Molecular vibrations of heptacirculene and molecular topology of monoheptapolyhexes

Journal of

MOLECULAR STRUCTURE

ELSEVIER

Journal of Molecular Structure 327 (1994) 121-130

Molecular vibrations of heptacirculene and molecular topology of monoheptapolyhexes* B.N. Cyvin, E. Brendsdal, J. Brunvoll, S.J. Cyvin* Department of Physical Chemistry, The University of Trondheim, N-7034 Trondheim-NTH,

Norway

Received 10 December 1993

Abstract

A normal coordinate analysis is reported for the molecular vibrations of CZ8Hi4heptacirculene. The corresponding molecular graph is a heptagon surrounded by seven hexagons and belongs to the polygonal system called monoheptapolyhexes. Topological properties of this system are studied with emphasis on the extremal monoheptapolyhexes (to which heptacirculene belongs).

1. Introduction

Completely condensed polycyclic conjugated hydrocarbons are of great interest in organic chemistry [l]. As chemical graphs [2] they are represented by polygonal systems, i.e. connected geometrical constructions of polygons, where any two polygons either share exactly one edge or are disjointed. Benzenoid hydrocarbons [ 1, 31 possess exclusively six-membered rings and are represented by benzenoid systems [3] consisting of hexagons. A polygonal system which consists of exactly one q-gon in addition to hexagons (if any), is referred to as a mono-q-polyhex [4]. These systems have many important chemical counterparts: biphenylenoid hydrocarbons [.5] correspond to monotetrapolyhexes (q = 4), fluorenoids/fluoranthenoids [6,7] to *Part 28 of the series Condensed Aromatics. Part 27 is: E. Brendsdal, J. Brunvoh, B.N. Cyvin and S.J. Cyvin, Spectrosc. Lett., 25 (1992) 911; Ref. 8 is Part 25. Dedicated to Professor C.N.R. Rao. * Corresponding author. 0022-2860/94/$07.00

0

monopentapolyhexes (q = 5), and benzenoids are simply polyhexes (q = 6). A subclass of the mono-q-polyhexes is formed by [qlcirculenes [S], systems with one q-gon circumscribed by hexagons. A [qlcirculene has the chemical formula C+HZq. Some of the corresponding molecules are known. Pentacirculene (C2aHi0) was first synthesized by Barth and Lawton [9] and given the name corannulene. Recently this bowl-shaped molecule has attracted considerable renewed interest; two new syntheses have been reported [lo, 111, and the bowl-to-bowl inversion has been studied [11,12]. Hexacirculene (CZ4Ht2) is the well known coronene [ 1,131. Heptacirculene (C2sHt4) has been synthesized by Yamamoto et al. [14]. In the first part of the present work, a normal coordinate analysis of the molecular vibrations of heptacirculene is reported, in continuation of the corresponding work on corannulene [S].

1994 Elsevier Science B.V. All rights reserved

SSDI 0022-2860(94)08152-8

122

B.N. Cyvin et al./Journal of Molecular Structure 327

X

E;’ _

-

2.562

z-

E;’

-

-

2.047

,, E,

-

-

1.601

E;’

-

-

A;’

l-

I.211

-

A,

1.000

-

E;’ I

121-130

Finally, a synthesis of octacirculene (CszHib) has been attempted [ 1.51. In the second part of this work, some topological properties of the class of monoheptapolyhexes are studied. Heptacirculene belongs to this class. Emphasis is laid on the subclass of extremal monoheptapolyhexes (see below for a definition). Heptacirculene also belongs to this subclass. Very little theoretical work has been done before on monoheptapolyhexes; only a few general formulations for mono-q-polyhexes [5] are available, and still less work of this kind is available for monoheptapolyhexes [ 161.

I.449

I.

(1994)

2. Heptacirculene

-

0.607 0.603

.

El -

2.1. Htickel molecular orbital analysis

The Htickel molecular orbitals of heptacirculene, when related to the symmetry group &,, exhibit the following symmetric structure

O-

rn=A;+3A;+4E1”+4E2/‘+4Ej

E;’ -HE;’

-+c

-+c

-0.460

-+I-

-0.831

*+I-

-1.147

(1)

A simple Hi.ickel molecular orbital (HMO) analysis was performed and resulted in the HMO energy levels as shown in Fig. 1.

-l-

E;’ -t+ 0;’

t-c t-c

E;’

t+-

-+c-

-1.821

E;’

--+

,-+

-2.304

E;'

.+-

-1.377 -1.639

-2-

0;’

-2.676

--I+

-3Fig. 1. Energy heptacirculene.

levels

from

the

simple

HMO

analysis

of

Fig. 2. Notation heptacirculene.

for the four types of C-C

bond

distance

in

B.N. Cyvin et al./Journal of Molecular Structure 327 (1994) 121-130

123

Table 1 Bond orders (P), C-C bond lengths (R in pm) and C-C force constants ( f in Nm-‘) for heptacirculene C-C bond

K :

P

0.505 0.555 0.531 0.751

R

f

Calc.

Exp. [14]

142.6 141.7 142.1 138.3

144.7-146.6 143.3-143.5 140.7-142.3 132.7-134.4

431.5 447.7 439.8 516.5

The coefficients of molecular orbitals were used to calculate the Coulson bond orders (P) for the four types of carbon-carbon bonds (see Fig. 2). The P values yielded calculated values of carbon-carbon bond distances (R) when following the same approach as in the analysis of corannulene [S]. Table 1 shows the values of P and R, together with experimental bond distances from an X-ray structural investigation [14]. The agreement is not very good, but satisfactory for our purposes. 2.2. Normal coordinate analysis The five-parameter approximation [ 13,17,18] for the force field of polycyclic aromatic hydrocarbons was applied to heptacirculene. Refined force constants for the carbon-carbon stretchings (counted as one parameter) were computed from the bond orders (P) according to an adaptation of Badger’s rule [19]. The numerical values are included in Table 1. The normal modes of vibration are distributed into the symmetry species according to rvib = 6A; + 5A; + 1lE; + 12E; + 12E; + 2AI’ + 3A; + 5E,” + 6E; + 6Ej

(2)

when referred to a hypothetical planar structure of DTh symmetry. For the more realistic saddleshaped structure (Fig. 3) of C, symmetry the distribution into symmetry species is 61A’ + 59A”. Complete sets of vibrational frequencies were computed for both of the models DTh and C,, using X-ray data from Yamamoto et al. [14] (see also Table 1). However, these data were not sufficient for a complete structure determination.

Fig. 3. The saddle-shaped molecular model of C, symmetry for heptacirculene.

Here it was assumed that all the 14 tertiary carbon atoms were situated on the saddle-shaped area x = (z2 - y*)/lO. The other atoms were placed in planes defined by the secondary carbon atoms and the midpoints of the carbon-carbon bonds of the seven-membered ring. The carbon-hydrogen distances were estimated to be 108 pm. For the C, model, all the 120 vibrational frequencies are theoretically active in both infrared and Raman. For the sake of brevity we do not give all these frequencies here. We have concentrated upon the frequencies which are expected to show up in the infrared with appreciable intensity, namely those which would be active according to the D,,, model (11 Ei + 3Ai). The results, which are shown in Table 2, include the correlations &(E/) + C,(A’ + A”) and &,(A;) + C,(A’). In conclusion, the predicted infrared frequencies are those in the second column of Table 2, but they are entered with more than physically significant decimals, in order to make the frequency shifts from the DTh to the C, symmetries visible throughout. These shifts are seen to be negligible (< 1 cm-‘) for the highest frequencies. So far there are no experimental frequencies available for comparison.

3. Monoheptapolyhexes

3.1. Invariants and extremal systems The number of hexagons (in addition to one heptagon) in a monoheptapolyhex, M7, shall

124 Table 2 Calculated infrared-active model of heptacirculene, for the C, model

B.N. Cyvin et al./Journal of Molecular Structure 327 (1994) 121-130

frequencies (in cm-‘) for the &, correlated with frequencies (in cm-‘)

(PZ;S) = (4h - n; + 7; 2h - ni + 7)

an@‘)

C&4’ + A”)

3037.43

1 3037.01 3037.02

3032.16

{ 3032.25 3032.28

1594.14

1 1594.53 1594.83

1493.47

1 1492.29 1493.33

1454.94

( 1453.05 1452.80

1251.63

1 1251.06 1249.08

1074.43

1 1071.18 1068.30

1046.78

( 1048.06 1042.26

826.81

1 854.06 854.47

667.48

1

702.02 693.16

286.32

1

313.13 323.28

&tSA;) 856.79 543.50 109.00

(ni)max = 2h + 3 - [( 12h + 9)1’2]

(5)

where [a] is used to denote the smallest integer not smaller than a. It should be emphasized that extremal M7 systems are not produced in general by the spiral walk, as is the case for the mono-qpolyhexes with q = 6 [20, 211, q = 5 [7] and q = 4 [5]. In fact, Cyvin’s conjecture [4] about mono-qpolyhexes, which is based on the spiral walk, is valid for qd 6, but is found to be in error for q > 6 [22]. 3.2. Chemical formulae

CAA’) 858.47 577.76 104.91

ni=n-2s+7

(4)

As extremal mono-q-polyhex (M7) is defined as having the maximum number of internal vertices for a given number of hexagons: nj = (ni)max(h). This definition [5] is compatible with the special cases of extremal benzenoids [20,21], fluor(anth)enoids [7] and biphenylenoids [5]. The function (ni),,,(h) for extremal M7 systems is given by [16]

presently be designated h. Its chemical formula, say C,H,, may alternatively be written (n;s). Another important invariant (in addition to h, n and s), is the number of internal vertices, denoted by ni. An internal vertex is, by definition, shared by three polygons. Thus, for instance, heptacirculene has ni = 7. It is assumed throughout that M7 is simply connected and therefore has no holes. For M7(n;s), i.e. a M7 system with the formula C,H,, you have: h = 1 (n - s)

Conversely, the formula of M7 with h hexagons and ni internal vertices is given by

(3)

From Eqs. (4) and (5) all the possible C,H, formulae for M7 systems can be constructed by taking h = 0, 1, 2,. . . , and ni = 0, 1, 2,. . . , (ni)max for every h. Here h = 0 corresponds to the heptagon alone (C7H7). These formulae are listed in Table 3, which extends infinitely downward while each row becomes longer and longer. For a given n, which C,H, formulae are possible for the M7 systems? This question has been answered previously [l l] with the result: 2[$(n-

1)+i(6n-6)“21

+2
-n (6)

where the possible values of n are the integers n = 7, 11, 14, 15, and n 2 17. For a given n (in the allowed domain), all the s values between the upper and lower bounds of (6) are realized, provided that the parities of n and s are equal: either both n and s are even, or both of them are odd. Now to ask the related question: for a given S, which C,H,

125

B.N. Cyvin et a/./Journal of Molecular Structure 327 (1994) 121-130

Table 3 Formulae for monoheptapolyhexes h

n, 0

0

1 2 3

1

2

3

4

5

6

I

8

9

10

C34H16

C33H15

C7H7 G1H9

%HII

C14H10

C19H13

ClSH12

C17H11

C23Hl5

C22H14

C21H13

C20H12

4 5 6 7 8

C27H17

C26Hl6

C25HI5

C24H14

C23Hl3

C31H19

C30H18

C29Hl7

C28H16

C27H15

C26Hl4

C25H13

C35H21

C34H20

C33H19

C32Hl8

C31H17

C30H16

C29H15

C2SH14

C39H23

C38H22

C37H21

C36H20

C35H19

Cdl8

C33H17

C32H16

C31H15

9

C43H25

C42H24

C4lH23

Cd22

C39H21

C38H20

C37H19

C36Hl8

C35H17

formulae are possible for the M7 systems? We have found s-6+2~s/2J
(7)

when ]aJ is used to denote the largest integer not larger than a. The possible values of s are the integers s = 7 and s 2 9. Here again, n and s should have the same parity. The formulae for extremal M7 systems are found at the extreme right end of each row in Table 3. These formulae have been designated (n”; sa) and are given by [l l] (n”; sa) = (2h + 4 + [( 12h + 9)q; 4 + [( 12h + 9)“*])

(8)

3.3. Generation of extremal systems An extremal monoheptapolyhex M7 emerges when a hexagon at the perimeter of an extremal benzenoid is expanded into a heptagon. Sometimes all the C,H, isomers of M7 systems may be obtained in that way. This is the case for hG6: the isomers of C7H7, CllH9, Ci4H1a, Cr~Htt, C2sHr2, C2sHi3 and C25,H13 are obtained from benzene, naphthalene, phenalene, pyrene, naphthanthrene, C2*Hi2 and coronene, respectively. In the case of C22H12 + C2sHt3 (h = 5), there are three well known isomers of extremal benzenoids (anthanthrene, benzo[ghi]perylene, triangulene) to

be taken into account. The expansion of pyrene, Cr6Hi0 + C1,Hll (h = 3), is illustrated in Fig. 4. In Fig. 5 the generation of extremal M7 systems is continued from h = 7 through h = 15. Again, the method of hexagon-to-heptagon expansion gives all the isomers in many cases, but not always. It may also happen that an extremal M7 contains heptacirculene as a subgraph. For C2sHi4 (h = 7), for instance, there are in the first place the four systems generated from benzo[bc]coronene. In addition comes heptacirculene itself. 3.4. Properties of extremal systems Any extremal monoheptapolyhex (M7) can be circumscribed arbitrarily many times, a property shared with the extremal benzenoids [21]. It is also true that any circumscribed extremal benzenoid is itself a (larger) extremal benzenoid [21]. However, the corresponding rule is not valid for extremal M7 systems in general.

Fig. 4. The two C17Hll monoheptapolyhex isomers, obtained by expansions of pyrene. The hexagons marked by asterisks are expanded, one at a time, to heptagons (indicated by inscribed numeral).

B.N. Cyvin et aLlJournal of Molecular Structure 327 (1994) 121-130

126

h= 14

‘413~18

h=

15

‘4gH18

06

Fig. 5. Generation of the extremal the numbers of isomers.

monoheptapolyhex

isomers with 7
Specific cases where circum-M7 also is extremal are found in Fig. 5: circum-C7H-I = C2sHi4, circum-CitHs = Cs6Hi6, circum-Ct4H10 = C4iH1,, circum-CiTH,i = (&His. But many counterexamples can be found: circum-Cz3Hi3 = Cs6H2s, circum-C3iHtS = &Hz2 and circum-(&HI6 = for example, are non-extremal. It is C75H23~

15 (see also the legend of Fig. 4). Encircled

numerals

indicate

interesting to notice that heptacirculene (C2sHL4) and circumheptacirculene (C&Hz,) are both extremal, but dicircumheptacirculene (C, i2H2s) is non-extremal, as in fact are all k-fold circumscribed heptacirculenes for k32. In other words, the k-fold circumscribed heptagon (C7H,) is extremal for k = 1, 2, but non-extremal for k > 2.

127

B.N. Cyvin et al./Journal of Molecular Structure 327 (1994) 121-130

Fig. 6. The smallest circular benzenoids with non-benzenoid successive circumscribings of these six characteristic forms.

No other extremal M7 can be found which remains extremal on double circumscribing. These seemingly confusing features are fully explained by theorems, which are treated in the following. But first a few mathematical expressions are needed. Assume that k-circum-M7(n0;s,,) = (k-circumM7) (Q;Q). Then the formula of the k-fold circumscribed M7 is given by (nk; s/J = (no + 2k.Q + 7k2; So + 7k)

(9)

A condition for M7 to be extremal is given in Eq. (5). We wish to make it more chemically oriented. On combining (5) with (3) you have ](6n - 6s + 9)“2] = s - 4

(10)

C7H7 included. All the larger circular benzenoids are obtained on

Theorem I.

Circumscribed 0’, circum-0’, is an extremal monoheptapolyhex. For the sake of convenience, we reproduce here the forms of the circular benzenoids [23, 251; see Fig. 6. All the above examples of circum-M7 which themselves are extremal monoheptapolyhexes, are seen to be consistent with Theorem 1. In order to catch up also C3,Hi5 (see Fig. 5), we should include the degenerate system (nonbenzenoid) among the circular benzenoids, which has the formula C7H7 (see Fig. 6). Proof of Theorem 1.

The formulae of circular benzenoids (including the non-benzenoid C7H7), O(N; S), are given by ~24, 251

which holds if and only if M7 (n;s) is an extremal monoheptapolyhex. The relation (10) makes it feasible to check this property directly from the chemical formula of M7. If by O(N; 5’) a circular benzenoid [23-251 with the formula CNHs is denoted, and it is assumed that one of its hexagons at the perimeter is expanded to a heptagon, the resulting monoheptapolyhex is O’(n’; s’), where

If it is assumed that 0’ is circumscribed, and Eq. (9) is applied with k = 1 to formula (13), the result for (circum-0’) (n; S) is

n’=N+l

(n;s) = (3S+ 10+2~~(S2-6S)];s+8)

s’=S+l

(11)

(N;S) = (S+2[@2-6S)J;S)

(12)

for S 2 6. When modified to O’(n’; s’) as described above, you obtain in accord with Eq. (11): (n’;s’) = (S+ 1 +2#S2

-6S)j;S+

1)

(13)

(14)

128

B.N. Cyvin et al./Journal of Molecular Structure 327 (1994) 121-130

Now we shall prove that n and s from (14) fit into Eq. (lo), i.e. [[12S+21+

12]+(S* -6S)]]“*]

=S+4

(15)

On the one hand, if an integer A satisfies (a) A x, then A = [xl. On the other hand, x - 1 < 1x1
in B, then H < Ho by virtue of the fundamental property of circular benzenoids (Ho = H,,, [23-2516. Since N = 2H + S - 2, you also have N < N , and N is at least two units less than No. Hence N=S+2&(S*-6S)J

-26

(23)

where S> 1 (actually 6 = 2,4,6, . . .). The following formula is arrived at for (circum-B’) (n;s) in the same way as in the proof of Theorem 1.

yields after some elementary manipulations (n;s)=(3S+10+2[&(S2-6S)]-26; [h(S” - 6S)j > &(S’ - 6s) - 1

(17)

which is always satisfied. Condition (b), i.e. S+4>[12S+21

+ 12]&(S*-6S)J]i’*

(18)

yields (19)

and consequently: ]A (S* - 6S)J < & (S* - 6s) + ff (2s - 5)

(20)

This relation is valid when 2S-520

The same approach Theorem 1 leads to

as under condition

(21)

2. Circumscribed B’, circum-B’, is a non-extremal monoheptapolyhex. Theorem

of theorem 2. A circular benzenoid with the same S as in B has

Proof

(25)

Since S> 1, 1x1 > x, a contradiction. Hence the coefficients (n and s) from Eq. (24) do not fit into Eq. (lo), and Theorem 2 is proved. It is unnecessary to inspect condition (b). 3. In the set of k-fold circumscribed 0’ (k-circum0’) for k > 1, two-fold circumscribed C7H7 is the only extremal system. Proof

of theorem

3.

The formula of O’(n’; $) is given in (13). For (k-circum-O’)(n; s) (n; s) = (7k2 + (2k + l)(S + 1) +2#S*-6S)j;S+7k+l)

(26)

as a generalization of (14), which is the special case of k = 1. On inserting from (26) into (10) the condition for k-circum-0’ to be extremal is obtained as: [[12kS + 3(14k* - 10k + 3)

lh(S*-6S)J+l

No = S+2[&(S2

(a) of

Theorem

i.e. when S >, 3, and therefore always in the range of interest for S, i.e. S 2 6. This completes the proof of Theorem 1. Denoted by B(N; S) is a benzenoid which is not circular, but can be circumscribed. B is modified to B’ by expanding one of its hexagons at the perimeter to a heptagon.

Ho=

(24)

[& (S* - 6S)J > & (S* - 6s) + S - 1

$(S*-6S)J&S*-4S-5

S+8)

-6S)J

(22)

indicating its Ho number of hexagons and No number of vertices. If H is the number of hexagons

+ lZ&(S*

- 6S)]]“*]

= S + 7k + 3

(27)

This is a generalization of Eq. (15). Now proceeding in the same way as under Theorem 1,

B.N. Cyvin et al.lJournal of Molecular Structure 327 (1994) 121-130

condition (a) yields: & (S2 - 6S)J > & (S2 - 6s) - 1 + & (k - 1)(2S + 7k - 19)

(28)

A necessary (but not sufficient) condition for (28) to be valid, is: &(k-

1)(2S+7k-

19) < 1

(29)

The case of k = 1 pertains to Theorem 1. Now the case of k # 1 is of interest, and in that case: S<6(k-l)-‘+1(19-7k)

129

with all circumscribed extremal benzenoids which first have been modified by expanding one of the hexagons at the perimeter to a heptagon. However, this procedure does not in general lead to extremal mono-Q-polyhexes for Q > 7. It is found that only octacirculene is extremal, while all the other extremal mono-Q-polyhexes for Q > 7 have the Q-gon at the perimeter. Second, it is demonstrated above that circumheptacirculene is an extremal system. No other circum[Q]circulene for Q > 7 has this property. Even circumoctacirculene is non-extremal.

(30)

A simple analysis shows that k > 2 is impossible, and fork = 2, S < 8.5. That leaves the possibilities S = 6, 7, 8. An inspection of Eq. (27) reveals that only S = 6 fulfils the requirement. This completes the proof of Theorem 3. It is unnecessary to consider condition (b); it must certainly be satisfied for k = 2, S = 6 as a consequence of our direct inspection of Eq. (27). Theorem 4.

Any k-fold circumscribed B’ is a non-extremal monoheptapolyhex. This is a generalization of Theorem 2. It can be proved along the same lines as above. We omit the details.

4. Conclusion In the first part of this work, the simple Hiickel molecular orbital analysis and a normal coordinate analysis of molecular vibrations of C2sHi4 heptacirculene are reported. From a topological viewpoint, the heptacirculene chemical graph, i.e. one heptagon surrounded by seven hexagons, is a unique system in some ways. First, heptacirculene is an extremal system, but not of the usual kind for mono-Q-polyhexes (Q > 6) with the Q-gon at its perimeter; the heptagon is circumscribed by hexagons. Heptacirculene shares this property with infinitely many extremal monoheptapolyhexes, actually

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130 [19] J.C.

B.N. Cyvin et al./Journal of Molecular Structure 327 (1994) 121-130

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[22] E.K. Lloyd, J. Math. Chem., 15 (1994) 73. [23] S.J. Cyvin, Theor. Chim. Acta, 81 (1992) 269. [24] J. Brunvoll, B.N. Cyvin and S.J. Cyvin, Top. Curr. Chem., 162 (1992) 181. [25] S.J. Cyvin, B.N. Cyvin and J. Brunvoll, Top. Curr. Chem., 166 (1993) 65.