Valence-Bond Determination of Bond Lengths of Polycyclic Aromatic

ARTICLE pubs.acs.org/JPCA

Valence-Bond Determination of Bond Lengths of Polycyclic Aromatic Hydrocarbons: Comparisons with Recent Experimental and Ab Initio Results Jerry Ray Dias* Department of Chemistry, University of Missouri, Kansas City, Missouri 64110-2499, United States

bS Supporting Information ABSTRACT: Pauling’s valence-bond (VB) method for determining bond lengths is compared to ten recent literature experimental and theoretical results and is shown to give comparable results. His method only requires computation of the number of Kekul
e (K) and Dewar structures (DS) of conjugated hydrocarbons. Both K and DS are obtained from the last two coefficients of the matching polynomial which is also used to obtain topological resonance energy (TRE). A molecular fragmentation method is given for determining DS of essentially disconnected polycyclic aromatic hydrocarbons (PAHs). Both Kekul
ean alternant and nonalternant PAHs, including essentially disconnected and non-Kekul
ean systems, have bond lengths that are easily determined by this method.

’ INTRODUCTION While Pauling’s 1980 paper1 has been cited over 30 times in the literature, in no case did any of these citations refer to its major theme involving the use of first-excited valence-bond (VB) resonance structures as well as unexcited VB resonance structures. While unexcited resonance structures (Kekul
e structures) are adequate for bond length evaluation of many nonradical conjugated molecules,2 those PAHs with essentially single (double) bonds require the inclusion of first-excited resonance structures (Dewar resonance structures). With the inclusion of the first-excited (Dewar) valence-bond resonance structures, the essentially single (double) bonds in the unexcited (Kekul
e) structures of conjugated hydrocarbons acquire some double (single) bond character. We will examine Pauling’s treatment for determining bond lengths of polycyclic aromatic hydrocarbons (PAHs). We will treat both alternant and nonalternant polycyclic aromatic systems and study how first-excited VB (Dewar) resonance structures affect the bond orders of essentially single bonds in PAHs. Basically, Pauling likened first-excited resonance structures as having a long bond which means that the electrons associated with this long bond are spin-paired. In benzene, besides the two unexcited resonance structures called Kekul
e structures (K = 2), there are three first-excited resonance structures with long bonds called Dewar structures (DS = 3). Disjoint diradicals do not have Kekul
e structures and require the use of Dewar resonance structures for determination of their bond lengths. Because bond lengths are important structural parameters, we will compare Pauling’s VB method for determining bond lengths in PAHs against recent experimental and theoretical results. For additional applications of valence-bond theory, the reader is referred to the recent overview paper by Shaik and Hiberty.3 r 2011 American Chemical Society

’ RESULTS AND DISCUSSION pπ-Bond Orders. Pauling defines bond number as being equal to 1 + pπ-bond order.1 Bond lengths in polycyclic aromatic hydrocarbons (PAHs) are intricately related to their pπ-bond orders. Ultimately both are measures of bond energies, the sum of which gives the energy of stabilization for the corresponding molecule. There are two important definitions of pπ-bond order between atoms r
s (prs). Coulson’s H€uckel MO definition4 over j occupied levels is

prs c ¼

∑ njcjr cjs

ð1Þ

where nj represents the number of electrons in level j and the coefficients of the bonded atoms r and s are cjr and cjs. Pauling’s original valence-bond (VB) definition is prs p ¼ K½G
ðers Þ=K½G

ð2Þ

where K[G] is the number of Kekul
e structures (K) of the PAH molecular graph (G) and K[G
(ers)] is the number Kekul
e structures of the subgraph generated from the molecular graph by deleting its r
s bond (edge) with its atoms (vertices) r and s;5 later on we will amend eq 2 by replacing K by SC (structure-count) for monoradicals. Here it is important to note that Pauling refers to Kekul
e structures as being unexcited VB resonance structures and Dewar structures as first-excited VB resonance structures. Also, Kekul
e structures correspond to one class of canonical structures.6 Pauling’s bond orders can be derived exactly from the eigenvalues and eigenvectors obtained from HMO calculations. If one weighs the terms in the Coulson’s bond order eq 1 by the inverse of the Received: August 5, 2011 Revised: October 10, 2011 Published: October 11, 2011 13619

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corresponding eigenvalues (ej), then prs p ¼

∑ njcjr cjs =ej

ð3Þ

is obtained.4 Both HMO and VB measure pπ-electronic energy and are devoid of steric information of the conjugated molecular systems. Essentially Disconnected Polycyclic Aromatic Hydrocarbons. Brunvoll, Cyvin, and Cyvin7 define essentially disconnected benzenoids as having some set of fixed bonds (single or double) separating benzenoid fragments in all their Kekul
e structures. In 1980, Pauling pointed out that in his valence-bond treatment that for benzenoid hydrocarbons with essentially single bonds (bonds that are single in all its Kekul
e Structures) like the two bonds in perylene, which basically separate two naphthalene fragments, one needed to include first-excited valence-bond structures (Dewar structures) as well as unexcited structures (Kekul
e structures) to reproduce the experimental bond lengths of perylene.1 With the inclusion of the first-excited valence-bond structures, the essentially single bonds in the unexcited structures of perylene-related benzenoid hydrocarbons acquire some double bond character. Recently, we employed8 Pauling’s VB strategy,1 which included first-excited VB resonance structures in addition to his unexcited VB structures in the solution to the C28H14 fluoranthenoid hydrocarbon example by Gutman and co-workers.9 The essential double bonds in their example require the evaluation of first-excited structures as well as the unexcited valence bond structures (Kekul
e structures) considered by them. Like the perylene example, with the inclusion of the firstexcited valence bond structures, the essential double bonds in the pentagonal ring of the unexcited structures (Kekul
e structures) of dibenzo[cd,mn]indeno[1,2,3-gf]pyrene acquires some single bond character.
Resonance Structures. The determination of the Kekule number of Kekul
e resonance structures for any benzenoid can be rapidly done using the John-Sachs theorem as described by Gutman and Cyvin.10 The John-Sachs theorem gives the Kekul
e structure counts of a benzenoid in terms of an npnp determinant, where np is the number of peaks, equal to the number of valleys; for example, pyrene can be oriented with one peak and one valley or alternatively with two peaks and two valleys. Gutman and Cyvin’s method10 only require simple matrix algebra and knowledge that the number of Kekul
e resonance structures for acenes is given by K = r + 1 (r = the number of hexagonal rings in a linear acene) and other catacondensed benzenoids by the Fibonacci-like numbers, as described by Gordon and Davidson.11 The Lecture Notes by Cyvin and Gutman for determining the number of Kekul
e resonance structures is also readily available.12 Randi
c has also streamlined Gutman and Cyvin’s John-Sachs method and used it to solve K = 540000 for superphenalene (C90H30) and K = 66998000 for supertriphenylene (C132H42).13 Dewar (First-Excited) Valence-Bond Resonance Structures. The extensive chemometric analysis of PAHs by Kiralj and Ferrerira14 used nearest and next nearest neighbor parameters as well as Pauling’s bond orders based on unexcited (Kekul
e) resonance structures but did not include first-excited (Dewar) resonance structures for benzenoids having formally (essentially) single bonds that have no Kekul
e structure with a double bond. One can obtain the number of Dewar resonance structures (DS) for PAHs from the last two coefficients (aacN
2 and aacN; note that these coefficients have opposite signs) of the matching

Figure 1. Recent literature examples of essentially disconnected alternant PAHs studied.

(acyclic) polynomials from the following equation: DS ¼ jaac N
2 þ ðNc =2Þaac N j ¼ jaac N
2 j
ðNc =2ÞK ð4Þ N

Note that for diradicals aac = K = 0 and the last term becomes zero. Listings of Dewar structures and characteristic and matching (acyclic) polynomials for benzenoid hydrocarbons having up to seven rings can be found in the work of Hosoya and co-workers.15 Disjoint diradical benzenoids have no Kekul
e structures but only first-excited (Dewar) resonance structures where long-range spin-pairing of the two electrons can be represented by Pauling’s long bonds. Numerous classes of diradical benzenoids and equations for determining their number of resonance structures (SC) can be found in the work of Dias and Cash.16 Wheland has also published some recursion equations for obtaining the number of resonance structures of each degree of excitation for the catacondensed benzenoids that also include Dewar resonance structures.17 Herein, we will give a molecular fragmentation method for determining the number of Dewar resonance structures for essentially disconnected PAHs. Bond Lengths of Benzenoid Hydrocarbons with Fragments Connected by Essentially Single Bonds. Pauling derived the following eq 518 assuming a potential function for a resonating bond is given as the sum of two parabolic functions: blðr
sÞ ¼ 1:504
0:17f1:84prs =ð0:84prs þ 1Þg in Å ð5Þ For prs = 0.3333, 0.50, and 1.0, this equation gives 1.422 (graphite, expt. 1.421), 1.394 (benzene, expt. 1.395 Å), and 1.334 Å (double bond, expt. 1.339 Å for ethene), respectively. We will directly use this equation. Pauling further assumed that a weighting factor of 0.03 for first-excited VB resonance structures relative to unexcited VB resonance structures.1 Pauling showed how to obtain the bond lengths in perylene (Figure 1) that agreed with those obtained by X-ray crystallography, which we now reproduce in Table 1. All the nomenclature and the bond numbering used herein is that of IUPAC. Our values for perylene differ slightly from Pauling’s published 13620

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Table 1. Bond Lengths of Perylene X-ray (Å)1

Pauling, unexcited (Å)

bond No.

Pauling, 1st excited (Å)

bond No.

Herndon4 (Å)

1
2

1.415

1.423

1.3333

1.412 (1.412)

1.391 (1.390)

1.422

2
3

1.369

1.370

1.6667

1.380 (1.379)

1.591 (1.610)

1.381

3
3a

1.398

1.423

1.3333

1.421 (1.421)

1.342 (1.339)

1.422

6a
6b

1.471

1.504

1.0

1.470 (1.470)

1.120 (1.120)

1.464

6a
6a1

1.428

1.423

1.3333

1.424 (1.427)

1.328 (1.322)

1.422

6
6a

1.394

1.370

1.6667

1.386 (1.386)

1.549 (1.558)

1.381

3a
3a1

1.421

1.423

1.3333

1.428 (1.427)

1.304 (1.322)

1.422

bond

Figure 2. Determination of the number of Dewar structures (DS) of perylene and its subgraph generated by localizing a double bond on its essentially single bond h.

values (in parentheses in Table 1)1 presumably because he used a slide-rule for the calculations whereas we used an electronic calculator to reproduce his result. In Figure 2, we show how to calculate the number of Dewar resonance structures for perylene. From Wheland and Hosoya’s tabulations15,17 or by use of eq 4, we know that naphthalene has DS = 16. Because bond h (bond 6a
6b) in perylene is essentially single in all its Kekul
e resonance structures, Dewar resonance structures for the two naphthalene units (DS = 16) can only occur separately each multiplied by the number of Kekul
e structures of the other, that is, 16  3 + 16  3. For Dewar structures to occur from one naphthalene moiety to the other, a double bond must now occur at one of the h bonds (bond 6a
6b or 12a
12b). Removing this localized h bond gives a disjoint diradical where each monoradical fragment has SC = 7, as determined by the zero-sum rule; each monoradical fragment corresponds to 1-propenylbenzene monradical having seven resonance structures, and they are disjoint because the electron of each fragment is unable to cross over to the other fragment. The product of SC = 7 gives the number of long bonds that can occur between these monoradical fragments, which correspond to the number of Dewar resonance structures DS[G
h] = 7  7 = 49. Thus, the total number of Dewar structures for perylene is DS = 16  3 + 16  3 + 2  49 = 194;

using the last two coefficients of the matching polynomial gives DS = 284
9  10 = 194, in agreement with Hosoya’s compilation.15 This procedure can be used to obtain the number of Dewar resonance structures for all essentially disconnected alternant benzenoid systems. We now further analyze Pauling’s VB method in regard to perylene. Pauling states, “I have accepted the usual value of 1.470 Å for the bonds g, which corresponds to 12% contribution of the excited structures and to the ratio 0.03.” Pauling’s g bonds (cf. with Table 5 in ref 1) correspond to our h (bond 6a
6b in Table 1) and q (bond 12a
12b) bonds in perylene. Our analysis [0.029  49/(9 + 0.029  98) = 0.12] shows that Pauling rounded off 0.029 to 0.03 and that only for the h bond (i.e., the essentially single bond) does he use 2  49 = 98 in the denominator to solve for the bond order. For VB determination of all other bonds of perylene, Pauling uses 0.03  194 in the denominator where DS = 194 for perylene itself; for perylene, this rounding off to 0.03 does not change the other bond values given in Table 1. To illustrate this point, we solve for the VB length of bond 1
2 (bond a in Figure 1). We need to determine all the Kekul
e (K) and Dewar structures (DS) when a double bond is localized at bond a in perylene. This is accomplished by deleting this bond with its carbon vertices (Figure 3). Using the procedure outlined above for determining the perylene DS value and h bond order, we can now easily compute the a bond order. Because removal of bond a (bond 1
2) from perylene gives a successor molecular graph DS[G
(a)] that has the original essentially single bonds separating an o-quinodimethane fragment (K = 1, DS = 7) from a naphthalene fragment (K = 3, DS = 16), Dewar resonance structures for the o-quinodimethane unit and naphthalene unit can only occur separately, each multiplied by the number of Kekul
e structures of the other, that is, 7  3 + 16  1. For Dewar structures to occur from the o-quinodimethane unit to the naphthalene unit, a double bond must now occur at one of the essentially single bonds joining these units. Removing either bond gives a disjoint diradical, as shown in Figure 3. Using the zero-sum rule and multiplying the SC of the monoradical fragments within each disjoint diradical gives their corresponding number Dewar resonance structures as DS = 35 and 21. Summing all these values together gives the number of Dewar resonance structures DS[G
(a)] = 7  3 + 16  1 + 35 + 21 = 93. Alternatively, using the last two coefficients of the matching polynomial of G
(a) gives DS[G
(a)] = 120
3  9 = 93. Thus, the bond order p12 for bond a (bond 1
2 in Table 1) is given by p12 = (3 + 0.03  93)/(9 + 0.03  194) = 0.391 using the weighting factor of 0.03 or p12 = (3 + 0.029  93)/(9 + 0.029  194) = 0.390 using the weighting factor of 0.029. Inputting either value into Pauling’s eq 5 gives the bond length for bond 1
2 as bl(1
2) = 1.412 Å. The remaining bond lengths of perylene were similarly determined. 13621

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Table 3. Bond Lengths of Teranthrene (Dianthra[1,9,8abcd:10 ,90 ,80 -jklm]coronene) bond

X-ray (Å)19

Pauling 1st excited (Å)

bond No.

1
18b

1.394

1.386

1.5502

1
2 2
3

1.398 1.375

1.415 1.378

1.3734 1.6080

3
3a

1.410

1.428

1.3025

3a
4

1.410

1.410

1.4015

3a
3a1

1.430

1.440

1.2487

3a1
3a2

1.418

1.414

1.3797

3a1
18b

1.417

1.417

1.3680

7a
7b

1.453

1.453

1.1898

7b
7b1 7b1
7b2

1.417 1.419

1.420 1.417

1.3485 1.3618

7a2
7b2

1.424

1.420

1.3479

7b
8

1.395

1.395

1.4940

8
9

1.369

1.398

1.4727

7b1
9a2

1.430

1.436

1.2679

Figure 3. Determination of the number of Dewar structures (DS) of the subgraph of perylene generated by localizing a double bond on bond a (bond 1
2).

Table 2. Bond Lengths of Bisanthrene (Phenanthro[1,10,9,8-opqra]perylene) bond

X-ray (Å)19

Pauling 1st excited (Å)

bond No.

1
14a

1.422

1.429

1.3020

1
2

1.357

1.376

1.6213

2
3

1.400

1.417

1.3622

3
3a 3a
3a1

1.383 1.436

1.384 1.426

1.5666 1.3130

3a1
14a

1.436

1.441

1.2418

14
14a

1.408

1.403

1.4453

3a1
3a2

1.415

1.411

1.3960

3a
3b

1.467

1.466

1.1337

3a2
3b2

1.447

1.442

1.2376

Thus, the original Pauling bond order eq 2 based solely on Kekul
e resonance structures is now amended in eq 6 to include both the number Kekul
e (K) and Dewar (DS) resonance structures as prs ¼ ðK½G
ðers Þ þ 0:03DS½G
ðers ÞÞ=ðK½G þ 0:03DS½GÞ

ð6Þ

without the p superscript, which is applicable to all the bonds of any PAH, except essentially single (double) bonds, that is, bonds that are single (double) in all the Kekul
e resonance structures. We now study the bond lengths in bisanthrene (C28H14, phenanthro[1,10,9,8-opqra]perylene) and teranthrene (C42H18) from Figure 1. In Tables 2 and 3, we compare the experimental bond lengths listed in a paper by Kubo and co-workers19 with those we computed by Pauling’s VB method. Figure 4 outlines the determination of the number of Dewar resonance structures for bisanthrene. From Hosoya and Wheland’s tabulations15,17 or from eq 4, we know that anthracene has DS = 48. Because bond d (bond 3a
3b) in bisanthrene is essentially single in all 16 of its

Figure 4. Determination of the number of Dewar structures (DS) of bisanthrene and its subgraphs generated by localizing a double bond on its essentially single bonds d and y.

Kekul
e resonance structures, Dewar resonance structures for the two anthracene units (K = 4, DS = 48) can only occur separately, each multiplied by the number of Kekul
e structures of the other, that is, 48  4 + 48  4. For Dewar structures to occur from one anthracene moiety to the other, a double bond must now occur at one of the d bonds (bond 3a
3b or 10a
10b) or central y bond (bond 3a2
3b2). Removing the localized d bond from G gives a disjoint diradical G
(d), where each monoradical fragment has SC = 12, as determined by the zero-sum rule. The product of SC = 12 gives the number of long bonds that can occur between these monoradical fragments that correspond to the number of Dewar resonance structures DS[G
d] = 12  12 = 144. Also, 13622

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Figure 5. Determination of the number of Dewar resonance structures of teranthrene (dianthra[1,9,8-abcd:10 ,90 ,80 -jklm]coronene) and its subgraphs formed by excision of its essentially single bonds and determination of the corresponding bond lengths. The matching polynomial was used to obtain the 128 and 176 values for the lower monoradicals in the above disjoint diradicals.

localizing a double bond at the centrally located essentially single bond labeled y (bond 3a2
3b2) in Figure 4 can only result in corresponding Dewar resonance structures. Removing this localized y bond gives a disjoint diradical G
(y) where each monoradical fragment has SC = 16, as determined by the zero-sum rule. Thus, the total number of Dewar structures for bisanthrene is DS = 48  4 + 48  4 + 2  144 + 256 = 384 + 544 = 928; using the last two coefficients of the matching polynomial of bisanthrene and eq 4 gives DS = 1152
16  14 = 928. From these numbers of Dewar resonance structures for bisanthrene, we calculate the following bond orders. For bond d (bond 3a
3b) and bond y (bond 3a2
3b2) we get pd = 0.03  144/ (16 + 0.03  544) = 0.13366 and py = 0.03  256/(16 + 0.03  544) = 0.23762. Using Pauling’s eq 5 on these bond orders gives a bond length of bl(3a
3b) = 1.466 Å and bl(3a2
3b2) = 1.442 Å; using a weighting factor of 0.029 instead of 0.03 gives bl(3a
3b) = 1.467 Å and bl(3a2
3b2) = 1.443 Å, respectively. These results agree completely with those given by Kubo and co-workers (Table 2).19 The remaining bond lengths of bisanthrene were computed similarly, but the denominator number of 544 for the number of Dewar resonance structures associated with localizing a double bond at one of the essentially single bonds is replaced by 928 for the total number of Dewar resonance structures associated with the bisanthrene parent (cf. with Figures S1
S3 in the Supporting Information). Table 2 shows that our VB results for bisanthrene nicely agree with those by Kubo and co-workers.19

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Figure 6. Clar’s goblet is an essentially disconnected diradical. Localizing a double bond at bond j (8a
8b) and excising it gives a disjoint tetraradical with a SC that is equivalent to the number of Pauling second-excited resonance structures with two long bonds from one fragment to the other.

Figure 5 summarizes our determination of the number of Dewar resonance structures and bond lengths of the essentially single bonds of teranthrene (dianthra[1,9,8-abcd:10 ,90 ,80 -jklm]coronene). Table 3 shows that our VB results for teranthrene have good agreement with those by Kubo and co-workers.19 Because teranthrene (C42H18) is too large for our matching polynomial program from Ramaraj and Balsubramanian20 to compute, we used the fragmentation method, as done for perylene and bisanthrene in Figures 2 and 4. Also, because the lower monoradical fragments belonging to the diradicals in Figure 5 have antiaromatic components, their SC cannot be determined by the zero-sum rule, and we had to determine them from the last coefficient in their matching polynomial. Unlike the prior three structures, Clar’s goblet is a nonKekul
ean benzenoid hydrocarbon. It is a concealed diradical that has been studied a number of times.16,21,22 Figure 6 gives our VB calculation of the bond order and bond length of the essentially single bond l (bond 8a
8b) in Clar’s goblet. Placing a double bond therein at l (bond 8a
8b) can only occur in the second-excited resonance structures in which two long bonds go from the upper fragment to the lower one. Localizing the double at l (bond 8a
8b) results in a disjoint tetraradical in which the upper and lower diradical fragments have SC = 101; the number of two long bonds from the upper to lower diradical fragments is (101)2 = 10201. In analogy to Pauling’s assumption, we assume a weight of 0.03 for these long bonds. Thus, the l (bond 8a
8b) bond order is p8a
8b = (0.03  101)2/(0.03  2704) = 0.1132, which gives a bond length of bl(8a
8b) = 1.472 Å, in good agreement with Pogodin and Agranat UB3LYP/6-31G* value of 1.478 Å.22 A further example of the VB calculation of the bond order and bond length of the bond j (bond 8
8a) for Clar’s goblet is given in Figure 7. Table 4 shows all our VB results have 13623

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Table 5. Bond Lengths of Dibenzo[cd,lm]perylene UB3LYP/6-31G** (Å)23

Pauling 1st excited (Å)

bond No.

2
3

1.390

1.395

1.4904

3
3a

1.403

1.404

1.4384

3a
3a1

1.425

1.428

1.3037

3a
4

1.430

1.442

1.2387

4
5

1.359

1.366

1.7039

5
5a

1.435

1.436

1.2678

5a
5a1

1.428

1.427

1.3114

3a1
5a1 5a
5b

1.433 1.423

1.420 1.415

1.3481 1.3726

gond

Figure 7. Determination of the bond order and bond length of the j (8
8a) bond in Clar’s goblet.

Table 4. Bond Lengths of Clar’s Goblet (Diphenaleno[2,1,9,8defg:20 ,10 ,90 ,80 -opqr]pentacene-4,13-diyl) UB3LYP/6-31G* (Å)22

Pauling 1st excited (Å)

bond No.

1
2

1.393

1.403

1.4423

2
3

1.392

1.385

1.5577

3
3a 3a
4

1.418 1.409

1.424 1.406

1.3269 1.4231

3a
3a1

1.436

1.439

1.2500

3a1
3a2

1.426

1.420

1.3462

3a2
8a1

1.431

1.427

1.3077

8a
8a1

1.436

1.427

1.3077

8a
8b

1.478

1.472a

1.1132

8
8a

1.385

1.382

1.5769

7a
8 7
7a

1.418 1.416

1.444 1.403

1.2308 1.4423

7a
7a1

1.422

1.424

1.3269

bond

a

Computed based on 2nd excited resonance structure with two long bonds.

good agreement with the UB3LYP/6-31G* values of Pogodin and Agranat for Clar’s Goblet.22 The VB bond lengths of dibenzo[cd,lm]perylene (Table 5) compare well with those values calculated at the DFT optimized geometry of D2h at B3LYP/6-311**// B3LYP/6-311** by Pododin and Agranat.23 While dibenzo[cd,lm]perylene is not a disconnected nor a nonalternant PAH, it has two phenalenyl moieties joined together in a way that it resembles benzo[cd]benzo[5,6]acenaphtho[1,2-k]fluoranthene, which is a nonalternant PAH of interest (Figure 8) in the next section on nonalternant PAHs. While in general the VB bond orders and bond lengths for these alternant PAHs (Tables 1
5) are in agreement with their comparative references to within (0.01 Å, there is some tendency to underestimate the bond orders and consequently overestimate the bond lengths in one or two isolated cases where the largest bond length variation occurs (+0.02 Å); these largest variations occurred for PAH e(2,2) and e(2,3) perimeter bond types. In every case, the essentially single bonds took on double bond character with the inclusion of Dewar resonance structures, and their bond lengths were well predicted.

Figure 8. Recent literature examples of related phenalenyl and nonalternant PAHs studied.

VB Determination of Bond Lengths in Nonalternant PAHs. Table 6 presents the VB determined bond lengths in fluoranthene (Figure 8), which compares well with the corresponding X-ray crystallographic (averaged) bond length results of Hazel, Jones, and Sowden.24 Also, the VB Table 6 results compare favorably with the Table 4 ab initio (B3LYP, PPP, and RHF) results of Scoles.25 The naphthalene moiety exhibits more bond alternation, and the bond lengths in the benzene moiety are more uniform, except for the bond (6b
10a) that is part of the fivemembered ring. Fluoranthene is a disconnected nonalternant PAH, and VB predicts well its essentially single bond length (bond 6a
6b, 1.476 Å, X-ray, vs 1.472 Å, VB; also, cf. with Figure S4, Supporting Information). Table 6 also gives the bond length results calculated by Herndon’s empirical equation.4 The synthesis and semiconductive behavior of the delocalized singlet biradical hydrocarbon, benzo[cd]benzo[5,6]acenaphtho[1,2-k]fluoranthene, has been described.26,27 The X-ray crystal structure of derivatives of benzo[cd]benzo[5,6]acenaphtho[1,2-k]fluoranthene have been recorded and the ab initio determination at various levels of theory of the bond lengths of the parent 13624

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Table 6. Bond Lengths of Fluoranthene X-ray (Å)24

Pauling, unexcited (Å)

bond No.

Pauling, 1st excited (Å)

bond No.

Herndon4 (Å)

1
2

1.411

1.423

1.3333

1.413

1.387

1.422

2
3

1.368

1.370

1.6667

1.380

1.592

1.381

3
3a

1.422

1.423

1.3333

1.421

1.339

1.422

8
9

1.379

1.394

1.5

1.399

1.467

1.402

bond

7
8

1.388

1.394

1.5

1.393

1.509

1.402

6b
7

1.384

1.394

1.5

1.401

1.457

1.402

6a
6b

1.476

1.504

1.0

1.472

1.112

1.464

3a1
6a 6
6a

1.411 1.367

1.423 1.370

1.3333 1.6667

1.425 1.386

1.322 1.551

1.422 1.381

3a
3a1

1.400

1.423

1.3333

1.427

1.308

1.422

6b
10a

1.417

1.394

1.5

1.411

1.399

1.402

Table 7. Bond Lengths of Benzo[cd]benzo[5,6]acenaphtho[1,2-k]fluoranthene bond

RB3LYP/6-31G* (Å)

26

Pauling 1st excited (Å)

bond No.

Table 8. Bond Lengths of Phenalenyl Monoradical bond

DFT (Å)31

Pauling’s 1st order (Å)

2
3

1.399

1.395

1.4920

1
2 3
3a

1.393 1.418

1.394 1.419

3
3a

1.409

1.406

1.4254

3a
3a1

1.431

1.429

3a
4

1.437

1.437

1.2594

4
5 5
5a

1.387 1.415

1.371 1.426

1.6645 1.3152

5a
5b

1.446

1.429

1.3007

5b
6

1.396

1.399

1.4674

5b
14a

1.453

1.453

1.1884

5a
5a1

1.417

1.427

1.3123

3a1
5a1

1.393

1.420

1.3478

3a
3a1

1.425

1.429

1.2993

Table 9. Bond Lengths of Triangulene Diradical DFT (Å)31

Pauling’s 1st order (Å)

X-ray (Å)a

1
2

1.393

1.394

1.374

3
3a

1.421

1.423

1.400

3a
3a1

1.434

1.431

1.424

3a
4

1.409

1.410

1.474b

1.424

1.423

1.419

bond

3a
4a 1

2

a

have been reported.28 Its VB computed bond lengths are shown to be comparable to the RB3LYP/6-31G* computed values in Table 7.28 If we compare bond 3a
3a1 (1.400 Å, X-ray, vs 1.427 Å, VB) in fluoranthene (Table 6) and bond 3a1
5a1 (1.393 Å, UB3LYP/6-31G**, vs 1.420 Å, VB) in benzo[cd]benzo[5,6]acenaphtho[1,2-k]fluoranthene (Table 7), both belonging to the exo double in a fulvene substructure, we speculate that this difference is attributable to angle/bond distortion associated with the fulvene substructure. This arises because VB does not take into account angle/bond strain energy; it is only a measure of pπ-electronic energy. These results should be compared to bond 3a1
5a1 (1.433 Å, UB3LYP/6-31G**, vs 1.420 Å, VB) in dibenzo[cd, lm]perylene (Table 5), which displays the opposite trend. While benzo[cd]benzo[5,6]acenaphtho[1,2-k]fluoranthene has DS = 2160, it has 800 first-excited valence-bond structures with its benzo group manifesting its full aromaticity, which is responsible for its singlet biradical character (Figure 8). Kubo and coworkers26 estimated the amount of biradical character of benzo[cd]benzo[5,6]acenaphtho[1,2-k]fluoranthene to be 30% from the LUMO occupation number of 0.30. Based on our above numbers, we estimate the biradical character to be [0.03  800/(18 + 0.03  2160)]  100 = 29%. The presence of two pentagonal rings in a PAH results in the appearance of 4n = 12 antiaromatic circuits.29 For benzo[cd]benzo[5,6]acenaphtho[1,2-k]fluoranthene there are 8 Kekul
e and 152 Dewar resonance structures with 4n = 12 antiaromatic circuits. If we exclude the Kekul
e and phenalenyl Dewar structures having 4n = 12

For 4,8-dioxo-4H,8H-dibenzo[cd,mn]pyren-12-olate tetrabutylammonium salt in ref 30. b Bond is adjacent to the C
O moiety.

antiaromatic circuits, then we estimate the biradical character to be [0.03  648/(10 + 0.03  2008)]  100 = 28%. Bond Lengths of Triangular Benzenoid Hydrocarbons. Phenalenyl and triangulene have been extensively studied experimentally and theoretically as radical, ionic, and derivative molecular species.30 Tables 8 and 9 summarize the bond length results for phenalenyl monoradical and triangulene diradical computed by both VB and ab initio plane wave based all valence electron DFT (density functional theory) calculations;31 the IUPAC nomenclature numbering system is used in Tables 8 and 9 and Figure 9. Figure 10 gives illustrative examples on how to compute the VB bond lengths by determining the SC for both the parent molecular graph and its successor subgraph generated by deleting a fixed double bond of interest and then using eq 2. For phenalenyl monoradical, there are 10 resonance structures (SC = 10) if we fix a double bond at bond 2
3 compared to 20 resonance structures (SC = 20) of the parent. The tail coefficient of the corresponding matching polynomials gives us these structure count (SC) values. Alternatively, one can use the zero-sum rule to determine the number of resonance structures belonging to the subgraph generated by deleting the double bond of interest. Thus, from eq 2, replacing K by SC, the 2
3 bond order (p23) for phenalenyl monoradical is p23 = SC[G
(e23)]/ SC[G] = 10/20 = 0.5, which gives a bond length of 1.394 Å using 13625

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Figure 9. Recent literature examples of triangular-shaped radical PAHs studied.

Figure 11. Decomposition of a C2v or higher symmetry shaped benzenoid molecular graph by operating on a central nonstarred peak per SC(G) = 2  SC[G
(e)] proves that all such graphs with an odd number of peaks must have an even SC value and the same apex VB length.

Figure 10. Illustrative determination of the bond lengths phenalenyl monoradical and triangulene diradical.

Pauling’s eq 5. As is apparent from Table 8, these VB results coincide almost exactly with those calculated by Philpott and coworkers.31 The results of triangulene in Table 9 are even more informative. For triangulene diradical, there are 153 resonance structures (SC = 153) if we fix a double bond at bond 2
3 compared to 306 resonance structures (SC = 306) of the parent. The tail coefficient of the corresponding matching polynomials gives us these structure count (SC) values. Because there are antiaromatic circuits in the subgraph of triangulene, we cannot use the zero-sum rule because it only gives the corrected structure count (CSC) value.32 Thus, the 2
3 bond order (p23) is p23 = SC[G
(e23)]/SC[G] = 153/306 = 0.5, which gives a bond length of 1.394 Å using Pauling’s eq 2. This result is exactly the same as was obtained for the 2
3 bond of phenalenyl. Figure 11 proves that the apex bonds of all triangulene-related benzenoids must give the same bond length at the VB level because SC[G] = 2  SC[G
(ers)]. This was previously proven by Cash and Dias, which explains the observation reported by Philpott and co-workers in both their abstract and conclusion; also, see Figure 5 in ref 21. To quote Philpott and co-workers, “At each apex the middle atom was on a local C2v site ... conjoined ... by the shortest bonds (CC ≈ 139.3 pm) in the molecule.” In general, each member of

the triangulene D3h one-isomer series has an odd number of peaks and operating on the central one, as done in Figure 11, must give the same bond length at the VB level because SC[G] = 2  SC [G
(ers)], resulting in a bond order of 0.500. Philpott and coworkers31 also obtained this same bond length result for the apex 2
3 bond of tetraagulene (Figure 9) and obtained 1.42 Å for the next succeeding perimeter 3
3a bond that, using a resonancetheoretic argument,33 should have a bond order of 0.3333, giving a bond length of 1.423 Å per Pauling’s eq 2. Consistent with the results of Philpott and co-workers, the more interior C
C bond lengths should also approach the value of 1.423 Å based on resonance-theoretic arguments. This is emphasized by the triangulene D3h one-isomer series (C22H12 f C52H18 f C94H24 f C148H30 f ...)34 each member that has a nonstarred vertexcentric carbon that can only accommodate one double bond at a time from any of its three nearest neighbors. Thus, these central carbon atoms all have a pπ-bond order of 1/3 and a VB bond length of 1.423 Å to each of their three nearest neighbors (cf. with bond 3a1
4a2 in Table 9).

’ CONCLUSION The inclusion of Dewar resonance structures does improve the VB determination of bond lengths in PAHs, especially those with essentially single (double) bonds in their Kekul
e resonance structures. Essentially disconnected PAHs can be fragmented into monoradical components that, if without antiaromatic circuits, can have their number of resonance structures determined by the zero-sum rule, the product of which gives their number of Dewar structures. Disjoint diradicals, like Clar’s goblet, have Dewar resonance structures where the Pauling long (formal) bonds join a starred position and a nonstarrred position, each occupied by a single electron that is long-range spinned-paired and, therefore, identifies such molecules as having a singlet ground state.22 Nondisjoint diradicals, like triangulene, have triplet ground states because their unpaired electrons always reside on starred positions, in compliance to Hund’s rule. The diverse examples of four Kekul
ean and four non-Kekul
ean AHs and two nonAHs, studied herein, demonstrate that Pauling’s VB method performs well in comparison with experimental and other ab initio calculations and gives another perspective. Bond lengths are important parameters in structure and 13626

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The Journal of Physical Chemistry A chemical reactivity. This work has relevance to two important indexes of aromaticity. Topological resonance energy (TRE) uses the matching polynomial to compute the pπ-energy of a cyclic conjugated system of the hypothetical reference molecule devoid of any cyclic contribution.35,36 The harmonic oscillator model of aromaticity (HOMA) is a geometry-based index that uses bond lengths of a cyclic conjugated system;37 the principle behind the HOMA index is the greater bond alternation in a cyclic conjugated system, the less aromatic it is. This work shows that greater bond alternation occurs on the perimeter of benzenoids. Pauling was a master at reducing complicated problems to their essence. His 1980 paper written 26 years after he received the Nobel Prize in Chemistry in 1954 and the excellent VB length prediction results presented herein for PAHs from the recent chemical literature is another testimony to his intellectual capacity.

’ ASSOCIATED CONTENT

bS

Supporting Information. Valence-bond determination of the bond length of all the bonds in bisanthrene and the number of Dewar resonance structures of fluoranthene. This material is available free of charge via the Internet at http://pubs. acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

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