Genealogy of Conjugated Acyclic Polyenes

molecules Article

Genealogy of Conjugated Acyclic Polyenes Haruo Hosoya Department of Information Sciences, Faculty of Science, Ochanomizu University (Emeritus) Otsuka 2-1-1, Bunkyo-ku, Tokyo 112-8610, Japan; [email protected] Academic Editor: Shunichi Fukuzumi Received: 30 April 2017; Accepted: 24 May 2017; Published: 29 May 2017

Abstract: Based on the total π-electron energies Eπ s of Hückel Molecular Orbital (HMO) method for all the possible isomers of conjugated acyclic polyenes (C2n H2n+2 ) up to n = 7, the structure–stability relation of the possible isomers was analyzed. It was shown that the mean length of conjugation L can roughly predict the ordering of stability among isomers, while the Z-index, or Hosoya-index, can almost perfectly reproduce their stability. Further, the genealogy of the conjugated acyclic polyene family was obtained by drawing systematic diagrams connecting these isomers of different n, and governed by several simple rules. Namely, the stability change of a given isomer in the genealogy connecting n and n + 1 polyenes can be classified into three different modes of vinyl addition (elongation, inner and outer branching) and horn growing, i.e., substitution of –HC=CH– moiety with –HC(=CH2 )–C(=CH2 )H–. By using the Z-index, we can extend this type of discussion to polyene radicals and even to “cross-conjugated” cyclic polyenes containing only one odd-membered cycle, such as radialene and fulvene. Keywords: total π-electronic energy of HMO; conjugated polyene; topological index; Hosoya index; structure-stability relation; Kekulé structure; mean length of conjugation; cross-conjugation; branching

1. Introduction The successful isolation and identification of the tremendously large family of organic compounds, even limited to hydrocarbons, seems to have established the logical structure of the kingdom of organic chemistry, which is opening its open gates to other fields of science spanning from biology to astronomy, information technology, and general physics. However, due to the scarcity of the isolated conjugated acyclic polyene molecules, even for the smallest members, the present status of organic chemists’ understanding of the structure–activity relationship and mathematics underlying the whole family of conjugated acyclic polyenes is rather low. Unfortunately, without paying attention to the essence of quantum theory, they are still playing with the old-style resonance theory originally proposed by the chemists in “pre-quantum chemistry age”. In order to steer towards the right direction, the present author has published several papers aimed at understanding and justifying the organic electron theory, mainly involving conjugated polyenes by using the graph-theoretical molecular orbital (GTMO) method. The classical concept of cross-conjugation, if properly appreciated with a slight modification, will play an important role for understanding the correct part of the conventional organic electron theory [1–6]. Fortunately, however, novel methods for synthesizing dendralenes (vide infra) and the related hydrocarbons have recently been discovered and several researchers have reconsidered the importance of the role of cross-conjugation in organic chemistry [7–12]. In the present paper, the genealogy and mathematical structure of the whole family of conjugated acyclic polyene molecules are explained in various levels of logic from high school chemistry (without wavefunction) to sophisticated mathematical chemistry (with perturbation theory) only by using the Molecules 2017, 22, 896; doi:10.3390/molecules22060896

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theory) only by using the Kekulé structure, total π-electron energy of the HMO method, and Kekulé structure, of the HMO method, and topological index Z (the so-called topological index total Z (theπ-electron so-called energy Hosoya-index) [13,14]. Hosoya-index) [13,14]. 2. Preliminary Discussions 2. Preliminary Discussions 2.1. 2.1. Planar Planar Conjugated Conjugated Acyclic Acyclic Polyene Polyene Isomers Isomers We are concerned concerned only only with with planar planar conjugated conjugated acyclic acyclic polyenes. polyenes. First, We are First, consider consider the the series series of of linear polyenes, i.e., ethylene 1, butadiene 2, linear hexatriene 3-1, etc., which are growing mostly linear polyenes, i.e., ethylene 1, butadiene 2, linear hexatriene 3-1, etc., which are growing mostly in in zigzag form, or all-trans conformation up to polyacetylene. The longer the chain length, the more zigzag form, or all-trans conformation up to polyacetylene. The longer the chain length, the more their π-electronic stability stability increases. increases. Above their π-electronic Above hexatriene, hexatriene, we we need need to to consider consider isomers, isomers, such such as as 3-methylene-1,4-pentadiene, or 3-dendralene 3-2, which is known also as the smallest entity 3-methylene-1,4-pentadiene, or 3-dendralene 3-2, which is known also as the smallest entity of of cross-conjugated hydrocarbon [15]. According to the analysis of electron diffraction, 3-2 takes a cross-conjugated hydrocarbon [15]. According to the analysis of electron diffraction, 3-2 takes a slightly slightly distorted ct (cis-trans) conformation [16]—in accordance with the results of molecular distorted ct (cis-trans) conformation [16]—in accordance with the results of molecular mechanics and mechanics and ab initio ab initio calculation [17].calculation [17]. 3-2

However, one easily conjecture this fact by estimating the repulsion non-bonded However, onecan can easily conjecture thisjust fact just by estimating the between repulsion between hydrogen atoms. Thus,atoms. one can develop ourdevelop naïve discussion the relative stabilities “almost non-bonded hydrogen Thus, one can our naïve on discussion on the relativeofstabilities planar” conjugated acyclic polyene isomers up to, say tetradecaheptaene, whose most crowded of “almost planar” conjugated acyclic polyene isomers up to, say tetradecaheptaene, whoseisomer most might be among 96 isomers. Here from nowHere on, we will notnow consider their complicated cis–trans crowded isomer its might be among itsand 96 isomers. and from on, we will not consider their conformations, helicalconformations, structures, and sophisticated isomer counting involving radicals and triple complicated cis–trans helical structures, and sophisticated isomer counting involving bonds [18–20]. radicals and triple bonds [18–20]. CH22 HC H22C

H C

C

CH22 CH H C

C

C

CH22

CH22

CH22

It may be a worthwhile exercise for students to obtain the possible isomer numbers for smaller It may be a worthwhile exercise for students to obtain the possible isomer numbers for smaller 22-type as given in Table 1 [21], because it is difficult to conjugated acyclic polyenes of “naïve” sp 2 conjugated acyclic polyenes of “naïve” sp -type as given in Table 1 [21], because it is difficult to check check these important numbers in any of the available standard textbooks of organic chemistry past these important numbers in any of the available standard textbooks of organic chemistry past and and present. Although it would be impossible to experimentally ascertain these numbers, we need to present. Although it would be impossible to experimentally ascertain these numbers, we need to know know both the topographical and mathematical structure of this half hypothetical, but half realistic both the topographical and mathematical structure of this half hypothetical, but half realistic kingdom kingdom of conjugated acyclic polyenes, the genealogy of which we are going to clarify in this of conjugated acyclic polyenes, the genealogy of which we are going to clarify in this paper. paper. Table 1. Number of isomers of conjugated acyclic polyenes C2n H2n+2 . Table 1. Number of isomers of conjugated acyclic polyenes C2n 2nH2n+2 2n+2.

n n No. of isomers No. of isomers

11

22

11

11

3 3 4 4 5 56 2 2 4 4 11 11 30

76 96 30

7 96

Recall we have have established Recall that that in in physical physical organic organic chemistry, chemistry, we established the the grand grand conceptual conceptual kingdom kingdom of of aromatic aromatic and and anti-aromatic anti-aromatic hydrocarbons hydrocarbons by by using using not not only only the the experimentally experimentally obtained obtained results results but but also assuming hypothetical hypotheticalaromatic aromaticand andanti-aromatic anti-aromatic compounds. The present author considers also assuming compounds. The present author considers the the logical base of this kingdom to still be shaky in this modern age, because logically and logical base of this kingdom to still be shaky in this modern age, because logically and mathematically mathematically it shouldon bethe constructed the world firm basis of the world of polyenes. conjugatedThe acyclic it should be constructed firm basis on of the of conjugated acyclic only polyenes. The only common understanding among a majority of chemists is that the linear polyene common understanding among a majority of chemists is that the linear polyene is the most stable and is the most stable is the stable among the isomers. dendralene is the and leastdendralene stable among theleast isomers. Although, on this structure–stability relation of conjugated acyclic polyenes, Gutman has shown that the isomer of the following type is the most stable among the branched isomers [22], this fact does not yet seem to be widely known to organic chemists.

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Although, on this structure–stability relation of conjugated acyclic polyenes, Gutman has shown Molecules 2017, 22,of 896 3 of 13 that the isomer the following type is the most stable among the branched isomers [22], this fact does Molecules 2017, 22, 896 3 of 13 not yet seem to be widely known to organic chemists.

On the other hand, Gineityte’s discussion is too sophisticated and specific to be followed by a majority chemists [11]. Gineityte’s discussion is too sophisticated and specific to be followed by a Onofthe other hand, On the other hand, Gineityte’sdiscussion discussionisistoo toosophisticated sophisticatedand andspecific specifictotobebefollowed followed On the hand, Gineityte’s byby a majority of other chemists [11]. a majority of chemists [11]. 2.2. Meanof Length of Conjugation majority chemists [11]. 2.2. Now Meanconsider Length ofthe Conjugation relative stability of two isomers of hexatrienes. Due to the short conjugated 2.2. Mean Length of Conjugation 2.2. Mean Length of Conjugation path of the consider π-electronic 3-2isomers is less stable than linear 3-1to[15]. about the Now the system, relative 3-dendralene stability of two of hexatrienes. Due the What short conjugated Now consider the relative stability of two isomers of hexatrienes. Due to the short conjugated relative stability of four isomers of octatetraene, 4-1~4-4? consider the relative stability of two isomers of stable hexatrienes. Due to short conjugated pathNow of the π-electronic system, 3-dendralene 3-2 is less than linear 3-1the [15]. What about the path Gutman’s of the π-electronic system, 3-dendralene 3-2Eis less stable than linear 3-1following [15]. What about the assertion can be supported by the π calculation in the order. path of the π-electronic 3-dendralene 3-24-1~4-4? is less stable [4,6] than as linear 3-1 [15]. What about the relative stability of foursystem, isomers of octatetraene, relative stability of four isomers of octatetraene, 4-1~4-4? relative stability assertion of four isomers octatetraene, 4-1~4-4? Gutman’s can be of supported by the Eπ calculation [4,6] as in the following order. Gutman’s assertion can be supported by the Eπ calculation [4,6] as in the following order. Gutman’s assertion can be supported by the Eπ calculation [4,6] as in the following order.

Except for the order within the middle two isomers, the number of tertiary carbon atoms, T, can Except for order within the middle twoisomers, isomers, thenumber number of tertiary carbon atoms, T, predict the relative their the stability. Gutman et al. also pointed out the important role ofT, T can in Except forthe theorder orderof within middle two the of tertiary carbon atoms, can predict the relative order their stability.two Gutman etal. al. also pointed out the important role of discussing relative stability ofthe conjugated acyclic polyenes [23]. Although the counting of Dewar Except for the order within middle isomers, the number of tertiary carbon atoms, T, predict thethe relative order ofoftheir stability. Gutman et also pointed out the important role ofTcan Tinin discussing the relative stability of conjugated acyclic polyenes [23]. Although the counting of Dewar structures (with a long bond connecting a pair of disjointed carbon atoms) can predict the correct predict the relative orderstability of theirofstability. Gutman et polyenes al. also pointed out the important roleofofDewar T in discussing the relative conjugated acyclic [23]. Although the counting structures (with a long bond connecting a pair of disjointed atoms) can the correct order order of stability unfortunately this notcarbon explained in anypredict elementary chemistry discussing the relative stability of conjugated acyclic polyenes [23]. Although the of Dewar structures (with a[10], long bond connecting amethod pair of is disjointed carbon atoms) cancounting predict the correct oforder stability [10], unfortunately this method not of explained any elementary chemistry textbooks. structures (with a long connecting aispair disjointed carbon atoms) predicttextbooks. thechemistry correct of stability [10],bond unfortunately this method is notinexplained in anycan elementary present author has shown the effectiveness of the “mean length of conjugation”, L [4,6], The present author has shown the effectiveness of the “mean length of conjugation”, L [4,6], for order of stability [10], unfortunately this method is not explained in any elementary chemistry textbooks. for predicting relative stability between and 4-3, which easily enumerated from the the predicting the the relative stability between 4-24-2 and 4-3, which isiseasily enumerated from textbooks. The present author has shown the effectiveness of the “mean length of conjugation”, L unique [4,6], for structure as exemplified in Figure 1. Here, each l is the largest number of C=C bonds in Kekulé structure as exemplified in Figure 1. Here, each l is the largest number of C=C bonds infor a The present author has shown the effectiveness of the “mean lengthenumerated of conjugation”, [4,6], predicting the relative stability between 4-2 and 4-3, which is easily from Lthe unique a linearly conjugated acyclic polyene moiety that consists of alternating l C=C and l–1 C–C bonds. linearly acyclic polyene that consists of lalternating lenumerated C=C and l–1of C–C predicting the relative stability between 4-2 4-3,each which easily from the unique Kekuléconjugated structure as exemplified inmoiety Figure 1.and Here, isisthe largest number C=Cbonds. bonds in a Kekulé structure as exemplified in Figure 1. Here, each l is the largest number of C=C bonds in linearly conjugated acyclic polyene moiety that consists of alternating l C=C and l–1 C–C bonds. a l = 3moiety that consists of alternating l C=C and l–1 C–C bonds. linearly conjugated acyclic polyene l=2 l=3 l=3 ll == 33

l=2 l=3 l=2 l=3 L = (3 + 2)/2 = 2.5

L = l(3= +3 3)/2 = 3 l=3 L = (3 + 2)/2 = 2.5 L mean = (3 + 3)/2 = 3of conjugation Figure 1. Calculation of L for 4-2 4-2 and 4-3, 4-3, both both with with T T= = 1. Figure 1. Calculation of mean length length of conjugation 1. L = (3L +for 2)/2 =and 2.5 L = (3 + 3)/2 = 3 Figure 1. Calculation of mean length of conjugation L for 4-2 and 4-3, both with T = 1. Figure 1. Calculation of mean 2.3. Hückel Molecular Orbital Method andlength Eπ of conjugation L for 4-2 and 4-3, both with T = 1.

2.3. In Hückel Orbital Method and Eπ stability among conjugated polyene isomers, let us turn orderMolecular to settle settle the the problem of relative relative In order to problem of stability among conjugated polyene isomers, let us turn 2.3. Hückel Molecular Orbital Method and Eπand most reliable technique at hand. The total π-electron to the the HMO method, which is is the easiest to HMO method, which the easiest and most reliable at hand. The total π-electron In order to settle the problem of relative stability amongtechnique conjugated polyene isomers, let us turn energy, E , is defined as the double of the sum of the occupied orbital energies, {x }s as in let us turn π n energy, E π , is defined as the double of the sum of the occupied orbital energies, {x n }s as in In order settle the problem ofeasiest relativeand stability polyene to the HMOto method, which is the most among reliableconjugated technique at hand. isomers, The total π-electron toenergy, the HMO method, the easiest reliable technique hand. The Eπ, is definedwhich as theisdouble of theand summost of occ the occupied orbital at energies, {xn}stotal as inπ-electron occ E = 2 x (1) energy, Eπ, is defined as the double of the sum occupied orbital energies, {xn}s as in = 2the Eππ of xn ∑ (1) occ n

 n=1 E = 2x 2x The {xn}s }s are are obtained obtained as the solution ofEthe= characteristic polynomial, PG(x) (x) ==0, 0,for for π

n

π

n =1 occ

n =1

n =1

n

n

G

(1) (1)

of (−1) the characteristic polynomial, PG(x) = 0, for The {xn}s are obtained as the solution ( )= = (2) PP (−1) N det( det(A − − xE)) (2) G ( x ) of the characteristic polynomial, PG(x) = 0, for The {xn}s are obtained as the solution ( ) = (−1) det( − ) (2) of molecular graph G, representing Pthe topology of the carbon atom skeleton of the polyene of molecular graph G, representing theP topology of the carbon skeleton of the polyene molecule, ( ) = (−1) det( − atom ) (2) molecule, wheregraph A andG,E are respectively adjacency matrix of G and unitskeleton matrix of N, of molecular representing thethe topology the unit carbon atom ofthe theorder polyene where A and E are respectively the adjacency matrix ofof G and matrix of the order N, the number the numberwhere of graph carbon atoms ofrespectively G, and the det means to take the determinant thematrix matrix given in theN, ofmolecule, molecular G, representing topology of the carbon skeleton ofofthe A and E are the adjacency matrix of Gatom and of unit thepolyene order parentheses. The adjacency matrix A is defined to be for G. molecule, where A and atoms E are respectively the adjacency matrix of G and unit of the order the number of carbon of G, and det means to take the determinant of matrix the matrix given in N, the the number ofThe carbon atoms matrix of G, and meanstotobetake parentheses. adjacency A isdet defined for the G. determinant of the matrix given in the parentheses. The adjacency matrix A is defined to be for G.

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of carbon atoms of G, and det means to take the determinant of the matrix given in the parentheses. The adjacency matrix A is defined to be for G. Molecules 2017, 22, 896 Molecules Molecules 2017, 2017, 22, 22, 896 896

of 13 444 of of 13 13

(

1 : i and j are neighbors Aij = (3) 0 :and are neighbors otherwise 1: A ij = (3) Aijij = (3) 0: otherwise In Table 2,πππ Eand PG (x) of the four isomers of octatetraene compared with T and L values. π and In 2, P of four isomers of are compared with T L In Table Table 2, E E and PGGG(x) (x) of the the four isomers of octatetraene octatetraene areare compared with T and and L values. values. Needless to say, regarding correlation L with is rather good. However, quantity Needless to regarding T, the correlation of L E good. However, the quantity Z, Needless to say, say, regarding T, T, thethe correlation of of L with with EπππEis isπ rather rather good. However, thethe quantity Z, Z, which will be explained below, is found to be perfectly correlated with E . The correlation coefficient π correlation coefficient which will be explained below, is found to be perfectly correlated with Eπππ. The ρ is almost unity for Z, while ρ is 0.981 for L. ρρ is almost unity for Z, while ρ is 0.981 for L. is almost unity for Z, while ρ is 0.981 for L. Table 2. Characteristic quantities of four isomers 4-1~4-4 of octatetraenes. Table Table 2. 2. Characteristic Characteristic quantities quantities of of four four isomers isomers 4-1~4-4 4-1~4-4 of of octatetraenes. octatetraenes. Isomer Molecular Graph Graph GG E(H&S) Z Z PG (x) PGGG(x) Isomer Molecular EEππππ E(H&S) T 8 6 4 2 8 6 4 2 66 +7x 4-1 9.518 9.538 – 2210x 4-1 9.518 9.538 3434 x88 − x7x− 15x+4415x – 10x +1 +10

LT 40

L 4

4-2 4-2

9.446 9.446

9.447 9.447

8 − 6 +44414x4 – 666 +7x 9x2 +11 3232 x888 −x7x 14x – 9x222 +1

31

3

4-3 4-3 4-3

9.409 9.409 9.409

9.408 9.408 9.408

2.5 1 2.5

2.5

4-4 4-4

9.332 9.332

9.308 9.308

8 − 6 +44414x 4– 666 + 222 +1 31 –– 8x 8x2 +111 3131 xx888 −−x7x 7x +7x14x 14x 8x +1 4 666 + 13x 222 +12 8 − 4– 2929 x888 −x7x – 7x 7x6 +4413x 7x +12

22

2

PPGGG(x) obtained by using Equation (x)Pcan can be obtained by by using Equation (5). (x)be can be obtained using Equation(5). (5). G

2.4. 2.4. Topological Topological Index Index 2.4. Topological Index In 1971, the present author proposed to define the topological index Z [13,24], which is now In called 1971, the author [25–28], proposed tograph defineG topological index [13,24], which is of now generally the Hosoya-index (See Table the graphs generally called thepresent Hosoya-index [25–28], for for graph Gthe (See Table 22 for for the Zmolecular molecular graphs of generally called the Hosoya-index [25–28], for graph G (See Table 2 for the molecular graphs octatetraenes) as the sum of the non-adjacent number, p(G,k), where the number of ways for of octatetraenes) as the sum of the non-adjacent number, p(G,k), where the number of ways for choosing choosing kk non-adjacent edges from G choosing non-adjacent edges from G as as k non-adjacent edges from G as /2 /2[]]N/2] [[NNN/2

p(Gp,(kG,) k) ∑ k =0

Z =Z =

==000 kkk=

(4) (4)

also found graphs (representing acyclic molecules), PG (x) expressed andand also found thatthat forfor treetree graphs (representing acyclic molecules), PGGG(x) cancan be be expressed justjust by by using p(G,k) numbers using thethe p(G,k) numbers as as /2 /2[]]N/2] [[NNN/2

k p,(kG,)xk) x N −2k Gtree) ∈ tree) ( −1)(−p1()G (G(∈ ∑ k =0

x) = PGGGP(x) G (=

kkk

N −−222kkk N N−

(5) (5)

==000 kkk=

Now, as seen in Table 2, the Z are equal to the sum of the absolute values of the Now, as as seen in in Table 2, all allall thethe Z values values areare equal to to thethe sum of of thethe absolute values of of thethe Now, seen Table 2, Z values equal sum absolute values coefficients of P coefficients ofGGG(x). PG (x). Originally, p(G,k) and Z proposed for analyzing the thermodynamic properties of the Originally, p(G,k) and Z were were proposed forfor analyzing thethe thermodynamic properties of of thethe Originally, p(G,k) and Z were proposed analyzing thermodynamic properties structural isomers ofofsaturated as inferred inferredfrom fromthe the close relation with structural isomers saturatedhydrocarbons. hydrocarbons. However, However, as close relation with HMO π HMO as (5), Z was found to be well correlated with E π αs π as (5), Z was found to be well correlated with Eπ αs Eπππ ∝ log Z (6) Eπ ∝ log Z (6) which was proven by using the perturbation theory of Longuet-Higgins [14,29]. When the relative whichamong was proven by hydrocarbons using the perturbation theory [14,29]. the relative stability isomeric is we are allowed relation [2,4,6], stability among isomeric hydrocarbons is discussed, discussed, we of areLonguet-Higgins allowed to to use use aa simpler simpler When relation [2,4,6], stability among isomeric hydrocarbons is discussed, we are allowed to use a simpler relation [2,4,6], Eπππ = a Z + b. (7) Eπvalue = a Zof + b. A convenient method for calculating the Z branched tree graphs will be explained in (7) Appendix A. Appendix A. A convenient method for calculating the Z value ofSchaad branched tree graphs will be explained In the same year as the debut of the Z-index, Hess and proposed an empirical recipe for in Appendix A. reproducing Eπππ for conjugated acyclic polyenes by adding the contribution of eight types of bonds, such such as as H H222C=CH, C=CH, HC=CH, HC=CH, etc., etc., which which gave gave tremendously tremendously good good results results as as shown shown by by the the E(H&S) E(H&S) values in Table 2 [30]. Their recipe gives rather reasonable results even for conjugated cyclic hydrocarbons hydrocarbons to to estimate estimate their their aromatic aromatic character character [31–33]. [31–33]. However, However, it it is is very very difficult difficult to to draw draw any any physico-chemical meaning of each parameter, and further, for larger hydrocarbons with N physico-chemical meaning of each parameter, and further, for larger hydrocarbons with N ≥≥ 10 10 the the discriminative power suddenly drops down, as will be shown later.

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In the same year as the debut of the Z-index, Hess and Schaad proposed an empirical recipe for reproducing Eπ for conjugated acyclic polyenes by adding the contribution of eight types of bonds, such as H2 C=CH, HC=CH, etc., which gave tremendously good results as shown by the E(H&S) values in Table 2 [30]. Their recipe gives rather reasonable results even for conjugated cyclic hydrocarbons to estimate their aromatic character [31–33]. However, it is very difficult to draw any physico-chemical meaning of each parameter, and further, for larger hydrocarbons with N ≥ 10 the discriminative power Molecules 2017, 22, 896 as will be shown later. 5 of 13 suddenly drops down, Molecules 2017, 22, 896 5 of 13 By scrutinizing Table 2, it can be inferred that the structure–stability relation for the conjugated By scrutinizing Table 2, it can be inferred that the structure–stability relation for the conjugated Bypolyenes scrutinizing Table 2, canrather be inferred that thestraightforward structure–stability relation theoptimistic conjugated acyclic polyenes is governed byit by rather simple and straightforward rules. This inference is acyclic is governed simple and rules. Thisfor optimistic inference is acyclic polyenes is governed by rather simple and straightforward rules. This optimistic inference is further further strengthened by checking the data for larger conjugated polyenes [4,6]. strengthened by checking the data for larger conjugated polyenes [4,6]. further strengthened by checking the data for larger conjugated polyenes [4,6]. Namely, for example, 2a,b the show the correlation Eπ’s of isomers eleven isomers of Namely, for example, Figure Figure 2a,b show correlation of E ’s of of eleven of decapentaenes, Namely, for example, Figure 2a,b show the correlation π of Eπ’s of eleven isomers of C10 H12, with L and Z. Their ρ values are,asrespectively, as high as0.999. 0.954 and 0.999. C10 H12 ,decapentaenes, with L and Z. Their ρ values are, respectively, high as 0.954 and decapentaenes, C10H12, with L and Z. Their ρ values are, respectively, as high as 0.954 and 0.999. Eπ Eπ

Eπ E π

LL

Z

(a)

(b)

(a)

Z

(b)

2. Correlation ’s of isomers of decapentaenes with L (a)with and ZL(b). FigureFigure 2. Correlation ofofEofEππE’s ofeleven eleven isomers of decapentaenes (a) and Z (b). Figure 2. Correlation π’s of eleven isomers of decapentaenes with L (a) and Z (b).

Further, for much larger conjugated acyclic polyenes with n = 12 and 14, the good correlation

Further, forEπmuch conjugated acyclic polyenes n =nrespectively, 12 14, the between Z much and does larger notconjugated change as shown in Figure 3a,b,with where, onegood and correlation three correlation Further, for larger acyclic polyenes with = and 12 and 14, the good between Z and E π does not change as shown in Figure 3a,b, where, respectively, one and three are found. Now weasare going to analyze the genealogy of respectively, conjugated acyclic betweenisospectral Z and Epairs does not change shown in Figure 3a,b, where, one and three π isospectral polyenes. pairs are found. Now we are going to analyze the genealogy of conjugated acyclic isospectral pairs are found. Now we are going to analyze the genealogy of conjugated acyclic polyenes. polyenes. Eπ

Eπ

Eπ

Z

Z

Eπ

Z

(a) Figure 3. Correlation between Z and Eπ (a) tetradecaheptaenes.

(b) Z for isomers of (a) 30 dodecahexaenes and (b) 96 (b)

Figure 3. Correlation between Z and Eπ for isomers of (a) 30 dodecahexaenes and (b) 96

Figure tetradecaheptaenes. 3. Correlation between Z and Eπ for isomers of (a) 30 dodecahexaenes and (b) 3. Results and Discussion 96 tetradecaheptaenes. 3. Results and Discussion 3.1. Vinyl Addition and Horn Growing

3. Results and WithDiscussion these results in mind, reconsider the relation between linear hexatriene 3-1 and four 3.1. Vinyl Addition and Horn Growing

octatetranes, three of which can be derived by adding a vinyl group, CH2=CH–, to 3-1. As seen in

3.1. Vinyl Addition and Horn Growing With these results in mind, theberelation linear 3-1at and Figure 4, the three isomers, 4-1, 4-2,reconsider and 4-3, can derived,between respectively, byhexatriene the addition the four positions 1 (red), 3 (blue), andcan 2 (green), counted from thea terminal carbon atom. Let to us 3-1. call As these 2=CH–, seen in octatetranes, three of which be derived by adding vinyl group, CH

With these results in mind, reconsider the relation between linear hexatriene 3-1 and four step-up4,growing processes, elongation, branching, outer branching, respectively. Figure the three isomers, 4-1, 4-2, inner and 4-3, can beand derived, respectively, by the addition at the octatetranes, three of which derived by adding a terminal vinyl group, 2 =CH–, positions 1 (red), 3 (blue),can andbe 2 (green), counted from the carbonCH atom. Let us to call3-1. theseAs seen in Figure 4, the three isomers, 4-1, 4-2, and 4-3, can be derived, respectively, by the addition at the step-up growing processes, elongation, inner branching, and outer branching, respectively. positions 1 (red), 3 (blue), and 2 (green), counted from the terminal carbon atom. Let us call these step-up growing processes, elongation, inner branching, and outer branching, respectively.

Molecules 2017, 2017, 22, Molecules 22, 896 896

of 13 13 66 of

Molecules 2017, 22, 896

6 of 13 9.518 Eπ

9.518 Eπ 4-1

L = 3L = 3

L

L

9.446 9.446

π

Z = Z13= 13

Z

Z

4

4

Eπ =E6.988 = 6.988

Molecules 2017, 22, 896

34

34

4-1

32 32

* * * * * * ** 3-13-1

4-2 4-2

6 of 13

3 3

9.518 Eπ 349.409 9.409 Z

* elongation * elongation 4-1

* inner branching * inner branching * Zhorn = 13growing

*

* * 3-1

L=3

31

2.5

4-3

* outer π branching * horn growing

4 31 L

4-3

*E outer branching = 6.988

9.446 2.5 329.372

*

4-2

3 29

4-4

* elongation

9.372 29

2 9.409

4-4

2

31 to 3-1 derive all the four inner branching Figure 4. Three kinds of* addition of a vinyl group and “horn growing”

Figure 4. Three kinds of addition of a vinyl group and “horn growing” to 3-1 derive all the four isomers 4-3 * outer branching 2.5 Figure 4. Three isomers of C8kinds H10. of addition of a vinyl group and “horn growing” to 3-1 derive all the four of C8 H10 . * horn growing 9.372 isomers of C8H10. 29 of a vinyl group, but can be The least stable 4-4 cannot be derived from 3-1 by the addition 4-4 The least stable cannot bebe derived from 3-1 by the addition of2aisomers vinyl butNcan be the derived derived from 3-24-4 by branching. Similarly, one can all the of a given from The least stable 4-4outer cannot derived from 3-1 byderive the addition of a group, vinyl group, but can be set of N–2 isomers by using the elongation and two types of branching. However, for understanding from 3-2 by outer branching. Similarly, one can derive all the isomers of a given N from the set of N–2 Figure 4. Three kinds of addition of a vinyl group and “horn growing” to 3-1 derive all the four derived from 3-2 by outer branching. Similarly, one can derive all the isomers of a given N from the the by whole genealogy structure–stability relation hidden there, for let understanding us consider a slightly isomers of C8elongation H10. and theand isomers using the two types of branching. However, the whole set of N–2 isomers by using the elongation and two types of branching. However, for understanding different growing type, such as the relation one shown in Figure 5, which already been given in Figure 4 genealogy the structure–stability hidden there, let us has consider alet slightly different growing the whole and genealogy and4-4the structure–stability relation us consider The stable cannot be derived from 3-1 by the hidden addition there, of a vinyl group, but can a be slightly and may beleast called “horn growing”.

type, suchgrowing as the one shown in Figure which in hasFigure already been given in Figure 4 and may be called different the one5,Similarly, shown 5, which already given derived fromtype, 3-2 bysuch outeras branching. one can derive all thehas isomers of a been given N fromin theFigure 4 “horn growing”. and may “horn setbe of called N–2 isomers bygrowing”. using the elongation and two types of branching. However, for understanding the whole genealogy and the structure–stability relation hidden there, let us consider a slightly different growing type, such as the one shown in Figure 5, which has already been given in Figure 4 and may be called “horn growing”.

Figure 5. Horn growing.

Now, we can draw the whole genealogy of three generations of conjugated acyclic polyenes, Figure 5. Horn growing. C6H8~C10H12, as in Figure 6, where Eπ values, Z-index (in italics), L, and T are given. Figure 5. Horn growing. Figure 5. Horn growing.

Z Now, we can draw the whole genealogy of threeEπ generations ofL conjugated acyclic polyenes, 12.053 89 Now, we can draw the whole genealogy of three generations of5 conjugated acyclic polyenes, T=0 C6H8~C10H12Now, , as inweFigure 6, where Eπ values, Z-index italics), L,ofand T are given. can draw the whole genealogy of three(in generations conjugated acyclic polyenes, C6 H8 ~CC10 H , as in Figure 6, where E values, Z-index (in italics), L, and T are given. 1210H12, as in Figure 6, where Eπ 6H8~C π values, Z-index (in italics), L, and T are given. 9.518

11.985

34

4

Eπ

Eπ 12.053

11.967 12.053

9.518

9.518

13

6.988

9.446

11.938 11.985

34

11.985

34

4

11.918

3

11.967 11.967 11.925 11.938

13

13 3

3

9.446

9.446 9.409

3

31 3 2.5

80

81

8076

11.925

11.843 11.875

6.899

2

9.409

2.5 2

31

11.764

T=1

3.5T=1

3.5

81

11.925 11.864

11.864 31 29

4

4

8380 83

8077

11.875

9.409 9.372

T=1 T=0

T=0

81 84

80

32.67

T=2

3

80 75

77

74

11.831 12

5

5 3.5

84

11.875 11.918

11.918

32

32

L

L

3

11.938 6.988

4

Z Z 89 8983

32 4

3

6.988

84

76

70

11.864 11.843

75

11.831

74

2.33

77 2.67

T=2

2.67 2

76

T=3

T=2

2.33

75 11.843 2.5 Figure 6. Genealogy of hexatrienes~decapentaene. Meaning of colors is the same as in Figure 4. 6.899

12

9.372

29

6.899

12

11.764 11.831

70

74

2

2.33

T=3

2

2

9.372

29

11.764

70

2 Figure 6. Genealogy of hexatrienes~decapentaene. Meaning of colors is the same as T=3 in Figure 4.

Figure 6. Genealogy of2hexatrienes~decapentaene. Meaning of colors is the same as in Figure 4. 2

Figure 6. Genealogy of hexatrienes~decapentaene. Meaning of colors is the same as in Figure 4.

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It canMolecules easily2017, be22,seen that all these numbers (except for the reverse feature of T)7 oflie almost in 896 13 parallel with each other. Our discussion can be strengthened by drawing a more quantitative diagram canwhere easily be seen that the all these numbers (except for the reverse feature of three T) lie almost in groups is such as FigureIt 7, although relative height of the ordinates for the isomer parallel with each other. Our discussion can be strengthened by drawing a more quantitative tentative, diagram their correlation diagrams within each isomer are drawn to the same scale. Here, we can such as Figure 7, where although the relative height of the ordinates for the three isomer see the four groups of almost parallel arrows the are growing their own colors. groups is tentative, their correlation diagramsrepresenting within each isomer drawn toprocess the sameby scale. Here, we can seethe the blue four groups of almost parallel arrows representing the processthat by their For example, see and green lines connecting C8 and C , indicating theown Eπ values of 10growing colors. For see theby blue and green lines connecting andfound C10, indicating that the a Eπsmall values range and different isomers ofexample, C10 derived inner branching from C8C8are to lie within of different isomers of C10 derived by inner branching from C8 are found to lie within a small range distinctively different from the isomers derived by outer branching. and distinctively different from the isomers derived by outer branching. Eπ Eπ

4

5

12.0

4 3.5

9.5

Eπ 7.0

3

3

3

11.9

9.4 2.5

2.67

Elongation Inner branching Outer branching Horn growing

2.33

6.9 2

6π

2 9.3

12

11.8

10π

8π 30

2 35

L

70

80

Z

Figure 7. Quantitative diagram demonstrating the systematic growing scheme of the genealogy of

Figure 7. Quantitative diagram demonstrating the systematic growing scheme of the genealogy of C6~C10 isomers. C6 ~C10 isomers. Very large destabilization caused by horn growing (black lines) is prominent in this diagram, while stabilization by elongation (red) and destabilization by outer branching (green) can also Very clearly large be destabilization caused by horn growing (black lines) is prominent in this diagram, perceived. On the other hand, the change by inner branching (blue) is less prominent, but while stabilization by elongation (red) andofdestabilization (green) can also clearly distinctive from the three other types growing. Actually,by weouter couldbranching draw this type of diagrams not given showing the other systematic genealogy of larger up to C14, which be perceived. On the hand, the change by isomers inner branching (blue)are is unfortunately less prominent, but distinctive here because their entangled look. Actually, we could draw this type of diagrams showing the from the three other of types of growing. Before going into more detailed discussion, we can summarize the global features of the systematic genealogy of larger isomers up to C14 , which are unfortunately not given here because of structure–stability relation in the genealogy of conjugated acyclic polyenes as follows:

their entangled look. (i) Relative stability among the isomers derived by elongation, branching, and horn growing can Before going into more detailed discussion, we can summarize the global features of the roughly be estimated according to their respective ΔT value in the reverse of this order. structure–stability relation in thebetween genealogy of conjugated asinner follows: (ii) ΔL can discriminate the relative stability ofacyclic isomerspolyenes derived by and outer branching.

(i)

Relative stability theouter isomers derived bytoelongation, branching, and horn can (iii) The lesser among stability of branching relative inner branching can be attributed to growing the short-range conjugation caused by the vinyl group addition, in contrast to the wide-range of the roughly be estimated according to their respective ∆T value in the reverse of this order. inner branching (elaborated upon later). (ii) ∆L can discriminate between the relative stability of isomers derived by inner and In any case, as a rough summary of (i)~(iii), we propose Table 3. outer branching. (iii) The lesser of outer branching relative inner naively branching can be to the Table stability 3. Energy change caused by the four types of growingto supported by the change of T attributed and L. short-range conjugation caused by the vinyl group addition, in contrast to the wide-range Type of Growing Energy Change ΔL ΔT of the inner branchingelongation (elaborated upon later). Stabilization 0 + inner branching

small change

+1

horn growing

big destabilization

+2

In any case, as a rough we propose Table 3.+1 outersummary branching of (i)~(iii),Destabilization

+, 0, – – ––

Table 3. Energy change caused by the four types of growing supported naively by the change of T and L. Type of Growing

Energy Change

∆T

∆L

elongation inner branching outer branching horn growing

Stabilization small change Destabilization big destabilization

0 +1 +1 +2

+ +, 0, – – ––

Molecules 2017, 22, 896 Molecules 2017, 22, 896 Molecules 2017, 22, 896

8 of 13 8 of 13 8 of 13

Although this is manifest at the level of elementary chemistry, the present author has never Although this this is is manifest manifest at at the the level level of of elementary elementary chemistry, chemistry, the the present present author author has has never never Although encountered this type of discussion in the literature of chemistry from educational to researchers’ encountered this type of discussion in the literature of chemistry from educational to researchers’ encountered this type of discussion in the literature of chemistry from educational to researchers’ levels. As mathematical chemist, the author has been struggling—considering the present status of levels. As As aaa mathematical mathematical chemist, chemist, the the author author has has been been struggling—considering struggling—considering the the present present status status of of levels. chemical education—to make beginners in the field of chemistry realize that they are in the midst of chemical education—to make beginners in the field of chemistry realize that they are in the midst of chemical education—to make beginners in the field of chemistry realize that they are in the midst of modern science modern science science modern 3.2. Discriminative Discriminative Power of Z 3.2. Discriminative Power of Z than Figure 4, we out theout Z–Ethe the Cfor Similar to tobut butmore morequantitative quantitative than Figure 4,can wesingle can single Z–Efor π plot the C12 π plot 12 isomers Similar to but more quantitative than Figure 4, we can single out the Z–Eπ plot for the C12 isomers anofisomer ofus C10call (let5*ushere) call 5* asin shown derived derived from an from isomer C10 (let as here) shown Figurein8.Figure 8. isomers derived from an isomer of C10 (let us call 5* here) as shown in Figure 8.

E Eππ

n tio gaation n o el ng elo

ΔT = 0 ΔT = 0

er g innnecrhinng in n hi brraanc b r # te g # # ouutecrhinng # # o n hi # branc bra

ΔT = 1 ΔT = 1 ΔT = 2 ΔT = 2

rnn ng horw i hroow ing gro g

* *

* *

Z Z

Figure 8. Z–Eπ plot derived (5*) plot of of eleven eleven isomers isomers of of C12 derived from from an an isomer isomer of C10 (5*) showing showing the the four 12 derived 10 (5*) Figure 8. Z–Eππ plot of eleven isomers of C12 from an isomer of C10 showing the four distinctive types of polyene growing. See the later discussion on the marks * and #. distinctive types of polyene growing. See the later discussion on the marks * and #.

In order to to supplementthe the discussionmore more quantitatively, we have prepared Figure 9, where In order wewe have prepared Figure 9, where for to supplement supplement thediscussion discussion morequantitatively, quantitatively, have prepared Figure 9, where 12 isomer the changed and/or added conjugated path caused by the vinyl addition and for each C eacheach C12 C isomer the changed and/or added conjugated pathpath caused by the addition and horn 12 isomer the changed and/or added conjugated caused byvinyl the vinyl addition and for horn growing is drawn by theline bentinline inbyred, by which ΔL can be calculated. The Z values of the growing is drawn by the bent red, which ∆L can be calculated. The Z values of the eleven horn growing is drawn by the bent line in red, by which ΔL can be calculated. The Z values of the are also given. Notice that as their stability (Eπ) shown in Figure 8 is well eleven C12 isomers C12 isomers are also are given. Notice that as their stability (Eπstability ) shown in 8 is in well correlated also given. Notice that as their (EFigure π) shown Figure 8 is with well eleven C12 isomers correlated with Z, it is very easy to locate the point for each isomer in Figure 8. Z, it is verywith easyZ,toitlocate the point for each in Figure 8. correlated is very easy to locate theisomer point for each isomer in Figure 8. T=2 group T=2 group

T=1 group T=1 group

ΔL Z ΔL Z +1 217 4 –> 5+1 217

2 –> 3 2 –> 3

4

6-4 +0.33 200 6-4 +0.33 200

4 –> 5

6-1 6-1

+0.5 212 +0.5 212

4 4

6-3 6-3 T=3 group T=3 group 2 2

2 2

2 2

–0.75 181 –0.753 181 3

6-10 –0.75 179 6-10 –0.75 179 2 3 3

6-11 6-11

2

2 2

br br i n i n a n a n ne ne ch ch r r in in gg

+0.5 209 +0.5 209

6-5 6-5

4 4

5* 5* gr grhohr or owown n in in gg

6-2 6-2

on n ati tio ngnga eloelo

4 –> 5 4 –> 5 2 –> 3 2 –> 3

ΔL Z ΔL 202 Z +0.33 +0.33 202 4

3 3

6-6 6-6 3 3

L=3 L=3 Z=81 Z=81 b out brraanocuhteerr nching ing

± 0 198 ± 0 198

6-7 6-7 2 2

± 0 197 ± 0 197 –0.33 196 –0.33 196

6-8 6-8 –0.33 193 –0.33 193

2 2

6-9 6-9

Figure 9. Diagram showing the change of conjugated paths (in red) in each C12 isomer caused by the Figure Diagram showing the change of paths (in (in red) in in each each C C12 isomer caused by Figure 9. 9.of Diagram showing the change of conjugated conjugated paths isomerare caused by the the 12isomer addition a vinyl group and horn growing to 5*. ΔL (in italics)red) and Z for each also given. addition addition of of aa vinyl vinyl group group and and horn horn growing growing to to 5*. 5*. ΔL ∆L (in (in italics) italics) and and Z Z for for each each isomer isomer are are also also given. given.

The C10 isomer 5* selected for Figures 8 and 9 is situated as the fourth most stable 10π polyene The C10 isomer 5* selected for Figures 8 and 9 is situated as the fourth most stable 10π polyene with L = 3 in Figure 7. By overlapping eleven diagrams similar to Figure 8 for all the isomers of C10, with L = 3 in Figure 7. By overlapping eleven diagrams similar to Figure 8 for all the isomers of C10, we can obtain the complete genealogy diagram for the relation among all the isomers of C10 and C12. we can obtain the complete genealogy diagram for the relation among all the isomers of C10 and C12.

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The C10 isomer 5* selected for Figures 8 and 9 is situated as the fourth most stable 10π polyene with L =2017, 3 in22, Figure C10 , Molecules 896896 7. By overlapping eleven diagrams similar to Figure 8 for all the isomers of 9 of Molecules 2017, 22, 913 of 13 Molecules 2017, 22, 896 9 of 13 we can obtain the complete genealogy diagram for the relation among all the isomers of C10 and C12 . However, Figures 8 and 9 by themselves revealvery veryimportant importantsecrets secretsunderlying underlying the the whole However, Figures 8 and 9 by themselves reveal However, Figures and themselves revealvery veryimportant importantsecrets secretsunderlying underlyingthe thewhole whole However, Figures 8 8and 9 9bybythemselves reveal genealogy together with interesting issues for grading the various theories or indices for discussing genealogy together with interesting issues grading the various theories indices discussing genealogy together with interesting issues forfor grading the various theories oror indices forfor discussing problem. this structure–stability this structure–stability problem. this structure–stability problem. Further, Figure 8 8disclosed thethe limitation of Hess–Schaad recipe [30] for reproducing the Eπthe value Further, Figure disclosed limitation of of Hess–Schaad recipe [30] forfor reproducing Eπ Eπ Further, Figure disclosed the limitation Hess–Schaad recipe [30] reproducing the Further, Figure 8 8disclosed the limitation of Hess–Schaad recipe [30] for reproducing the Eπ of value conjugated acyclic polyenes. If one calculates the E of the two and three isomers marked with value of of conjugated acyclic polyenes. If one calculates the E π of the two and three isomers marked π conjugated acyclic polyenes. If one calculates the E π of the two and three isomers marked value of conjugated acyclic polyenes. If one calculates the Eπ of the two and three isomers marked *with and Figure 8, respectively, the same the values canvalues be obtained each group. This redundancy *# in and # in Figure 8, 8, respectively, same cancan beinobtained in in each group. This with and Figure respectively, the same values obtained each group. This with * *and # #ininFigure 8, respectively, the same values can bebeobtained in each group. This already occurs for two pairs in the eleven isomers of C , indicating that the Hess–Schaad recipe is redundancy already occurs for two pairs in the eleven isomers of C 10, indicating that the Hess– 10 redundancyalready alreadyoccurs occursforfortwo twopairs pairsininthe theeleven elevenisomers isomersofofC10 C,10indicating , indicatingthat thatthe theHess– Hess– redundancy only applicable small polyenes. for the three kinds for of for elongated C isomers in FigureC8, Schaad recipe istois only applicable toActually, small polyenes. Actually, thethe three kinds of of elongated 12 12kinds Schaad recipe only applicable small polyenes. Actually, three elongated C12 Schaad recipe is only applicable totosmall polyenes. Actually, for the three kinds of elongated C12 their stability difference is so large that it should not be overlooked. isomers in Figure 8, their stability difference is so large that it should not be overlooked. isomers in Figure 8, their stability difference is so large that it should not be overlooked. isomers in Figure 8, their stability difference is so large that it should not be overlooked. If one uses the recursion formula explained in Appendix A, theA, Z values 6-1~6-3 can be obtained one uses the recursion formula explained in in Appendix thethe Z of values of of 6-1~6-3 cancan be one uses the recursion formula explained Appendix values 6-1~6-3 If Ifone uses the recursion formula explained in Appendix A,A,the Z Zvalues of 6-1~6-3 can bebe byobtained a “back envelope” calculation as in Figure 10, where bent lines are reproduced from Figure 9 obtained byof a “back of envelope” calculation as in Figure 10, where bent lines are reproduced from a “back envelope” calculation Figure where bent lines are reproduced from obtained byby a “back ofof envelope” calculation asas inin Figure 10,10, where bent lines are reproduced from for calculating the L values, which cannot discriminate the different stability between 6-2 and 6-3. Figure 9 for calculating thethe L values, which cannot discriminate thethe different stability between 6-26-2 Figure calculating values, which cannot discriminate different stability between Figure 9 9forforcalculating the L Lvalues, which cannot discriminate the different stability between 6-2 However, one can guess that this stability difference might come from the difference between the “long and 6-3. However, one can guess that this stability difference might come from the difference and 6-3. However, one can guess that this stability difference might come from the difference and 6-3. However, one can guess that this stability difference might come from the difference range elongation” and “short range elongation” caused by the vinyl addition. On theaddition. other hand, between thethe “long range elongation” and “short range elongation” caused byby thethe vinyl OnOn between “long range elongation” and “short range elongation” caused vinyl addition. between the “long range elongation” and “short range elongation” caused by the vinyl addition. On 6-1 obtains the largest stabilization by “double elongation”. Similar results can be obtained by selecting thethe other hand, 6-16-1 obtains thethe largest stabilization byby “double elongation”. Similar results cancan be other hand, obtains largest stabilization “double elongation”. Similar results the other hand, 6-1 obtains the largest stabilization by “double elongation”. Similar results can bebe many other polyene isomers for the group with the same T value. Thus, we can safely assert that obtained by selecting many other polyene isomers for the group with the same T value. Thus, we can obtained selecting many other polyene isomers the group with the same value. Thus, we can obtained byby selecting many other polyene isomers forfor the group with the same TT value. Thus, we can safely assert that safely assert that safely assert that (iv) Z-index can accurately discriminate the stability difference among the same type of growing (iv)(iv)Z-index can accurately discriminate thethe stability difference among thethe same type of of growing process, while ∆L cannot. Z-index can accurately discriminate stability difference among same type growing (iv) Z-index can accurately discriminate the stability difference among the same type of growing while ΔL cannot. (v) process, Deep understanding of the structure–stability relation of conjugated acyclic polyenes can be process, while cannot. process, while ΔLΔL cannot. (v)(v)Deep understanding of the structure–stability relation of conjugated acyclic polyenes can be be by the complementary discussion withrelation their T, L, and Z values,acyclic even without thecan help Deepunderstanding understanding thestructure–stability structure–stability relation conjugated acyclic polyenes can (v) obtained Deep ofofthe ofof conjugated polyenes be obtained by the complementary discussion with their T, L, and Z values, even without the help of a computer. obtained the complementary discussion with their and values, even without the help obtained byby the complementary discussion with their T,T, L,L, and ZZ values, even without the help of of a computer. a computer. of a computer.

2-->3 2-->3 2-->3

L=L 4 L = 4= 4

4-->5 4-->5 4-->5 6-1 6-1 6-1

2 2 2

2-->3 2-->3 2-->3

L=L 3.5= 3.5

L = 3.5 4-->5 4-->5 4-->5

6-2 6-2 6-2 L=L 3.5= 3.5 L = 3.5

4 4 4

double elongation double elongation double elongation . Z . Z = 13x13 + 2x3x8 = 217 = Z6=. Z + Z 6.Z + 2 Z3 . Z5 . Z = 13x13 + 2x3x8 = 217 Z . . . 6 6 2 3 = Z6 Z6 + Z2 Z3 Z5 =513x13 + 2x3x8 = 217 single longlong range elongation single range elongation single long range elongation . Z . Z = 5x34 + 1x2x21 = 212 = Z4=. Z + Z 8.Z + 1 Z2 . Z7 . Z = 5x34 + 1x2x21 = 212 Z = Z4 . Z48 +8 Z1 . Z12 . Z2 7 =75x34 + 1x2x21 = 212

single shortshort range elongation single range elongation single short range elongation

. . = Z6=. Z 6 .+Z Z+ 1 Z4 . Z 5 .= 13x13 + 1x5x8 = 209 .Z . 1 Z . 4 Z5 = 13x13 + 1x5x8 = 209 6 +6 Z Z = Z6 Z 6 1 Z4 Z5 = 13x13 + 1x5x8 = 209

6-3 6-3 6-3

Figure 10.10. Back-of-envelope calculation of Z-indices of 6-1~6-3, 6-1~6-3, where Znn Z means the Z path graph Figure 10. calculation of of where Z the ZZof of graph Figure Back-of-envelope calculation of Z-indices 6-1~6-3, where n means Z of path graph Figure 10.Back-of-envelope Back-of-envelope calculation ofZ-indices Z-indices ofof 6-1~6-3, where Zn means means thethe ofpath path graph S n-th Fibonacci number (See Appendix A). First cut the graph at the dashed straight line, then Snn,,Sor or n-th Fibonacci number (See Appendix A). First cut the graph at the dashed straight line, then cut , or n-th Fibonacci number (See Appendix First graph dashed straight line, then Sn, nor n-th Fibonacci number (See Appendix A).A). First cutcut thethe graph at at thethe dashed straight line, then cut at the dashed round curve to apply the recursion formula. Bent lines are the same as in Figure at the dashed round round curve to apply the recursion formula. Bent lines are theare same assame in Figure 9. 9. 9. cut at the dashed curve to apply the recursion formula. Bent lines the as in Figure cut at the dashed round curve to apply the recursion formula. Bent lines are the same as in Figure 9.

3.3. Beyond Conjugated Acyclic Polyenes 3.3. Beyond Conjugated Acyclic Polyenes 3.3. Beyond Conjugated Acyclic Polyenes 3.3. Beyond Conjugated Acyclic Polyenes Up to now, we have discussed only the “stable” conjugated acyclic polyenes. Then, consider thethe Up to now, we have discussed only the “stable” conjugated acyclic polyenes. Then, consider Up now, we have discussed only the “stable” conjugated polyenes. Then, consider Up toto now, we have discussed only the “stable” conjugated acyclic polyenes. Then, consider the 6-3 6-3 6-3

, , ,

6-4 6-4 6-4

andand , ,and , , and

7-1 7-1 7-1

, etc.? , etc.? , etc.? Actually,

relevant radicals, such as as the relevant radicals, such , etc.? Actually, relevant radicals, such Actually, relevant radicals, such as Actually, the Zs of and 6-4 are calculated to 10, inin parallel with their low EπEvalues of π values the ZsZs of 6-3 6-36-3 and 6-46-4 areare calculated tobe be11 11and and 10,respectively, respectively, parallel with their low values the and calculated be and respectively, in parallel with their low values the Zs ofof 6-3 and 6-4 are calculated toto be 1111 and 10,10, respectively, inwe parallel with their low EπEπ 6.159 and 6.000 relative to their singlet isomers, 6-1 and 6-2. While can systematically discuss the of of 6.159 and 6.000 relative to to their singlet isomers, 6-16-1 and 6-2.6-2. While wewe cancan systematically discuss 6.159 and 6.000 relative their singlet isomers, and While systematically discuss of 6.159 and 6.000 relative to their singlet isomers, 6-1 and 6-2. TWhile wewe canare systematically discuss stability of all these 6π conjugated systems by using their Z and values, confronted with the thethe stability of all these 6π conjugated systems by using their Z and T values, we are confronted with stability of all these 6π conjugated systems by using their Z and T values, we are confronted with the stability of all these 6π conjugated systems by using their Z and T values, we are confronted with difficulty of counting L values for radicals. However, try to expand definition of L to conjugated the difficulty of of counting L values forfor radicals. However, trytry toour expand ourour definition of of L to the difficulty counting L values radicals. However, to expand definition the difficulty of counting L values for radicals. However, try to expand our definition of L Ltoto conjugated acyclic polyene radicals as as in Figure 11,11, where wewe take thethe average L of thethe length of of conjugated acyclic polyene radicals Figure where take average length conjugated acyclic polyene radicals as ininFigure 11, where we take the average L Lofofthe length of conjugation l for allall thethe possible paths in in thethe given radical. This time, thethe value of of l composed of of b CC conjugation l for possible paths given radical. This time, value l composed b CC conjugation l for all the possible paths in the given radical. This time, the value of l composed of b CC bonds is tentatively chosen as as (b +(b1)/2, which is consistent with what hashas already been defined forfor bonds is tentatively chosen + 1)/2, which is consistent with what already been defined bonds is tentatively chosen as (b + 1)/2, which is consistent with what has already been defined for stable (singlet) conjugated acyclic polyene molecules. stable (singlet) conjugated acyclic polyene molecules. stable (singlet) conjugated acyclic polyene molecules.

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acyclic polyene radicals as in Figure 11, where we take the average L of the length of conjugation l for all the possible paths in the given radical. This time, the value of l composed of b CC bonds is tentatively chosen as (b + 1)/2, which is consistent with what has already been defined for stable (singlet) conjugated Molecules 2017, 22, 896 acyclic polyene molecules. 10 of 13 l = 1.5

l = 2.5

L = (1.5 + 2.5 + 2.5)/3 = 2.17

l = 2.5

Figure 11. 11. Counting L for for aa conjugated conjugated acyclic acyclic polyene polyene radical. radical. Figure Counting of of L

Now, family of ofCC77HH9 9conjugated conjugatedradicals. radicals. See Figure where of the Now, consider the family See Figure 12,12, where EπEvalues of the six π values six isomers are plotted against Z and their L values are written down in italics. Only one isomer isomers are plotted against Z and their L values are written down in italics. Only one isomer situated situated as an is outlier is in quartet ground all others in doublet ground state plotted as an outlier in quartet ground state, state, whilewhile all others in doublet ground state areare plotted onona astraight straightline. line. 2.5

8.0 Eπ

3.5

2

7.5

7.0 2

16

18

20

Z

Figure Z–Eπ six isomers isomers of of C C77H H99conjugated conjugatedradicals, radicals,among amongwhich whichonly onlyone oneisomer isomerisisa π plot of six Figure 12. 12. Z–E aquartet quartetradical. radical.

Since all bebe derived from thethe hexatriene family by all these theseheptatrienyl heptatrienylradical radicalfamily familymembers memberscan can derived from hexatriene family methyl addition to either of 6-1~6-4, we can draw diagrams suchsuch as those in Figures 6 and6 7and in the by methyl addition to either of 6-1~6-4, we can draw diagrams as those in Figures 7 incase the of vinyl addition. Although the results are not given here, the global features of the genealogy of this case of vinyl addition. Although the results are not given here, the global features of the genealogy case arecase very similar and ourand findings (i)~(v) can be applied. of this are very similar our findings (i)~(v) can be applied. In this way, molecules and radicals is shown to be way, the the genealogy genealogy of of conjugated conjugated acyclic acyclic polyene polyene molecules governed by rather simple rules, which can be explained roughly by naive chemical tools as T and L, but supported profoundly profoundly by by the the Z-index. Z-index. Although present author has already analyzedanalyzed the conceptthe of aromaticity anti-aromaticity Although the the present author has already concept ofandaromaticity and for conjugated cyclic compounds by using the modified Z-index in line with the present analysis [4], anti-aromaticity for conjugated cyclic compounds by using the modified Z-index in line with the the results obtained in the present analysis would be helpful forwould supplementing remolding the present analysis [4], the results obtained in the present analysis be helpful and for supplementing previous theory.the This work istheory. in progress. and remolding previous This work is in progress. However, before tackling tackling this thisbig bigproblem, problem,i.e. i.e.aromaticity, aromaticity,one oneneeds needs extend definition toto extend thethe definition of of conjugated acyclic polyenes change conventional definition of cross-conjugation. conjugated acyclic polyenes andand alsoalso change the the conventional definition of cross-conjugation. See See Figure which demonstrates that following twopairs pairsofofconjugated conjugatedpolyenes, polyenes, namely namely (a) Figure 13, 13, which demonstrates that thethe following two dendralene and (odd) radialene, and (b) aa certain certain kind kind of of mono-branched mono-branched conjugated conjugated polyene and fulvene, are approaching the same limits, respectively. This means that both of these two pairs of conjugated polyenes should belong to the same family, family, or or “conjugated “conjugated acyclic acyclic polyenes”. polyenes”. If the term “acyclic” oneone maymay rephrase it as it “conjugated mono-Kekulenoid polyenes” or simply “acyclic” isisnot notfavorable, favorable, rephrase as “conjugated mono-Kekulenoid polyenes” or as “conjugated polyenes”. In any event, should anrepel even-membered cycle. Also, keep in mind simply as “conjugated polyenes”. In anywe event, werepel should an even-membered cycle. Also, keep that onlythat a single cycle is allowed, since a couple disjoint odd-membered cycles in mind only odd-membered a single odd-membered cycle is allowed, since a of couple of disjoint odd-membered contributes a smalla amount of aromatic or anti-aromatic character to thetoπ-electron system [14]. [14]. cycles contributes small amount of aromatic or anti-aromatic character the π-electron system

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dendralene odd radialene

mono-branched conjugated polyene

same limit

same limit

fulvene

Figure 13. 13. Infinitely Infinitely large large odd odd radialenes radialenes and and fulvenes, fulvenes, respectively, respectively, converge converge to to the the limit limit of ofcertain certain Figure cross-conjugated acyclic polyenes. cross-conjugated acyclic polyenes.

As already inferred in the above discussion, one may notice that radialenes and fulvenes are As already inferred in the above discussion, one may notice that radialenes and fulvenes are automatically joining the club of “cross-conjugated” hydrocarbons. That is, the conventional automatically joining the club of “cross-conjugated” hydrocarbons. That is, the conventional definition definition of cross-conjugation indicates such “a compound possessing three unsaturated groups, of cross-conjugation indicates such “a compound possessing three unsaturated groups, two of which, two of which, although conjugated to a third unsaturated center, are not conjugated to each other” although conjugated to a third unsaturated center, are not conjugated to each other” [15]. However, [15]. However, as Hopf already declared [7], let us use the term cross-conjugation in a broader sense as Hopf already declared [7], let us use the term cross-conjugation in a broader sense than defined above. than defined above. Namely, if at least one C=C double bond in a molecule is conjugated with more Namely, if at least one C=C double bond in a molecule is conjugated with more than two conjugated than two conjugated paths, that molecule has a cross-conjugated π-electron system. Then, paths, that molecule has a cross-conjugated π-electron system. Then, triafulvene, the smallest fulvene triafulvene, the smallest fulvene with only four π-electrons, is an important member of the with only four π-electrons, is an important member of the cross-conjugated polyenes and can join the cross-conjugated polyenes and can join the enlarged family of conjugated acyclic polyenes [6]. enlarged family of conjugated acyclic polyenes [6]. The present author has already pointed out that by applying the idea of Z and L to conjugated The present author has already pointed out that by applying the idea of Z and L to conjugated polyene networks, we can obtain mathematical support for and point out the limitation to the polyene networks, we can obtain mathematical support for and point out the limitation to the conventional organic electron theory, especially the use of the “curved arrow” originally proposed conventional organic electron theory, especially the use of the “curved arrow” originally proposed by by organic chemists without any knowledge of quantum mechanics. However, here, we do not organic chemists without any knowledge of quantum mechanics. However, here, we do not expand the expand the scope of the discussion to these issues. The interested readers can refer to the relevant scope of the discussion to these issues. The interested readers can refer to the relevant papers [1,2,4,6]. papers [1,2,4,6]. Conflicts Conflictsof ofInterest: Interest:The Theauthors authorsdeclare declareno noconflict conflictofofinterest. interest.

Appendix Simple Recipe Recipe for for Calculating Calculating the Appendix A. A. Simple the Z-Index Z-Index of of Branched Branched Conjugated Conjugated Polyenes Polyenes Since and Z-indices were defined in 1971 [13] and been discussed so often, Sincethe thep(G,k) p(G,k)numbers numbers and Z-indices were defined in 1971 [13] have and have been discussed so here, simplea recipe the “back-of-envelope calculation” of Z for a of treeZgraph introduced. often,a here, simplefor recipe for the “back-of-envelope calculation” for a will tree be graph will be First, remember that the Zs of path thegraph carbonSatom skeleton of linear polyene or radical N , or N, or the carbon atom skeleton of linear introduced. First, remember that graph the ZsSof path with N carbon atoms, are nothing but the Fibonacci numbers as given Table A1.asThese easily polyene or radical with N carbon atoms, are nothing but the Fibonacciinnumbers givenZs incan Table A1. be derived by easily using be thederived recursion These Zs can byrelation, using the recursion relation, N = ZN–1 + ZN–2, ZZ N = ZN-1 + ZN-2 , together with the initial values of Z1 = 1 and Z2 = 2. Note also that by rotating Table A1, or the p(G,k) together with initial values of Z1 = 1 andby Z245 = 2. Note also by rotating A1,will or the p(G,k) values for thethe SN family, counter-clockwise degrees, the that famous Pascal’sTable triangle appear. values for the SN family, counter-clockwise by 45 degrees, the famous Pascal’s triangle will appear. One given inin Figure A1, where G–lG–l is the subgraph of Onemore moreimportant importanttip tipisisthe therecursion recursionformula formula given Figure A1, where is the subgraph G by deleting an arbitrary edge l, and l,GΘl the one obtained from G–lfrom by further ofobtained G obtained by deleting an arbitrary edge andis GΘl is the one obtained G–l bydeleting further all the edges which were adjacent l. If a graph several components either of the deleting all the edges which weretoadjacent to l.isIfdivided a graphinto is divided into several by components by above the Zprocesses, value of the resultant is the productgraph of the is Zsthe of either cutting of the processes, above cutting the Z valuemulticomponent of the resultantgraph multicomponent all the components in Figure A1. Then, by the Then, Z values the two product of the Zs ofasallshown the components as shown in adding Figure A1. by of adding thesubgraphs, Z values ofG–l the and we obtain value ofobtain G. two GΘl, subgraphs, G–l the andZGΘl, we the Z value of G.

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Table A1. p(G,k) and Z-index of path graph SN .

Table A1. p(G,k) and Z-index of Spath graph SN. N. Table p(G,k) and Z-index path Table A1. p(G,k) andof of path graph SN. Table A1. p(G,k) and Z-index of path graph SSN N Table A1. A1. p(G,k) and Z-index ofZ-index path graph graph N.. Table of SSN .. p(G,k) TableA1. A1.p(G,k) p(G,k)and andZ-index Z-indexp(G,k) ofpath pathgraph graph N p(G,k) p(G,k)ZN p(G,k) G = SN NG p(G,k) N ZN N NN G S N N NG G ===== S SN N G = SkN= 0 N p(G,k) k22=10 33 1 442 2Z 4 Z4N 1p(G,k) ZNNN 3 G S N N k = 0 k = 0 1 k == 00 11 k 22= 0 33 1 44 2 ZZNN 3 3 4 G = SN N k 1 ●● ● k = 0 1 21 3 4 11 1 1 1111 1 ● ● 1111 1 1 1 ●● 111 1 11 1 2 2 222 1 1 1 1 2 1 11 2 22 1 111 2 1 2 1 2 22 111 1 221 12 3 332 3 2 3 3 3 11 222 1 333 2 3 33 3 11 4 444 1 1 5553 1 1 333 11113 3 4 5 1 3 11 44 5 1 4 5 111 1 443 331 4 3 31 885 5 555 4 3 11 4 3 88 5 44 3131 55 8 8 4 3 13 5 8 6 11 66 11 1 1 55 13 5 6 1 6 666 5 6 1 13 1 1 5 6 1 6 1 13 5 16 155 66 13 6 13 11 6 111 13 6 10 77 21 21 1016 4444 21 4 111 6666 10 7 777 10 10 21 6 10 4 7 10 21 21 4 1 1046 11 1034 15 111 34 7 21 77 88 7 15 15 10 10 11 34 8 34 11 17 10 15 107 15 115 34 1010 1 34 8 88 1 1 77 15 8 1 8

G G G G G G A FF A A FF A AA F F llll ll

G G

H H H H H H

A A A A A A

7

IIII II

H F FF F H F F

15

10

1

34

12 of 13 12 of 13

ZN 1 2 3 5 8 13 21 34

I

AI FF A A FF A A FF A F A F B E B B EA B BF A E AE FB B E E E B B A E B F E E B E B E BB EE B A E B F E E B E l B C D B C E D B l C BD D B C E C D C D C EE D D C EC C B D C D D E C C D C C DD B C D CG DD CG–l D G l G–l G G G G G–l C D G G llll G–l C D C G DG G–l C D G G l G–l C l) =DZ(H) Z(I) C Z(C) D Z(D) Z(F) Z (G) = Z (G– ) + Z (G + Z(A) Z(B) Z(C) Z(D) Z(E) l Z(F) Z (G) = Z (G– ) + Z (G l) = Z(H) Z(I) + Z(A) Z(B) Z(E) l ZZ (G) Z(C) Z(F) G l (G) = = ZZ (G– (G– ll))) + + ZZ (G (G l) l)G= = Z(H) Z(H) Z(I) Z(I) + + Z(A) Z(A) Z(B) Z(B) Z(C) Z(D) Z(D) Z(E) Z(E) Z(F) G–l Z(F) ZZ(G) Z(C) Z(D) Z(E) (G)==ZZ(G– (G–l l )++ZZ(G (G l)G l)== Z(H) Z(H)Z(I) Z(I) ++ Z(A) Z(A)Z(B) Z(B) G–lZ(C) Z(D) Z(E) Z(F)G l

Figure formula for obtaining Z of by deleting of l. Figure A1. Recursion formula of G by two-step deleting of edge l.l. Figure A1. A1. Recursion Recursion formula for obtaining ZZ(H) of G GZ(I) by two-step two-step deleting of edge edge Z (G) = Z (G–for Z (G l) =Z + Z(A) Z(B) Z(C) Z(D) Z(E) Z(F) l ) +obtaining Figure formula of by deleting of l.l. Z (G) = Z (G–for ) +obtaining Z (G l) = ZZ Z(H) Z(I) +two-step Z(A) Z(B) Z(C) Z(D) Z(E) FigureA1. A1.Recursion Recursion formula obtaining ofGG bytwo-step deleting ofedge edgeZ(F) lfor

Figure A1. Recursion formula for obtaining Z of G by two-step deleting of edge l. Let us apply this recursion formula to with vertices in A2. First cut Figure A1. Recursion formula forG obtaining of G bygiven two-step deleting of edge l. Let us apply this recursion formula formula to aaa graph graph Gobtaining with ten tenZZ vertices given in Figure Figure A2.of First cut Figure A1. Recursion forG of G bygiven two-step deleting edge l. Let this formula to ten in Figure A2. First cut Letus usapply apply thisrecursion recursion formula to dashed agraph graph Gwith with tenvertices vertices given inthe Figure A2.cut First cut the edge in the middle (the vertical straight line) to obtain G–l; then, do second (pair the edge edge in in the the middle middle (the (the vertical vertical straight straight dashed dashed line) line) to to obtain obtain G–l; G–l; then, then, do do the the second second cut cut (pair (pair the the edge in the (the straight dashed to sum obtain G–l; then, do thegiven second cut (pair Let usmiddle apply thistovertical recursion formula toproduct aline) graph G with ten vertices in Figure A2. First cut cut of the curved dashed lines) obtain GΘl. Take the of all the components according to Let us apply this recursion formula to a graph G with ten vertices given in Figure A2. First First of the curved dashed lines) to obtain GΘl. Take the product sum of all the components according to A2. Let us apply this recursion formula to a graph G with ten vertices given in Figure cut of the curved dashed lines) to obtain GΘl. Take the product sum of all the components according to of the curved dashed lines) to obtain GΘl. Take the product sum of all the components according to the edge in the middle (the vertical straight dashed line) to obtain G–l; then, do the second cut (pair Figure A1, and we obtain Z = 77 for G. theA1, edge inwe the middle (the vertical straight dashed dashed line) line) to to obtain obtain G–l; G–l; then, then, do do the the second second cut cut (pair (pair Figure A1, andin obtain Z == 77 77 forvertical G. Figure and obtain Z G. the edge the middle straight Figure A1, andwe we obtain Z lines) =(the 77for forto G.obtain of the curved dashed GΘl. Take the product sum of all the components according to

++++++

++++++

of the the curved curved dashed dashed lines) lines) to to obtain obtain GΘl. GΘl. Take Take the the product product sum sum of of all all the the components components according according to to of Figure A1, and we obtain Z = 77 for G. Figure A1, A1, and and we we obtain obtain Z Z == 77 77 for for G. G. Figure = =

2. 2. 2. 2. 2. 3. 3. 3. 3. 3. 4. 4. 4. 4. 4. 5. 5. 5. 5. 5. 6. 6. 6. 6. 6. 7. 7. 7. 7. 7. 8. 8. 8. 8. 8. 9. 9. 9. 9. 9.

++

1. 1. 1. 1. 1.

++

References References References References

++++++

++

== = SS44 xx SS66 ++ SS11xx SS22 xx SS22 xx SS33 = == SSS444 xxxSSS666 +++SSS111xxxSSS222 xxxSSS222 xxxSSS333 = 55 xx4 13 13 + +6 = 1 x12 x 22 x 32= 77 773 = 13 + +=11 xx 22 xx 22 xx 33 = = 77 == 555xxx13 13 +11xx22xx22xx33==77 77 Figure A2. A2. Example Example of of calculating calculating Z Z of of aa tree tree graph graph with with ten ten vertices. vertices. Figure Figure A2. Example of calculating Z of a tree graph with ten x tree S6 + graph S1x S2with x S2 xten S vertices. 4 xa Figure ==ZZSSof4of S2 x ten S33 vertices. FigureA2. A2.Example Exampleof ofcalculating calculating aStree vertices. 6 + Sgraph 1x S2 xwith 13 + 1 x 2 x 2 x 3 = 77 == 55 xx 13 + 1 x 2 x 2 x 3 = 77

Hosoya, H. H. Mathemaical Mathemaical foundation of organic organic electron theory—How theory—How π-electrons flow in conjugated conjugated Figure A2. Example of electron calculating Z of a treedo graph with ten vertices. Hosoya, foundation of π-electrons flow in Hosoya, H. H. Mathemaical Mathemaical foundation of organic organic electron theory—How theory—How do π-electrons flow in conjugated conjugated Figure A2. Example of calculating Z of a treedo graph with ten vertices. Hosoya, foundation electron Hosoya, J. H.Mol. Mathemaical foundationof of organic electron theory—Howdo doπ-electrons π-electronsflow flowin in conjugated systems? Struct. (THEOCHEM) 1999, 461/462, 473–482. systems? J. Mol. Struct. (THEOCHEM) 1999, 461/462, 473–482. systems? J. Mol. Struct. (THEOCHEM) 1999, 461/462, 473–482. systems? (THEOCHEM) 1999, systems?J.J.Mol. Mol.Struct. Struct.to (THEOCHEM) 1999,461/462, 461/462,473–482. 473–482. quantum-mechanical aspects of π-electron Hosoya, References Hosoya, H. From how why. Graph-theoretical verification of Hosoya, H. H. From From how how to to why. why. Graph-theoretical Graph-theoretical verification verification of of quantum-mechanical quantum-mechanical aspects aspects of of π-electron π-electron References Hosoya, H. From how to why. Graph-theoretical verification of References Hosoya, in H.conjugated From how to why. Graph-theoretical verification ofquantum-mechanical quantum-mechanicalaspects aspectsof ofπ-electron π-electron behavior systems. Bull. Chem. Soc. Jpn. 2003, 76, 2233–2252. behavior in conjugated systems. Bull. Chem. Soc. Jpn. 2003, 76, 2233–2252. behavior in conjugated systems. Bull. Chem. Soc. Jpn. 2003, 76, 2233–2252. behavior in systems. Bull. Chem. Soc. Jpn. 2003, 76, 2233–2252. behavior inconjugated conjugated systems. Bull. Chem.the Soc. Jpn. 2003, 76, 2233–2252. 1. Hosoya, H. Mathemaical foundation of organic electron theory—How do π-electrons flow in conjugated Hosoya, H. Aromaticity index can predict stability of polycyclic conjugated hydrocarbons. Monatsh. Hosoya, H. index predict stability of polycyclic conjugated Monatsh. 1. Hosoya, H. foundation organic do flow Hosoya, H. Aromaticity Aromaticity index can can predict the theof stability of electron polycyclic conjugated hydrocarbons. hydrocarbons. Monatsh. Hosoya, H. Mathemaical Mathemaical foundation of organic electron theory—How theory—How doπ-electrons π-electrons flow in in conjugated conjugated 1. Hosoya, H. Aromaticity index can of conjugated Hosoya, H.136, Aromaticity index(THEOCHEM) canpredict predictthe thestability stability ofpolycyclic polycyclic conjugatedhydrocarbons. hydrocarbons.Monatsh. Monatsh. systems? J. Mol. Struct. 1999, 461/462, 473–482. Chem. 2005, 1037–1054. Chem. 2005, 136, 1037–1054. Chem. 2005, 136, 1037–1054. systems? J. Mol. Struct. (THEOCHEM) 1999, 461/462, 473–482. systems? J. Mol. Struct. (THEOCHEM) 1999, 461/462, 473–482. [CrossRef] Chem. 2005, 136, 1037–1054. Chem. 2005, 136, Hosoya, H. can we explain stability of hydrocarbon and heterosubstituted 2. Hosoya, H.1037–1054. From how to the why. Graph-theoretical verification of quantum-mechanical aspects of π-electron Hosoya, H. How we explain the stability of conjugated hydrocarbon and heterosubstituted networks Hosoya, H. How How can wehow explain the stability of conjugated conjugatedverification hydrocarbonof and heterosubstituted networks networks 2. Hosoya, H. From to Graph-theoretical quantum-mechanical aspects 2. Hosoya, H.can From towhy. why. Graph-theoretical verification of quantum-mechanical aspectsof ofπ-electron π-electron Hosoya, H. How can we explain the of conjugated hydrocarbon and heterosubstituted Hosoya, H. How can wehow explain thestability stability of conjugated hydrocarbon and heterosubstitutednetworks networks by topological descriptors? Curr. Comp. Aid. Drug Des. 2010, 6, 225–234. behavior in conjugated systems. Bull. Chem. Soc. 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Aromaticity index can can predict the to stability of polycyclic conjugated hydrocarbons. Monatsh. 3. Hosoya, Aromaticity index predict the stability ofofofmathematics? polycyclic conjugated Hosoya, H.How How can mathematical chemistry contribute tothe thedevelopment development mathematics? Int. J.Philos. Philos. hydrocarbons. Chem. (HYLE) 2013 19, 87–105. Chem. (HYLE) 2013 19, 87–105. Chem.Chem. (HYLE) 2013136, 19, 87–105. 87–105. 2005, 1037–1054. Chem. (HYLE) 2013 19, Chem. 136, 1037–1054. Chem. (HYLE) 2013 19,2005, 87–105. Monatsh. Chem. 136, 1037–1054. [CrossRef] Hosoya, H. 2005, Cross-conjugation at the the heart of of understanding understanding the electronic electronic theory theory of of organic organic chemistry. chemistry. Hosoya, H. at the Hosoya, H. Cross-conjugation Cross-conjugation at explain the heart heartthe of stability understanding the electronic electronic theory of of organic organic chemistry. 4. Hosoya, H. 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