A New Yardstick for Benzenoid Polycyclic Aromatic ... - ACS Publications

Oct 22, 2008 - The newly introduced signature of benzenoids (a sequence of six real numbers si with i = 6−1) shows the composition of the π-electro...
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J. Phys. Chem. A 2008, 112, 11769–11776

11769

A New Yardstick for Benzenoid Polycyclic Aromatic Hydrocarbons Matevz Pompe† Faculty of Chemistry and Chemical Technology, UniVersity of Ljubljana, AsˇkercˇeVa 5, 1001 Ljubljana, SloVenia

Milan Randic´‡ National Institute of Chemistry, HajdrihoVa 19, SI-1000 Ljubljana, SloVenia

Alexandru T. Balaban* Texas A&M UniVersity at GalVeston, 5007 AVenue U, GalVeston, Texas 77551, and Visiting Scientist at the Faculty of Chemistry and Chemical Technology, UniVersity of Ljubljana, AsˇkercˇeVa 5, 1001 Ljubljana, SloVenia ReceiVed: June 7, 2008; ReVised Manuscript ReceiVed: July 22, 2008

The newly introduced signature of benzenoids (a sequence of six real numbers si with i ) 6-1) shows the composition of the π-electron partition by indicating the number of times all rings of the benzenoid are assigned 6, 5, 4, 3, 2, or 1 π-electrons. It allows the introduction of a new ordering criterion for such polycyclic aromatic hydrocarbons by summing some of the terms in the signature. There is an almost perfect linear correlation between sums s6 + s5 and s4 + s3 for isomeric cata- or peri-fused benzenoids, so that one can sort such isomers according to ascending s6 + s5 or to descending s4 + s3 sums (the resulting ordering does not differ much and agrees with that based on increasing numbers of Clar sextets and of Kekule´ structures). Branched cata-condensed benzenoids have higher s6 + s5 sums than isomeric nonbranched systems. For nonisomeric peri-condensed benzenoids, both sums increase with increasing numbers of benzenoid rings and decrease with the number of internal carbon atoms. Other partial sums that have been explored are s6 + s5 + s3 and s6 + s2 + s1, and the last one appears to be more generally applicable as a parameter for the complexity of benzenoids and for ordering isomeric benzenoids. Partition of π-Electrons in Rings of Benzenoids and Benzenoid Signatures In a recent series of papers, the partition of π-electrons in rings of conjugated polycyclic hydrocarbons was discussed on the basis of the following convention: a shared CdC bond common to two condensed rings allows one π-electron to be assigned to each of these rings, whereas a nonshared CdC bond contributes with two π-electrons to the corresponding ring.1-16 Upon counting for all resonance structures (assumed to have equal weights) how many π-electrons are assigned to each ring and upon dividing this count by the number K of resonance structures, one obtains the ring partition of all P π-electrons of the benzenoid. For a given Kekule´an benzenoid, one can produce an array with h rows corresponding to each hexagonal ring showing the number of times each of the h benzenoid rings is assigned 6, 5, 4, 3, 2, 1, or 0 π-electrons on seven columns denoted by R6 through R0. The row sum divided by K provides the abovementioned partition, but one can also look at the column sums (the Ri sequence of seven integers with i ) 6-0). Obviously, ΝiRi ) Kh. Upon dividing each Ri value by K, one obtains the ri sequence with seven rational numbers, also with i ) 6-0. Again, obviously, Νiri ) h. The benzenoid signature (or the si * To whom correspondence should be addressed. E-mail: balabana@ tamug.edu. † E-mail: [email protected]. ‡ Visitor Emeritus, Department of Mathematics & Computer Science, Drake University, Des Moines, Iowa. E-mail: [email protected].

Figure 1. The two isomeric octaperifusenes investigated by Clar and Zander, with their partitions and Ri sequences.

sequence) is a sequence of six real numbers obtained from the ri sequence by taking into account the number of π-electrons contributed by each of the h rings, si ) i × ri; with the corollary, we arrive at the total number of π-electrons P from a different summation, namely, P ) ∑isi.12-16 For catafusenes, P ) 4h + 2, and for perifusenes P ) 4h - a + 2 (where a denotes the number of internal carbon atoms). As an example, in Figure 1, one can see the corresponding partitions and Ri sequences, indicated in boldface characters in the last rows for each one of the two isomeric peri-condensed benzenoids with h ) 8 (of Clar-Zander fame) having different reactivities.17-19 Sums of entries in each row equal the number of resonance structures. One can observe that rings with Clar sextets have a higher value for the partition (bold italic numbers)

10.1021/jp805020h CCC: $40.75  2008 American Chemical Society Published on Web 10/23/2008

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TABLE 1: Signatures and Partial Sums of Nonisoarithmic Catafusenes with h ) 2-6 Benzenoid Rings comp.

h

K

s6

s5

s4

s3

s2

s1

s6 + s5

s4 + s3

s2 + s1

s6521

s621

s653

2c-1 3c-2 3c-3 4c-4 4c-5 4c-6 4c-7 5c-8 5c-9 5c-10 5c-11 5c-12 5c-13 5c-14 5c-15 6c-16 6c-17 6c-18 6c-19 6c-20 6c-21 6c-22 6c-23 6c-24 6c-25 6c-26 6c-27 6c-28 6c-29 6c-30 6c-31 6c-32 6c-33

2 3 3 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

3 4 5 5 7 8 9 6 9 10 11 12 13 13 14 7 11 13 14 15 16 17 17 18 19 19 19 20 21 22 22 23 24

4.000 3.000 6.000 2.400 5.143 7.500 8.667 2.000 4.667 4.200 6.545 7.000 9.231 7.846 9.857 1.714 4.364 3.692 6.000 5.600 6.750 6.000 7.412 8.333 8.526 8.842 6.947 9.000 10.857 9.545 11.182 11.739 12.000

3.333 5.000 4.000 6.000 6.429 3.750 6.667 6.667 7.778 9.000 5.909 7.500 3.846 9.231 6.071 7.143 8.636 10.385 7.143 8.000 9.375 10.000 10.588 6.111 7.105 8.158 11.842 8.500 3.810 10.227 5.682 6.304 8.333

2.667 6.000 2.400 9.600 4.571 4.000 1.333 13.333 7.556 6.800 6.545 4.000 4.923 3.692 3.429 17.143 10.909 9.846 9.714 9.067 6.000 6.353 6.824 7.333 5.684 5.895 6.105 6.000 6.095 3.273 5.091 4.174 2.667

0.000 0.000 1.200 0.000 1.286 2.250 0.333 0.000 1.333 1.200 2.455 2.500 3.231 0.231 1.500 0.000 1.364 1.154 2.571 2.800 2.625 2.471 0.176 3.333 3.632 1.895 0.158 1.350 4.286 1.500 2.727 2.348 1.000

0.000 0.000 0.400 0.000 0.571 0.500 0.667 0.000 0.667 0.800 0.545 1.000 0.769 0.615 0.857 0.000 0.727 0.923 0.571 0.533 1.250 1.176 0.588 0.889 1.053 0.947 0.526 0.800 0.952 1.091 1.091 1.130 1.500

0.000 0.000 0.000 0.000 0.000 0.000 0.333 0.000 0.000 0.000 0.000 0.000 0.000 0.385 0.286 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.412 0.000 0.000 0.263 0.421 0.300 0.000 0.364 0.227 0.304 0.500

7.333 8.000 10.000 8.400 11.572 11.250 15.334 8.667 12.445 13.200 12.454 14.500 13.077 17.077 15.928 8.857 13.000 14.077 13.143 13.600 16.125 16.000 18.000 14.444 15.631 17.000 18.789 17.500 14.667 19.772 16.864 18.043 20.333

2.667 6.000 3.600 9.600 5.857 6.250 1.666 13.333 8.889 8.000 9.000 6.500 8.154 3.923 4.929 17.143 12.273 11.000 12.285 11.867 8.625 8.824 7.000 10.666 9.316 7.790 6.263 7.350 10.381 4.773 7.818 6.522 3.667

0.000 0.000 0.400 0.000 0.571 0.500 1.000 0.000 0.667 0.800 0.545 1.000 0.769 1.000 1.143 0.000 0.727 0.923 0.571 0.533 1.250 1.176 1.000 0.889 1.053 1.210 0.947 1.100 0.952 1.455 1.318 1.434 2.000

7.333 8.000 10.400 8.400 12.143 11.750 16.334 8.667 13.112 14.000 12.999 15.500 13.846 18.077 17.071 8.857 13.727 15.000 13.714 14.133 17.375 17.176 19.000 15.333 16.684 18.210 19.736 18.600 15.619 21.227 18.182 19.477 22.333

4.000 3.000 6.400 2.400 5.714 8.000 9.667 2.000 5.334 5.000 7.090 8.000 10.000 8.846 11.000 1.714 5.091 4.615 6.571 6.133 8.000 7.176 8.412 9.222 9.579 10.052 7.894 10.100 11.809 11.000 12.500 13.173 14.000

7.333 8.000 11.200 8.400 12.858 13.500 15.667 8.667 13.778 14.400 14.909 17.000 16.308 17.308 17.428 8.857 14.364 15.231 15.714 16.400 18.750 18.471 18.176 17.777 19.263 18.895 18.947 18.850 18.953 21.272 19.591 20.391 21.333

than rings without Clar sextets. The first octaperifusene has K ) 45 resonance structures and is a “sextet resonant benzenoid” with rings that either have a Clar sextet (indicated by the Armit-Robinson circle) or are “empty rings”. With five Clar

sextets, this benzenoid is unreactive toward dienophiles. By contrast, as shown by Clar and Zander,17 the second benzenoid with K ) 37 Kekule´ valence structures has only four Clar sextets so that its Clar formula has also rings with conjugated double bonds and does react with dienophiles. The corresponding ri and si sequences are 0.889, 3.067, 1.156, 0.489, 1.111, 1.033, 0.267 and 5.333, 15.333, 4.662, 1.467, 2.222, 1.022, respectively, for the first benzenoid with five Clar sextets and 0.757, 2.541, 1.622, 1.162, 1.027, 0.730, 0.162 and 4.541, 12.703, 6.486, 3.486, 2.054, 0.730, respectively, for the second benzenoid with four Clar sextets. We have investigated how to compress the information contained in the benzenoid signatures into a single number with low degeneracy that could serve as a parameter for expressing the complexity of a benzenoid and for ordering isomeric benzenoids. The simplest way for arriving at such a number is by means of partial sums of terms in the benzenoid signature. The number of Clar sextets and the number K of Kekule´ valence structures (resonance structures) reflect the stability of benzenoids, but they are integers with high degeneracies. There is a correlation between these numbers manifested also in the fact that the number of Clar sextets is highest for fully resonant benzenoids among isomers; therefore, it should be sufficient to look at correlations with K for the targeted ordering parameter.

Figure 2. Dualists of catafusenes with h ) 2-6 benzenoid rings. Isokekule´an (but not isoarithmic) benzenoids are on the same horizontal line.

In several recent papers, the structures of cata- and pericondensed benzenoid with up to eight benzenoid rings were displayed.1-16 Whenever the dualists of the benzenoids have notations that differ only in interchanging digits 1 and 2, most properties of such benzenoids (called isoarithmic benzenoids), including the number of resonance structures and the electron

Benzenoid Polycyclic Aromatic Hydrocarbons

J. Phys. Chem. A, Vol. 112, No. 46, 2008 11771

TABLE 2: Signatures of Nonisoarithmic Heptacatafusenes and Their Partial Sums no.

K

RE (eV)

s6

s5

s4

s3

s2

s1

s65

s43

s21

s653

s621

7c-34 7c-35 7c-36 7c-38 7c-37 7c-39 7c-40 7c-42 7c-43 7c-44 7c-46 7c-48 7c-49 7c-51 7c-53 7c-56 7c-57 7c-59 7c-62 7c-70 7c-41 7c-45 7c-47 7c-50 7c-52 7c-54 7c-55 7c-58 7c-60 7c-61 7c-63 7c-64 7c-65 7c-66 7c-67 7c-68 7c-69 7c-71 7c-72 7c-73 7c-74 7c-75 7c-76 7c-77

8 13 16 17 17 19 20 22 23 23 24 25 25 26 27 29 29 30 31 34 21 24 25 26 27 28 29 30 30 31 31 32 32 33 33 34 34 35 35 36 37 38 40 41

2.131 2.900 3.210 3.279 3.273 3.279 3.396 3.562 3.704 3.734 3.743 3.857 3.926 3.898 3.924 4.030 4.107 4.199 4.143 4.238 3.515 3.722 3.798 3.831 3.884 3.983 4.006 4.038 4.085 4.089 4.133 4.146 4.182 4.211 4.208 4.255 4.258 4.277 4.307 4.329 4.379 4.411 4.528 4.557

1.500 4.154 3.375 5.647 3.176 5.053 6.600 5.455 7.826 5.739 5.000 7.440 8.160 7.615 7.556 9.931 7.862 10.000 10.258 12.529 7.143 8.250 6.480 8.539 8.000 6.000 8.069 10.200 9.400 10.258 8.516 10.875 8.625 10.909 10.909 10.765 11.117 12.686 11.143 13.000 13.297 13.421 12.000 14.341

7.500 9.231 11.250 7.941 11.765 9.211 10.500 11.364 7.392 11.957 12.500 8.400 8.800 9.231 9.630 6.035 11.035 6.667 7.258 3.824 11.429 9.375 13.200 9.808 10.556 14.465 11.035 7.834 12.167 7.903 12.742 8.594 12.813 8.788 9.394 11.030 10.736 5.572 9.887 5.834 6.216 8.027 13.000 8.781

21.000 14.462 13.250 13.176 12.941 12.210 8.800 9.454 10.435 8.348 8.667 9.760 8.000 8.154 8.000 8.552 5.793 7.467 6.581 7.176 10.286 9.000 9.280 9.231 8.444 8.572 8.552 7.467 5.333 8.000 5.548 6.625 5.750 6.667 4.970 4.941 5.176 6.172 4.914 6.000 5.406 4.210 2.400 3.512

0.000 1.385 1.125 2.647 1.059 3.000 2.700 2.455 3.391 2.478 2.500 3.360 3.840 3.923 3.556 4.448 3.724 4.800 4.548 5.294 0.143 2.125 0.120 1.269 1.778 0.107 1.241 3.100 1.500 2.516 1.548 2.344 1.406 2.182 3.000 1.765 0.882 4.029 2.571 3.583 3.487 2.210 0.600 1.317

0.000 0.769 1.000 0.588 1.059 0.526 1.400 1.273 0.957 1.478 1.333 1.040 1.200 1.077 1.259 1.034 1.586 1.067 1.355 1.176 0.571 1.000 0.480 0.769 0.889 0.429 0.690 1.200 1.200 1.032 1.290 1.250 1.000 1.091 1.455 1.176 1.529 1.371 1.143 1.333 1.297 1.684 1.300 1.415

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.429 0.250 0.440 0.385 0.333 0.429 0.414 0.200 0.400 0.290 0.355 0.313 0.406 0.364 0.273 0.324 0.559 0.171 0.371 0.257 0.297 0.447 0.700 0.634

9.000 13.385 14.625 13.588 14.941 14.263 17.100 16.818 15.217 17.696 17.500 15.840 16.960 16.846 17.185 15.966 18.896 16.667 17.516 16.353 18.572 17.625 19.680 18.346 18.555 20.465 19.103 18.034 21.567 18.161 21.258 19.469 21.438 19.697 20.303 21.794 21.853 18.257 21.030 18.834 19.513 21.447 25.000 23.122

21.000 15.846 14.375 15.824 14.000 15.210 11.500 11.909 13.826 10.826 11.167 13.120 11.840 12.077 11.556 13.000 9.517 12.267 11.129 12.471 10.428 11.125 9.400 10.500 10.222 8.679 9.793 10.567 6.833 10.516 7.097 8.969 7.156 8.849 7.970 6.706 6.059 10.200 7.486 9.583 8.892 6.421 3.000 4.829

0.000 0.769 1.000 0.588 1.059 0.526 1.400 1.273 0.957 1.478 1.333 1.040 1.200 1.077 1.259 1.034 1.586 1.067 1.355 1.176 1.000 1.250 0.920 1.154 1.222 0.857 1.103 1.400 1.600 1.323 1.645 1.563 1.406 1.455 1.727 1.500 2.088 1.543 1.514 1.590 1.595 2.132 2.000 2.049

9.000 14.770 15.750 16.235 16.000 17.264 19.800 19.274 18.609 20.174 20.000 19.200 20.800 20.769 20.742 20.414 22.621 21.467 22.064 21.647 18.715 19.750 19.800 19.616 20.334 20.572 20.345 21.134 23.067 20.677 22.806 21.813 22.844 21.879 23.303 23.560 22.735 22.287 23.601 22.417 23.000 23.658 25.600 24.439

1.500 4.923 4.375 6.235 4.235 5.579 8.000 6.728 8.783 7.217 6.333 8.480 9.360 8.692 8.815 10.965 9.448 11.067 11.613 13.705 8.143 9.500 7.400 9.693 9.222 6.858 9.173 11.600 11.000 11.580 10.161 12.438 10.031 12.364 12.637 12.265 13.205 14.228 12.657 14.590 14.891 15.552 14.000 16.390

distribution, are exactly the same. Therefore, for further discussions, we will examine only one of any set of isoarithmic benzenoids. Starting with benzenoids with small numbers h of benzenoid rings, we present in Table 1 the signatures and partial sums of nonisoarithmic catafusenes with up to six benzenoid rings. The analogous Table 2 contains the same data for heptacatafusenes.15,16 The notation for each compound contains (in boldface characters) the number of benzenoid rings followed by c for catafusenes and p for perifusenes, a hyphen, and a number that corresponds to the notation in previous papers that had presented structures, partitions, and sequences in detail.15,16 In Table 3, we present signatures and partial sums for nonisoarithmic perifusenes with up to seven benzenoid rings having contiguous internal vertices. In the column labeled P of Tables 3 and 6, boldface numbers correspond to peri-condensed benzenoids having contiguous internal vertices. Their numbers of internal vertices are easily computed as a ) 4h + 2 - P. Their molecular formula is CPHP+2-2h. Figures 2 (catafusenes) and 3 (perifusenes) display structures of benzenoids with up to six benzenoid rings via their dualists, in which vertices correspond to centers of rings and edges

connect vertices corresponding to condensed rings (i.e., rings sharing a CC bond). The ordering of structures in these figures and tables is first according to increasing h and then according to increasing K values. The structures and numbering of heptafusenes may be found in references published earlier.15,16 Properties of Partial Sums of Terms in Benzenoid Signatures Among all possible partial sums of terms in benzenoid signatures, Tables 2 and 3 contain for each compound sums s6 + s5, s4 + s3, s2+ s1, s6 + s5 + s3, and s6 + s2+ s1 (denoted for brevity by s65, s43, s21, s653, and s621, respectively). One can observe that whereas upon going from 6c-17 to 6c-33 or from 7p-19 to 7p-69, s21 increases about twice and has small values, s65 and s43 have larger values, and s65 also increases about twice and s43 decreases about three times. Most interestingly, in a plot of these two last sums (s65 versus s43) for catafusenes, there is an almost perfect linear correlation (for hexacatafusenes, the coefficient of determination is R2 ) 0.998), as seen in Figures 4 and 5. There is an explanation about this correlation, namely, the way that the P π-electrons are distributed. Of course, for a

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TABLE 3: Signatures and Partial Sums of Perifusenes with h ) 4-7 Benzenoid Rings no.

h

K

s6

s5

s4

s3

s2

s1

P

s65

s43

s21

s6521

s653

s621

4p-1 5p-2 5p-3 5p-4 6p-5 6p-6 6p-7 6p-8 6p-9 6p-10 6p-11 6p-12 6p-13 6p-14 6p-15 6p-16 6p-17 6p-18 7p-19 7p-20 7p-21 7p-22 7p-23 7p-24 7p-25 7p-26 7p-27 7p-28 7p-29 7p-30 7p-31 7p-32 7p-33 7p-34 7p-35 7p-36 7p-37 7p-38 7p-39 7p-40 7p-41 7p-42 7p-43 7p-44 7p-45 7p-46 7p-47 7p-48 7p-49 7p-50 7p-51 7p-52 7p-53 7p-54 7p-55 7p-56 7p-57 7p-58 7p-59 7p-60 7p-61 7p-62 7p-63 7p-64 7p-65 7p-66 7p-67 7p-68 7p-69

4 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7

6 9 9 11 9 10 12 12 13 14 14 15 15 16 16 17 17 20 9 12 14 15 15 15 15 16 16 17 18 18 19 19 20 20 20 21 21 21 21 21 21 21 22 22 23 23 23 23 23 24 24 24 24 24 25 25 25 25 26 27 27 27 27 28 28 29 29 30 31

0.000 0.000 2.000 2.727 0.000 0.000 1.500 1.500 3.692 4.286 0.000 2.400 3.600 4.500 1.875 3.882 4.941 5.400 0.000 1.500 1.714 2.400 2.400 1.200 1.200 3.000 2.250 3.176 0.000 0.000 3.789 2.842 1.800 3.900 0.000 4.286 3.429 1.714 2.571 3.429 4.000 1.429 5.455 2.182 5.739 4.435 3.652 2.870 2.348 4.500 3.750 5.250 6.500 6.500 4.800 5.760 4.080 2.640 6.000 7.333 6.222 6.000 3.556 5.786 6.429 6.000 4.552 7.400 6.581

8.333 13.333 7.778 10.909 13.333 7.000 11.667 9.583 6.923 7.500 10.714 13.667 8.333 10.000 13.438 10.294 10.294 13.500 13.333 11.667 6.429 13.667 10.667 12.667 10.667 10.000 6.875 8.529 10.000 12.500 10.000 9.474 9.500 12.000 7.500 12.381 13.095 15.952 10.714 10.952 11.667 14.762 7.727 10.455 7.826 12.174 12.391 12.391 12.391 12.917 13.542 8.125 9.167 9.167 14.000 9.600 12.800 13.000 9.615 10.000 10.741 10.926 14.444 10.536 10.714 12.586 16.034 12.500 12.903

4.000 5.333 5.333 3.273 5.333 7.200 8.667 7.667 7.385 6.000 5.714 5.067 5.867 5.000 5.750 5.412 4.706 2.400 9.333 8.667 8.857 5.067 12.267 11.467 10.667 12.000 7.750 9.882 7.111 8.000 7.579 6.947 7.800 8.400 7.200 7.238 7.810 7.238 8.381 6.667 7.238 8.952 7.455 6.727 6.957 7.130 7.826 7.826 5.739 6.667 6.667 7.167 7.000 6.333 4.800 6.720 7.200 5.120 7.231 5.481 5.333 5.333 5.333 6.143 5.429 4.552 4.828 4.267 4.387

3.000 0.000 4.000 1.636 4.000 6.600 0.750 4.250 5.077 4.929 3.643 1.200 5.000 2.813 1.500 2.824 2.471 0.600 4.000 5.000 7.714 5.000 1.200 1.200 4.400 1.500 7.500 5.471 7.333 3.500 5.053 4.737 4.800 1.950 6.900 2.143 1.714 1.286 5.000 5.429 3.286 1.429 6.000 4.500 6.000 2.609 2.348 3.261 3.000 2.250 2.250 6.125 3.625 4.000 2.400 4.080 2.400 2.640 3.462 3.333 3.667 4.000 2.778 3.643 3.536 2.379 0.517 1.600 1.935

0.667 0.889 0.889 1.091 1.333 1.200 1.000 1.000 0.923 1.286 1.571 1.200 1.200 1.375 1.000 1.294 1.176 1.400 1.333 1.167 1.286 1.867 1.067 1.067 1.067 1.125 1.625 0.941 1.556 1.556 1.579 1.579 1.800 1.300 2.100 1.524 1.524 1.333 1.333 1.524 1.524 0.952 1.364 1.727 1.478 1.217 1.391 1.391 1.913 1.167 1.333 1.333 1.333 1.750 1.520 1.600 1.040 1.920 1.308 1.407 1.704 1.407 1.481 1.571 1.500 2.000 1.310 1.533 1.548

0.000 0.444 0.000 0.364 0.000 0.000 0.417 0.000 0.000 0.000 0.357 0.467 0.000 0.313 0.438 0.294 0.412 0.700 0.000 0.000 0.000 0.000 0.400 0.400 0.000 0.375 0.000 0.000 0.000 0.444 0.000 0.421 0.300 0.450 0.300 0.429 0.429 0.476 0.000 0.000 0.286 0.476 0.000 0.409 0.000 0.435 0.391 0.261 0.609 0.500 0.458 0.000 0.375 0.250 0.480 0.240 0.480 0.680 0.385 0.444 0.333 0.333 0.407 0.321 0.393 0.483 0.759 0.700 0.645

16 20 20 20 24 22 24 24 24 24 22 24 24 24 24 24 24 24 28 28 26 28 28 28 28 28 26 28 26 26 28 26 26 28 24 28 28 28 28 28 28 28 28 26 28 28 28 28 26 28 28 28 28 28 28 28 28 26 28 28 28 28 28 28 28 28 28 28 28

8.333 13.333 9.778 13.636 13.333 7.000 13.167 11.083 10.615 11.786 10.714 16.067 11.933 14.500 15.313 14.176 15.235 18.900 13.333 13.167 8.143 16.067 13.067 13.867 11.867 13.000 9.125 11.705 10.000 12.500 13.789 12.316 11.300 15.900 7.500 16.667 16.524 17.666 13.285 14.381 15.667 16.191 13.182 12.637 13.565 16.609 16.043 15.261 14.739 17.417 17.292 13.375 15.667 15.667 18.800 15.360 16.880 15.640 15.615 17.333 16.963 16.926 18.000 16.322 17.143 18.586 20.586 19.900 19.484

7.000 5.333 9.333 4.909 9.333 13.800 9.417 11.917 12.462 10.929 9.357 6.267 10.867 7.813 7.250 8.236 7.177 3.000 13.333 13.667 16.571 10.067 13.467 12.667 15.067 13.500 15.250 15.353 14.444 11.500 12.632 11.684 12.600 10.350 14.100 9.381 9.524 8.524 13.381 12.096 10.524 10.381 13.455 11.227 12.957 9.739 10.174 11.087 8.739 8.917 8.917 13.292 10.625 10.333 7.200 10.800 9.600 7.760 10.693 8.814 9.000 9.333 8.111 9.786 8.965 6.931 5.345 5.867 6.322

0.667 1.333 0.889 1.455 1.333 1.200 1.417 1.000 0.923 1.286 1.928 1.667 1.200 1.688 1.438 1.588 1.588 2.100 1.333 1.167 1.286 1.867 1.467 1.467 1.067 1.500 1.625 0.941 1.556 2.000 1.579 2.000 2.100 1.750 2.400 1.953 1.953 1.809 1.333 1.524 1.810 1.428 1.364 2.136 1.478 1.652 1.782 1.652 2.522 1.667 1.791 1.333 1.708 2.000 2.000 1.840 1.520 2.600 1.693 1.851 2.037 1.740 1.888 1.892 1.893 2.483 2.069 2.233 2.193

9.000 14.666 10.667 15.091 14.666 8.200 14.584 12.083 11.538 13.072 12.642 17.734 13.133 16.188 16.751 15.764 16.823 21.000 14.666 14.334 9.429 17.934 14.534 15.334 12.934 14.500 10.750 12.646 11.556 14.500 15.368 14.316 13.400 17.650 9.900 18.620 18.477 19.475 14.618 15.905 17.477 17.619 14.546 14.773 15.043 18.261 17.825 16.913 17.261 19.084 19.083 14.708 17.375 17.667 20.800 17.200 18.400 18.240 17.308 19.184 19.000 18.666 19.888 18.214 19.036 21.069 22.655 22.133 21.677

11.333 13.333 13.778 15.272 17.333 13.600 13.917 15.333 15.692 16.715 14.357 17.267 16.933 17.313 16.813 17.000 17.706 19.500 17.333 18.167 15.857 21.067 14.267 15.067 16.267 14.500 16.625 17.176 17.333 16.000 18.842 17.053 16.100 17.850 14.400 18.810 18.238 18.952 18.285 19.810 18.953 17.620 19.182 17.137 19.565 19.218 18.391 18.522 17.739 19.667 19.542 19.500 19.292 19.667 21.200 19.440 19.280 18.280 19.077 20.666 20.630 20.926 20.778 19.965 20.679 20.965 21.103 21.500 21.419

0.667 1.333 2.889 4.182 1.333 1.200 2.917 2.500 4.615 5.572 1.928 4.067 4.800 6.188 3.313 5.470 6.529 7.500 1.333 2.667 3.000 4.267 3.867 2.667 2.267 4.500 3.875 4.117 1.556 2.000 5.368 4.842 3.900 5.650 2.400 6.239 5.382 3.523 3.904 4.953 5.810 2.857 6.819 4.318 7.217 6.087 5.434 4.522 4.870 6.167 5.541 6.583 8.208 8.500 6.800 7.600 5.600 5.240 7.693 9.184 8.259 7.740 5.444 7.678 8.322 8.483 6.621 9.633 8.774

plot of s6521 versus s43, one would have a perfect correlation with R2 ) -1 because s6521 + s43 ) P, the number of all π-electrons. Then, one has to consider that s21 adds a relatively small contribution to s65.

Such linear correlations between sums s6 + s5 and s4 + s3 are general, and they occur on practically parallel lines that depend on the number of rings (and for perifusenes, also on the number of internal vertices). As illustrated, Figure 4 presents

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TABLE 4: Correlation Coefficient R between K Values and Partial Sums of Signature Terms s6 ) -s54321 s5 ) -s64321 s4 ) -s65321 s3 ) -s65421 s2 ) -s65431 s1 ) -s65432 s21 ) -s6543 s31 ) -s6542 s41 ) -s6532 s51 ) -s6432 s61 ) -s5432 s32 ) -s6541 s42 ) -s6531 s52 ) -s6431s62 ) -s5431 s43 ) -s6521 s53 ) -s6421 s63 ) -s5421 s54 ) -s6321 s64 ) -s5321 s65 ) -s4321 s321 ) -s654 s421 ) -s653 s521 ) -s643 s621 ) -s543 s431 ) -s652 s531 ) -s642 s631 ) -s542 s541 ) -s652 s641 ) -s532 s651 ) -s432

7-cata-all

7-cata-br.

7-cata-nonbr.

7-peri-all

7-peri-2-int.

7-peri-4-int.

8-peri-4-int.

0.940 -0.161 -0.935 0.237 0.610 0.620 0.821 0.364 -0.927 -0.102 0.946 0.346 -0.941 -0.072 0.943 -0.850 -0.036 0.849 -0.876 -0.154 0.837 0.459 -0.935 -0.016 0.951 -0.853 0.041 0.872 -0.854 -0.083 0.831

0.889 -0.369 -0.916 0.394 0.778 0.293 0.831 0.164 -0.917 -0.344 0.900 0.523 -0.904 -0.281 0.889 -0.696 -0.294 0.799 -0.824 0.111 0.628 0.582 -0.909 -0.256 0.900 -0.711 -0.263 0.817 -0.808 0.140 0.615

0.926 -0.361 -0.916 0.939 0.678 0.678 0.939 -0.916 -0.361 0.926 0.972 -0.928 -0.235 0.950 -0.778 0.250 0.936 -0.953 -0.331 0.790 0.971 -0.928 -0.235 0.950 -0.778 0.250 0.937 -0.930 -0.331 0.790

0.777 0.205 -0.764 -0.348 0.273 0.592 0.559 -0.304 -0.720 0.248 0.812 -0.293 -0.788 0.238 0.799 -0.788 -0.117 0.402 -0.377 0.092 0.780 -0.244 -0.743 0.280 0.832 -0.780 -0.025 0.471 -0.318 0.163 0.716

0.813 0.043 -0.786 -0.190 0.450 0.579 0.685 -0.120 -0.748 0.099 0.847 -0.115 -0.299 0.094 0.821 -0.755 -0.133 0.493 -0.587 -0.040 0.745 -0.036 -0.760 0.146 0.853 -0.758 -0.047 0.557 -0.527 0.039 0.743

0.538 0.804 -0.728 -0.849 0.027 0.797 0.503 -0.851 -0.601 0.816 0.631 -0.840 -0.801 -0.801 0.568 -0.910 0.150 -0.530 0.629 0.036 0.940 -0.838 -0.669 0.811 0.657 -0.916 0.323 -0.474 0.667 0.175 0.934

0.485 0.815 -0.922 -0.867 0.786 0.892 0.947 -0.859 -0.885 0.830 0.623 -0.828 -0.902 0.842 0.585 -0.953 0.383 -0.539 0.370 -0.687 0.948 -0.812 -0.854 0.854 0.695 -0.957 0.576 -0.461 0.462 -0.545 0.945

an analogous correlation for catafusenes with h ) 2-6, and Figure 5 displays the same correlation for hexaperifusenes (only the upper points of the preceding figure). In Figure 4, one can recognize the four parallel directions corresponding to the benzenoids with h ) 3-6. In Figure 6, one can see the many points corresponding to hexaperifusenes with two internal vertices and two points on a parallel direction for the two hexaperifusenes with four internal vertices, namely 6p-6 and 6p-11. There is also segregation according to whether the catafusene has a branched dualist or a nonbranched dualist. As one can see in Figure 7A and B, the nonbranched catafusenes are concentrated in the upper left part of the straight line, whereas the branched isomers are concentrated on the lower left end of the same line. Correlations between Numbers K of Resonance Structures and Partial Sums of Signature Terms We have explored the correlation between K values and all possible partial sums of terms in the signatures of representative sets of benzenoids, having a sufficient number of isomeric compounds in each set. For the set of catafusenes with seven benzenoid rings, Table 4 presents selections from the Pearson product-moment correlation coefficient R for single, pair, and triplet sums. One should take into account the equality (with changed R sign) between the 15 pair sums and the quadruplet sums, as well as that between half of the 20 triplet sums and the remaining half. Because for nonbranched catafusenes all s1 terms are zero, there is no number in the corresponding site of Table 4. A relatively small set of perifusenes with eight benzenoid rings and four internal contiguous vertices is also included in Table 4. It can be seen that the R values for singlets s6, s2, and s1 are always positive, for s4, they are always negative, and for s5 and

s3, they change sign upon passing from catafusenes to perifusenes. Among doublet sums, s21, s61, s62, and s65 lead always to positive R values, s41, s42, and s43 lead to negative R values, and other doublet sums cause sign changes. One can understand these observations by remembering that s6 and s4 (with opposite signs) have larger values than other partial sums, whereas s1, s2, s3, and s5 have relatively smaller values. Another conclusion based on looking at the entries in Table 4 (any heptacatafusene and heptaperifusenes with less than four internal carbon atoms) is that the triplet sum s6 + s2 + s1 (denoted as s621 for brevity, and similarly for other partial sums) with boldface entries in Table 4 can be considered to be the most general parameter for the complexity of all benzenoids, although for certain other sets of isomeric compounds, other partial sums (such as s65 or s653) may yield higher correlation coefficients. Correlations between Resonance Energies and Partial Sums of Signature Terms In addition to numbers of Clar structures or resonance structures, a more refined and less degenerate property of benzenoids is the resonance energy (RE). There are several types and corresponding programs for obtaining resonance energy values. In Table 2, one can see the RE values (in eV) for heptacatafusenes calculated according to the conjugated circuits method.20 For all nonisoarithmic heptacatafusenes, there is an excellent correlation between RE and ln K values,21 as seen in Figure 8. Upon plotting RE values for all (except the three least and most branched ones 7c-34, 7c-41, and 7c-70) heptacatafusenes against partial sums of signature terms s621 (Figure 9) and s653 (Figure 10), satisfactory correlations are obtained. Although the latter is slightly better than the former, taking into

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Figure 5. Plot of s4 + s3 versus s6 + s5 for hexacatafusenes indicated in Figure 2 by 6c-x and here by x (R2 ) 0.9962, y ) 27.122 - 1.1357x).

Figure 6. Plot of sums s4 + s3 versus s6 + s5 for hexaperifusenes indicated in Figure 3 by 6p-x and here by x.

Figure 3. Dualists of perifusenes with h ) 4-6 benzenoid rings. Isoarithmic benzenoids are on the same horizontal line.

Figure 7. Plots of sums s3 + s4 versus s5 + s6 for nonbranched (A, upper part) and branched heptacatafusenes (B, lower part). Figure 4. Plot of sums s3 + s4 versus s5 + s6 for catafusenes with h ) 2-6 carbon atoms. In Figure 2, the benzenoids were denoted by hc-x, but here, they are indicated only by x for h ) 2-5.

account the information of the preceding section, we will concentrate henceforth on s621 as the preferred ordering parameter. A Novel Yardstick and the Corresponding Ordering of Benzenoids In a previous paper,13 for the set of all 33 possible nonisoarithmic octaperifusenes with four contiguous internal vertices,

the ordering induced by s6521 or s652 was found to approximate fairly well that induced by K values; the partial sum s621 did not lead to satisfactory results. It was mentioned there that for catafusenes, the latter parameter provided better results than the two former ones. In the present paper, we are now able to substantiate this claim, concentrating on s621. Among the various partial sums indicated in Table 4, we have to choose one to encode the complexity of benzenoids and to allow an ordering of isomeric benzenoids. The chosen parameter

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TABLE 5: Heptacatafusenes Ordered by s621 (Separately for Nonbranched and Branched) no.

K

s6 + s5

s4 + s3

s2 + s1

s6 + s2 + s1

7c-34 7c-37 7c-36 7c-35 7c-39 7c-38 7c-46 7c-42 7c-44 7c-40 7c-48 7c-51 7c-43 7c-53 7c-49 7c-57 7c-56 7c-59 7c-62 7c-70 7c-54 7c-47 7c-41 7c-55 7c-52 7c-45 7c-50 7c-65 7c-63 7c-60 7c-61 7c-58 7c-68 7c-66 7c-64 7c-67 7c-72 7c-69 7c-76 7c-71 7c-73 7c-74 7c-75 7c-77

8 17 16 13 19 17 24 22 23 20 25 26 23 27 25 29 29 30 31 34 28 25 21 29 27 24 26 32 31 30 31 30 34 33 32 33 35 34 40 35 36 37 38 41

9.000 14.941 14.625 13.385 14.263 13.588 17.500 16.818 17.696 17.100 15.840 16.846 15.217 17.185 16.960 18.896 15.966 16.667 17.516 16.353 20.465 19.680 18.572 19.103 18.555 17.625 18.346 21.438 21.258 21.567 18.161 18.034 21.794 19.697 19.469 20.303 21.030 21.853 25.000 18.257 18.834 19.513 21.447 23.122

21.000 14.000 14.375 15.846 15.210 15.824 11.167 11.909 10.826 11.500 13.120 12.077 13.826 11.556 11.840 9.517 13.000 12.267 11.129 12.471 8.679 9.400 10.428 9.793 10.222 11.125 10.500 7.156 7.097 6.833 10.516 10.567 6.706 8.849 8.969 7.970 7.486 6.059 3.000 10.200 9.583 8.892 6.421 4.829

0.000 1.059 1.000 0.769 0.526 0.588 1.333 1.273 1.478 1.400 1.040 1.077 0.957 1.259 1.200 1.586 1.034 1.067 1.355 1.176 0.857 0.920 1.000 1.103 1.222 1.250 1.154 1.406 1.645 1.600 1.323 1.400 1.500 1.455 1.563 1.727 1.514 2.088 2.000 1.543 1.590 1.595 2.132 2.049

1.500 4.235 4.375 4.923 5.579 6.235 6.333 6.728 7.217 8.000 8.480 8.692 8.783 8.815 9.360 9.448 10.965 11.067 11.613 13.705 6.858 7.400 8.143 9.173 9.222 9.500 9.693 10.031 10.161 11.000 11.580 11.600 12.265 12.364 12.438 12.637 12.657 13.205 14.000 14.228 14.590 14.891 15.552 16.390

must be general, even if it is not always the one with the highest correlation coefficient with K or RE. In Table 4, one sees that the sum of terms s6 + s2+ s1 (s621), which is displayed in boldface characters, looks to be the best

Figure 9. Plot of sum s621 versus RE for all but three nonisoarithmic heptacatafusenes.

Figure 10. Plot of sum s653 versus RE for all but three nonisoarithmic heptacatafusenes.

choice because it always affords positive correlation coefficients with K, and in most cases, it has the highest R values. Now for looking at the ordering induced by this parameter, let us consider Tables 5 and 6 with cata- and peri-condensed heptafusenes. For nonbranched heptacatafusenes, shown in the first half of Table 5, it can be observed that the ordering induced by s621 does not differ much from that induced by K; it does, however, differ a little more for branched catafusenes shown in the second half of Table 5. Also, the ordering of heptaperifusenes induced by s621 differs fairly frequently from that induced by K, but the general trends with the lowest and highest values are not different in any of the systems examined. In Tables 3 and 6, one may observe for compounds 7p-20 and 7p-24 the first case of degeneracy among partial sums for terms in signatures. Such degeneracies are to be expected because the terms in signatures are rational numbers; therefore, degeneracies may occur for some combinations of si terms. Conclusions

Figure 8. Plot of ln K versus resonance energies (RE in eV) for all nonisoarithmic heptacatafusenes.

A study is presented on how to compress into one index of complexity the benzenoid signatures, which represent the sequence of six rational numbers into which the number P of π-electrons is divided in nonisoarithmic cata- and pericondensed Kekule´an benzenoids. Partial sums of terms in benzenoid signatures may serve as parameters for characterizing the complexity of benzenoids. It was found that the Pearson correlation coefficients R with the number K of resonance structures are always positive for terms s6, s2, and s1 of the signature; for s4, they are always negative, and for s5 and s3,

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TABLE 6: Perifusenes Ordered by h and Then by s621 no.

h

K

P

s65

s43

s21

s621

4p-1 5p-2 5p-3 5p-4 6p-6 6p-5 6p-11 6p-8 6p-7 6p-15 6p-12 6p-9 6p-13 6p-16 6p-10 6p-14 6p-17 6p-18 7p-19 7p-29 7p-30 7p-25 7p-35 7p-20 7p-24 7p-42 7p-21 7p-38 7p-23 7p-27 7p-33 7p-39 7p-28 7p-22 7p-44 7p-26 7p-48 7p-32 7p-49 7p-40 7p-58 7p-31 7p-37 7p-47 7p-63 7p-51 7p-57 7p-34 7p-41 7p-46 7p-50 7p-36 7p-52 7p-67 7p-55 7p-43 7p-45 7p-56 7p-64 7p-59 7p-62 7p-53 7p-61 7p-65 7p-66 7p-54 7p-69 7p-60 7p-68

4 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7

6 9 9 11 10 9 14 12 12 16 15 13 15 17 14 16 17 20 9 18 18 15 20 12 15 21 14 21 15 16 20 21 17 15 22 16 23 19 23 21 25 19 21 23 27 24 25 20 21 23 24 21 24 29 25 22 23 25 28 26 27 24 27 28 29 24 31 27 30

16 20 20 20 22 24 22 24 24 24 24 24 24 24 24 24 24 24 28 26 26 28 24 28 28 28 26 28 28 26 26 28 28 28 26 28 28 26 26 28 26 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28

8.333 13.333 9.778 13.636 7.000 13.333 10.714 11.083 13.167 15.313 16.067 10.615 11.933 14.176 11.786 14.500 15.235 18.900 13.333 10.000 12.500 11.867 7.500 13.167 13.867 16.190 8.143 17.667 13.067 9.125 11.300 13.286 11.706 16.067 12.636 13.000 15.261 12.316 14.739 14.381 15.640 13.789 16.524 16.043 18.000 17.292 16.880 15.900 15.667 16.609 17.417 16.667 13.375 20.586 18.800 13.182 13.565 15.360 16.321 15.615 16.926 15.667 16.963 17.143 18.586 15.667 19.484 17.333 19.900

7.000 5.333 9.333 4.909 13.800 9.333 9.357 11.917 9.417 7.250 6.267 12.462 10.867 8.235 10.929 7.813 7.176 3.000 13.333 14.444 11.500 15.067 14.100 13.667 12.667 10.381 16.571 8.524 13.467 15.250 12.600 13.381 15.353 10.067 11.227 13.500 11.087 11.684 8.739 12.095 7.760 12.632 9.524 10.174 8.111 8.917 9.600 10.350 10.524 9.739 8.917 9.381 13.292 5.345 7.200 13.455 12.957 10.800 9.786 10.692 9.333 10.625 9.000 8.964 6.931 10.333 6.323 8.815 5.867

0.667 1.333 0.889 1.455 1.200 1.333 1.929 1.000 1.417 1.438 1.667 0.923 1.200 1.588 1.286 1.688 1.588 2.100 1.333 1.556 2.000 1.067 2.400 1.167 1.467 1.429 1.286 1.810 1.467 1.625 2.100 1.333 0.941 1.867 2.136 1.500 1.652 2.000 2.522 1.524 2.600 1.579 1.952 1.783 1.889 1.792 1.520 1.750 1.810 1.652 1.667 1.952 1.333 2.069 2.000 1.364 1.478 1.840 1.893 1.692 1.741 1.708 2.037 1.893 2.483 2.000 2.194 1.852 2.233

0.667 1.333 2.889 4.182 1.200 1.333 1.928 2.500 2.917 3.313 4.067 4.615 4.800 5.470 5.572 6.188 6.529 7.500 1.333 1.556 2.000 2.267 2.400 2.667 2.667 2.857 3.000 3.523 3.867 3.875 3.900 3.904 4.117 4.267 4.318 4.500 4.522 4.842 4.870 4.953 5.240 5.368 5.382 5.434 5.444 5.541 5.600 5.650 5.810 6.087 6.167 6.239 6.583 6.621 6.800 6.819 7.217 7.600 7.678 7.693 7.740 8.208 8.259 8.322 8.483 8.500 8.774 9.184 9.633

they change sign upon passing from catafusenes to perifusenes. Upon searching all such possible partial sums for cata- and pericondensed benzenoids with up to seven benzenoid rings, it was observed that plots of pairwise sums s6 + s5 versus s4 + s3

afford linear correlations with intercepts depending on the branching of catafusenes and on the numbers of internal vertices for perifusenes. Among all other partial sums, it appears that s621, that is, s6 + s2 + s1, represents the best parameter for this purpose because it presents a satisfactory correlation with the number K of Kekule´ structures (and therefore also with the number of Clar sextets). Among all benzenoids with up to seven benzenoid rings, only one pair of heptaperifusenes with the same value for this parameter was found. Acknowledgment. The authors acknowledge the financial support of the Slovenian Research Agency through the Research Grants P1-0153, BI-US/06-07-025 and 1000-07-780002, of the Slovenian Human Resources Development and Scholarship AdFutura. References and Notes (1) Randic´, M.; Balaban, A. T. Partitioning of π-electrons in rings of polycyclic conjugated hydrocarbons. Part 1. Catacondensed benzenoids. Polycyclic Aromat. Compd. 2004, 24, 173–193. (2) Balaban, A. T.; Randic´, M. Partitioning of π-electrons in rings of polycyclic benzenoid hydrocarbons. Part 2. Catacondensed coronoids. J. Chem. Inf. Comput. Sci. 2004, 44, 50–59. (3) Balaban, A. T.; Randic´, M. Partitioning of π-electrons in rings of polycyclic conjugated hydrocarbons. Part 3. Perifusenes. New J. Chem. 2004, 28, 800–806. (4) Vukicˇevic´, D.; Randic´, M.; Balaban, A. T. Partitioning of π-electrons in rings of polycyclic conjugated hydrocarbons. Part 4. Benzenoids with more than one geometric Kekule structure corresponding to the same algebraic Kekule´ structure. J. Math. Chem. 2004, 36, 271–279. (5) Balaban, A. T.; Randic´, M. Partitioning of π-electrons in rings of polycyclic conjugated hydrocarbons. Part 5. Nonalternant compounds. J. Chem. Inf. Comput. Sci. 2004, 44, 1701–1707. (6) Gutman, I.; Randic´, M.; Balaban, A. T.; Furtula, B.; Vucˇkovic´, V. π-Electron contents of rings in the double-hexagonal chain homologous series (pyrene, anthanthrene and other acenoacenes). Polycyclic Aromat. Compd. 2005, 25, 215–226. (7) Randicˇ, M.; Balaban, A. T. Partitioning of -electrons in rings for Clar structures of benzenoid hydrocarbons. J. Chem. Inf. Model. 2006, 46, 57–64. (8) Balaban, A. T.; Furtula, B.; Gutman, I.; Kovacˇevic´, R. Partitioning of π-electrons in rings of aza-derivatives of polycyclic benzenoid hydrocarbons. Polycyclic Aromat. Compd. 2007, 27, 51–63. (9) Balaban, A. T.; Randicˇ, M. Partitioning of π-electrons in rings of polycyclic conjugated hydrocarbons. Part 6. Comparison with other methods for estimating the local aromaticity of rings in polycyclic benzenoids. J. Math. Chem. 2005, 37, 443–453. (10) Balaban, A. T.; Gutman, I.; Stankovic´, S. Effect of heteroatoms on partitioning of π-electrons in rings of catafusenes. Polycyclic Aromat. Compd. 2008, 28, 85–97. (11) Balaban, A. T.; Randic´, M.; Vukicˇevic´, D. Partition of π-electrons between faces of polyhedral carbon aggregates. J. Math. Chem. 2008, 43, 773–779. (12) Balaban, A. T.; Randic´, M. Perfect matchings in polyhexes, or recent graph-theoretical contributions to benzenoids. J. UniVers. Comput. Sci. 2007, 13, 1514–1539. (13) Balaban, A. T.; Randic´, M.; Pompe, M. π-Electron partitions, signatures, and Clar structures of selected benzenoid hydrocarbons. J. Phys. Chem. A 2008, 112, 4148–4157. (14) Balaban, A. T.; Randic´, M. Correlations between various ways of accounting for the distribution of π-electrons in benzenoids. New J. Chem. 2008, 32, 1071–1078. (15) A. T., Balaban.; Randic´, M. Ring signatures for benzenoids with up to seven rings. Part 1. cata-Condensed systems. Int. J. Quantum Chem. 2008, 108, 865–897. (16) Balaban, A. T.; Randic´, M. Ring signatures for benzenoids with up to seven rings. Part 2. peri-Condensed systems. Int. J. Quantum Chem. 2008, 108, 898–926. (17) Clar, E.; Zander, M. 1:12-2:3-10:11-Tribenzoperylene. J. Chem. Soc. 1958, 1861–1865. (18) Clar, E. The Aromatic Sextet; Wiley: London, 1972. (19) Dias, J. R. Handbook of Polycyclic Hydrocarbons, Part A. Benzenoid Hydrocarbons; Elsevier: Amsterdam, 1987. (20) Randic´, M. Aromaticity of polycyclic conjugated hydrocarbons. Chem. ReV. 2003, 103, 3449–3606. (21) Swinborne-Sheldrake, R.; Herndon, W. C.; Gutman, I. Tetrahedron Lett. 1975, 16, 755–758.

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