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A periodic table for polycyclic aromatic hydrocarbons
137 (1986) 9-29 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
Journal of Molecular Structure (Theochem),
A PERIODIC TABLE FOR POLYCYCLIC AROMATIC HYDROCARBONS Part IX. Isomer Enumeration and Properties of Radical Strictly PeriCondensed Polycyclic Aromatic hydrocarbons
JERRY
RAY DIAS
Department of Chemistry, University of Missouri, Kansas City, MO 64110 (U.S.A.) (Received 6 February 1985)
ABSTRACT Isomer enumeration of strictly per&condensed benzenoid (PAHG) radicals is simplified by enumerating their excised internal structures. The number of many radical benzenoid isomers are enumerated for the first time. Structural properties of this important class of high-spin hydrocarbons has been detailed. It is shown that for a given number of formula carbon atoms, the strictly per&condensed benzenoid hydrocarbon structures are the most stable, and the possible pyrolytic formation of strictly peri-condensed benzenoid hydrocarbons via condensation of other benzenoid hydrocarbons can only occur with the evolution of hydrogen. If the excised internal structure of a strictly peri-condensed PAH6 is not a cross conjugated diradical species and it has a HMO eigenvalue of zero, then its corresponding strictly per&condensed PAH6 structure will also have an eigenvalue of zero. INTRODUCTION
Although, previous research emphasized the more stable, even-carbon benzenoid hydrocarbons [ 1, 21, it was pointed out that all the principles derived therein could be extended to odd-carbon benzenoid hydrocarbons [ 31. It is the purpose of this work to now consider this class of compounds. High-spin hydrocarbons have been referred to as organic ferromagnets [4] and the existence of a class of (very) high-spin hydrocarbons has been predicted based on a number of different theories [5]. Macromolecular benzenoid hydrocarbons represent a class of compounds having a wide range of high-spin congeners. In fact, all odd-carbon benzenoid hydrocarbons are at least monoradical and many even-carbon benzenoid hydrocarbons have diradical (and higher) isomers. Also, many odd-carbon radical benzenoid hydrocarbons are isoconjugate to even-electron heterocyclic compounds and therefore function as models for these compounds [6] . For example, phenalenyl (1, Fig. 3) is isoconjugate to cyc1[3.3.3]azine (C*H,N) which has a nitrogen atom replacing the internal third degree carbon of the former, going one step further, the anion of phenalenyl (C,,H<) is isoelectronic to cyc1[3.3.3] azine. The published synthesis and properties of cyc1[3.3.3] azine show it to be moderately reactive, as one might anticipate from its electronic relationship to the phenalenyl radical [ 71. 0166-1280/86/$03.50
0 1986 Elsevier Science Publishers B.V.
10
Two new concepts for enumeration of benzenoid hydrocarbons (PAHGs) have been recently presented [ 1, 21. These concepts evolved only after the discovery of a formula periodic table for benzenoid hydrocarbons (Table 1) which organized all even-carbon PAH6s into a unified framework according to two graph theoretical invariants, d, and N,,. In determining d, and N,, one focusses on the internal edges (a-C-C bonds) and internal vertices (carbon atoms), respectively, in a carbon o-bond molecular graph in which the pn and C-H bonds and the carbon and hydrogen atoms have been suppressed; d, is the net number of disconnections or connections (zero or negative disconnections) of internal edges and N,, is the number of internal third degree carbon vertices in such a molecular graph. For example, both anthracene and phenanthrene (CL4H10) have two internal edges each between two fused rings (d, = 1) and have no internal third degree vertices (N,, = 0); whereas, both anthanthrene and benzo[ghi]perylene (C22H12) have all their internal edges connected (d, = 0) and four internal third degree vertices (N,, = 4). All benzenoid PAH6 structures having the same d, value will have formulae found in the same column of Table 1, and all PAH6 structures having the same N,, value will have formulae found in the same row of Table 1. Thus, every position on Table 1 has unique coordinates (d,, IV,,), and all PAH6 isomers of the same formula must obey the following simple equation: d, + N,, = r - 2 where r is the number of rings. The first concept was that all benzenoid PAH6 isomers of a particular formula could be recursively enumerated from benzenoid isomers having the corresponding formulae immediately above it and immediately to the left of it in Table 1. This is illustrated in Scheme I for the enumeration of
C14”10
C6H2
Cl6HIO
Scheme I: Illustration of recursive enumeration.
-
%8%2
GAO(~)
Nc/NH > 3
-
I
C&L
-
%Hm
C&,z
C,,H,dO)
‘Q-Ml)
G&,(12)
Forbidden region
‘G.,%
C&,,(2) C,,%o -
C&-I,,(O) C.&I,@) %Hm %J%z
2 < NJNH
*No. of diradical isomers are indicated by the number in parentheses.
C,oH,z -
‘G.,%(l)
Leastreactive
% reactive
C,,H,,KO C&,,(l) C,,H,,(4) C,,Hm CaHzo CJ-I,, C,o%,
Intermediate Reactive
C&-W’) GG-IM(~) C&I,, C&Z0 W-h -
Most
‘&,HJl) C,,R,J5) C.,oH,, C.w.% C&I,, C&L %,I%, -
C,,H,,(l)
C,,Hlo(0)
(PAHG) (NC = ~NH - 14 - 24ja
C&L.(l) C,,R,s C&m C&L -
Formula periodic table for benzenoid polycyclic aromatic hydrocarbons
TABLE 1
C,,%,(6) CA-I,, CJ-Im C&-L C,.S%G CC.,% %A~ -
C,,H,,(l)
‘&H,,(O)
‘&$I,, C&m C,,%z W%a %.CPH,S ‘L.%B -
C,&,,(9)
Cz$Jl)
C,$,,(63) C.&m C&-I,, C&L C&z, CwHzs -
C,,H@)
-4 -6 -6 -10 -12 -14
-2
NIc
12
anthracene/phenanthrene (C14H10) and pyrene (C16H10) from naphthalene (C&,Hs). The second concept was that strictly per&condensed PAH6 structures have formulae only along the left-hand extreme diagonal edge of Table 1 and their isomers can be determined by enumerating the number of isomers associated with their excised internal structures. Strictly pericondensed benzenoid hydrocarbons have no cata-condensed appendages and have all their internal edges mutually connected. Thus the number of nonradical isomers of CZZHIZ is two, anthanthrene and benzo[ghi] perylene, and the corresponding number of circumscribing isomers (C&H18) is two as shown in Fig. 1 (cf. Fig. 2). RESULTS AND DISCUSSION
Throughout this paper only the C-C u-bond molecular graph G will be drawn and all the pn and C-H bonds and the carbon and hydrogen atoms will be omitted. Each carbon atom vertex of this molecular graph has associated with it a 71 orbital. All the molecular graphs G of this study are alternant (bipartite) graphs such that the vertices can be partitioned into two disjoint sets with no vertex of one set joined by an edge (C-C u-bond) of G to any other vertex of the same set. These disjoint sets are termed the and the number of sites starred and unstarred, or * and 0 sets, respectively,
C22H 12, Anthanthrene
C22H12, Benzo(ghi)perylene 2-Factorable
C52Hi8
C52HI8
C73H21
2-Factorable Clrcum(30janthanthrene
Cwcum(30)benzo(ghdperylene
Fig. 1. The two nonradical isomers of C&H,,
Dwcum(36,
24)phenalenyl
radxal
and C,,H,,.
Fig. 2. Two successive excisions of internal structures from C&H,, leads to the phenalenyl radical.
13
in each is denoted by N, and No where N, > No. In the classical structure theory, bonding structures are merely drawn where the number of px-bonds along the edges of the molecular graph G is maximized. If the structure thus obtained is the superposition of a l-factor subgraph of G on the molecular graph G, then the ground-state spin is S = 0 [l].Alternatively, if some Aelectrons are not contained in such nearest-neighbor bonds and, since one end of a pn-bond must be on a starred vertex and the other end of an unstarred vertex, then the ground-state spin is S > (l/2) (N, + NO) - (No. pn-bonds) = l/2 (N* + N,) -No = l/2 (N, - No). In Hiickel MO theory, the Coulson-Rushbrooke pairing theorem gives the number of occupied nonbonding MO’s as being equal or greater than N, --No with a 2cN *-N o)fold spin degeneracy [ 5, 61. Application of Hund’s rule also leads to the prediction that S > l/2 (N, --No). The 43 polycyclic conjugated hydrocarbon isomers of anthracene/ phenanthrene or the 420 polycyclic conjugated hydrocarbon isomers of pyrene are called isovalent isomers since the number of second- and thirddegree carbon vertices in their corresponding u-bond skeletons are conserved among each isomeric set [2]. All benzenoid isomeric sets are isovalent. If one considers all the C-C u-bond graph isomers of 2,2-dimethylbutane, 2,3-dimethylbutane, 2-methylpentane, 3-methylpentane, and n-hexane only 2-methylpentane and 3-methylpentane are isovalent isomers where both have three first-degree, two second-degree, and one third-degree carbon vertices. A summary of the notation used throughout this paper is presented in an appendix at the end; also see refs. 1 and 2 for further details. Enumeration of strictly per&condensed radical benzenoid hydrocarbons Figure 3 presents the u-bond molecular graphs of all the isomers corresponding to the odd carbon PAH6s having formulae of C13H9, CIpH1l, CZ5H13, C27H13, C&HIS, &H15, (&HI,, C&HI,, and Cd5H1,. The number of isomers of the odd carbon strictly peri-condensed PAH6s are itemized by the numbers in parentheses in Table 2; the number of C53H19 isomers should be regarded as tentative. The number of isomers of the even carbon diradical PAH6s are itemized by the numbers in parentheses in Table 1, many of which were previously published [ 8, 91. Strictly per&condensed benzenoid hydrocarbons have all their internal edges mutually connected and have formulae found at the extreme left-hand diagonal boundaries of Tables 1 and 2. Consider the phenalenyl radical (1, Fig. 3) which has a formula of C13H9 in Table 2. This PAH6 structure (1) has a twelve carbon atom perimeter engulfing a single third-degree carbon vertex which corresponds to a methyl excised internal structure 1'.No other arrangement of 13 vertices into a system of hexagonal rings is possible. Circumscribing 24 carbon vertices around 1 gives a twelve ring benzenoid structure having a formula of C3,H1$ with 13 internal third-degree vertices (N,, = 13). Since no other hexagonal arrangement of these 13 internal third-
14 TABLE 2 Formula periodic table for odd-benzenoid PAH8
da -
-8 Forbidden
-7
-6
-5
4
-8
-2
-1
region
(i,%(1) C,&,,(9) 2
> 3
C,,%,(l) C,,H,,(55)
-
C&&) -
C&I,,(4) -
C,@,,(3) -
w-II,,(l) C,,H,,U5) -
‘&H,,(2) C,,H,,(34)
C,,H,,(15) -
I
aThe number in parentheses is the number of benzenoid isomers of that formula.
degree vertices is possible, the corresponding C3,H15 benzenoid structure has only one possible structure. Similarly, if this C&HI5 benzenoid structure is circumscribed by 36 more carbon vertices, the only possible polyhex arrangement of 27 rings is obtained and corresponds to a benzenoid formula to the of C,~HU. Figure 2 shows the polyhex molecular graph corresponding &HZ1 PAH6 structure. Conversely, two successive excisions of this &HZ1 PAH6 structure shown in Fig. 2 would give the phenalenyl structure 1.Thus, the series of polyhexagonal structures with corresponding formulas of C13H9, C37Hl5, C73H21, Cl21H27, - . . have only one benzenoid isomer. Structure 2 in Fig. 3 is the only possible arrangement of 5 hexagonal rings with 19 vertices. Circumscribing successively 28, 40, 52, . . . carbon vertices around structure 2 leads to the polyhexagonal molecular graphs with corresponding benzenoid formulae of Cg7H17, C87HZ3, C139H29, . . . , respectively, which represents another one-isomer series. Structure 6 in Fig. 3 is the only possible arrangement of 8 hexagonal rings with 27 vertices, and circumscribing successively 32, 44, 56, . . . carbon vertices around 6 leads to benzenoid structures with formulae of Cs9H19, C103H25, C159H31, . . . , respectively, which represents another one-isomer series. These results are summarized in Table 3. All the formulae corresponding to these three strictly pericondensed benzenoid one-isomer series are located at the extreme left-hand edge of Table 2. The PAH6 structure 2 in Fig. 3 has a sixteen carbon atom perimeter surrounding three third-degree internal carbon vertices corresponding to an ally1 radical excised internal structure 2’ in Fig. 4. Also, the PAH6 structure 6 has a twenty carbon atom perimeter circumscribing a benzyl excised internal structure 6’. All PAH6 structures having the same number of hydrogens are found in a linear diagonal array of formulae in Tables 1 and 2 and have the
15
Nrc 0
1
2
3
4
5
6
C,,H,W C1#,1(1) C,,H,3(3)
C,,%(l)
C*,H,,(6)
C&,&W
C,,H,3(4) C,,H,,(21)
c,,w25) C#,,U54)
C,,H,,(144) -
C,,H,,(106) ~,~~,~(625) -
C,,H,,(453) -
C,,H,,(1966) -
'&H&5) -
1 3 5 7 9 11 13 15 17 19 21 23 25 21 -
-
same perimeter length (4, = 2NH - 6 = constant for NH = constant). For example, the formulae C1,Hll and C 19H 11 in Table 2 both have benzenoid structures with a 16 carbon atom perimeter, and the formulae C21H13, CZ3H13,CZ5H13,and C2,H13 all correspond to benzenoid structures having a 20 carbon atom perimeter. In a system of polyhexagonal graphs, the maximum number of internal third-degree carbon vertices that can be contained in a 12 carbon atom peripheral cycle is one (l), in a 16 carbon atom peripheral cycle it is three (2), and in a 20 carbon atom peripheral it is seven (6); these corresponding structures belong to the three one-isomer series and have formulae appearing on the left-hand edge of Table 2. All the terminal formulae found on the left-hand boundary of Tables 1 and 2 correspond to PAH6 structures containing the maximum number of internal third-degree carbon vertices possible for these polyhexagonal systems. Table 3 summarizes the graphical characteristics of these three one-isomer series; note that the number of formula hydrogens in each of these series become successively incremented by six since Nn = Npe + 6 [2] . An odd PAH6 structure has a 4n pn electron circuit as an outer circumference. Other isomeric series include %Hw %Hx, C113H27, . . . where each of these formulae have 9 corresponding radical PAH6 isomers, C3gH15, C&,HZ1,C115H27,. . . having 2 radical PAH6 radical isomers, Cq3H1,, CslH23, C131H29,. . . having 15 radical PAH6 isomers, (&Hi,, Cs3H23,&3H2$,, . . . having 3 radical PAH6 iSOmerS, and CVHW, C&I257 . . . having 4 corresponding radical PAH6 isomers. Enumeration
of excised internal structures
Figure 4 presents all the corresponding (primed) excised internal structures of all the associated PAH6 isomers shown in Fig. 3. The set of excised
16
@
94
v 63
66
97 C‘EHl7
99
62
#b 63
99
Fig. 3. Strictly peri-condensed odd-carbon vertex benzenoid hydrocarbons.
internal structures corresponding to a set of PAH6 isomers will have the same number of vertices (NJ, edges (q), and rings and will have six hydrogens less (AiV, = --6) than their associated PAH6 structures that would be obtained by circumscribing carbon atoms around their perimeter; the excised internal structure 1' (CH3’) is six hydrogens less than its corresponding
17 TABLE 3 One-isomer series of strictly peri-condensed odd-carbon PAH6s Formula
.I’
2’
3’
4’
5’
6’
QP
4,
r
12 24 36 48
1 13 37 73
3 12 27 48
16 28 40 52
3 19 47 87
5 16 33 56
20 32 44 56
7 27 59 103
8 21 40 65
7’
5’
9’
IO’
II’
12’
13’
14’
IS’
16’
Fig. 4. Excised internal structures of strictly peri-condensed odd vertex benzenoid hydrocarbons.
PAH6 structure 1 (C13H9) which would be obtained from it by circumscribing twelve carbon atoms around it. The ally1 radical 1' (C,H,) is the excised internal structure of 2 (C19Hll). The various conformations of the pentadienyl radical (C&H;) having at least a two carbon atom gap within the s-cis-1,3-butadiene substructures (as in 4’) are the excised internal structures of 3 and 4. The benzyl radical (6’, C,H,) is the excised internal structure of 6 (C&H,,). The various conformations and isomers of phenylallyl (C9H9,
18
7-X’) with at least two carbon atom gaps within the s-c&1,3-butadiene substructures are the excised internal structures of PAH6 structures 7- to 15 (C3JH15); note that both 15’ and 15 are tri-radicals. Both (Y-and P-methylenylnaphthalene (CliHg, 16’ and 17’) are excised internal structures of the benzo-ovalenes 16 and 17 (&H15). The phenalenyl radical 1 (C13H9) is the only excised internal structure of a PAH6 having a formula of C3,HL5. All the above excised internal structures enumerated strictly per&condensed PAH6s found at the extreme left-hand diagonal edge of Table 2. Only some of the PAH6 structures having a formula adjacent to these boundary formulae are strictly per&condensed. Consider the PAI-IG structures having the formula C41H1,. Structures 18-50 are strictly peri-condensed PAH6s with the corresponding internal structures of 18’ to 50’. Structure 51 has a cata-condensed appendage and is the only PAH6 benzenoid of the formula C 41H 17 which is not strictly pericondensed; this structure has no excised internal structure isomeric to the others (18’ to 50’) as the internal edge associated with the cata-condensed apendage is not connected to the rest of the internal edges of structure 51. Structure 51 can only be derived by attaching a &Hz unit to one edge of the highly symmetrical and only PAH6 isomer of C37H15 (Scheme I). There are 15 excised internal structures (C15Hll, 52’-66’) corresponding to 15 strictly per&condensed PAH6s (C&HI,, 52-66) and three isomeric methylenylpyrene excised internal structures ( C1,H 11, 67’-69’) corresponding to PAH6s structures 67-69 (&HI,). Methylenylcoronene and the three isomers 3 to 5 (CZ5H13, Fig. 3) are the only excised internal structures of the strictly per&condensed PAH6s having the formula of C5,H19 (Table 2). Some of the excised internal structures corresponding to the strictly per&condensed even-carbon PAH6s in Table 1 are presented in Fig. 5. Note that within this definition of strictly peri-condensed, naphthalene and all the cata-condensed PAH6s can be regarded, in a formal sense, as strictly peri-condensed PAH6s with zero or null excised internal structures. Thus within this definition, all boundary formulae in Table 1 correspond to benzenoid structures having mutually connected excised internal structures. However, in this work the term “strictly peri-condensed PAHGs” has been restricted to those benzenoid hydrocarbon structures without catacondensed appendages and having non-zero, mutually connected excised internal structures. Thus cata-condensed PAH6s have zero excised internal PAH6s have constructures, n3 = 0, and Ni, = 0, and strictly peri-condensed nected excised internal structures, n4 = 0, and a maximum N,, for a given perimeter q, if it has a formula on the extreme left-hand diagonal edge of Tables 1 or 2. Recognition of per-i-condensed diradical benzenoids If the trimethylenemethane diradical is ultimately obtained upon successive excision and pruning of a specified even carbon peri-condensed benzenoid
19
Fig. 5. Diradical excised internal structures of strictly formulae of C,,H,,, C,,H,,, C,,H,,, C,,H,,, and G,H,,.
peri-condensed
PAH6s having
hydrocarbon having a formula below the IV,, = 2 row series in Table 1, then that original even carbon benzenoid hydrocarbon is also a diradical. Trimethylenemethane diradical is the excised internal structure of triangulene (CZ2H12) and is also obtained by pruning off the even-carbon fragments of ethene from 2-methylenepentadienyl diradical and butadiene from 1,3-benzoquinodimethane. Note that the cata-condensed PAH6s do not have any internal third-degree carbon vertices and therefore cannot have any diradical isomers. Since trimethylenemethane diradical has four carbon atoms, no PAH6 in the IV,, = 2 row series or above can have trimethylenemethane diradical as an excised internal structure. However, dibenzo[de, hi] naphthacene (CZ4H14) is a diradical and belongs to the N,, = 2 row series of
20
Table 1; this molecule has a disconnected excised internal structure composed of two fragments each made up of an odd number of carbon atoms specifically, in this case, of methyl radicals. Thus a necessary, but not sufficient, alternative requirement for a PAH6 with cata-condensed appendages to be a diradical is that its excised internal structure be disconnected and composed each of an odd number of carbon atoms; recall that non-radical perylene has a disconnected excised internal structure. In summary, a strictly per&condensed PAH6 will be a diradical if its ultimate excised internal structure is trimethylenemethane diradical. A per&condensed PAH6 with cata-condensed appendages will be a diradical if its ultimate excised internal structure is trimethylenemethane diradical (Type 1 diradical) or may be a diradical if its excised internal structure is disconnected with each fragment composed of an odd number of carbon atoms (Type 2 diradical); note that in the latter case, if the PAH6 is not a diradical it will instead be a l-factorable isomer having an exceptionally small number of Kekule structures. The probability of diradicals of this second type decreases as one moves from right to left in Table 1. Table 4 summarizes the distribution of these two diradical types in some typical even PAH6s. Figure 6 presents some typical benzenoid polyradicals of which the C&H1, and C44H22 structures are composite of Type 1 and Type 2 diradicals. The first CS3H1, PAH6 in Fig. 6 is a triradical and the second is a monoradical which is characteristic of odd carbon PAHGs; structure 15 and its excised internal structure are related triradicals. The first (&HZ2 PAH6 is a tetraradical and the second is a benzenoid is a tetraradical with an excised internal diradical. The &Hls structure which can be pruned of two 1,3-butadiene substructures leaving two trimethylenemethane diradicals. Note that formulae in the N,, = 1 row series of Table 2 can only have corresponding monoradical PAH6 structures, and and the formulas in the N,, = 3 row series can only have monoradical Type 2 triradical PAH6 structures. Only the C&HI8 structure in Fig. 6 has a connected excised internal structure and is, therefore, a strictly pericondensed benzenoid hydrocarbon. It is informative to consider the maximally starred diradical structures of triangulene (Type 1 diradical) and dibenzo[de, hilnaphthacene (Type 2 diradical) versus the nonradical structure of dibenzo[de, mn] naphthacene shown in Fig. 7. These diradical structures each possess two electrons which travel only on the starred sites and are unable to spin-pair since electronpairing can only occur between adjacent electrons, one on a starred site and the other on an unstarred one. From these examples it is evident that whenever an even carbon benzenoid PAH6 has more starred internal third-degree vertices than nonstarred ones, then that benzenoid will be a diradical or polyradical (N, -No = N,,, -iVIco). Note that the N,, = 2 row series in Table 1 has only Type 2 benzenoid diradicals. In Fig. 7 some representative polycyclic conjugated isomers of triangulene and dibenzo [de, hi] naphthacene with other ring sizes are also presented. All benzenoid hydrocarbons having formulae found in Table 1 will have only closed shell (non-radical)
21 TABLE 4 Isomer characteristics of the benxenoid PAHs Formula
No. of nonradical isomers 1 2 1 5 3 2 12 1 13 9 31 8 62 3 58 >123 1 46 288 37 333 > 446 21 >1352 10 3 1 22 7 2 1 20 12 3
No. of diradicai isomers 0 0 0 0 0
1 0 0 1 1 0 1 6 1 9 0 0 9 41 5 63 0 4 > 249 3 1 0 9 2 1 0 12 3 1
aTrimethylenemethane diradical types. diradicai types. CCata-condensed PAHG.
% Diradicai Type 1 diradicaV 0 0 0 0 0
33.3 0 0 7.7 10.0 0 11.1 8.8 25.0 13.4 0 0 16.3 12.5 11.9 16.9 0 16.0 15.5 23.1 25.0 0 29.0 22.2 33.3 0 37.5 20.0 25.0 b Disconnected
Type 2 diradicalsb
-
-
1
0 -
1 0 -
0 1 -
1 0 1 6 -
0 6 0 3 -
8 0 5 24 -
1 41 0 39 -
4 0 3 1 -
0 > 249 0 0 -
9 2 1 -
0 0 0 -
12 3 1
0 0 0
-
excised
internal
structure
polycyclic conjugated hydrocarbon isomers possessing odd ring sizes. Also, if the excised internal structure of a strictly peri-condensed PAH6 possesses an eigenvalue of zero and is not a cross conjugated species, then the strictly peri-condensed PAH6 structure also has an eigenvalue of zero.
22
ho C33H17
Ca4Hz2
Cq4Hz2
Trmdical
C33H,7
Monorad~cal
Tetroradwal
Dwadml
Fig. 6. Representative
high-spin benzenoid
polyradicals.
Periodic properties of odd vertex PAH6 All odd-carbon vertex benzenoid formulae in the first row (NIc = 1 row series) or Table 2 have corresponding polyhex structures with only one internal third-degree vertex (e.g., l), in the second row (IV,, = 3 row series) they have three internal third-degree vertices (e.g., 2), in the third row (IV,, = 5 row series) they have five third-degree vertices (e.g., 3-5), and so forth. All odd benzenoid formulae in the column headed by d,= 0 in Table 2 have corresponding polyhex structures with totally connected internal edges, in the d, = 1 column they have their internal edges with a single disconnection, and so forth. The formulae in Table 2, along a positively sloping diagonal with the same number of hydrogen atoms, have corresponding polyhex structures with the same circumference of q, = 2Nn 6 = 4 n (n = 3, 4, . . .). For example, benzanthyl radical (C1,HII) and 2 (C19Hll) both have a perimeter length of q, = 16 and the structures having the formulae of CZ1H13, CZ3H13, CZ5H13 (3-5), and C&H13 (6) have a perimeter length of qp = 20. Unlike even carbon PAH6s [ 11, odd PAH6s have no l-factor or 2-factor subgraphs or Hamiltonian circuits, and they have no total resonant sextet isomers; these items are consistent with the lower stability of odd-PAH6s compared with even ones. All odd-PAH6 structures, with only strictly peri-condensed isomers, have formulae found only on the extreme left hand boundary of Table 2.
23
*&* &o& **+ */ ts&@l *A*,*‘;* * * +;* Triongulene
*’
\’
* */
.b
Dibenzo[de, h/]naphihacene
* Dlbenzo[de, mn]naphthacene
Fig. 7. Isomers of triangulene and dibenzonaphthacene.
There are four quadrants to Table 2 which are determined by the range of values in the ratio of N,/Nn for the possible formulae of PAHs. The intersection of these quadrants occurs at the nonexistent formula of CO&IOO. The horizontal line is defined by the linear array of formulae having N,/Nn = 2, and the vertical line is defined by the linear array of formulae having NJNn = 3 (cf. formula C57H19 in Table 2). In the upper left-hand corner quadrant is the forbidden region where no reasonable formula can exist, and the boundary diagonal line passes through the nonexistent C._,H,,,, formula array which is parallel to the positively sloping diagonal linear array of formulae all having the same number of hydrogens (e.g., CxyH1,). The lower left-hand quadrant contain formulae where NJN, > 3, the lower right-hand quadrant has formulae such that 2 < NJN, < 3, and the upper right hand quadrant contain formulae where NJN, < 2. In the rows of the Formula Periodic Table for odd-benzenoid PAH6s (Table 2), the number of formula carbons (N,) increases from left to right according to the odd residue classes of congruent modulo 4. For example, in the N,, = 1 and N,, = 3 row series, the number of formula carbons follow N, = 1 (mod 4) and N, = 3 (mod 4), respectively; these two respective relationships successively repeats for every other row in Table 2. In the columns of Table 2, the number of formula carbons increases from top to bottom according to the even residue classes
24
of congruent modulo 6. In the d, = 0, d, = 1, and d, = 2 column series, the number of formula carbons are given by N, = 1 (mod 6), N, = 5 (mod 6), and N, = 3 (mod 6), respectively; this sequence of congruent modulo 6 relationships successively repeats for all the columns in Table 2. The average pn electron density of a PAH is defined by N,/a = 2N,/ (3N, - Nn ). All PAHs having formulas belonging to the d, = -7 column series (N, = 3 Nn ) have an isoelectrodensity (i.e., the same electron density) of NC/q = 0.75. Any formula column to the left of the d, = -7 column have NC/q values which approach 0.75 from smaller values, and any formula column to the right have NC/q values which approach 0.75 from larger values as the NJN, ratio approaches 3. The theoretical average pn electron density of graphite is NC/q = 0.667. On average, all PAH6 structures having formulae belonging to the left side of the d, = -7 column of Table 2 are less reactive than those structures having formulae to the right of this column. All the odd PAH6 structures in Table 2 have at least one eigenvalue of zero value (i.e., the HOMO occurs at e = 0). If NH is held constant, then a comparison of the pn energy of all PAH6 structures having the same graph theoretical circumference (c = qP = 2 NH - 6) can be made. These PAH6s lie along a formula diagonal on Tables 1 and 2 having a positive slope. Consider the C22H12, and C24H12 located in Table 1 sloping down series GsH~z, GoHu, from right to left from C24H12 to C1,Hll and the series C1,Hll and C19Hll in Table 2. For a constant graphical circumference as the number of carbon TABLE 5 Comparison of the total pn energies of benzenoid PAH6s of the same graphical circumference Formula
Even PAH6s C,s% W-b %H,, W-$,
0 2 4 6
C,,H,, G$-h, ‘U-L
0 2 8
-1
C,,H,,
10
-2
Odd PAH6s C,J-b, C,,H,, CL% W-L
range for isomer seta
En(p)
r
2
4
1
6
0
6
-1 3 2
7
24.9306-25.2745 26.220-28.3361 31.2529-31.4251 34.5718
5 6
30.5940--30.9990 33.7932-34.1644
9 10
1
1
3
1
0 2
3
1
4 5 5 6
aRef. 22. bData for the most stable isomer, Naphtho[ bcdlcoronene,
43.1197-> 46.4915
43.3000b
23.6818 26.6964 29.2465--29.3473 32.4313-32.5095
was not available.
25
atoms increases in both series, the pn energy increases by approximately 1.558 per carbon atom (AN,, in Table 5). Holding d, + Ni,, p3, and r simultaneously constant as the number of carbon atoms increases results in a pn energy increase of approximately 1.338 per carbon atom; this is illustrated by comparing CzoHlz vs. CZZH14,CZ2HIZ vs. CZ4H14,and C19Hll vs. CZ1H13 in Table 5. If N,, is held fixed as the number of rings increases, then the pn energy increases by approximately 1.440 per carbon atom. Holding the number of formula carbons constant while the graphical circumference is allowed to increase (Nn increases) results in a decrease of approximately 0.410 per four bonds of increase in circumference, and this is illustrated by comparing the E, values of C22H12 vs. CzzH14 and C24H12 vs. C24H14 in Table 5. Increasing the number of carbon atoms while holding the graphical circumference constant, therefore, leads to a more rapid increase in overall pn energy of PAH6s. For any given number of carbon atoms, the strictly peri-condensed benzenoid hydrocarbons are the most stable. The last coefficient (a,) of the characteristic polynomial has a positive sign for odd PAH6 structures having formulae in the N, = 1 (mod 4) row series and a negative sign for PAH6 structures having formulae in the N, = 3 (mod 4) row series in Table 2. Each position in Tables 1 and 2 has unique coordinates (d,, N,,). There is associated with every position in Tables 1 and 2 a set of graphical invariants or properties that remain unchanged for all isomeric polycyclic u-bond graphs corresponding to polycyclic conjugated hydrocarbon structures. The set of known graphical invariants (GI) for isomeric PAH6s is summarized by GWAW={Q,~~
+
no
+
2r6,a%,d,
+
NIc,Nc,N,,N,,
+
Npc,q,
r}
Each of these graphical invariants can be expressed as a function of the two ultimate (independent) graphical invariants of N, and q, e.g., d, + N,, = r-2=q - N, - 1. From the relationships of d, + N,, - r + 2 = 0, r > 2, and N,, > 0, one must conclude that the superposition of Tables 1 and 2 represents a truncated digitized plane having x, y, z-coordinates of (d,, Nrc, r) in 3-dimensional space; the normal to this plane has direction numbers of cos 01 = -l/43, cos p = -l/43, and cos y = l/43, and the perpendicular distance of this digitized plane from the origin is 2/d3. The periodic table for benzenoid PAH6s (Tables 1 and 2) organizes the set of benzenoid formulae and associated structures into a partial order with hierarchy relationships, i.e., Tables 1 and 2 may be regarded as a relation R on the set PAHG. The formula CloHs is a least minimal element of Table 1, and the cata-condensed and strictly per&condensed formulae at the top of all the columns in Table 1 are minimal elements. No maximal or greatest elements exist in Tables 1 or 2. Each element of Tables 1 and 2 can be represented by an ordered-pair of numbers corresponding to the formula subscripts; e.g., CloHs may be represented by (10, 8). Vector addition of any two ordered-pairs from the N,, = 0 and 2 row series gives an ordered-pair not existing in Table 1, similarly vector addition of any ordered-pair from
26
the N,, = 0 row series with any ordered-pair in the N,, = 4 row series gives an ordered-pair not existing in Table 1; all other vector additions result in ordered-pairs that belong to Table 1 and are called “allowed.” Vector addition of any two ordered-pairs from the iV,, = 1 row series in Table 2, and vector addition of any ordered-pair in the N,, = 1 row series with any ordered-pair from the N,, = 3 row series gives an ordered-pair not existing in Table 1. Note that allowed vector addition of two ordered-pairs comprised of odd numbers from Table 2 would give an ordered-pair comprised of even numbers which must belong to Table 1. Similarly, if a vector addition of an ordered-pair from Table 1 with an ordered-pair from Table 2 gives an ordered-pair belonging to Table 2, it is referred to as an allowed vector addition. In the allowed vector addition of ordered-pairs from Tables 1 and 2, the total number of vertices (N,), the number of second-degree vertices (Nn), the number of third-degree vertices Q3 = N,, + N,,), and the number of u-bond edges (9) of the associated PAH6 structures are each conserved; in this process the number of rings decreases by one and the sum of d, + N,, increases by one in the corresponding final set of structures compared to the initial ones. Thus, in the vector addition of (10, 8) + (28, 14) = (38, 22), naphthalene (a = 11, r = 2) and tribenzo[cd, ghi, lm] perylene (q = 35, r = 8) would go to a set of C&HZ2 PAH6 structures having q = 46 and r = 9, the total pn-energy (E,) change for this hypothetical condensation process ranges from AE, = 0.48p for conversion to the more stable tetrabenzo[a, c, 1, nlpentacene to AE, = -0.530 for conversion to the less stable nonaphene. In the vector addition of (16, 10) + (22, 12) = (38,22), pyrene(q = 19, r = 4) and anthanthrene/benzo[ghi]perylene (q = 27, r = 6) go to the same set of C3sHZZ PAH6 structures having q = 46 and r = 9, and in this hypothetical condensation process AE, ranges from 0.07 to -0.670 for formation of the same two respective PAH6 products. Similarly, in the vector addition of (19, 11) + (19, 11) = (38, 22), two dibenzo[def, jh]phenanthryl radicals (2 in Fig. 3, q = 23, r = 5) go to the same set of C&HZ2 PAH6 structures having q = 46 and r = 9; in this hypothetical condensation process, A E, ranges from --0.14 to + 0.430 for the same two respective PAH6 products. From these examples, it can be seen that, in the absence of steric factors, there is less thermodynamic advantage for the condensation of even-PAH6s compared to odd-PAH6s while holding N,, N,, and q fixed. Since in this vector addition process N, and q are conserved quantities, McClelland’s approximate relationship of E, < (2qN,)‘/’ is applicable. No vector addition of any two ordered-pairs from Tables 1 and 2 can give an ordered-pair corresponding to a strictly peri-condensed PAH6 structure with a formula on the extreme left-hand edge of Tables 1 and 2. The vector addition of any two ordered-pairs from the N,, = 6 row series leads to an orderedpair also belonging to the same row series in Table 1, and the vector addition of any two ordered-pairs belonging to the d, = -7 column series gives an ordered-pair belonging to the same column series in Tables 1 and 2. Pyrolytic induced condensation of benzenoid hydrocarbons having formulae found in
27
the N,, = 0, 1, and 2 row series of Tables 1 and 2 to other benzenoid hydrocarbons cannot occur without the loss of Hz, and pyrolytic formation of strictly peri-condensed benzenoid hydrocarbons via condensation of other benzenoid hydrocarbons cannot occur without co-formation of HZ. Whenever a benzenoid excised internal structure is 2-factorable, has a bay region, and/or has an eigenvalue of zero, then the corresponding strictly peri-condensed benzenoid hydrocarbon formed by circumscribing the excised internal structure with a perimeter of carbon atoms and incrementing by six hydrogens will also be 2-factorable, have a bay region, and/ or have an eigenvalue of zero. For example benzo[ghi]perylene (CZ2H12) and its larger CSzHia daughter benzenoid (Fig. 1) both have one bay region and are 2-factorable, whereas anthanthrene and its C&HI4 daughter do not. All odd-carbon, strictly per&condensed PAH6 structures have E = 0, and all even-carbon, strictly peri-condensed PAH6 structures that have diradical (or polyradical) excised internal structures and can be ultimately pruned to trimethylenemethane diradical have t = 0. He teroa tom containing analogs of odd carbon PAH6s Although, odd carbon vertex PAH6s are radical pn systems of lesser stability than the even vertex PAHGs, they undoubtedly occur as pyrolytic intermediates and have heterocyclic analogs [lo]. A major constituent of carbon black is GH-benzo[cd]pyrene-6-one which probably derives via 2 (Fig. 3) [lo]. The carbocation of 2 has been characterized [ 111. A number of oxygenated natural products related to 1 and 2 have been isolated and synthesized [12,13]. A recent study of the equilibrium acidities of phenalene, benzanthrene, and GH-benzo[cd] pyrene yielded pK values of 18.49 for the formation of the carbanion of 1, 21.43 for the benzanthryl carbanion (C1,H;l), and 19.91 for the carbanion of 2 [14] ; this observed relative order of acidity is nicely predicted by PM0 theory [ 151. The synthesis and properties of 1,3,4,6,7,9_hexaazacycl[ 3.3.31 azine has been reported, and this readily hydrolyzed compound is isoelectronic to the phenalenyl carbanion but has a nitrogen replacing a C-H at each starred position [ 161. The syntheses of 1-azaphenalene, 1-oxaphenalene and 1-thiaphenalene have been reported [ 17, 181; these heteroatom analogs of the phenalenyl carbanion are all isoelectronic to the following disulfide bridged structure.
CONCLUSION
Overlapping Tables 1 and 2 results grand unification of formula/structure
in a composite table leading to the properties of all benzenoid hydro-
28
carbons possessing both even and odd number of carbon vertices. Although the emphasis of this work has been on benzenoid hydrocarbons, it is believed that polyhexes represent a fundamental structure type of nature [ 191. Consequently, the relationships derived in this series will go beyond the realm of these compounds. The single-particle state associated with the unpaired electron in macromolecular free radicals is called a topological soliton [ 201. On each side of a soliton, there is a change of phase in the R wave-function and the amplitude vanishes at the center of the soliton domain giving what is called a phase kink. These neutral solitons are paramagnetic and play a role in conductive polymers. The results of this work should contribute toward the eventual understanding of the conductive mechanisms in graphite-like materials [ 211. APPENDIX
NC NH 4, 4%
PAH6 lP1 =p = N,
P3
I&I= 4 41 4,
r 7
net tree disconnections of internal graph edges (positive values) or connections (negative values, called negative disconnection). total number of carbon atoms in a PAH. total number of hydrogen atoms in a PAH. number of internal carbon atoms in a PAH having a degree of 3. number of peripheral carbon atoms in a PAH having a degree of 3. polycyclic aromatic hydrocarbon containing exclusively fused hexagonal rings; also referred to as benzenoid and polyhex. total number of graph points. number of graph points (vertices) having a degree of 3. number of graph edges (lines or C-C bonds). number of internal graph edges. number of peripheral graph edges, number of rings. number of rings obtained upon deletion of all internal third degree vertices from a PAH6 u-bond graph.
REFERENCES 1 J. R. Dias, Nouv. J. Chim., 9 (1986) 125. 2 J. R. Dias, J. Chem. Inf. Comput. Sci., 24 (1984) 124; Match 14 (1983) 83. 3 J. R. Dias, Can. J. Chem., 62 (1984) 2914. 4 A. Ovchinnikov, Theor. Chim. Acta (Berlin), 47 (1978) 297. 5 D. J. Klein, C. Nelin, S. Alexander and F. Matsen, J. Chem. Phys., 77 (1982) 3101. 6 M. J. S. Dewar, The MO Theory of Organic Chemistry, McGraw-Hill, NY, 1969, Chap. 9. 7 D. Farguhar, T. Gough and D. Leaver, J. Cbem. Sot., Perkin Trans. 1: (1976) 341. 8 N. Trinajstic, 2. Jericevic, J. V. Knop, W. R. Miiller and K. Szymanski, Pure Appl. Chem., 65 (1983) 379; J. V. Knop, K. Szmanski, Z. Jericevii: and N. Trinajstic, J. Comput. Chem., 4 (1983) 23, and reference therein. 9 A. T. Balaban and F. Harary, Tetrahedron, 24 (1968) 2505. 10 C, Tintel, M. van der Brugge, J. Lugtenburg, Reel. Trav. Chim. Pays-Bas, 102 (1983) 220; I. C. Lewis and C. A. Kovac, Carbon, 16 (1978) 426. 11 D. Reid and W. Bonthrone, J. Chem. Sot., (1965) 5920. 12 J. Brooks and G. Morrison, J. Chem. Sot., Perkin Trans. 1: (1972) 421. 13 J. Kingston and L. Weiler, Can. J. Chem., 55 (1977) 785.
29
14 A. Streitwieser, J. Word, F. Guibe and J. Wright, J. Org. Chem., 46 (1981) 2588. 15 M. J. S. Dewar and R. C. Dougherty, The PM0 Theory of Organic Chemistry, Plenum Press, NY, 1975. 16 R. S. Hosmane, M. Rossman and N. J. Leonard, J. Am. Chem. Sot., 104 (1982) 5497. 17 S. O’Brien and D. C. Smith, J. Chem. Sot., (1963) 2907. 18 L. Y. Chiang and J. Meinwald, J. Org. Chem., 46 (1981) 4060. 19 R. Zallen, The Physics of Amorphous Solids, Wiley, NY, 1983. 20 J. C. W. Chien, J. Polym. Sci. Lett., 19 (1981) 249. 21 N. Tyutyulkov, 0. E. Polansky, P. Schuster, S. Karabunarliev and C. Ivanov, Theor. Chim. Acta (Berlin), 67 (1985) 211. 22 R. Zahradnik and J. Pancir, HMO Energy Characteristics, Plenum Press, NY, 1970.
Journal of Molecular Structure (Theochem),
A PERIODIC TABLE FOR POLYCYCLIC AROMATIC HYDROCARBONS Part IX. Isomer Enumeration and Properties of Radical Strictly PeriCondensed Polycyclic Aromatic hydrocarbons
JERRY
RAY DIAS
Department of Chemistry, University of Missouri, Kansas City, MO 64110 (U.S.A.) (Received 6 February 1985)
ABSTRACT Isomer enumeration of strictly per&condensed benzenoid (PAHG) radicals is simplified by enumerating their excised internal structures. The number of many radical benzenoid isomers are enumerated for the first time. Structural properties of this important class of high-spin hydrocarbons has been detailed. It is shown that for a given number of formula carbon atoms, the strictly per&condensed benzenoid hydrocarbon structures are the most stable, and the possible pyrolytic formation of strictly peri-condensed benzenoid hydrocarbons via condensation of other benzenoid hydrocarbons can only occur with the evolution of hydrogen. If the excised internal structure of a strictly peri-condensed PAH6 is not a cross conjugated diradical species and it has a HMO eigenvalue of zero, then its corresponding strictly per&condensed PAH6 structure will also have an eigenvalue of zero. INTRODUCTION
Although, previous research emphasized the more stable, even-carbon benzenoid hydrocarbons [ 1, 21, it was pointed out that all the principles derived therein could be extended to odd-carbon benzenoid hydrocarbons [ 31. It is the purpose of this work to now consider this class of compounds. High-spin hydrocarbons have been referred to as organic ferromagnets [4] and the existence of a class of (very) high-spin hydrocarbons has been predicted based on a number of different theories [5]. Macromolecular benzenoid hydrocarbons represent a class of compounds having a wide range of high-spin congeners. In fact, all odd-carbon benzenoid hydrocarbons are at least monoradical and many even-carbon benzenoid hydrocarbons have diradical (and higher) isomers. Also, many odd-carbon radical benzenoid hydrocarbons are isoconjugate to even-electron heterocyclic compounds and therefore function as models for these compounds [6] . For example, phenalenyl (1, Fig. 3) is isoconjugate to cyc1[3.3.3]azine (C*H,N) which has a nitrogen atom replacing the internal third degree carbon of the former, going one step further, the anion of phenalenyl (C,,H<) is isoelectronic to cyc1[3.3.3] azine. The published synthesis and properties of cyc1[3.3.3] azine show it to be moderately reactive, as one might anticipate from its electronic relationship to the phenalenyl radical [ 71. 0166-1280/86/$03.50
0 1986 Elsevier Science Publishers B.V.
10
Two new concepts for enumeration of benzenoid hydrocarbons (PAHGs) have been recently presented [ 1, 21. These concepts evolved only after the discovery of a formula periodic table for benzenoid hydrocarbons (Table 1) which organized all even-carbon PAH6s into a unified framework according to two graph theoretical invariants, d, and N,,. In determining d, and N,, one focusses on the internal edges (a-C-C bonds) and internal vertices (carbon atoms), respectively, in a carbon o-bond molecular graph in which the pn and C-H bonds and the carbon and hydrogen atoms have been suppressed; d, is the net number of disconnections or connections (zero or negative disconnections) of internal edges and N,, is the number of internal third degree carbon vertices in such a molecular graph. For example, both anthracene and phenanthrene (CL4H10) have two internal edges each between two fused rings (d, = 1) and have no internal third degree vertices (N,, = 0); whereas, both anthanthrene and benzo[ghi]perylene (C22H12) have all their internal edges connected (d, = 0) and four internal third degree vertices (N,, = 4). All benzenoid PAH6 structures having the same d, value will have formulae found in the same column of Table 1, and all PAH6 structures having the same N,, value will have formulae found in the same row of Table 1. Thus, every position on Table 1 has unique coordinates (d,, IV,,), and all PAH6 isomers of the same formula must obey the following simple equation: d, + N,, = r - 2 where r is the number of rings. The first concept was that all benzenoid PAH6 isomers of a particular formula could be recursively enumerated from benzenoid isomers having the corresponding formulae immediately above it and immediately to the left of it in Table 1. This is illustrated in Scheme I for the enumeration of
C14”10
C6H2
Cl6HIO
Scheme I: Illustration of recursive enumeration.
-
%8%2
GAO(~)
Nc/NH > 3
-
I
C&L
-
%Hm
C&,z
C,,H,dO)
‘Q-Ml)
G&,(12)
Forbidden region
‘G.,%
C&,,(2) C,,%o -
C&-I,,(O) C.&I,@) %Hm %J%z
2 < NJNH
*No. of diradical isomers are indicated by the number in parentheses.
C,oH,z -
‘G.,%(l)
Leastreactive
% reactive
C,,H,,KO C&,,(l) C,,H,,(4) C,,Hm CaHzo CJ-I,, C,o%,
Intermediate Reactive
C&-W’) GG-IM(~) C&I,, C&Z0 W-h -
Most
‘&,HJl) C,,R,J5) C.,oH,, C.w.% C&I,, C&L %,I%, -
C,,H,,(l)
C,,Hlo(0)
(PAHG) (NC = ~NH - 14 - 24ja
C&L.(l) C,,R,s C&m C&L -
Formula periodic table for benzenoid polycyclic aromatic hydrocarbons
TABLE 1
C,,%,(6) CA-I,, CJ-Im C&-L C,.S%G CC.,% %A~ -
C,,H,,(l)
‘&H,,(O)
‘&$I,, C&m C,,%z W%a %.CPH,S ‘L.%B -
C,&,,(9)
Cz$Jl)
C,$,,(63) C.&m C&-I,, C&L C&z, CwHzs -
C,,H@)
-4 -6 -6 -10 -12 -14
-2
NIc
12
anthracene/phenanthrene (C14H10) and pyrene (C16H10) from naphthalene (C&,Hs). The second concept was that strictly per&condensed PAH6 structures have formulae only along the left-hand extreme diagonal edge of Table 1 and their isomers can be determined by enumerating the number of isomers associated with their excised internal structures. Strictly pericondensed benzenoid hydrocarbons have no cata-condensed appendages and have all their internal edges mutually connected. Thus the number of nonradical isomers of CZZHIZ is two, anthanthrene and benzo[ghi] perylene, and the corresponding number of circumscribing isomers (C&H18) is two as shown in Fig. 1 (cf. Fig. 2). RESULTS AND DISCUSSION
Throughout this paper only the C-C u-bond molecular graph G will be drawn and all the pn and C-H bonds and the carbon and hydrogen atoms will be omitted. Each carbon atom vertex of this molecular graph has associated with it a 71 orbital. All the molecular graphs G of this study are alternant (bipartite) graphs such that the vertices can be partitioned into two disjoint sets with no vertex of one set joined by an edge (C-C u-bond) of G to any other vertex of the same set. These disjoint sets are termed the and the number of sites starred and unstarred, or * and 0 sets, respectively,
C22H 12, Anthanthrene
C22H12, Benzo(ghi)perylene 2-Factorable
C52Hi8
C52HI8
C73H21
2-Factorable Clrcum(30janthanthrene
Cwcum(30)benzo(ghdperylene
Fig. 1. The two nonradical isomers of C&H,,
Dwcum(36,
24)phenalenyl
radxal
and C,,H,,.
Fig. 2. Two successive excisions of internal structures from C&H,, leads to the phenalenyl radical.
13
in each is denoted by N, and No where N, > No. In the classical structure theory, bonding structures are merely drawn where the number of px-bonds along the edges of the molecular graph G is maximized. If the structure thus obtained is the superposition of a l-factor subgraph of G on the molecular graph G, then the ground-state spin is S = 0 [l].Alternatively, if some Aelectrons are not contained in such nearest-neighbor bonds and, since one end of a pn-bond must be on a starred vertex and the other end of an unstarred vertex, then the ground-state spin is S > (l/2) (N, + NO) - (No. pn-bonds) = l/2 (N* + N,) -No = l/2 (N, - No). In Hiickel MO theory, the Coulson-Rushbrooke pairing theorem gives the number of occupied nonbonding MO’s as being equal or greater than N, --No with a 2cN *-N o)fold spin degeneracy [ 5, 61. Application of Hund’s rule also leads to the prediction that S > l/2 (N, --No). The 43 polycyclic conjugated hydrocarbon isomers of anthracene/ phenanthrene or the 420 polycyclic conjugated hydrocarbon isomers of pyrene are called isovalent isomers since the number of second- and thirddegree carbon vertices in their corresponding u-bond skeletons are conserved among each isomeric set [2]. All benzenoid isomeric sets are isovalent. If one considers all the C-C u-bond graph isomers of 2,2-dimethylbutane, 2,3-dimethylbutane, 2-methylpentane, 3-methylpentane, and n-hexane only 2-methylpentane and 3-methylpentane are isovalent isomers where both have three first-degree, two second-degree, and one third-degree carbon vertices. A summary of the notation used throughout this paper is presented in an appendix at the end; also see refs. 1 and 2 for further details. Enumeration of strictly per&condensed radical benzenoid hydrocarbons Figure 3 presents the u-bond molecular graphs of all the isomers corresponding to the odd carbon PAH6s having formulae of C13H9, CIpH1l, CZ5H13, C27H13, C&HIS, &H15, (&HI,, C&HI,, and Cd5H1,. The number of isomers of the odd carbon strictly peri-condensed PAH6s are itemized by the numbers in parentheses in Table 2; the number of C53H19 isomers should be regarded as tentative. The number of isomers of the even carbon diradical PAH6s are itemized by the numbers in parentheses in Table 1, many of which were previously published [ 8, 91. Strictly per&condensed benzenoid hydrocarbons have all their internal edges mutually connected and have formulae found at the extreme left-hand diagonal boundaries of Tables 1 and 2. Consider the phenalenyl radical (1, Fig. 3) which has a formula of C13H9 in Table 2. This PAH6 structure (1) has a twelve carbon atom perimeter engulfing a single third-degree carbon vertex which corresponds to a methyl excised internal structure 1'.No other arrangement of 13 vertices into a system of hexagonal rings is possible. Circumscribing 24 carbon vertices around 1 gives a twelve ring benzenoid structure having a formula of C3,H1$ with 13 internal third-degree vertices (N,, = 13). Since no other hexagonal arrangement of these 13 internal third-
14 TABLE 2 Formula periodic table for odd-benzenoid PAH8
da -
-8 Forbidden
-7
-6
-5
4
-8
-2
-1
region
(i,%(1) C,&,,(9) 2
> 3
C,,%,(l) C,,H,,(55)
-
C&&) -
C&I,,(4) -
C,@,,(3) -
w-II,,(l) C,,H,,U5) -
‘&H,,(2) C,,H,,(34)
C,,H,,(15) -
I
aThe number in parentheses is the number of benzenoid isomers of that formula.
degree vertices is possible, the corresponding C3,H15 benzenoid structure has only one possible structure. Similarly, if this C&HI5 benzenoid structure is circumscribed by 36 more carbon vertices, the only possible polyhex arrangement of 27 rings is obtained and corresponds to a benzenoid formula to the of C,~HU. Figure 2 shows the polyhex molecular graph corresponding &HZ1 PAH6 structure. Conversely, two successive excisions of this &HZ1 PAH6 structure shown in Fig. 2 would give the phenalenyl structure 1.Thus, the series of polyhexagonal structures with corresponding formulas of C13H9, C37Hl5, C73H21, Cl21H27, - . . have only one benzenoid isomer. Structure 2 in Fig. 3 is the only possible arrangement of 5 hexagonal rings with 19 vertices. Circumscribing successively 28, 40, 52, . . . carbon vertices around structure 2 leads to the polyhexagonal molecular graphs with corresponding benzenoid formulae of Cg7H17, C87HZ3, C139H29, . . . , respectively, which represents another one-isomer series. Structure 6 in Fig. 3 is the only possible arrangement of 8 hexagonal rings with 27 vertices, and circumscribing successively 32, 44, 56, . . . carbon vertices around 6 leads to benzenoid structures with formulae of Cs9H19, C103H25, C159H31, . . . , respectively, which represents another one-isomer series. These results are summarized in Table 3. All the formulae corresponding to these three strictly pericondensed benzenoid one-isomer series are located at the extreme left-hand edge of Table 2. The PAH6 structure 2 in Fig. 3 has a sixteen carbon atom perimeter surrounding three third-degree internal carbon vertices corresponding to an ally1 radical excised internal structure 2’ in Fig. 4. Also, the PAH6 structure 6 has a twenty carbon atom perimeter circumscribing a benzyl excised internal structure 6’. All PAH6 structures having the same number of hydrogens are found in a linear diagonal array of formulae in Tables 1 and 2 and have the
15
Nrc 0
1
2
3
4
5
6
C,,H,W C1#,1(1) C,,H,3(3)
C,,%(l)
C*,H,,(6)
C&,&W
C,,H,3(4) C,,H,,(21)
c,,w25) C#,,U54)
C,,H,,(144) -
C,,H,,(106) ~,~~,~(625) -
C,,H,,(453) -
C,,H,,(1966) -
'&H&5) -
1 3 5 7 9 11 13 15 17 19 21 23 25 21 -
-
same perimeter length (4, = 2NH - 6 = constant for NH = constant). For example, the formulae C1,Hll and C 19H 11 in Table 2 both have benzenoid structures with a 16 carbon atom perimeter, and the formulae C21H13, CZ3H13,CZ5H13,and C2,H13 all correspond to benzenoid structures having a 20 carbon atom perimeter. In a system of polyhexagonal graphs, the maximum number of internal third-degree carbon vertices that can be contained in a 12 carbon atom peripheral cycle is one (l), in a 16 carbon atom peripheral cycle it is three (2), and in a 20 carbon atom peripheral it is seven (6); these corresponding structures belong to the three one-isomer series and have formulae appearing on the left-hand edge of Table 2. All the terminal formulae found on the left-hand boundary of Tables 1 and 2 correspond to PAH6 structures containing the maximum number of internal third-degree carbon vertices possible for these polyhexagonal systems. Table 3 summarizes the graphical characteristics of these three one-isomer series; note that the number of formula hydrogens in each of these series become successively incremented by six since Nn = Npe + 6 [2] . An odd PAH6 structure has a 4n pn electron circuit as an outer circumference. Other isomeric series include %Hw %Hx, C113H27, . . . where each of these formulae have 9 corresponding radical PAH6 isomers, C3gH15, C&,HZ1,C115H27,. . . having 2 radical PAH6 radical isomers, Cq3H1,, CslH23, C131H29,. . . having 15 radical PAH6 isomers, (&Hi,, Cs3H23,&3H2$,, . . . having 3 radical PAH6 iSOmerS, and CVHW, C&I257 . . . having 4 corresponding radical PAH6 isomers. Enumeration
of excised internal structures
Figure 4 presents all the corresponding (primed) excised internal structures of all the associated PAH6 isomers shown in Fig. 3. The set of excised
16
@
94
v 63
66
97 C‘EHl7
99
62
#b 63
99
Fig. 3. Strictly peri-condensed odd-carbon vertex benzenoid hydrocarbons.
internal structures corresponding to a set of PAH6 isomers will have the same number of vertices (NJ, edges (q), and rings and will have six hydrogens less (AiV, = --6) than their associated PAH6 structures that would be obtained by circumscribing carbon atoms around their perimeter; the excised internal structure 1' (CH3’) is six hydrogens less than its corresponding
17 TABLE 3 One-isomer series of strictly peri-condensed odd-carbon PAH6s Formula
.I’
2’
3’
4’
5’
6’
QP
4,
r
12 24 36 48
1 13 37 73
3 12 27 48
16 28 40 52
3 19 47 87
5 16 33 56
20 32 44 56
7 27 59 103
8 21 40 65
7’
5’
9’
IO’
II’
12’
13’
14’
IS’
16’
Fig. 4. Excised internal structures of strictly peri-condensed odd vertex benzenoid hydrocarbons.
PAH6 structure 1 (C13H9) which would be obtained from it by circumscribing twelve carbon atoms around it. The ally1 radical 1' (C,H,) is the excised internal structure of 2 (C19Hll). The various conformations of the pentadienyl radical (C&H;) having at least a two carbon atom gap within the s-cis-1,3-butadiene substructures (as in 4’) are the excised internal structures of 3 and 4. The benzyl radical (6’, C,H,) is the excised internal structure of 6 (C&H,,). The various conformations and isomers of phenylallyl (C9H9,
18
7-X’) with at least two carbon atom gaps within the s-c&1,3-butadiene substructures are the excised internal structures of PAH6 structures 7- to 15 (C3JH15); note that both 15’ and 15 are tri-radicals. Both (Y-and P-methylenylnaphthalene (CliHg, 16’ and 17’) are excised internal structures of the benzo-ovalenes 16 and 17 (&H15). The phenalenyl radical 1 (C13H9) is the only excised internal structure of a PAH6 having a formula of C3,HL5. All the above excised internal structures enumerated strictly per&condensed PAH6s found at the extreme left-hand diagonal edge of Table 2. Only some of the PAH6 structures having a formula adjacent to these boundary formulae are strictly per&condensed. Consider the PAI-IG structures having the formula C41H1,. Structures 18-50 are strictly peri-condensed PAH6s with the corresponding internal structures of 18’ to 50’. Structure 51 has a cata-condensed appendage and is the only PAH6 benzenoid of the formula C 41H 17 which is not strictly pericondensed; this structure has no excised internal structure isomeric to the others (18’ to 50’) as the internal edge associated with the cata-condensed apendage is not connected to the rest of the internal edges of structure 51. Structure 51 can only be derived by attaching a &Hz unit to one edge of the highly symmetrical and only PAH6 isomer of C37H15 (Scheme I). There are 15 excised internal structures (C15Hll, 52’-66’) corresponding to 15 strictly per&condensed PAH6s (C&HI,, 52-66) and three isomeric methylenylpyrene excised internal structures ( C1,H 11, 67’-69’) corresponding to PAH6s structures 67-69 (&HI,). Methylenylcoronene and the three isomers 3 to 5 (CZ5H13, Fig. 3) are the only excised internal structures of the strictly per&condensed PAH6s having the formula of C5,H19 (Table 2). Some of the excised internal structures corresponding to the strictly per&condensed even-carbon PAH6s in Table 1 are presented in Fig. 5. Note that within this definition of strictly peri-condensed, naphthalene and all the cata-condensed PAH6s can be regarded, in a formal sense, as strictly peri-condensed PAH6s with zero or null excised internal structures. Thus within this definition, all boundary formulae in Table 1 correspond to benzenoid structures having mutually connected excised internal structures. However, in this work the term “strictly peri-condensed PAHGs” has been restricted to those benzenoid hydrocarbon structures without catacondensed appendages and having non-zero, mutually connected excised internal structures. Thus cata-condensed PAH6s have zero excised internal PAH6s have constructures, n3 = 0, and Ni, = 0, and strictly peri-condensed nected excised internal structures, n4 = 0, and a maximum N,, for a given perimeter q, if it has a formula on the extreme left-hand diagonal edge of Tables 1 or 2. Recognition of per-i-condensed diradical benzenoids If the trimethylenemethane diradical is ultimately obtained upon successive excision and pruning of a specified even carbon peri-condensed benzenoid
19
Fig. 5. Diradical excised internal structures of strictly formulae of C,,H,,, C,,H,,, C,,H,,, C,,H,,, and G,H,,.
peri-condensed
PAH6s having
hydrocarbon having a formula below the IV,, = 2 row series in Table 1, then that original even carbon benzenoid hydrocarbon is also a diradical. Trimethylenemethane diradical is the excised internal structure of triangulene (CZ2H12) and is also obtained by pruning off the even-carbon fragments of ethene from 2-methylenepentadienyl diradical and butadiene from 1,3-benzoquinodimethane. Note that the cata-condensed PAH6s do not have any internal third-degree carbon vertices and therefore cannot have any diradical isomers. Since trimethylenemethane diradical has four carbon atoms, no PAH6 in the IV,, = 2 row series or above can have trimethylenemethane diradical as an excised internal structure. However, dibenzo[de, hi] naphthacene (CZ4H14) is a diradical and belongs to the N,, = 2 row series of
20
Table 1; this molecule has a disconnected excised internal structure composed of two fragments each made up of an odd number of carbon atoms specifically, in this case, of methyl radicals. Thus a necessary, but not sufficient, alternative requirement for a PAH6 with cata-condensed appendages to be a diradical is that its excised internal structure be disconnected and composed each of an odd number of carbon atoms; recall that non-radical perylene has a disconnected excised internal structure. In summary, a strictly per&condensed PAH6 will be a diradical if its ultimate excised internal structure is trimethylenemethane diradical. A per&condensed PAH6 with cata-condensed appendages will be a diradical if its ultimate excised internal structure is trimethylenemethane diradical (Type 1 diradical) or may be a diradical if its excised internal structure is disconnected with each fragment composed of an odd number of carbon atoms (Type 2 diradical); note that in the latter case, if the PAH6 is not a diradical it will instead be a l-factorable isomer having an exceptionally small number of Kekule structures. The probability of diradicals of this second type decreases as one moves from right to left in Table 1. Table 4 summarizes the distribution of these two diradical types in some typical even PAH6s. Figure 6 presents some typical benzenoid polyradicals of which the C&H1, and C44H22 structures are composite of Type 1 and Type 2 diradicals. The first CS3H1, PAH6 in Fig. 6 is a triradical and the second is a monoradical which is characteristic of odd carbon PAHGs; structure 15 and its excised internal structure are related triradicals. The first (&HZ2 PAH6 is a tetraradical and the second is a benzenoid is a tetraradical with an excised internal diradical. The &Hls structure which can be pruned of two 1,3-butadiene substructures leaving two trimethylenemethane diradicals. Note that formulae in the N,, = 1 row series of Table 2 can only have corresponding monoradical PAH6 structures, and and the formulas in the N,, = 3 row series can only have monoradical Type 2 triradical PAH6 structures. Only the C&HI8 structure in Fig. 6 has a connected excised internal structure and is, therefore, a strictly pericondensed benzenoid hydrocarbon. It is informative to consider the maximally starred diradical structures of triangulene (Type 1 diradical) and dibenzo[de, hilnaphthacene (Type 2 diradical) versus the nonradical structure of dibenzo[de, mn] naphthacene shown in Fig. 7. These diradical structures each possess two electrons which travel only on the starred sites and are unable to spin-pair since electronpairing can only occur between adjacent electrons, one on a starred site and the other on an unstarred one. From these examples it is evident that whenever an even carbon benzenoid PAH6 has more starred internal third-degree vertices than nonstarred ones, then that benzenoid will be a diradical or polyradical (N, -No = N,,, -iVIco). Note that the N,, = 2 row series in Table 1 has only Type 2 benzenoid diradicals. In Fig. 7 some representative polycyclic conjugated isomers of triangulene and dibenzo [de, hi] naphthacene with other ring sizes are also presented. All benzenoid hydrocarbons having formulae found in Table 1 will have only closed shell (non-radical)
21 TABLE 4 Isomer characteristics of the benxenoid PAHs Formula
No. of nonradical isomers 1 2 1 5 3 2 12 1 13 9 31 8 62 3 58 >123 1 46 288 37 333 > 446 21 >1352 10 3 1 22 7 2 1 20 12 3
No. of diradicai isomers 0 0 0 0 0
1 0 0 1 1 0 1 6 1 9 0 0 9 41 5 63 0 4 > 249 3 1 0 9 2 1 0 12 3 1
aTrimethylenemethane diradical types. diradicai types. CCata-condensed PAHG.
% Diradicai Type 1 diradicaV 0 0 0 0 0
33.3 0 0 7.7 10.0 0 11.1 8.8 25.0 13.4 0 0 16.3 12.5 11.9 16.9 0 16.0 15.5 23.1 25.0 0 29.0 22.2 33.3 0 37.5 20.0 25.0 b Disconnected
Type 2 diradicalsb
-
-
1
0 -
1 0 -
0 1 -
1 0 1 6 -
0 6 0 3 -
8 0 5 24 -
1 41 0 39 -
4 0 3 1 -
0 > 249 0 0 -
9 2 1 -
0 0 0 -
12 3 1
0 0 0
-
excised
internal
structure
polycyclic conjugated hydrocarbon isomers possessing odd ring sizes. Also, if the excised internal structure of a strictly peri-condensed PAH6 possesses an eigenvalue of zero and is not a cross conjugated species, then the strictly peri-condensed PAH6 structure also has an eigenvalue of zero.
22
ho C33H17
Ca4Hz2
Cq4Hz2
Trmdical
C33H,7
Monorad~cal
Tetroradwal
Dwadml
Fig. 6. Representative
high-spin benzenoid
polyradicals.
Periodic properties of odd vertex PAH6 All odd-carbon vertex benzenoid formulae in the first row (NIc = 1 row series) or Table 2 have corresponding polyhex structures with only one internal third-degree vertex (e.g., l), in the second row (IV,, = 3 row series) they have three internal third-degree vertices (e.g., 2), in the third row (IV,, = 5 row series) they have five third-degree vertices (e.g., 3-5), and so forth. All odd benzenoid formulae in the column headed by d,= 0 in Table 2 have corresponding polyhex structures with totally connected internal edges, in the d, = 1 column they have their internal edges with a single disconnection, and so forth. The formulae in Table 2, along a positively sloping diagonal with the same number of hydrogen atoms, have corresponding polyhex structures with the same circumference of q, = 2Nn 6 = 4 n (n = 3, 4, . . .). For example, benzanthyl radical (C1,HII) and 2 (C19Hll) both have a perimeter length of q, = 16 and the structures having the formulae of CZ1H13, CZ3H13, CZ5H13 (3-5), and C&H13 (6) have a perimeter length of qp = 20. Unlike even carbon PAH6s [ 11, odd PAH6s have no l-factor or 2-factor subgraphs or Hamiltonian circuits, and they have no total resonant sextet isomers; these items are consistent with the lower stability of odd-PAH6s compared with even ones. All odd-PAH6 structures, with only strictly peri-condensed isomers, have formulae found only on the extreme left hand boundary of Table 2.
23
*&* &o& **+ */ ts&@l *A*,*‘;* * * +;* Triongulene
*’
\’
* */
.b
Dibenzo[de, h/]naphihacene
* Dlbenzo[de, mn]naphthacene
Fig. 7. Isomers of triangulene and dibenzonaphthacene.
There are four quadrants to Table 2 which are determined by the range of values in the ratio of N,/Nn for the possible formulae of PAHs. The intersection of these quadrants occurs at the nonexistent formula of CO&IOO. The horizontal line is defined by the linear array of formulae having N,/Nn = 2, and the vertical line is defined by the linear array of formulae having NJNn = 3 (cf. formula C57H19 in Table 2). In the upper left-hand corner quadrant is the forbidden region where no reasonable formula can exist, and the boundary diagonal line passes through the nonexistent C._,H,,,, formula array which is parallel to the positively sloping diagonal linear array of formulae all having the same number of hydrogens (e.g., CxyH1,). The lower left-hand quadrant contain formulae where NJN, > 3, the lower right-hand quadrant has formulae such that 2 < NJN, < 3, and the upper right hand quadrant contain formulae where NJN, < 2. In the rows of the Formula Periodic Table for odd-benzenoid PAH6s (Table 2), the number of formula carbons (N,) increases from left to right according to the odd residue classes of congruent modulo 4. For example, in the N,, = 1 and N,, = 3 row series, the number of formula carbons follow N, = 1 (mod 4) and N, = 3 (mod 4), respectively; these two respective relationships successively repeats for every other row in Table 2. In the columns of Table 2, the number of formula carbons increases from top to bottom according to the even residue classes
24
of congruent modulo 6. In the d, = 0, d, = 1, and d, = 2 column series, the number of formula carbons are given by N, = 1 (mod 6), N, = 5 (mod 6), and N, = 3 (mod 6), respectively; this sequence of congruent modulo 6 relationships successively repeats for all the columns in Table 2. The average pn electron density of a PAH is defined by N,/a = 2N,/ (3N, - Nn ). All PAHs having formulas belonging to the d, = -7 column series (N, = 3 Nn ) have an isoelectrodensity (i.e., the same electron density) of NC/q = 0.75. Any formula column to the left of the d, = -7 column have NC/q values which approach 0.75 from smaller values, and any formula column to the right have NC/q values which approach 0.75 from larger values as the NJN, ratio approaches 3. The theoretical average pn electron density of graphite is NC/q = 0.667. On average, all PAH6 structures having formulae belonging to the left side of the d, = -7 column of Table 2 are less reactive than those structures having formulae to the right of this column. All the odd PAH6 structures in Table 2 have at least one eigenvalue of zero value (i.e., the HOMO occurs at e = 0). If NH is held constant, then a comparison of the pn energy of all PAH6 structures having the same graph theoretical circumference (c = qP = 2 NH - 6) can be made. These PAH6s lie along a formula diagonal on Tables 1 and 2 having a positive slope. Consider the C22H12, and C24H12 located in Table 1 sloping down series GsH~z, GoHu, from right to left from C24H12 to C1,Hll and the series C1,Hll and C19Hll in Table 2. For a constant graphical circumference as the number of carbon TABLE 5 Comparison of the total pn energies of benzenoid PAH6s of the same graphical circumference Formula
Even PAH6s C,s% W-b %H,, W-$,
0 2 4 6
C,,H,, G$-h, ‘U-L
0 2 8
-1
C,,H,,
10
-2
Odd PAH6s C,J-b, C,,H,, CL% W-L
range for isomer seta
En(p)
r
2
4
1
6
0
6
-1 3 2
7
24.9306-25.2745 26.220-28.3361 31.2529-31.4251 34.5718
5 6
30.5940--30.9990 33.7932-34.1644
9 10
1
1
3
1
0 2
3
1
4 5 5 6
aRef. 22. bData for the most stable isomer, Naphtho[ bcdlcoronene,
43.1197-> 46.4915
43.3000b
23.6818 26.6964 29.2465--29.3473 32.4313-32.5095
was not available.
25
atoms increases in both series, the pn energy increases by approximately 1.558 per carbon atom (AN,, in Table 5). Holding d, + Ni,, p3, and r simultaneously constant as the number of carbon atoms increases results in a pn energy increase of approximately 1.338 per carbon atom; this is illustrated by comparing CzoHlz vs. CZZH14,CZ2HIZ vs. CZ4H14,and C19Hll vs. CZ1H13 in Table 5. If N,, is held fixed as the number of rings increases, then the pn energy increases by approximately 1.440 per carbon atom. Holding the number of formula carbons constant while the graphical circumference is allowed to increase (Nn increases) results in a decrease of approximately 0.410 per four bonds of increase in circumference, and this is illustrated by comparing the E, values of C22H12 vs. CzzH14 and C24H12 vs. C24H14 in Table 5. Increasing the number of carbon atoms while holding the graphical circumference constant, therefore, leads to a more rapid increase in overall pn energy of PAH6s. For any given number of carbon atoms, the strictly peri-condensed benzenoid hydrocarbons are the most stable. The last coefficient (a,) of the characteristic polynomial has a positive sign for odd PAH6 structures having formulae in the N, = 1 (mod 4) row series and a negative sign for PAH6 structures having formulae in the N, = 3 (mod 4) row series in Table 2. Each position in Tables 1 and 2 has unique coordinates (d,, N,,). There is associated with every position in Tables 1 and 2 a set of graphical invariants or properties that remain unchanged for all isomeric polycyclic u-bond graphs corresponding to polycyclic conjugated hydrocarbon structures. The set of known graphical invariants (GI) for isomeric PAH6s is summarized by GWAW={Q,~~
+
no
+
2r6,a%,d,
+
NIc,Nc,N,,N,,
+
Npc,q,
r}
Each of these graphical invariants can be expressed as a function of the two ultimate (independent) graphical invariants of N, and q, e.g., d, + N,, = r-2=q - N, - 1. From the relationships of d, + N,, - r + 2 = 0, r > 2, and N,, > 0, one must conclude that the superposition of Tables 1 and 2 represents a truncated digitized plane having x, y, z-coordinates of (d,, Nrc, r) in 3-dimensional space; the normal to this plane has direction numbers of cos 01 = -l/43, cos p = -l/43, and cos y = l/43, and the perpendicular distance of this digitized plane from the origin is 2/d3. The periodic table for benzenoid PAH6s (Tables 1 and 2) organizes the set of benzenoid formulae and associated structures into a partial order with hierarchy relationships, i.e., Tables 1 and 2 may be regarded as a relation R on the set PAHG. The formula CloHs is a least minimal element of Table 1, and the cata-condensed and strictly per&condensed formulae at the top of all the columns in Table 1 are minimal elements. No maximal or greatest elements exist in Tables 1 or 2. Each element of Tables 1 and 2 can be represented by an ordered-pair of numbers corresponding to the formula subscripts; e.g., CloHs may be represented by (10, 8). Vector addition of any two ordered-pairs from the N,, = 0 and 2 row series gives an ordered-pair not existing in Table 1, similarly vector addition of any ordered-pair from
26
the N,, = 0 row series with any ordered-pair in the N,, = 4 row series gives an ordered-pair not existing in Table 1; all other vector additions result in ordered-pairs that belong to Table 1 and are called “allowed.” Vector addition of any two ordered-pairs from the iV,, = 1 row series in Table 2, and vector addition of any ordered-pair in the N,, = 1 row series with any ordered-pair from the N,, = 3 row series gives an ordered-pair not existing in Table 1. Note that allowed vector addition of two ordered-pairs comprised of odd numbers from Table 2 would give an ordered-pair comprised of even numbers which must belong to Table 1. Similarly, if a vector addition of an ordered-pair from Table 1 with an ordered-pair from Table 2 gives an ordered-pair belonging to Table 2, it is referred to as an allowed vector addition. In the allowed vector addition of ordered-pairs from Tables 1 and 2, the total number of vertices (N,), the number of second-degree vertices (Nn), the number of third-degree vertices Q3 = N,, + N,,), and the number of u-bond edges (9) of the associated PAH6 structures are each conserved; in this process the number of rings decreases by one and the sum of d, + N,, increases by one in the corresponding final set of structures compared to the initial ones. Thus, in the vector addition of (10, 8) + (28, 14) = (38, 22), naphthalene (a = 11, r = 2) and tribenzo[cd, ghi, lm] perylene (q = 35, r = 8) would go to a set of C&HZ2 PAH6 structures having q = 46 and r = 9, the total pn-energy (E,) change for this hypothetical condensation process ranges from AE, = 0.48p for conversion to the more stable tetrabenzo[a, c, 1, nlpentacene to AE, = -0.530 for conversion to the less stable nonaphene. In the vector addition of (16, 10) + (22, 12) = (38,22), pyrene(q = 19, r = 4) and anthanthrene/benzo[ghi]perylene (q = 27, r = 6) go to the same set of C3sHZZ PAH6 structures having q = 46 and r = 9, and in this hypothetical condensation process AE, ranges from 0.07 to -0.670 for formation of the same two respective PAH6 products. Similarly, in the vector addition of (19, 11) + (19, 11) = (38, 22), two dibenzo[def, jh]phenanthryl radicals (2 in Fig. 3, q = 23, r = 5) go to the same set of C&HZ2 PAH6 structures having q = 46 and r = 9; in this hypothetical condensation process, A E, ranges from --0.14 to + 0.430 for the same two respective PAH6 products. From these examples, it can be seen that, in the absence of steric factors, there is less thermodynamic advantage for the condensation of even-PAH6s compared to odd-PAH6s while holding N,, N,, and q fixed. Since in this vector addition process N, and q are conserved quantities, McClelland’s approximate relationship of E, < (2qN,)‘/’ is applicable. No vector addition of any two ordered-pairs from Tables 1 and 2 can give an ordered-pair corresponding to a strictly peri-condensed PAH6 structure with a formula on the extreme left-hand edge of Tables 1 and 2. The vector addition of any two ordered-pairs from the N,, = 6 row series leads to an orderedpair also belonging to the same row series in Table 1, and the vector addition of any two ordered-pairs belonging to the d, = -7 column series gives an ordered-pair belonging to the same column series in Tables 1 and 2. Pyrolytic induced condensation of benzenoid hydrocarbons having formulae found in
27
the N,, = 0, 1, and 2 row series of Tables 1 and 2 to other benzenoid hydrocarbons cannot occur without the loss of Hz, and pyrolytic formation of strictly peri-condensed benzenoid hydrocarbons via condensation of other benzenoid hydrocarbons cannot occur without co-formation of HZ. Whenever a benzenoid excised internal structure is 2-factorable, has a bay region, and/or has an eigenvalue of zero, then the corresponding strictly peri-condensed benzenoid hydrocarbon formed by circumscribing the excised internal structure with a perimeter of carbon atoms and incrementing by six hydrogens will also be 2-factorable, have a bay region, and/ or have an eigenvalue of zero. For example benzo[ghi]perylene (CZ2H12) and its larger CSzHia daughter benzenoid (Fig. 1) both have one bay region and are 2-factorable, whereas anthanthrene and its C&HI4 daughter do not. All odd-carbon, strictly per&condensed PAH6 structures have E = 0, and all even-carbon, strictly peri-condensed PAH6 structures that have diradical (or polyradical) excised internal structures and can be ultimately pruned to trimethylenemethane diradical have t = 0. He teroa tom containing analogs of odd carbon PAH6s Although, odd carbon vertex PAH6s are radical pn systems of lesser stability than the even vertex PAHGs, they undoubtedly occur as pyrolytic intermediates and have heterocyclic analogs [lo]. A major constituent of carbon black is GH-benzo[cd]pyrene-6-one which probably derives via 2 (Fig. 3) [lo]. The carbocation of 2 has been characterized [ 111. A number of oxygenated natural products related to 1 and 2 have been isolated and synthesized [12,13]. A recent study of the equilibrium acidities of phenalene, benzanthrene, and GH-benzo[cd] pyrene yielded pK values of 18.49 for the formation of the carbanion of 1, 21.43 for the benzanthryl carbanion (C1,H;l), and 19.91 for the carbanion of 2 [14] ; this observed relative order of acidity is nicely predicted by PM0 theory [ 151. The synthesis and properties of 1,3,4,6,7,9_hexaazacycl[ 3.3.31 azine has been reported, and this readily hydrolyzed compound is isoelectronic to the phenalenyl carbanion but has a nitrogen replacing a C-H at each starred position [ 161. The syntheses of 1-azaphenalene, 1-oxaphenalene and 1-thiaphenalene have been reported [ 17, 181; these heteroatom analogs of the phenalenyl carbanion are all isoelectronic to the following disulfide bridged structure.
CONCLUSION
Overlapping Tables 1 and 2 results grand unification of formula/structure
in a composite table leading to the properties of all benzenoid hydro-
28
carbons possessing both even and odd number of carbon vertices. Although the emphasis of this work has been on benzenoid hydrocarbons, it is believed that polyhexes represent a fundamental structure type of nature [ 191. Consequently, the relationships derived in this series will go beyond the realm of these compounds. The single-particle state associated with the unpaired electron in macromolecular free radicals is called a topological soliton [ 201. On each side of a soliton, there is a change of phase in the R wave-function and the amplitude vanishes at the center of the soliton domain giving what is called a phase kink. These neutral solitons are paramagnetic and play a role in conductive polymers. The results of this work should contribute toward the eventual understanding of the conductive mechanisms in graphite-like materials [ 211. APPENDIX
NC NH 4, 4%
PAH6 lP1 =p = N,
P3
I&I= 4 41 4,
r 7
net tree disconnections of internal graph edges (positive values) or connections (negative values, called negative disconnection). total number of carbon atoms in a PAH. total number of hydrogen atoms in a PAH. number of internal carbon atoms in a PAH having a degree of 3. number of peripheral carbon atoms in a PAH having a degree of 3. polycyclic aromatic hydrocarbon containing exclusively fused hexagonal rings; also referred to as benzenoid and polyhex. total number of graph points. number of graph points (vertices) having a degree of 3. number of graph edges (lines or C-C bonds). number of internal graph edges. number of peripheral graph edges, number of rings. number of rings obtained upon deletion of all internal third degree vertices from a PAH6 u-bond graph.
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