Peierl’s distortion and correlation effects predict that the infinite polyacetylene chain converge to bond alternation values close to 1.36 and 1.43 Å for the double and single bonds, respectively, which is close to the average of 1.3939 Å.28 Setting the single and double bond order equations equal (pdouble = psingle) in Figures 6 and 7 gives 1 + 0.03(1/16)[Nc2 − 8Nc +12] = 0.03(1/16)[Nc2 − 4], which upon solving one obtains Nc ≈ 70, or about 35 conjugated double bonds are required before bond length equalization begins to occur in the middle region of polyacetylene. This value is close to the chain length values studied by several research groups.29−31 Adapting Krygowski’s and co-workers’ bond length alternation equation [GEO = (257.7/n)∑(Rav − Ri)2],32 we compare the bond length alternation for the bonds 1−6 with those from 7 to 12 (Figure 5, both having n = 5 bonds) in docosa-1,3,5,7,9,11,13,15,17,19,21-undecaene. This gives GEO(1−6) = 0.442 and GEO(7−12) = 0.241 for the MP2/ cc-pVTZ bond lengths or GEO(1−6) = 0.708 and GEO(7− 12) = 0.140 for the VB-DS bond lengths; note that GEO = 0

highly correlative linear regression for dendralenes is obtained for a plot of HMO total pπ-electronic energy (Eπ in β units) versus DS (Eπ = 2.4312DS + 2.0388, R2 = 1). Polyacetylene. 1,3-Butadiene is the smallest member of both the polyene series in Figure 2. The smallest member of a homologous series invariably have properties that are more deviant from the trend being compared relative to the higher members. A thorough MP2/cc-pVTZ ab initio study of bond lengths of trans-polyenes up to 11 conjugated double bonds by Schmaltz and Griffin is noteworthy.26 In Figure 5 the bonds lengths of docosa-1,3,5,7,9,11,13,15,17,19,21-undecaene calculated by both the ab initio MP2/cc-pVTZ and our valencebond (VB) method that uses the number of Dewar resonance structures (DS). Both sets of bond length calculations compare reasonably well. While our semiempirical VB method is only qualitative compared to ab initio methods because it does not incorporate electron correlation, Peierl’s distortion, nor does it differentiate between cis- and trans-polyacetylene, but it does have the advantage of allowing the formulation of analytical and recursion equations not possible otherwise. This example shows that bond length alternation between the formal carbon− carbon single and double bonds diminish toward the middle of the conjugated chain. The double bond on the terminus does not change much from compared to 1,3-butadiene (1.3401 Å by MP2/cc-pVTZ and 1.3367 Å by VB-DS). Inclusion of Dewar resonance structures result in bond shortening of essentially fixed single bonds and lengthening of essentially fixed double bonds. We expect that the carotene tetraterpenes which have 11 linearly conjugated trans-double bonds to have a similar bond length pattern. Five pairs of strongly subspectral 10829



Article

Figure 9. Dimethylenedihydronaphthalene isomers and other related 6-double bond molecules and their pπ-electronic properties. The Hamiltonian paths are shown by the squiggly lines for the naphthoquinododimethide isomers. The first two molecules have Hamiltonian paths that include all the double bonds in succession, and the last two have no Hamiltonian path.

that exhibited moderate aromaticity and diatropicity, although they only have a single classical structure with no aromatic conjugated circuits (i.e., they are devoid of any sextet rings, whatsoever).39 While it is true that the tail coefficient in the characteristic polynomial is devoid of contributing Sachs cyclic graphs leading to K = 1, there are Sachs cyclic graphs contributing to the other coefficients, suggesting that this should lead to some small to moderate cyclic resonance energy. Maximum aromaticity in an alternant even carbon polycyclic conjugated hydrocarbon can only occur when the tail coefficient in its characteristic polynomial corresponds to a set of covering (4n + 2) cyclic and K2 Sachs graph components. A comparison of BRE (upper number) and CRE (lower number) data are given by the decimal numbers within the individual rings for the series in Figure 8. Recall that both BRE = CCE = 0.273 for benzene and that Aihara and co-workers have shown that BRE is an indicator of local aromaicity.40 As an example, one could say that 3,6-dimethylenecyclohexa-1,4diene in Figure 8 is about 22% times as aromatic as is benzene because 0.0612/0.273 = 0.224 even though it has only one classical structure (i.e., K = 1). Thus, it is evident by the larger BRE and CCE numbers given within the rings of the molecules in the series shown in Figure 8 that the rings most remote to the exo-double bonds are the most aromatic by these measures. The upper two series in Figure 8 were also theoretically studied in regard to diradical character.41 Further support that conjugated hydrocarbons with a preponderance of fixed double bonds within rings exhibit a nonstandard mode of cyclic conjugation not associated with a Kekulé or Clar structure is gained by examining the middle series in Figure 8. One topological (Hamiltonian) pathway exists from one exo-double bond to the other via the sequence of double bonds along the perimeter of the conjugated members in this series; Hamiltonian paths are single paths that pass through all the molecular graph carbon vertices without disruption. This Hamiltonian path is stitched together by cross-conjugated links to form rings. Each cross-link should

for no bond length alternation. This shows that bond length alternation is decreasing as one moves toward the middle section of docosa-1,3,5,7,9,11,13,15,17,19,21-undecaene. Comparison of theory with experimental results for polyacetylene faces many challenges such as Peierl’s distortion, coiling and conformation, interchain interaction, and nonuniform chain length just to mention a few. Conjugation in Ring Systems with One Classical Structure. The radialenes like the middle cross-conjugated series in Figure 2 have DS values corresponding to the number of single bonds joining the exo-double bonds. Unlike the crossconjugated and linear polyenes, radialenes have no end group effect and their double bonds are locked into a cis relationship. The best studied radialene is [6]radialene. The X-ray crystal structure of its hexamethyl derivative has essentially D3d symmetry.33 The lack of aromaticity in [6]radialene has been demonstrated by various measures. Its HOMA = −2.41,34 BRE = 0.009β,35 and absence of ring current.36 The experimental bond lengths (1.337 and 1.495 Å) of [6]radialene and the B3LYP computed bond lengths (1.343 and 1.493 Å)37 agree well with the values (1.339 and 1.496 Å) calculated by our valence-bond DS method.11 A highly correlative linear regression for radialenes is obtained for a plot of HMO total pπ-electronic energy (Eπ in β units) versus DS (Eπ = 2.4351DS − 0.0281, R2 = 1). While Figure 2 displays purely conjugated series of polyenes, Figure 8 now presents series having rings displaying some type enhanced conjugation, though each member of these series have only one Kekulé structure (K = 1). Gutman early on used his conjugated circuit energy (CCE) [also called energy effect, ef(G,Z)]17 to show that benzenoid hydrocarbons with a preponderance of fixed double bonds within rings exhibited a nonstandard mode of cyclic conjugation not associated with Kekulé or Clar structures.38 Subsequently, Aihara and Makino used BREs, CREs, magnetic resonance energies, and ring currents relative to benzene to show that the two upper series in Figure 8 have remote rings from the exo-methylene groups 10830



Article

Figure 10. Illustration of the origin of ring enhanced conjugation is shown by comparing the Sachs graphs for 5,6-dimethylenecyclohexa-1,3-diene, which has only one classical structure. While the tail coefficient in the characteristic polynomial is devoid of cylic Sachs graphs leading to K = 1, there is a cyclic graph in the prior coefficient (bold hexagon above); the corresponding Sachs graphs are shown above where the ones that have associated Dewar structures have long bonds represented by curved double headed arrows. The ring enhanced conjugation arises from the third graph (first graph, second row), which is an additional graph not present in the Hamiltonian octatetraene path; the third and fourth graphs are equivalent to the two Kekule’ resonance structures in benzene and correspond to the conjugation components of a circuit of six π-electrons.

resonance structures, and the largest total pπ-electronic energy as measured by Eπ. A number of topological parameters are relevant to the three isomeric series in Figure 8. All the double bonds in the polyacene-2,x-quinododimethides (uppermost series in Figure 8) are located in trans-1,3-butadiene subunits (fragments); no cis-1,3-butadiene subunits exist in this series. The polyacene2,3-quinododimethides (middle series in Figure 8) have both cis- and trans-1,3-butadiene subunits with the number of number of cis-1,3-butadiene subunits being equal to the number of rings plus one (r + 1). The lower zigzag 1,xquinododimethides also have cis-1,3-butadiene subunits within the rings. According to the topological measure of stability (TMS) index43 for PAHs, the more cis-1,3-butadiene fragments, the more stable is the polycyclic system. We note here that both the acyclic cross-conjugated and linear polyenes have all scis and all s-trans conformations accessible to them (via coiling for the longer systems), though the former is displayed as all scis and the latter as all s-trans in Figure 2. However, the radialenes only have all cis-1,3-butadiene units because of their ring systems. Three Levels of Conjugation in Systems with One Classical Structure. There are four degrees of conjugation that with each increase in degree progressively become more stable as measured by their equations for the number of Dewar resonance structures; these degrees correspond to the degree of their corresponding analytical equations for their number of Dewar resonance structures. They are (1) cross-conjugation, (2) linear conjugation, (3) ring enhanced conjugation, and (4) cyclic Kekulé resonance conjugation. The first three degree levels of conjugation are illustrated by the series given in

only add one additional Dewar resonance structure per link. On comparison of analytical third-degree DS equation for the middle series in Figure 8 with that of the second degree analytical DS equation for the acyclic linear polyenes in Figure 2, we conclude that there must an additional conjugation enhancement relative to the linear conjugation mode. Both the bottom two series have members that have a single Hamiltonian path that includes all the double bonds with the rings being formed by stitching with cross-links. Figure 9 compares all the nonradical dimethylenedihydronaphalene (naphthoquinododimethides) isomers and other 6-double bond related molecules. The general pattern revealed here is if all the double bonds are covered by a Hamiltonian path, the more Dewar resonance structures possible. Like the members in the two series in Figure 2 where the linear acyclic polyenes (upper series) have Hamiltonian paths and are more conjugated and more stable than the corresponding members in the cross-conjugated series (second series), which do not have Hamiltonian paths; the members of the uppermost series in Figure 8 do not have Hamiltonian paths and are less conjugated and stable than the corresponding members in the lower two series, which have Hamiltonian paths that do cover all their double bonds. The conjugation energy at the HMO and self-consistent field (SCF) (VB) levels of theory for the first two members of the uppermost series and the first five members of the middle series (Figure 8) can be found in columns labeled as RE in Tables I and III, respectively, of a paper by Gleicher and co-workers.42 Thus, we generalize and say when a set of isomers with only fixed double bonds are compared, the isomers with Hamiltonian paths have the fewest cross-links, the more conjugation and number of Dewar 10831



Article

Figure 11. All possible arrangements of three double bonds in octa-1,3,5,7-tetraene with its five Dewar resonance structures labeled with curved double headed arrows. Note that the characteristic and matching polynomials are identical for acyclic polyenes and that octa-1,3,5,7-tetraene is equivalent to the Hamiltonian path in benzene-2,3-quinododimethide in Figure 10. The non-Dewar structures are given by the number of double bonds times the number of Kekule’ structures.

Figures 2 and 8 and the last degree by our prior published DS analytical expression for the linear polyacene series.23 The first three degree levels are characteristic of systems herein with solely fixed single and double bonds. The fourth degree level includes cyclic delocalization of the pπ-electrons via Kekulé resonance. The middle cross-conjugated polyene series in Figure 2 is minimally conjugated with three conjugated modes per double bond as shown in Figure 3 and has a first degree DS analytical expression. The upper linear acyclic polyene series in Figure 2 is conjugated throughout its length as shown in Figure 3 and has a second-degree DS analytical expression. Unlike the cross-conjugated members, the latter linear acyclic polyene series has every double bond involved in extensive two-way conjugation, except the end ones, leading to greater overall conjugation and consequently greater stability. As represented by the ring series in Figure 8, except for two exo-double bonds, all the other double bonds are involved with two- and threeway conjugation with third-degree DS analytical expressions and are even more conjugated and stable due to the topology associated with rings. Finally, as illustrated by the fourth-degree DS analytical expression for the linear polyacenes previously published,23 all the rings are involved with mobile resonating ring circuits having superimposed aromatic stabilization due to Kekulé resonance structures. While it is true that the latter degree also exhibits conjugation our focus in the paper is on conjugated systems only having one classical structure, i.e., K = 1, because we desire to sort out the pure effect of conjugation without the complication of Kekulé cyclic resonance structures. In general, we can now say that the more Dewar resonance structures an isomeric polyene has the more conjugated and the more stable is its pπ-electronic energy. Figures 10−13 present our analysis of these degrees of conjugation for the first three members of the middle polyacene-2,3-quinododimethide series in Figure 8. Consider now benzene-2,3-quinododimethide (5,6-dimethylenecyclohexa-1,3-diene, K = 1), which has Eπ = 9.9540 β and DS = 7

for which the Sachs graphs were illustrated in Figure 1. Figure 10 now presents an explanation for ring enhancement of conjugation. Comparing both these figures, their homomorphic relationships should be evident. The C6 graph (bold hexagon graph) at the top is a resonant sextet component belonging to the characteristic polynomial and the third and fourth graphs are its conjugation components separated out by the matching polynomial. This is an important conceptual point that cannot be overemphasized! In determining the a6 coefficient in the characteristic and matching polynomials in Figure 10, note the homology that exists between cyclic C6 and the two arrangements of the three K2 components within the hexagonal ring. Within the Sachs graph framework, the C6 graph is the aromatic contribution, and the two Kekulé arrangements are the conjugation contributions. In general for even AHs, this homology between (4n + 2) cyclic graphs and K2 components always occurs. This homology is why use of Kekulé resonance structures in resonance theoretic methods has been so successful because explicit reference to Sachs cyclic graphs has not usually been necessary. Thus, the third and fourth graphs in Figure 10 being homologous to the cyclic graph above represent ring-enhanced conjugation. As emphasized in Figure 1, the Sachs graph K2 and cyclic components must be superimposable on the original molecular graph C−C skeleton. The Dewar long bonds shown as curved double headed arrows in Figure 10 denote interaction between pπ-electrons located on a starred and nonadjacent unstarred carbons sites not covered by the Sachs graph components. This in essence is equivalent to the origin of the equation for DS values using the last two matching polynomial coefficients given in the Theory section above. In Figure 10, one can count the number of Dewar structures giving DS = 7. Figure 11 gives the Dewar resonance structures for octa1,3,5,7-tetraene, are which equivalent to the Hamiltonian path possessed by 5,6-dimethylenecyclohexa-1,3-diene (benzene-2,3quinododimethide) in Figure 10. A component equivalent to 10832



Article

Figure 12. Additional resonance structures (ΔDS) due to ring formation leads to enhanced conjugation. The upper Dewar resonance structures (ΔDS = 5) are additional ones for naththalene-2,3-quinododimehtide not counted in the linear C12 conjugated polyene Hamiltonian path; the four lower ones were already counted in the Hamiltonian path. The difference between the upper four Dewar structures is the vertical double bond does not reside on the left-hand end.

Figure 13. Additional Dewar resonance structures (ΔDS) due to ring formation leads to enhanced conjugation. The Dewar resonance structures (ΔDS = 14) are additional ones for anthracene-2,3-quinododimethide not counted in the linear C16 conjugated polyene Hamiltonian path.

DS = (1/6)[2r 3 + 3r 2 + r ] + (1/8)Nc(Nc − 2)

the third graph in Figure 10 is not present in Figure 11. The analytical expression for the number of Dewar resonance structures (DS) for the polyacene-2,3-quinododimethide series in Figure 8 is given by DS = (1/6)[2r 3 + 15r 2 + 19r + 6]

(2)

For benzene-2,3-quinododimethide in Figure 10, eq 1 gives DS = 7, and the first term of eq 2 gives ΔDS = 1 in agreement with results from Figures 10 and 11. We now will examine the ring enhancement of conjugation for the next two members of the polyacene-2,3-quinododimethide series in Figure 8. Figure 12 presents our analysis of these degrees of conjugation for the members of the middle polyacene-2,3quinododimethide series in Figure 8. In Figure 12, the upper four structures represent five that are the Dewar resonance structures responsible for ring conjugation enhancement in naphthalene-2,3-quinododimethane because of the Kekulé-like

(1)

where r = number of rings, and the Dewar long bonds are shown by the double headed arrows. Equation 1 can be decomposed into two parts, where the first part represents the ring conjugation enhancement and the second part represents the number of Dewar resonance structures belonging to the spanning Hamiltonian path as follows: 10833



Article

Figure 14. Comparison of Dewar resonance structures for three C6H6 isomers: two nonalternants and one alternant with a 4n-ring.

conjugation in the first ring, three sextet-like conjugation in the second ring, and one sextet-like conjugation in the third ring. TRE of Polycyclic Conjugated Hydrocarbons with One Classical Structure. The success of TRE as a measure of ASE is due to the subtraction of conjugation energy belonging to the PAH. This is done by excluding cyclic Sachs graphs to generate a (matching) polynomial belonging to a hypothetical conjugated reference molecule. Since the characteristic and matching polynomials are one and the same for nonring polyenes, their topological resonance energy is zero (TRE = 0). Benzene-2.3-quinododimethide has a TRE = 0.059β because of the cyclic Sachs graph shown in Figure 10 (second graph at the top) makes a small contribution to give it a modest aromaticity and associated diatropic ring current, though it only has one classical structure. Figure 9 gives TRE for other ring-enhanced conjugated systems compared to nonring polyenes. The TRE for the members in the upper two series in Figure 8 can be found in the work of Aihara and Makino.39 Nonalternant Conjugated Hydrocarbons with a Single Classical Structure. Up to now, we have emphasized conjugated alternant hydrocarbons without odd size or 4nrings. Figure 14 now presents isomeric conjugated systems with odd size rings, i.e., two nonAHs, and one AH with 4n-rings. In comparison to [3]dendralene (DS = 2, Eπ = 6.899 β), [3]radialene has gained another Dewar structure because of ring formation (DS = 3) and 1,2-dimethylenylcyclobutene has gained two Dewar structures with antiaromatic-like 4n-Kekule structures because of 4n-ring formation. From another perspective, 1,2-dimethylenylcyclobutene in comparison with 1,3,5-hexatriene (DS = 3, Eπ = 6.988 β) has gained an additional Dewar structure because of the cross-link on its 1,3,5hexatriene-like Hamiltonian path to form a 4n-ring. In comparison with 1,3,5-hexatriene, fulvene has gained two additional Dewar structures because of the cross-link on its

ring contributions; note the numbers below each is the number of Kekulé-like contributing structures not already counted by the second term in eq 2. Solution of eq 2 for two rings gives DS = 5 + 15 = 20, where there are five long bond Dewar resonance structures contributing to conjugation enhancement beyond the 15 Dewar resonance structures belonging to its Hamiltonian path as shown in Figure 12. The Kekulé-like structures below these were already counted by the second term in eq 2 for the number of Dewar resonance structures in the Hamiltonian path. To see this more completely, as the vertical double bond at the right in the first naphthalene substructure above moves toward the left in each of the possible three Kekulé-like structures, in the end gives the Kekulé-like structure shown immediately below. Figure 13 similarly gives the 14 [DS = 14 + 28 = 42 per eq 2] Dewar resonance structures responsible for ring conjugation enhancement in anthracene-2,3-quinododimentane. What this example analysis shows is that formation of rings itself leads to additional long bond Dewar resonance structures possessing Kekulé-like structures in the remnant portion of the conjugated polyene, which would otherwise only have a single classical structure. This example analysis was made easier by exploiting the existence of a Hamiltonian path. A similar but slightly more complicated analysis of the upper series in Figure 8 uses the longest polyene path which is branched. The origin for the pattern of moderate ring aromaticity described by Aihara and Makino39 for anthracene-2,3quinododimethane as quantitated by the BRE and CCE values given in rings (Figure 8) is now made apparent in Figure 13. In all 14 of the Dewar structures, as one goes from left to right, the first ring displays Kekulé-like conjugation in all 14 of the Dewar structures, in the second ring 9 of the Dewar structures, and in the third ring only 1 of the Dewar structures. Alternatively, in the 14 Dewar structures, one can count nine sextet-like 10834



Article

Notes

1,3,5-hexatriene-like Hamiltonian path to form a pentagonal ring. Alternant conjugated hydrocarbons only have Dewar resonance structures with long bonds connecting a starred position to a nonadjacent unstarred position. If we delete bonds forming odd size rings in a nonalternant hydrocarbon and maximally star the resulting alternant hydrocarbon, we can formally use that starred pattern in the precursor nonalternant hydrocarbon. Such a provisionally starred nonalternant hydrocarbon, then can also have long bonds connecting a starred position to another starred position as illustrated for fulvene compared to the equivalent 1,3,5-hexatriene listed at the bottom in Figure 14. To summarize, when comparing a set of conjugated isomers having a single classical structure, nonalternant hydrocarbons have more Dewar resonance structures than alternant hydrocarbons because they have additional long bond contributors. Conjugated alternant hydrocarbons with a single classical structure having (4n + 2) rings lead to are more conjugated and stable systems, and those alternant hydrocarbons with 4n-rings are less stable.

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS This work was supported in part by a grant from by the UM Board of Curators (K0906077).

■

CONCLUSION We have shown that the number of Dewar resonance structures is a good index to the extent of conjugation and its contribution to stabilization in a conjugated hydrocarbon. Our VB approach has allowed us to derive a number of analytical equations, not otherwise possible. Through these equations we have identified several levels of conjugation and other aspects. There are three levels of conjugation for conjugated polyenes that have a single classical structure. They are cross-conjugated, linear conjugated, and ring-enhanced conjugated polyene systems. In the first case, the double bonds have minimal conjugation limited to nearest neighbors, in the second the double bonds have extended two-way conjugation, and in the third the presence of rings leads to additional Kekulé-like conjugation in certain long bond Dewar resonance structures. For ring conjugated systems, stability is enhanced if the rings are (4n + 2) alternants or odd size nonalternants and stability is diminished if the rings are 4n. The concept of conjugation itself does not distinguish between cis and trans nor does the number of Dewar structures. We can now compare a set of conjugated polyenes with the same number of conjugated double bonds each having a single classical structure and conceptually understand that the ones with more Dewar resonances structures are more conjugated and electronically more stable, if devoid of 4n-rings. While it is true that the tail coefficient in the characteristic polynomial of polycyclic conjugated hydrocarbons with one classical structure is devoid of contributing Sachs cyclic graphs leading to K = 1, there are Sachs cyclic graphs contributing to the other coefficients and there are certain long bond Dewar resonance structures that have Kekulé-like conjugation, suggesting that there should be some small to moderate cyclic resonance energy. Ring-enhanced conjugation results in a more rapid decrease in the HOMO−LUMO gap in π-electron conjugation of two-dimensional polymers compared to linear polymers.44 Polycyclic conjugated isomers with a single classical structure having their double bonds arranged in a single Hamiltonian path are more conjugated.

■

REFERENCES

(1) Schleyer, P. v. R. Introduction: Aromaticity. Chem. Rev. 2001, 101, 1115−1566. (2) Schleyer, P. v. R. Introduction: DelocalizationPi and Sigma. Chem. Rev. 2005, 105, 3433−3947. (3) Exner, O. Inductive Effect: Theory and Quantitative Assessment. J. Phys. Org. Chem. 1999, 12, 265−274. (4) Alabugin, I. V.; Gilmore, K. M.; Peterson, P. W. Hyperconjugation. WIREs Comput. Mol. Sci. 2011, 1, 109−141. (5) Wu, J. I-C.; Schleyer, P. von Rague Hyperconjugation in Hydrocarbons: Not just a “mild sort of conjugation. Pure Appl. Chem. 2013, 85, 921−940. (6) Trinajstić, N. Chemical Graph Theory, 2nd ed.; CRC Press: Boca Raton, FL, 1992; pp 161. (7) Jones, M., Jr.; Fleming, S. A. Organic Chemistry, 4th ed.: W.W. Norton & Co., Inc.: New York, 2010. (8) Pauling, L. Bond Numbers and Bond Lengths in Tetrabenzo[de,no,st,c1d1]heptacene and Other Condensed Aromatic Hydrocarbons: A Valence-Bond Treatment. Acta Crystallogr. 1980, B36, 1898−1901. (9) Aihara, J. Resonance Energies of Benzenoid Hydrocarbons. J. Am. Chem. Soc. 1977, 99, 2048−2053. Aihara, J. A New Definition of Dewar-Type Resonance Energies. J. Am. Chem. Soc. 1976, 98, 2750− 2759. (10) Balasubramanian, K. Computer Generation of the Characteristic Polynomials of Chemical Graphs. J. Comput. Chem. 1984, 5, 387−394. Ramaraj, R.; Balasubramanian, K. Computer Generation of Matching Polynomials of Chemical Graphs and Lattices. J. Comput. Chem. 1985, 6, 122−141. (11) Dias, J. R. Valence-Bond Determination of Bond Lengths of Polycyclic Aromatic Hydrocarbons: Comparisons with Recent Experimental and Ab Initio Results. J. Phys. Chem. A 2011, 115, 13619−13627. (12) Aihara, J. General Rules for Constructing HMO Characteristic Polynomials. J. Am. Chem. Soc. 1976, 98, 6840−6844. (13) Dias, J. R. An Example Molecular Orbital Calculation Using the Sachs Graph Method. J. Chem. Educ. 1992, 69, 695−700. (14) Aihara, J. Topological Resonance Energy, Bond Resonance Energy, and Circuit Resonance Energy. J. Phys. Org. Chem. 2008, 21, 79−85. (15) Bosanac, S.; Gutman, I. Z. Naturforsch. 1977, 32a, 10−12. Gutman, I.; Bosanac, S. Tetrahedron 1977, 33, 1809−1812. (16) Aihara, J. Circuit Resonance Energy: A Key Quantity That Links Energetic and Magnetic Criteria of Aromaticity. J. Am. Chem. Soc. 2006, 128, 2873−2879. (17) Gutman, I.; Petrovic, V.; Dias, J. R. Cyclic Conjugation in Total Resonant Sextet Benzenoid Hydrocarbons. Polycycl. Aromat. Comp. 2000, 18, 221−229. (18) Dias, J. R. Aromaticity and Cyclic Conjugation of Three Pairs of Strongly Subspectral Series of Conjugated Molecules as Models for Graphene Armchair and Zigzag Edges. Mol. Phys. 2010, 108, 3425− 3429. (19) Craig, N. C.; Groner, P.; McKean, D. C. Equilibrium Structures for Butadiene and Ethylene: Compelling Evidence for π-Electron Delocalization in Butadiene. J. Phys. Chem. A 2006, 110, 1461−1469. (20) Feller, D.; Craig, N. C.; Matlin, A. R. Ab Initio Structures for 90°−Twisted s-trans-1,3-Butadiene and Cyclooctatetraene: The Naked sp2−sp2 Bond. J. Phys. Chem. A 2008, 112, 2131−2133. (21) Hosoya, H. Factorization and Recursion Relations of the Matching and Characteristic Polynomials of Periodic Polymer Networks. J. Math. Chem. 1991, 7, 289−305.

AUTHOR INFORMATION

Corresponding Author

*Phone: (816) 234-2284; e-mail: [email protected]. 10835



Article

(44) Gutzler, R.; Perepichka, D. F. π-Electron Conjugation in Two Dimensions. J. Am. Chem. Soc. 2013, 135, 16585−16594.

(22) Dias, J. R. Properties and Relationships of Conjugated Polyenes Having a Reciprocal Eigenvalue Spectrum. Dendralenes and Radialenes. Croat. Chem. Acta 2004, 77, 325−330. (23) Dias, J. R. Valence-Bond determination of Diradical Character of Polycyclic Aromatic Hydrocarbons: From Acenes to Rectangular Benzenoids. J. Phys. Chem. A 2013, 117, 4716−4725. (24) Wodrich, M. D.; Wannere, C. S.; Mo, Y.; Jarowski, P. D.; Houk, K. N.; Schleyer, P.v.R. The Concept of Protobranching and Its Many Paradigm Shifting Implications for Energy Evaluation. Chem.Eur. J. 2007, 13, 7731−7744. (25) Palmer, M. H.; Blair-Fish, J. A.; Sherwood, P. Equilibrium Molecular Structures of Cross-Conjugated Alkenes with Reference to [3]- and [4]-Dendralenes from ab initio Studies with Various Conformations. J. Mol. Struct. 1997, 412, 1−18. (26) Schmaltz, T. G.; Griffin, L. L. How long are the ends of polyene chains? J. Chem. Phys. 2009, 131, 224301−1−224301−8. (27) Dias, J. R. Analysis of π−Electronic Structures of Small Alternant Hydrocarbons to Infinitely Large Polymeric Strips: The Aufbau Principle and End Group Effects. Int. J. Quantum Chem. 1999, 74, 721−733. (28) Choi, C. H.; Kertesz, M.; Karpfen, A. The Effects of Electron Correlation of the Degree of Bond Alternation and Electronic Structure of Oligomers of Polyacetylene. J. Chem. Phys. 1997, 107, 6712−6721. Yang, Y.; Olishevski, P.; Kertesz, M. Bandgap Calculations for Conjugated Polymers. Synth. Met. 2004, 141, 171− 177. (29) Brivio, G. P.; Mulazzi, E. Absorption and Resonant Raman Scattering from trans-(CH)x. Chem. Phys. Lett. 1983, 95, 555−5670. (30) Townsend, P. D.; Pereira, C. M.; Bradley, D. D. C.; Horton, M. E.; Friend, R. H. Increase in Chain Conjugation Length in Highly Oriented Durham-Route Polyacetylene. J. Phys. C. Solid State Phys. 1985, 19, L283−L289. (31) Tabata, M.; Sone, T.; Sadahiro, Y.; Yokotya, K.; Nozaki, Y. Pressure-Induced Cis to Trans Isomerization of Aromatic Polyacetylenes Prepared by Using Rh Complex Catalyst: A Control of πConjugation Length. J. Polym. Sci., Part A: Polym. Chem. 1998, 36, 217−223. (32) Krygowski, T. M.; Cyranski, M. K. Structural Aspects of Aromaticity. Chem. Rev. 2001, 101, 1385−1419. (33) Marsh, W.; Dunitz, J. D. Structure of Hexaethylidenecyclohexane by X-ray Analysis. Helv. Chim. Acta 1975, 38, 707−712. (34) Krygowski, T. M.; Crański, M. K. Two Faces of the Structural Aspects of Aromaticity. Phys. Chem. Chem. Phys. 2004, 6, 249−255. (35) Aihara, J.; Ishida, T.; Kanno, H. Bond Resonance Energy as an Indicator of Local Aromaticity. Bull. Chem. Soc. Jpn. 2007, 80, 1518− 1521. (36) Domene, C.; Fowler, P. W.; Jenneskens, L. W.; Steiner, E. On the Lack of Ring-Current Aromaticity of (Heteroatom) [N]Radialenes and their Dianions. Chem.Eur. J. 2007, 13, 269−276. (37) Shanshal, M.; Kubba, R. M. Bonding, Electron Densities and Vibration Frequencies of [3]-, [4]-, [5]-, and [6]Radialenes; a B3LYP Density Functional Study. Z. Naturforsch. 2006, 61a, 95−98. (38) Gutman, I. Nonaromatic Benzenoid Hydrocarbons. Pure Appl. Chem. 1990, 62, 429−434. (39) Aihara, J.; Makino, M. Aromaticity and Diatropicity of Polyacenequinododimethides. Chem.Asian J. 2008, 3, 585−591. (40) Aihara, J.; Ishida, T.; Kanno, H. Bond Resonance Energy as an Indicator of Local Aromaticity. Bull. Chem. Soc. Jpn. 2007, 80, 1518− 1521. (41) Radenković, S.; Marković, S.; Kuć, R.; Stanković, N. The Diradical Character of Polyacenequinododimethides. Monatsh. Chem. 2011, 142, 1013−1019. (42) Gleicher, G. J.; Newkirk, D. D.; Arnold, J. C. Calculations on Quinoid Compounds. I. The Quinododimethides. J. Am. Chem. Soc. 1973, 95, 2526−2531. (43) Ciesielski, A.; Krygowski, T. M.; Cyrański, M. K. Why Are the Kinked Polyacenes More Stable than the Straight Ones? A topological Study and Introduction of a NewTopological Index of Aromaticity. J. Chem. Inf. Model. 2008, 48, 1358−1366. 10836


A Quantitative Metric for Conjugation in Polyene Hydrocarbons Having

Recommend Documents