Theoretical Study on Oligoacenes and Polycyclic Aromatic

ARTICLE pubs.acs.org/JPCA

Theoretical Study on Oligoacenes and Polycyclic Aromatic Hydrocarbons Using the Restricted Active Space Self-Consistent Field Method Fumihiko Aiga* Corporate Research and Development Center, Toshiba Corporation, 1, Komukai-Toshiba-cho, Saiwai-ku, Kawasaki 212-8582, Japan

bS Supporting Information ABSTRACT: This is the first study, to my knowledge, to report the optimized geometries and vibrational frequency analysis for oligoacenes (naphthalene, anthracene, naphtacene, and pentacene) and polycyclic aromatic hydrocarbons (PAHs; perylene, phenanthrene, and picene) by using the restricted active space self-consistent field (RASSCF) method. For naphthalene, both the complete active space self-consistent field (CASSCF) and RASSCF calculations were performed. As a result, it was confirmed that the RASSCF, with its small computational costs, is appropriate for oligoacenes and PAHs. It should be noted that, for anthracene and perylene, the optimized geometries under D2h symmetry were not the minimum energy points, whereas the optimized geometries under Cs symmetry were the minimum energy points. For naphthalene, anthracene, naphtacene, pentacene, and phenanthrene, the calculated bond lengths and infrared absorption spectra by the RASSCF were in good agreement with the experimental values.

was used. In the RASSCF calculation for each system with n carbon atoms, all n π electrons were treated as the active electrons and distributed among n/2 bonding π orbitals and n/2 antibonding π* orbitals. The active space was restricted so as to include the configuration state functions up to two electron excitations from the reference determinant. Therefore, the Gaussian keyword “CASSCF(n, n, RASSCF(0,0,2,n/2))” was stated. In this paper, we call it the RASSCF(2). For oligoacenes and perylene, the optimized geometries under D2h symmetry using the RASSCF were obtained and vibrational frequency analysis was performed using numerical RASSCF derivatives. For anthracene and perylene, two imaginary frequencies were found under D2h symmetry. Therefore, the optimized geometries under Cs symmetry were obtained and vibrational frequency analysis was performed. For phenanthrene and picene, the optimized geometries under C2v symmetry using the RASSCF were obtained, and vibrational frequency analysis was performed using numerical RASSCF derivatives. For naphthalene, the CASSCF geometry optimization and the vibrational frequency analysis were also performed. For anthracene, the CASSCF geometry optimization without the vibrational frequency analysis was performed. For anthracene and phenanthrene, the RASSCF calculations with the configuration state functions up to four electron excitations

1. INTRODUCTION Graphene-based materials have been the focus of extensive experimental and theoretical efforts.1 Nanographenes2 are known as polycyclic aromatic hydrocarbons (PAHs),3 and oligoacenes4 are PAHs consisting of linearly fused benzene rings. The recent electronic structure calculations for these compounds suggest that their ground states have an open-shell singlet multiradical character.5 8 To deal with the open-shell singlet wave functions properly, the calculations based on the multiconfiguration wave functions are necessary, and such calculations have been made for these compounds.5,7,9 11 To my knowledge, however, no reports on the vibrational frequency analysis at the optimized geometries by using multiconfiguration wave functions for these compounds can be found. This paper reports the optimized geometries and vibrational frequency analysis for oligoacenes and PAHs by using the restricted active space self-consistent field (RASSCF)12 method. In section 2, the computational details are described. Section 3 shows the results for oligoacenes (naphthalene, anthracene, naphtacene, and pentacene), perylene, phenanthrene, and picene, followed by concluding remarks in section 4. 2. COMPUTATIONAL DETAILS The structures of PAH molecules investigated in this paper are illustrated in Figure 1, where hydrogen atoms are excluded, and the symbols for bonds are defined. The calculations were carried out with the Gaussian 03 package,13 and the 6-31G(d) basis set r 2011 American Chemical Society

Received: September 24, 2011 Revised: December 4, 2011 Published: December 28, 2011 663

dx.doi.org/10.1021/jp2092225 | J. Phys. Chem. A 2012, 116, 663–669

The Journal of Physical Chemistry A

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Table 2. Calculated Vibrational Frequencies (cm 1) and Intensities (km/mol) of Infrared Absorption Spectroscopy for Naphthalene freq

Figure 1. Structures of PAH molecules investigated in this paper. Hydrogen atoms are excluded. The symbols for bonds are defined.

Table 1. Calculated Bond Lengths of Naphthalene (Å) in Comparison with the Experimental Valuesa

a

CASSCF

RASSCF(2)

expb

a

1.422

1.421

1.421

b

1.373

1.367

1.378

c

1.427

1.426

1.425

d

1.416

1.411

1.426

int

irrep

CASSCF

RASSCF(2)

CASSCF

RASSCF(2)

b3u

187.8

194.7

1.7

1.6

b1u

384.9

387.2

0.8

1.0

b3u

514.6

525.5

14.9

15.8

b2u

664.8

668.0

3.9

4.5

b3u b1u

824.1 852.5

849.2 856.1

115.2 0.4

118.6 0.3

b3u

999.6

1032.4

2.6

2.1

b2u

1051.9

1053.0

1.4

1.5

b2u

1186.3

1185.2

2.6

4.2

b1u

1225.5

1233.3

1.6

1.7

b2u

1290.3

1297.7

0.3

0.2

b3u

1372.6

1379.2

5.1

5.9

b2u b1u

1447.6 1533.0

1461.1 1537.0

1.1 2.7

1.2 3.0

b2u

1652.7

1661.1

6.7

7.7

b1u

1744.1

1767.2

6.8

6.2

b1u

3351.1

3351.8

11.7

9.4

b2u

3351.1

3352.4

2.7

2.1

b1u

3370.1

3371.1

87.8

90.1

b2u

3382.2

3382.8

79.5

78.0

The symbols for bonds are defined in Figure 1. b Ref 14.

[CASSCF(n, n, RASSCF(0,0,4,n/2)) ] were performed. In this paper, we call it the RASSCF(4).

3. RESULTS 3.1. Naphthalene. For naphthalene, the geometry optimizations using the CASSCF(10,10) and CASSCF(10,10,RASSCF(0,0,2,5)) [RASSCF(2)] were performed under D2h symmetry, and the optimized geometries were confirmed to be the minimum energy points by the vibrational frequency analysis. In Table 1, the calculated bond lengths for the optimized geometries are given in comparison with the experimental values.14 The calculated bond lengths in both the CASSCF and RASSCF(2) are found to be in good agreement with the experimental values. In Table 2, the calculated vibrational frequencies and the intensities of the infrared (IR) absorption spectra are given. In Figure 2a, the calculated IR spectra by the CASSCF in comparison with the ones by the RASSCF(2) are illustrated. The calculated IR spectra by the CASSCF in comparison with the experimental ones15 are illustrated in Figure 2b, where the calculated vibrational frequencies are scaled by 0.9.16 It was found that the IR spectra by the RASSCF(2) are similar to the ones by the CASSCF and that the calculated IR spectra by the CASSCF are in good agreement with the experimental ones. I have designated the two orbitals with occupancies closest to 1: the “highest occupied natural orbitals” (HONO) with

Figure 2. (a) Calculated IR spectra of naphtharene by CASSCF in comparison with the ones by RASSCF(2). (b) Calculated IR spectra of naphtharene by CASSCF in comparison with the experimental ones in ref 14. The calculated vibrational frequencies are scaled by 0.9. 664

dx.doi.org/10.1021/jp2092225 |J. Phys. Chem. A 2012, 116, 663–669


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Table 3. Calculated Coefficient for the Wave Function of the Main Configuration C0, Occupation Number n(HONO) and n(LUNO), the Measures of the Unpaired Electrons N(T) and N(H) for Naphthalene, Anthracene, Naphtacene, Pentacene, Perylene, Phenanthrene, and Picene

Table 5. Calculated Bond Lengths (Å) and Bond Angles (degree) in Comparison with the Experimental Valuesa

C0 n(HONO) n(LUNO) N(T) N(H)

RASSCF(2)

RASSCF(4)

CASSCF(cc)

expb

a

1.43405

1.43430

1.43584

1.431

j b

1.43778 1.35697

1.43435 1.36329

1.43583 1.36729

1.431 1.396

naphthalene

CASSCF

0.90

1.87

0.14

1.57

0.82

i

1.35559

1.36325

1.36729

1.396

0.93 0.92

1.92 1.92

0.09 0.08

1.03 1.15

0.53 0.59

k

1.35558

1.36325

1.36729

1.396

anthracene

RASSCF(2) RASSCF(2)

p

1.35698

1.36328

1.36729

1.396

RASSCF(4)

0.87

1.85

0.16

2.08

1.09

c

1.43807

1.43872

1.44016

1.434

CASSCF (cc) 0.86

1.84

0.17

2.25

1.19

h

1.44180

1.43881

1.44016

1.434

naphtacene

RASSCF(2)

0.90

1.92

0.09

1.46

0.75

1

1.44180

1.43879

1.44016

1.434

pentacene

RASSCF(2)

0.89

1.92

0.08

1.62

0.83

perylene

RASSCF(2)

0.94

1.95

0.05

0.96

0.49

o d

1.43812 1.41585

1.43873 1.42784

1.44016 1.43014

1.434 1.437

phenanthrene RASSCF(4)

0.90

1.88

0.12

1.99

1.04

g

1.42569

1.42762

1.43010

1.437

picene

0.90

1.94

0.06

1.53

0.78

e

1.39636

1.39972

1.40263

1.403 1.403

RASSCF(2)

Table 4. Calculated Energies (hartree) of Anthracene at the Optimized Geometries under D2h and Cs Symmetrya D2h

Cs

ΔE

RASSCF(2)

536.1245

536.1249

0.3

RASSCF(4)

536.1450

536.1717

16.8

CASSCF(cc)

536.1706

536.2129

26.5

a

ΔE (kcal/mol) is the difference of the energy under D2h with the one for Cs.

occupancy greater than 1 and the “lowest unoccupied natural orbitals” (LUNO) with occupancy less than 1.9 Table 3 shows the calculated coefficient for the wave function of the main configuration C0, occupation number of HONO n(HONO), and LUNO n(LUNO). These values by the RASSCF(2) in Table 3 are found to be similar to those by the CASSCF, as well as the bond lengths and the IR spectra. Therefore, we can consider that the RASSCF(2), with its small computational costs, is appropriate for PAHs. In Table 3, the measures of the unpaired electrons N(T) defined by Takatsuka17 and N(H) defined by HeadGordon18 are also given. N(T) is defined as N(T) = Σini(2 ni), where ni is the occupation number of the i-th natural orbital and the summation is over all natural orbitals. On the other hand, N(H) is defined as N(H) = Σi min(ni,(2 ni)). In Figure S1, the distributions of the probability amplitudes of HONO and LUNO are illustrated. 3.2. Anthracene. For anthracene, the geometry optimization using the CASSCF(14,14,RASSCF(0,0,2,7)) [RASSCF(2)] with the 6-31 g(d) basis set was performed under D2h symmetry, but two imaginary frequencies were found in the vibrational frequency analysis. Therefore, considering the vibrational vectors for the imaginary frequencies, geometry optimization using the RASSCF(2) was performed under Cs symmetry. The optimized geometry under Cs symmetry was confirmed to be the minimum energy point by the vibrational frequency analysis. The geometry optimization using the RASSCF(2) with the cc-pVDZ basis set was performed under D2h symmetry, and similarly, two imaginary frequencies were found; under Cs symmetry, the minimum energy point was obtained. Next, the geometry optimization using the RASSCF(4) with the 6-31 g(d) basis set was performed under D2h symmetry, and similarly, two imaginary

a

f

1.39228

1.39985

1.40266

m

1.39231

1.39990

1.40266

1.403

n

1.39657

1.39978

1.40263

1.403

— ab

120.274

120.326

120.315

— ap — bc

120.272 120.913

120.332 120.937

120.315 120.921

— po

120.908

120.938

120.921

— cd

118.813

118.737

118.764

— od

118.820

118.730

118.763

— de

119.311

119.218

119.225

— dn

119.309

119.221

119.225

— ef

121.519

121.562

121.549

— nm — fg

121.511 119.174

121.556 119.221

121.550 119.225

— mg

119.175

119.222

119.225

— gh

118.695

118.740

118.764

— gl

118.698

118.735

118.764

— hi

120.890

120.933

120.921

—k

120.888

120.933

120.921

— ij

120.415

120.327

120.315

— kj

120.414

120.332

120.315


frequencies were found, and under Cs symmetry, the minimum energy point was obtained. The CASSCF geometry optimization with cc-pVDZ basis set was also performed under D2h and Cs symmetry. We call the CASSCF calculation with the cc-pVDZ basis set CASSCF(cc). The vibrational frequency analysis using the CASSCF method was not performed. In Table 4, the calculated energies (in hartree) at the optimized geometries under both D2h and Cs symmetry are given, and ΔE (in kcal/mol), which is the difference of the energy for D2h with the one for Cs, is also shown. In each wave function [RASSCF(2), RASSCF(4), CASSCF(cc)], the optimized geometry under Cs symmetry is revealed to be more stable than that under D2h symmetry. Cizek and Paldus19 pointed out that the symmetry of the minimum energy point of anthracene is lower than the D2h symmetry in the context of the instability of the Hartree Fock wave functions. They suggested that the symmetry of the geometry of the minimum energy point of anthracene is the C2v symmetry, in contrast to the results in this paper. 665



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Table 6. Calculated Bond Lengths (Å) of Naphthacene and Pentacene in Comparison with the Experimental Valuesa naphtacene

a

pentacene b

RASSCF(2)

expb

a

1.451

1.43

b

1.346

1.35

1.42 1.44

c d

1.452 1.447

1.42 1.44

1.379

1.39

e

1.370

1.38

1.413

1.40

f

1.424

1.40

1.426

1.46

g

1.433

1.45

h

1.396

1.39

RASSCF(2)

exp

a

1.445

1.46

b

1.350

1.38

c d

1.447 1.438

e f g


Figure 3. (a) Calculated IR spectra of anthracene by RASSCF(2) and RASSCF(4). (b) Calculated IR spectra of anthracene by RASSCF(4) in comparison with the experimental ones in ref 14. The calculated vibrational frequencies are scaled by 0.9.

In Table 5, the calculated bond lengths and bond angles for the optimized geometries under Cs symmetry are given in comparison with the experimental values.14 The calculated bond lengths in the CASSCF(cc), RASSCF(4), and RASSCF(2) are found to be in good agreement with the experimental values. It should be noted that the distortions of the optimized geometry under Cs symmetry compared with the D2h symmetry are quite small. In Table S1, the calculated vibrational frequencies and the intensities of the IR absorption spectra are given. In Figure 3a, the calculated IR spectra by the RASSCF(4) in comparison with the one by RASSCF(2) are illustrated, and the calculated IR spectra by the RASSCF(4) in comparison with the experimental ones15 are illustrated in Figure 3b, where the calculated vibrational frequencies are scaled by 0.9.16 It can be found that the IR spectra by the RASSCF(2) are similar to the ones by the RASSCF(4) and that the calculated IR spectra by RASSCF(4) are in good agreement with the experimental ones. Table 3 shows the calculated coefficient for the wave function of the main configuration C0, occupation number n(HONO) and n(LUNO), as well as the measures of the unpaired electrons N(T) and N(H). In Figure S1, the distributions of the probability amplitudes of HONO and LUNO are illustrated. 3.3. Naphtacene and Pentacene. For naphtacene, the geometry optimization using the CASSCF(18,18,RASSCF(0,0,2,9)) [RASSCF(2)] was performed under D2h symmetry, while for pentacene, the geometry optimization in the CASSCF(22,22, RASSCF(0,0,2,11)) [RASSCF(2)] was performed under D2h symmetry. For both naphtacene and pentacene, the optimized geometries were confirmed to be the minimum energy points by the vibrational frequency analysis. Cizek and Paldus19 pointed out that the symmetry of the minimum energy point of oligoacenes

Figure 4. Calculated IR spectra of naphtacene (a) and pentacene (b) by RASSCF(2) in comparison with the experimental ones in ref 14. The calculated vibrational frequencies are scaled by 0.9.

with odd numbers of benzene is lower than the D2h symmetry in the context of the instability of the Hartree Fock wave functions. In this work, however, the symmetry of the geometry of the minimum energy point of pentacene is the D2h symmetry. In Table 6, the calculated bond lengths for the optimized geometries are given in comparison with the experimental values.14 The calculated bond lengths in the RASSCF(2) are found to be in good agreement with the experimental values. In Tables S2 and S3, the calculated vibrational frequencies and the intensities of the IR absorption spectra are given for naphtacene and pentacene, respectively. In Figure 4a,b, the calculated IR spectra by the RASSCF(2) are illustrated in comparison with the experimental ones15 for naphtacene and pentacene, respectively. The calculated vibrational frequencies are scaled by 0.9.16 The calculated 666



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Table 7. Calculated Bond Lengths (Å) of Perylene, Phenanthrene, and Picene in Comparison with the Experimental Valuesa perylene

a

phenanthrene b

RASSCF(4)

expc

a

1.409

1.398

b

1.379

1.383

1.423

c

1.419

1.405

1.411

1.421

d

1.464

1.448

b f

1.359 1.357

1.374 1.363

e f

1.411 1.416

1.404 1.457

o

1.349

1.370

g

1.377

1.381

s

1.355

1.371

h

1.443

1.390

c

1.412

1.386

i

1.354

1.382

e

1.427

1.410

p

1.423

1.394

r

1.417

1.408

k h

1.373 1.386

1.387 1.401

a b

1.410 1.370

m

1.385

1.406

c

1.420

u

1.376

1.394

d

1.458

j

1.425

1.428

e

1.426

i

1.441

1.428

f

1.356

w

1.437

1.406

g

1.368

v

1.430

1.439

h

1.416

d q

1.409 1.411

1.421 1.426

i j

1.404 1.435

1

1.489

1.479

k

1.349

x

1.486

1.463

1

1.443

RASSCF(2)

exp

a

1.403

1.420

g

1.432

1.409

n

1.419

t

picene RASSCF(2)

m

1.398

n

1.447

Figure 5. (a) Calculated IR spectra of perylene by RASSCF(2) in comparison with the experimental ones in ref 14. The calculated vibrational frequencies are scaled by 0.9. (b) Calculated IR spectra of phenanthrene by RASSCF(4) in comparison with the experimental ones in ref 14. The calculated vibrational frequencies are scaled by 0.9.

The symbols for bonds are defined in Figure 1. b Ref 20. c Ref 14.

IR spectra by the RASSCF(2) are found to be in good agreement with the experimental ones. Table 3 shows the calculated coefficient for the wave function of the main configuration C0, occupation number n(HONO) and n(LUNO), as well as the measures of the unpaired electrons N(T) and N(H). In Figure S1, the distributions of the probability amplitudes of HONO and LUNO are illustrated. 3.4. Perylene. A perylene molecule has both zigzag edges and armchair edges, which is a good model system of the nanographenes. For perylene, the geometry optimization using the CASSCF(20,20,RASSCF(0,0,2,10)) [RASSCF(2)] was performed under D2h symmetry, but four imaginary frequencies were found by the vibrational frequency analysis. Therefore, considering the vibrational vectors for the imaginary frequencies, geometry optimization in the RASSCF(2) was performed under Cs symmetry. The optimized geometry under Cs symmetry was confirmed to be the minimum energy point by the vibrational frequency analysis. The energy of the optimized geometry under Cs symmetry is 19.9 kcal/mol lower than that under D2h symmetry. In Table 7, the calculated bond lengths for the optimized geometry are given in comparison with the experimental values.20 The symmetry of the experimental geometry determined by the molecular crystal20 is also broken from D2h. It should be noted that the calculated

Figure 6. Distributions of the probability amplitudes of HONO and LUNO for perylene (RASSCF(2)), phenanthrene (RASSCF(4)), and picene (RASSCF(2)). 667


The Journal of Physical Chemistry A distortions of the optimized geometry under Cs symmetry compared with the D2h symmetry are fairly large. In Table S4, the calculated vibrational frequencies and the intensities of the IR absorption spectra are given. In Figure 5a, the calculated IR spectra by the RASSCF(2) are illustrated in comparison with the experimental ones.15 The calculated vibrational frequencies are scaled by 0.9.16 The calculated IR spectra are found to be in good agreement with the experimental ones. Table 3 shows the calculated coefficient for the wave function of the main configuration C0, occupation number n(HONO) and n(LUNO), as well as the measures of the unpaired electrons N(T) and N(H). In Figure 6, the distributions of the probability amplitudes of HONO and LUNO are illustrated. It shows that the symmetry of the natural orbitals is largely broken from the D2h symmetry. 3.5. Phenanthrene and Picene. For phenanthrene, the geometry optimization using the CASSCF(14,14,RASSCF(0,0,2,7)) [RASSCF(2)] was performed under C2v symmetry, but the geometry optimization was not converged, whereas, the geometry optimization was converged by using the CASSCF(14,14,RASSCF(0,0,4,7)) [RASSCF(4)] under C2v symmetry. The optimized geometry was confirmed to be the minimum energy point by the vibrational frequency analysis. In Table 7, the calculated bond lengths for the optimized geometries are given in comparison with the experimental values.14 The calculated bond lengths are found to be in good agreement with the experimental values. In Table S5, the calculated vibrational frequencies and the intensities of the IR absorption spectra are given. In Figure 5b, the calculated IR spectra by the RASSCF(2) are illustrated in comparison with the experimental ones.15 The calculated vibrational frequencies are scaled by 0.9.16 The calculated IR spectra are found to be in good agreement with the experimental ones. Picene has attracted considerable attention because of its superconductivity.21,22 For picene, the geometry optimization using the CASSCF(22,22,RASSCF(0,0,2,11)) [RASSCF(2)] was performed under C2v symmetry. As a result, the optimized geometry was confirmed to be the minimum energy point by the vibrational frequency analysis. In Table 7, the calculated bond lengths for the optimized geometries are given. The symmetry of the experimental geometry determined by the molecular crystal23 is broken from C2v. In Table S6, the calculated vibrational frequencies and the intensities of the IR absorption spectra are given. In Figure S2, the calculated IR spectra by the RASSCF(2) are illustrated in comparison with the ones by the density functional theory (DFT) [B3LYP/6-31 g(d)] performed in this work. It can be found that the difference of the calculated IR spectra by RASSCF(2) is qualitatively large with the ones by B3LYP/6-31 g(d). To my knowledge, there is no report of the experimental IR spectra for picene. For phenanthrene and picene, the calculated coefficient for the wave function of the main configuration C0, occupation number n(HONO) and n(LUNO), as well as the measures of the unpaired electrons N(T) and N(H) are given in Table 3. In Figure 6, the distributions of the probability amplitudes of HONO and LUNO are illustrated. It can be found that the symmetry of the LUNO for phenanthrene is broken from the C2v symmetry. Casanova and Head-Gordon10 have reported that the symmetry of the natural orbitals for the oligoacenes with more than 14 benzenes is broken by their spin-flip RASSCF calculations on the fixed D2h geometries by the DFT geometry optimizations.

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4. CONCLUSIONS This paper reports the optimized geometries and vibrational frequency analysis for naphthalene, anthracene, naphtacene, pentacene, perylene, phenanthrene, and picene by using the RASSCF wave functions. The remarkable points are as follows: (1) For naphthalene, both the CASSCF and RASSCF were performed; it was confirmed that the RASSCF, with its small computational costs, is appropriate for PAHs. The calculated bond lengths and IR spectra were in good agreement with the experimental ones. (2) For anthracene, the optimized geometry under D2h symmetry was not the minimum energy point, whereas the optimized geometry under Cs symmetry was the minimum energy point. The distortions of the optimized geometry under Cs symmetry compared with the D2h symmetry were quite small. The calculated bond lengths and IR spectra by the RASSCF were in good agreement with the experimental ones. (3) For perylene as well as anthracene, the optimized geometry under D2h symmetry was not the minimum energy point. The optimized geometry under Cs symmetry was the minimum energy point. The calculated IR spectra by the RASSCF were in good agreement with the experimental ones. (4) For naphtacene, pentacene, and phenanthrene, the calculated bond lengths and IR spectra by the RASSCF were in good agreement with the experimental ones. ’ ASSOCIATED CONTENT

bS

Supporting Information. Calculated vibrational frequencies and intensities of IR spectroscopy for anthracene, naphtacene, pentacene, perylene, phenanthrene, and picene. The distributions of the probability amplitudes of HONO and LUNO for naphthalene, anthracene, naphtacene, and pentacene. Calculated IR spectra of picene. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT The author would like to thank Drs. S. Itoh, T. Tada, Y. Hiraoka, R. Yoshimura, T. Yoshida, and Y. Nishida for their helpful comments and suggestions. ’ REFERENCES (1) Geim, A. K. Science 2009, 324, 1530–1534. (2) Mullen, K.; Rabe, J. P. Acc. Chem. Res. 2008, 41, 511–520. (3) Wu, J.; Pisula, W.; Mullen, K. Chem. Rev. 2007, 107, 718–747. (4) Bettinger, H. F. Pure Appl. Chem. 2010, 82, 905–915. (5) Bendikov, M.; Duong, H. M.; Starkey, K.; Houk, K. N.; Carter, E. A.; Wudl, F. J. Am. Chem. Soc. 2004, 126, 7416–7417. (6) Jiang, D.; Sumpter, B. G.; Dai, S. J. Chem. Phys. 2007, 126, 134701. (7) Jiang, D.; Sumpter, B. G.; Dai, S. J. Chem. Phys. 2007, 127, 124703. (8) Hod, O.; Barone, V.; Scuseria, G. E. Phys. Rev. B 2008, 77, 035411. 668



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