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Computational and Theoretical Chemistry 977 (2011) 163–167

Contents lists available at SciVerse ScienceDirect

Computational and Theoretical Chemistry
journal homepage: www.elsevier.com/locate/comptc

Singlet–triplet (S0 ? T1) excitation energies of the [4 n] rectangular graphene
nanoribbon series (n = 2–6): A comparative theoretical study
Sierra Rayne a,⇑, Kaya Forest b
a

Chemologica Research, PO Box 74, 318 Rose Street, Mortlach, Saskatchewan, Canada S0H 3E0
Department of Environmental Engineering, Saskatchewan Institute of Applied Science and Technology, Palliser Campus, PO Box 1420, 600 6th Avenue NW, Moose Jaw,
Saskatchewan, Canada S6H 4R4

b

a r t i c l e

i n f o

Article history:
Received 4 September 2011
Received in revised form 17 September 2011
Accepted 20 September 2011
Available online 29 September 2011

Keywords:
Rectangular graphene nanoribbons
Singlet–triplet excitation energies
Density functional methods
Theoretical benchmarking

a b s t r a c t
Singlet–triplet (S0 ? T1) well-to-well (WWES–T) and vertical (VES–T) excitation energies of the [4 n]
rectangular graphene nanoribbon series (n = 2–6) were estimated using various semiempirical,
Hartree–Fock (HF), density functional (DFT), and second order Moller–Plesset perturbation theory methods with the assumption of a closed-shell singlet state. Significant model chemistry dependent variability
in theoretically obtained WWES–T/VES–T is evident for the rectangular graphene nanoribbons. With the
exception of the B2PLYP density functional (which, along with the mPW2PLYP functional, combines exact
HF exchange with an MP2-like correlation to the DFT calculation), all DFT, semiempirical, and HF methods investigated predict the onset of a negative WWES–T/VES–T (ground state triplet) starting somewhere
between the [4 3] through [4 6] derivatives, with most functionals predicting a transition from a singlet to triplet ground state between the [4 4] and [4 5] rectangular graphene nanoribbons. Consistent
with previous work on the n-acene series, MP2 WWES–T/VES–T estimates have a significant positive systematic bias and HF estimates have substantial negative systematic biases. Extrapolation of the B2PLYP
results, which are in excellent agreement with prior FPA-QZ VES–T estimates, for any [m n] rectangular
graphene nanoribbon derivatives predicts a vanishingly small singlet–triplet gap at the polymeric limit
(m ? 1 and/or n ? 1).
Ó 2011 Elsevier B.V. All rights reserved.

Graphene nanoribbons are thin strips of graphene that can be
viewed as unrolled single-walled carbon nanotubes. Based on a
variety of theoretical and experimental investigations, these
materials are foreseen as promising building blocks in nanoelectronics [1–10]. In recent work, Deleuze and co-workers have
employed high-level calculations with theoretical justifications to
show [11–13] that members of the [2 n; n = 1–11] and [4 n;
n = 2–6] graphene nanoribbon series are ground state closed shell
singlets with positive singlet–triplet excitation energies (ES–T) that
are expected to approach zero as n ? 1. Our prior study [14]
examined the ES–T of the [2 n] nanoribbon series (n = 1–10) from
benzene to decacene using various levels of Hartree–Fock, density
functional, Moller–Plesset perturbation, composite, coupled cluster, and quadratic configuration interaction theory and a range of
basis set sizes. We found that only the B2PLYP [15] density functional (which, along with the mPW2PLYP [16] functional, combines
exact HF exchange with an MP2-like correlation to the DFT calculation), appears to correctly predict positive ES–T for the n-acene

⇑ Corresponding author. Tel.: +1 306 690 0573.
E-mail address: rayne.sierra@gmail.com (S. Rayne).
2210-271X/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.comptc.2011.09.021

series out to a vanishingly small singlet–triplet gap at the polymeric limit.
In the present communication, we extend our previous ES–T
findings to the [4 n] rectangular graphene nanoribbon series
(n = 2–6; Fig. 1). All singlet state calculations presented herein
used a closed-shell consistent with the conclusions from Hajgato
et al. [11,12] and Huzak et al. [13] and the naming conventions
from Hod et al. [17]. All calculations employed Gaussian 09
[18]. Molecular structures were visualized using Gabedit 2.2.12
[19] and Avogadro 1.01 (http://avogadro.openmolecules.net/).
Singlet and triplet geometries of the [4 n] rectangular graphene
nanoribbon series were optimized at the B3LYP/6-31G(d) [20–28]
level. Corresponding frequency calculations at this level of theory
confirmed all structures were minima on their respective potential energy surfaces and are absent imaginary frequencies. Subsequent single point energy calculations using a broad range of
semiempirical, Hartree–Fock, density functional, and second order
Moller–Plesset perturbation theory (MP2 [29–33]) methods were
conducted using the cc-pVDZ [34,35] basis set (i.e., x/cc-pVDZ//
B3LYP/6-31G(d); semiempirical calculations are at the x//B3LYP/
6-31G(d) level of theory). Both well-to-well (WWES–T) and

Author's personal copy

164

S. Rayne, K. Forest / Computational and Theoretical Chemistry 977 (2011) 163–167

Fig. 1. Structures of the [4 n] rectangular graphene nanoribbon series (n = 2–6):
(1) perylene [4 2], (2) bisanthene [4 3], (3) tetrabenzo[bc,ef,kl,no]coronene
[4 4], (4) [4 5], and (5) [4 6] .

vertical (VES–T) singlet–triplet (S0–T1) excitation energies were
calculated.
A wide range of model chemistry dependent WWES–T and
VES–T were obtained, varying from +12.7 to +83.3/+26.2 to
+90.5 (WWES–T/VES–T) kcal/mol for 1, 22.5 to +80.0/ 13.2 to
+85.4 kcal/mol for 2, 52.5 to +92.5/ 46.7 to +96.0 kcal/mol
for 3, 75.4 to +116.3/ 72.2 to +118.7 for 4, and 93.0 to
+156.5/ 91.4 to +158.4 kcal/mol for 5. In all cases, HF estimates
represent WWES–T/VES–T minima and MP2 values represent the
corresponding maxima. Interestingly, while all semiempirical,
Hartree–Fock, and density functional methods exhibited declining WWES–T/VES–T with increasing [4 n] nanoribbon size, MP2
calculations suggest the opposite trend, and (after a slight decline of several kcal/mol between the [4 2] and [4 3] members) increase significantly from the [4 3] through [4 6]
derivatives.
Using a regression of high-level FPA-QZ VES–T estimates for the
[2 n; n = 1–11] nanoribbon series against their B3LYP/cc-pVDZ
VES–T and HOMO–LUMO band gap estimates on the [4 n;
n = 2–6] nanoribbon series, Huzak et al. [13] provided the following range of FPA-QZ extrapolated VES–T estimates (values in kcal/
mol): [4 2], 45.9–48.4; [4 3], 25.0–25.5; [4 4], 10.9–12.2;
[4 5], 2.6–4.8; and [4 6], 1.4 to 1.2. Similar to our prior work
on the [2 n; n = 1–10] nanoribbon series [14], the only model
chemistry examined that is in both quantitative and qualitative
agreement with the FPA-QZ VES–T estimates is the B2PLYP functional. Taking the average of the two FPA-QZ VES–T estimates for
each compound given by Huzak et al. [13], the mean signed, mean
absolute, and root mean squared deviations between our
B2PLYP(D)/cc-pVDZ//B3LYP/6-31G(d) VES–T estimates and the
FPA-QZ data of Huzak et al. [13] are 1.5, 1.9, and 2.2 kcal/mol,
respectively. Consistent with the FPA-QZ predictions of Huzak
et al. [13], the B2PLYP(D)/cc-pVDZ//B3LYP/6-31G(d) calculations
predict a vanishingly small (but never negative) singlet–triplet
energy gap for the [4 n] rectangular graphene nanoribbons as
n ? 1.
Although the WWES–T/VES–T prediction performance of the
mPW2PLYP functional was not as satisfactory as its B2PLYP counterpart, the mPW2PLYP functional is the only other model chemistry examined that predicts near-zero (and not substantially
negative [e.g., semiempirical, HF, all other DFT methods] or excessively positive [e.g., MP2]) WWES–T/VES–T for the larger [4 n]
graphene nanoribbons. No significant difference in WWES–T or
VES–T was observed by including the empirical dispersion correction [36] on either the B2PLYP or mPW2PLYP functionals. Frequency calculations conducted at the B3LYP/6-31G(d) level

indicate that WWES–T at this level of theory ([4 2], 32.9;
[4 3], 11.1; [4 4], 2.4; [4 5], 9.9; and [4 6], 13.6; values in kcal/mol) are about 1 kcal/mol lower than the corresponding
adiabatic singlet–triplet excitation energies (AES–T; [4 2], 34.9;
[4 3], 12.5; [4 4], 1.4; [4 5], 9.1; and [4 6], 12.8; values in kcal/mol). Thus, WWES–T at the B2PLYP(D)/cc-pVDZ//
B3LYP/6-31G(d) level can be reasonably taken to approximate
AES–T, with AES–T expected to be about 1 kcal/mol higher than
the WWES–T.
In our previous theoretical ES–T work on the [n]acene series
from benzene through decacene [14], we found that a single
exponential decay function having a y-axis offset of the general
form y = ae bx + c (where a, b, and c are constants, x is the number of acene units, and y is the ES–T) provided high quality
regression fits (i.e., r > 0.996, but more typically r 0.9999)
across a wide range of density functionals, as well as the HF
and MPn (n = 2–4) methods. We conducted similar regression
analyses on the estimated WWES–T/VES–T for the [4 n;
n = 2–6] rectangular graphene nanoribbon series (Table 1).
Because the predicted WWES–T/VES–T values increase with
increasing nanoribbon length using the MP2 method, this model
chemistry is not amenable to such a statistical analysis. Instead,
the observed WWES–T/VES–T trend for the [4 n] rectangular
graphene nanoribbon series at the MP2 level suggests that
WWES–T/VES–T ? 1 as n ? 1.
The B2PLYP(D) and mPW2PLYP(D) WWES–T/VES–T are also not
best fit using such a single exponential function (Fig. 2). These
functionals plateau at WWES–T/VES–T of about 0.0/1.0 and 3.5/
2.5 kcal/mol, respectively, and the single exponential decays
likely fail to account for this lower plateau apparently reached
between n = 5–6. While the regression approach predicts
WWES–T/VES–T of 2.9 ± 1.8/ 2.9 ± 1.9 and 6.8 ± 1.8/ 7.1 ±
2.1 kcal/mol (error bars are asymptotic standard errors) for the
B2PLYP(D) and mPW2PLYP(D) functionals, respectively, at
the polymeric limit as n ? 1, these are likely underestimates of
the true method performance by several kcal/mol each. Based
on the nature of the WWES–T/VES–T trends for these two model
chemistries between n = 4–6, the respective polymeric limit
WWES–T/VES–T for the B2PLYP(D) and mPW2PLYP(D) functionals
on the [4 n] rectangular graphene nanoribbon series are probably about 0 and 3 kcal/mol.
By comparison, the other model chemistries considered display highly model chemistry dependent polymeric limit extrapolated WWES–T/VES–T (excluding the MP2 method, which – as
discussed above – predicts WWES–T/VES–T ? 1 as n ? 1) ranging
between about 9 to 295 and 10 to 245 kcal/mol, respectively. Consequently, extrapolation of the B2PLYP results from
the current work and Ref. [14], in conjunction with the findings
from Deleuze and co-workers [11–13], suggests a vanishingly
small singlet–triplet gap at the polymeric limit (m ? 1 and/or
n ? 1) for any [m n] rectangular graphene nanoribbon
derivative.

Acknowledgements
This work was made possible by the facilities of the Western
Canada Research Grid (WestGrid: Project 100185), the Shared
Hierarchical Academic Research Computing Network (SHARCNET:
Project sn4612), and Compute/Calcul Canada.

Appendix A. Supplementary material
Supplementary data associated with this article can be found, in
the online version, at doi:10.1016/j.comptc.2011.09.021.

a

b

[29–33]
[15,36]
[15]
[16,36]
[16]
[48–50]
[48–50]
[51–53]
[51,52,54,55]
[56–58]
[56–58]
[59]
[60]
[21,22,61]
[53]
[20–22]
[20,62]
[63]
[20,64–66]
[67]
[68–74]
[21,22,75]
[21,22,67,76]
[54,55,77]
[60]
[54,55,75]
[78,79]
[60]
[80]
[21,22,48–50,81]
[82]
[21,22,48–50,83]
[84]
[85–88]
[89]
[42]
[43–47]
[40,41]
[90]
[37–39]
[90]

MP2
B2PLYPD
B2PLYP
mPW2PLYPD
mPW2PLYP
HFS
XAlpha
OTPSS
OPBE
HCTH/147
HCTH/407
VSXC
M06L
O3LYP
TPSSh
B3LYP
B3P86
X3LYP
B3PW91
B1B95
HSE06
mPW1LYP
B1LYP
PBE0
M06
mPW1PBE
B98
M062X
M05
BHandH
M06HF
BHandHLYP
CAM-B3LYP
LC-wPBE
wB97XD
PM6
PDDG
PM3
wB97X
AM1
wB97
HF

83.3
40.1
40.1
39.7
39.7
32.2
32.7
33.3
33.3
32.8
32.9
32.6
33.3
34.4
33.1
34.5
34.4
34.5
34.1
36.9
33.8
34.4
34.3
34.2
33.9
34.0
34.9
41.7
30.7
36.4
49.5
34.0
36.6
36.3
37.9
19.7
18.3
18.1
38.4
16.4
38.4
12.7

80.0
18.0
18.0
16.9
16.9
12.5
12.5
13.1
13.0
12.8
12.8
12.0
12.4
12.9
11.5
12.2
11.8
12.0
11.6
13.4
10.9
11.5
11.4
10.9
11.0
10.7
12.3
15.8
7.6
9.6
19.6
7.7
10.7
7.2
11.6
1.3
5.0
5.2
10.4
6.9
9.6
22.5

[4 3]

92.5
5.5
5.4
3.6
3.5
1.5
1.2
1.7
1.6
1.5
1.5
0.2
0.2
0.1
1.4
1.6
2.1
2.0
2.3
1.4
3.2
3.0
3.1
3.7
3.6
4.0
1.7
1.6
7.4
8.5
2.1
10.5
7.3
14.4
6.8
18.7
23.3
23.7
10.0
25.6
11.8
52.5

[4 4]

116.3
0.3
0.3
2.4
2.5
4.0
4.3
4.0
4.1
4.2
4.2
5.8
6.0
6.7
8.3
9.3
9.8
9.9
10.0
9.8
10.9
11.2
11.3
12.0
12.0
12.2
9.6
12.3
15.9
19.6
16.9
21.7
18.5
29.2
18.8
32.3
37.0
37.5
23.6
39.8
26.4
75.4

[4 5]
156.5
0.1
0.1
3.3
3.3
5.5
6.1
6.4
6.5
6.6
6.7
7.9
8.7
9.8
11.4
12.9
13.4
13.6
13.7
13.9
14.5
15.2
15.3
15.9
16.1
16.2
16.9
17.9
20.2
25.4
25.7
27.7
36.0
38.2
40.8
42.4
47.0
47.7
49.2
50.6
53.3
93.0

[4 6]
b
n/aa
n/ab
n/ab
n/ab
n/ab
0.682 ± 0.050
0.678 ± 0.043
0.645 ± 0.028
0.648 ± 0.028
0.642 ± 0.028
0.641 ± 0.025
0.647 ± 0.040
0.620 ± 0.030
0.591 ± 0.027
0.589 ± 0.028
0.553 ± 0.028
0.558 ± 0.027
0.546 ± 0.027
0.553 ± 0.026
0.534 ± 0.026
0.563 ± 0.028
0.533 ± 0.027
0.533 ± 0.027
0.540 ± 0.028
0.524 ± 0.027
0.539 ± 0.027
0.447 ± 0.039
0.468 ± 0.025
0.517 ± 0.028
0.467 ± 0.026
0.379 ± 0.021
0.452 ± 0.028
0.213 ± 0.095
0.364 ± 0.022
0.128 ± 0.117
0.235 ± 0.013
0.276 ± 0.009
0.271 ± 0.010
0.089 ± 0.117
0.254 ± 0.008
0.077 ± 0.113
0.228 ± 0.015

WWES–T
[4 1]
a
n/aa
n/ab
n/ab
n/ab
n/ab
160.5 ± 13.7
163.6 ± 11.9
157.6 ± 7.5
158.7 ± 7.3
155.5 ± 7.2
155.8 ± 6.4
162.1 ± 10.8
160.0 ± 7.9
160.6 ± 6.9
161.1 ± 7.1
162.5 ± 6.8
165.1 ± 6.9
163.2 ± 6.8
163.8 ± 6.5
169.2 ± 6.5
168.0 ± 7.3
165.0 ± 6.6
165.0 ± 6.6
168.5 ± 7.0
164.1 ± 6.5
168.6 ± 6.9
150.1 ± 7.4
181.3 ± 6.1
165.4 ± 6.7
187.8 ± 6.5
207.3 ± 4.3
184.1 ± 6.5
187.1 ± 20.8
203.0 ± 4.0
242.6 ± 115.3
163.9 ± 1.7
170.3 ± 0.9
171.6 ± 0.9
335.0 ± 292.8
175.0 ± 0.9
387.0 ± 400.2
280.2 ± 3.8
1
n/ab
n/ab
n/ab
n/ab
8.7 ± 1.1
9.4 ± 0.9
10.0 ± 0.7
10.0 ± 0.7
10.2 ± 0.7
10.3 ± 0.6
11.7 ± 1.0
12.9 ± 0.8
14.7 ± 0.9
16.4 ± 0.9
19.1 ± 1.1
19.5 ± 1.1
20.1 ± 1.1
19.9 ± 1.0
21.1 ± 1.2
20.6 ± 1.1
22.3 ± 1.2
22.4 ± 1.2
22.9 ± 1.2
23.6 ± 1.3
23.2 ± 1.2
26.6 ± 2.5
29.3 ± 1.8
28.0 ± 1.4
37.3 ± 1.9
47.5 ± 2.8
40.4 ± 2.1
86.5 ± 37.0
61.4 ± 3.1
151.3 ± 134.9
82.5 ± 3.6
79.7 ± 1.9
81.6 ± 2.2
243.7 ± 313.2
88.8 ± 2.0
294.9 ± 420.4
164.6 ± 7.6

c
n/aa
n/ab
n/ab
n/ab
n/ab
0.9995
0.9996
0.9998
0.9998
0.9998
0.9999
0.9997
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9996
0.9998
0.9998
0.9998
0.9999
0.9998
0.997
0.9999
0.995
0.99995
0.99997
0.99997
0.995
0.99998
0.996
0.99993

r
90.5
46.9
46.9
46.7
46.7
38.7
38.3
39.3
39.3
38.8
38.7
39.1
39.6
41.0
40.0
41.6
41.3
41.7
41.2
43.8
40.9
41.8
41.8
41.5
41.1
41.3
42.2
49.7
38.3
44.3
58.9
42.7
45.1
46.4
46.5
31.0
30.5
30.0
47.9
28.3
48.7
26.2

[4 2]
85.4
22.4
22.4
21.6
21.6
16.7
16.4
17.1
17.0
16.8
16.7
16.1
16.5
17.2
16.1
16.8
16.5
16.7
16.3
18.0
15.8
16.4
16.3
15.9
15.9
15.7
17.1
21.2
12.9
15.4
26.0
13.8
16.5
14.4
17.5
6.6
4.2
3.8
17.0
1.9
16.8
13.2

[4 3]
96.0
8.2
8.2
6.5
6.5
3.9
3.6
4.1
4.0
3.8
3.8
2.5
2.7
2.7
1.3
1.2
0.8
0.8
0.6
1.5
0.2
0.1
0.1
0.6
0.5
0.8
1.1
1.8
4.0
4.7
2.0
-6.6
3.5
9.7
3.0
13.7
17.2
17.7
5.6
19.9
7.1
46.7

[4 4]
118.7
1.8
1.8
0.8
0.8
2.9
3.2
2.8
2.9
3.1
3.1
4.7
4.8
5.4
6.9
7.8
8.3
8.4
8.5
8.3
9.3
9.7
9.7
10.3
10.3
10.5
8.1
10.3
14.1
17.3
14.4
19.5
16.3
26.3
16.5
29.3
33.3
33.9
21.0
36.5
23.5
72.2

[4 5]
158.4
0.9
0.9
2.4
2.4
5.1
5.7
6.0
6.1
6.2
6.3
7.5
8.2
9.3
10.8
12.2
12.7
12.9
12.9
13.1
13.7
14.4
14.5
15.1
15.2
15.3
16.2
16.8
19.2
24.1
24.1
26.4
34.7
36.3
39.4
40.5
44.7
45.5
47.5
48.6
51.5
91.4

[4 6]

The increasing trend of WWES–T and VES–T suggests infinitely large ES–T as n ? 1, and is not amenable to a regression analysis of the form y = ae bx + c.
As shown in Fig. 2, the WWES–T/VES–T trends for the B2PLYP(D) and mPW2PLYP(D) functionals are not well modeled by a regression analysis of the form y = ae bx + c.

Ref.

Method

[4 2]
n/aa
n/ab
n/ab
n/ab
n/ab
170.9 ± 12.5
168.0 ± 10.4
166.1 ± 6.2
167.1 ± 6.1
164.9 ± 7.1
164.2 ± 6.1
175.2 ± 10.3
171.2 ± 7.1
173.1 ± 6.2
174.1 ± 7.1
177.7 ± 6.5
177.8 ± 7.4
177.8 ± 8.0
177.8 ± 6.9
178.0 ± 11.3
178.0 ± 8.2
178.2 ± 11.5
178.3 ± 9.1
178.4 ± 10.1
178.4 ± 7.9
178.5 ± 8.2
178.5 ± 12.5
198.6 ± 6.2
179.4 ± 7.0
201.6 ± 6.7
228.5 ± 4.9
201.9 ± 7.1
204.5 ± 17.0
224.5 ± 4.8
248.4 ± 80.0
188.0 ± 2.3
196.7 ± 1.9
197.7 ± 2.0
319.9 ± 176.5
201.7 ± 2.1
357.0 ± 224.7
310.8 ± 4.3

VES–T
[4 1]
a
n/aa
n/ab
n/ab
n/ab
n/ab
0.631 ± 0.044
0.619 ± 0.038
0.598 ± 0.023
0.600 ± 0.023
0.596 ± 0.027
0.594 ± 0.023
0.610 ± 0.036
0.582 ± 0.026
0.557 ± 0.023
0.553 ± 0.027
0.529 ± 0.025
0.530 ± 0.028
0.526 ± 0.031
0.528 ± 0.026
0.506 ± 0.045
0.524 ± 0.031
0.514 ± 0.045
0.510 ± 0.035
0.509 ± 0.040
0.507 ± 0.031
0.504 ± 0.033
0.489 ± 0.051
0.454 ± 0.024
0.487 ± 0.028
0.442 ± 0.027
0.377 ± 0.022
0.433 ± 0.029
0.223 ± 0.084
0.358 ± 0.024
0.148 ± 0.106
0.247 ± 0.017
0.267 ± 0.017
0.261 ± 0.017
0.110 ± 0.105
0.251 ± 0.015
0.100 ± 0.101
0.234 ± 0.017

b

1
n/ab
n/ab
n/ab
n/ab
9.5 ± 1.3
10.3 ± 1.1
10.9 ± 0.8
10.9 ± 0.7
11.1 ± 0.9
11.2 ± 0.8
12.5 ± 1.2
13.7 ± 1.0
15.7 ± 1.0
17.5 ± 1.1
20.0 ± 1.2
20.2 ± 1.4
20.6 ± 1.5
20.7 ± 1.3
21.4 ± 2.5
21.6 ± 1.6
22.3 ± 2.4
22.8 ± 1.9
23.4 ± 2.2
23.7 ± 1.7
24.1 ± 1.8
24.4 ± 3.1
30.3 ± 2.0
29.3 ± 1.8
38.8 ± 2.4
48.4 ± 3.3
41.9 ± 2.7
86.6 ± 32.7
63.0 ± 4.0
139.6 ± 100.0
83.4 ± 5.1
84.8 ± 4.4
87.2 ± 4.6
210.0 ± 197.3
93.7 ± 4.7
245.0 ± 245.6
168.1 ± 8.9

c

n/aa
n/ab
n/ab
n/ab
n/ab
0.9996
0.9997
0.9999
0.9999
0.9998
0.9999
0.9997
0.9998
0.9999
0.9998
0.9999
0.9998
0.9998
0.9998
0.9995
0.9997
0.9995
0.9997
0.9996
0.9997
0.9997
0.9993
0.9998
0.9998
0.9998
0.9999
0.9998
0.998
0.9998
0.996
0.9999
0.9999
0.99991
0.996
0.99992
0.996
0.99991

r

Table 1
Well-to-well (WWES–T) and vertical (VES–T) singlet–triplet excitation energies at 298.15 K and 1 atm for the [4 n] rectangular graphene nanoribbon series (n = 2–6) using various model chemistries at the x/cc-pVDZ//B3LYP/6-31G(d)
[20–28,34,35] level of theory (the four semiempirical methods [AM1 [37-39], PM3 [40,41], PM6 [42], and PDDG [43–47]] considered are at the x//B3LYP/6-31G(d) [20–28] level of theory). Values are in kcal/mol. Regression analysis
parameters using a single exponential decay function having a y-axis offset of the general form y = ae bx + c (x = n, y(x) = WWES–T/VES–T in kcal/mol; c = WWES–T/VES–T at the polymeric limit) are also provided.

Author's personal copy

S. Rayne, K. Forest / Computational and Theoretical Chemistry 977 (2011) 163–167
165

Author's personal copy

166

S. Rayne, K. Forest / Computational and Theoretical Chemistry 977 (2011) 163–167

Fig. 2. Trends in calculated: (a) WWES–T and (b) VES–T for the B2PLYP(D) (solid line) and mPW2PLYP(D) (dashed line) functionals using a single exponential decay function
having a y-axis offset of the general form y = ae bx + c.

References
[1] M. Fujita, K. Wakabayashi, K. Nakada, K. Kusakabe, Peculiar localized state at
zigzag graphite edge, J. Phys. Soc. Jpn. 65 (1996) 1920–1923.
[2] K. Nakada, M. Fujita, G. Dresselhaus, M. Dresselhaus, Edge state in graphene
ribbons: nanometer size effect and edge shape dependence, Phys. Rev. B 54
(1996) 17954–17961.
[3] K. Wakabayashi, M. Fujita, H. Ajiki, M. Sigrist, Electronic and magnetic
properties of nanographite ribbons, Phys. Rev. B 59 (1999) 8271–8282.
[4] V. Barone, O. Hod, G.E. Scuseria, Electronic structure and stability of
semiconducting graphene nanoribbons, Nano Lett. 6 (2006) 2748–2754.
[5] M.Y. Han, B. Ozyilmaz, Y. Zhang, P. Kim, Energy band-gap engineering of
graphene nanoribbons, Phys. Rev. Lett. 98 (2007) 206805–206808.
[6] Y.W. Son, M.L. Cohen, S.G. Louie, Half-metallic graphene nanoribbons, Nature
444 (2006) 347–349.
[7] Y.W. Son, M.L. Cohen, S.G. Louie, Energy gaps in graphene nanoribbons, Phys.
Rev. Lett. 97 (2006) 216803–216806.
[8] X. Li, X. Wang, L. Zhang, S. Lee, H. Dai, Chemically derived, ultrasmooth
graphene nanoribbon semiconductors, Science 319 (2008) 1229–1232.
[9] W.Y. Kim, K.S. Kim, Prediction of very large values of magnetoresistance in a
graphene nanoribbon device, Nat. Nanotechnol. 3 (2008) 408–412.
[10] F. Schwierz, Graphene transistors, Nat. Nanotechnol. 5 (2010) 487–496.
[11] B. Hajgato, D. Szieberth, P. Geerlings, F. De Proft, M.S. Deleuze, A benchmark
theoretical study of the electronic ground state and of the singlet–triplet split
of benzene and linear acenes, J. Chem. Phys. 131 (2009) 224321–224338.
[12] B. Hajgato, M. Huzak, M.S. Deleuze, Focal point analysis of the singlet–triplet
energy gap of octacene and larger acenes, J. Phys. Chem. A 115 (2011)
9282–9293.
[13] M. Huzak, M.S. Deleuze, B. Hajgato, Half-metallicity and spin-contamination of
the electronic ground state of graphene nanoribbons and related systems: an
impossible compromise?, J Chem. Phys. 135 (2011) 104704–104722.
[14] S. Rayne, K. Forest, A comparison of density functional theory (DFT) methods
for estimating the singlet–triplet (S0–T1) excitation energies of benzene and
polyacenes, Comput. Theor. Chem. (2011), doi:10.1016/j.comptc.2011.08.010.
[15] S. Grimme, Semiempirical hybrid density functional with perturbative secondorder correlation, J. Chem. Phys. 124 (2006) 034108.
[16] T. Schwabe, S. Grimme, Towards chemical accuracy for the thermodynamics of
large molecules: new hybrid density functionals including non-local
correlation effects, Phys. Chem. Chem. Phys. 8 (2006) 4398.
[17] O. Hod, V. Barone, G.E. Scuseria, Half-metallic graphene nanodots: a
comprehensive first-principles theoretical study, Phys. Rev. B 77 (2008)
354111–354116.
[18] M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman,
G. Scalmani, V. Barone, B. Mennucci, G.A. Petersson, et al., Gaussian 09,
Revision B.01, Gaussian, Inc., Wallingford, CT, 2010.
[19] A.R. Allouche, Gabedit: a graphical user interface for computational chemistry
softwares, J. Comput. Chem. 32 (2011) 174–182.
[20] A.D. Becke, Density-functional thermochemistry. III. The role of exact
exchange, J. Chem. Phys. 98 (1993) 5648–5652.
[21] C. Lee, W. Yang, R.G. Parr, Development of the Colle–Salvetti correlation energy
formula into a functional of the electron density, Phys. Rev. B 37 (1988)
785–789.
[22] B. Miehlich, A. Savin, H. Stoll, H. Preuss, Results obtained with the correlation
energy density functionals of Becke and Lee, Yang and Parr, Chem. Phys. Lett.
157 (1989) 200–206.
[23] R. Ditchfield, W.J. Hehre, J.A. Pople, Self-consistent molecular-orbital methods.
IX. An extended Gaussian-type basis for molecular-orbital studies of organic
molecules, J. Chem. Phys. 54 (1971) 724–728.
[24] W.J. Hehre, R. Ditchfield, J.A. Pople, Self-consistent molecular orbital methods.
XII. Further extensions of Gaussian-type basis sets for use in molecular orbital
studies of organic molecules, J. Chem. Phys. 56 (1972) 2257–2261.
[25] P.C. Hariharan, J.A. Pople, Accuracy of AHn equilibrium geometries by single
determinant molecular orbital theory, Mol. Phys. 27 (1974) 209–214.

[26] P.C. Hariharan, J.A. Pople, The influence of polarization functions on molecular
orbital hydrogenation energies, Theor. Chem. Acc. 28 (1973) 213–222.
[27] M.S. Gordon, The isomers of silacyclopropane, Chem. Phys. Lett. 76 (1980)
163–168.
[28] M.M. Francl, Self-consistent molecular orbital methods. XXIII. A polarizationtype basis set for second-row elements, J. Chem. Phys. 77 (1982) 3654.
[29] M. Head-Gordon, J.A. Pople, M.J. Frisch, MP2 energy evaluation by direct
methods, Chem. Phys. Lett. 153 (1988) 503–506.
[30] S. Saebo, J. Almlof, Avoiding the integral storage bottleneck in LCAO
calculations of electron correlation, Chem. Phys. Lett. 154 (1989) 83–89.
[31] M.J. Frisch, M. Head-Gordon, J.A. Pople, Direct MP2 gradient method, Chem.
Phys. Lett. 166 (1990) 275–280.
[32] M.J. Frisch, M. Head-Gordon, J.A. Pople, Semi-direct algorithms for the MP2
energy and gradient, Chem. Phys. Lett. 166 (1990) 281–289.
[33] M. Head-Gordon, T. Head-Gordon, Analytic MP2 frequencies without fifth
order storage: theory and application to bifurcated hydrogen bonds in the
water hexamer, Chem. Phys. Lett. 220 (1994) 122–128.
[34] T.H. Dunning, Gaussian basis sets for use in correlated molecular calculations.
I. The atoms boron through neon and hydrogen, J. Chem. Phys. 90 (1989)
1007–1023.
[35] R.A. Kendall, T.H. Dunning, R.J. Harrison, Electron affinities of the first-row
atoms revisited. Systematic basis sets and wave functions, J. Chem. Phys. 96
(1992) 6796–6806.
[36] T. Schwabe, S. Grimme, Double-hybrid density functionals with long-range
dispersion corrections: higher accuracy and extended applicability, Phys.
Chem. Chem. Phys. 9 (2007) 3397–3406.
[37] M.J.S. Dewar, W. Thiel, Ground states of molecules. 38. The MNDO method.
Approximations and parameters, J. Am. Chem. Soc. 99 (1977) 4899–4907.
[38] M.J.S. Dewar, M.L. McKee, H.S. Rzepa, MNDO parameters for third period
elements, J. Am. Chem. Soc. 100 (1978) 3607.
[39] M.J.S. Dewar, E.G. Zoebisch, E.F. Healy, J.J.P. Stewart, Development and use of
quantum mechanical molecular models. 76. AM1: a new general purpose
quantum mechanical molecular model, J. Am. Chem. Soc. 107 (1985) 3902–
3909.
[40] J.J.P. Stewart, Optimization of parameters for semiempirical methods I.
Method, J. Comput. Chem. 10 (1989) 209–220.
[41] J.J.P. Stewart, Optimization of parameters for semiempirical methods II.
Applications, J. Comput. Chem. 10 (1989) 221–264.
[42] J. Stewart, Optimization of parameters for semiempirical methods V:
modification of NDDO approximations and application to 70 elements, J.
Mol. Model. 13 (2007) 1173–1213.
[43] M.P. Repasky, J. Chandrasekhar, W.L. Jorgensen, PDDG/PM3 and PDDG/MNDO:
improved semiempirical methods, J. Comput. Chem. 23 (2002) 1601–1622.
[44] I. Tubert-Brohman, C.R.W. Guimaraes, M.P. Repasky, W.L. Jorgensen, Extension
of the PDDG/PM3 and PDDG/MNDO semiempirical molecular orbital methods
to the halogens, J. Comput. Chem. 25 (2004) 138–150.
[45] I. Tubert-Brohman, C.R.W. Guimaraes, W.L. Jorgensen, Extension of the PDDG/
PM3 semiempirical molecular orbital method to sulfur, silicon, and
phosphorus, J. Chem. Theory Comput. 1 (2005) 817–823.
[46] J. Tirado-Rives, W.L. Jorgensen, Performance of B3LYP density functional
methods for a large set of organic molecules, J. Chem. Theory Comput. 4 (2008)
297–306.
[47] K.W. Sattelmeyer, J. Tirado-Rives, W.L. Jorgensen, Comparison of SCC-DFTB
and NDDO-based semiempirical molecular orbital methods for organic
molecules, J. Phys. Chem. A 110 (2006) 13551–13559.
[48] P. Hohenberg, W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136 (1964)
B864–B871.
[49] W. Kohn, L.J. Sham, Self-consistent equations including exchange and
correlation effects, Phys. Rev. 140 (1965) A1133–A1138.
[50] J.C. Slater, The Self-Consistent Field for Molecular and Solids, Quantum Theory
of Molecular and Solids, vol. 4, McGraw-Hill, New York, 1974.
[51] N.C. Handy, A.J. Cohen, Left–right correlation energy, Mol. Phys. 99 (2001)
403–412.

Author's personal copy

S. Rayne, K. Forest / Computational and Theoretical Chemistry 977 (2011) 163–167
[52] W.M. Hoe, A. Cohen, N.C. Handy, Assessment of a new local exchange
functional OPTX, Chem. Phys. Lett. 341 (2001) 319–328.
[53] J.M. Tao, J.P. Perdew, V.N. Staroverov, G.E. Scuseria, Climbing the density
functional ladder: nonempirical meta-generalized gradient approximation
designed for molecules and solids, Phys. Rev. Lett. 91 (2003) 146401.
[54] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made
simple, Phys. Rev. Lett. 77 (1996) 3865–3868.
[55] J.P. Perdew, K. Burke, M. Ernzerhof, Errata: generalized gradient approximation
made simple, Phys. Rev. Lett. 78 (1997) 1396.
[56] F.A. Hamprecht, A.J. Cohen, D.J. Tozer, N.C. Handy, Development and
assessment of new exchange-correlation functionals, J. Chem. Phys. 109
(1998) 6264–6271.
[57] A.D. Boese, N.L. Doltsinis, N.C. Handy, M. Sprik, New generalized gradient
approximation functionals, J. Chem. Phys. 112 (2000) 1670–1678.
[58] A.D. Boese, N.C. Handy, A new parametrization of exchange-correlation
generalized gradient approximation functionals, J. Chem. Phys. 114 (2001)
5497–5503.
[59] T. Van Voorhis, G.E. Scuseria, A novel form for the exchange-correlation energy
functional, J. Chem. Phys. 109 (1998) 400–410.
[60] Y. Zhao, D. Truhlar, The M06 suite of density functionals for main group
thermochemistry, thermochemical kinetics, noncovalent interactions, excited
states, and transition elements: two new functionals and systematic testing of
four M06-class functionals and 12 other functionals, Theor. Chem. Acc. 120
(2008) 215–241.
[61] A.J. Cohen, N.C. Handy, Dynamic correlation, Mol. Phys. 99 (2001) 607–615.
[62] J.P. Perdew, Density-functional approximation for the correlation energy of the
inhomogeneous electron gas, Phys. Rev. B 33 (1986) 8822–8824.
[63] X. Xu, W.A. Goddard, The X3LYP extended density functional for accurate
descriptions of nonbond interactions, spin states, and thermochemical
properties, Proc. Natl. Acad. Sci. USA 101 (2004) 2673–2677.
[64] J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C.
Fiolhais, Atoms, molecules, solids, and surfaces: applications of the generalized
gradient approximation for exchange and correlation, Phys. Rev. B 46 (1992)
6671.
[65] J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C.
Fiolhais, Erratum: atoms, molecules, solids, and surfaces: applications of the
generalized gradient approximation for exchange and correlation, Phys. Rev. B
48 (1993) 4978.
[66] J.P. Perdew, K. Burke, Y. Wang, Generalized gradient approximation for the
exchange-correlation hole of a many-electron system, Phys. Rev. B 54 (1996)
16533.
[67] A.D. Becke, Density-functional thermochemistry. IV. A new dynamical
correlation functional and implications for exact-exchange mixing, J. Chem.
Phys. 104 (1996) 1040–1046.
[68] J. Heyd, G. Scuseria, Efficient hybrid density functional calculations in solids:
assessment of the Heyd–Scuseria–Ernzerhof screened Coulomb hybrid
functional, J. Chem. Phys. 121 (2004) 1187–1192.
[69] J. Heyd, G.E. Scuseria, Assessment and validation of a screened Coulomb hybrid
density functional, J. Chem. Phys. 120 (2004) 7274–7280.
[70] J. Heyd, J.E. Peralta, G.E. Scuseria, R.L. Martin, Energy band gaps and lattice
parameters evaluated with the Heyd–Scuseria–Ernzerhof screened hybrid
functional, J. Chem. Phys. 123 (2005) 174101–174108.

167

[71] J. Heyd, G.E. Scuseria, M. Ernzerhof, Erratum: ‘‘Hybrid functionals based on a
screened Coulomb potential’’, J. Chem. Phys. 124 (2006) 219906.
[72] A.F. Izmaylov, G. Scuseria, M.J. Frisch, Efficient evaluation of short-range
Hartree–Fock exchange in large molecules and periodic systems, J. Chem. Phys.
125 (2006) 104103–104110.
[73] A.V. Krukau, O.A. Vydrov, A.F. Izmaylov, G.E. Scuseria, Influence of the
exchange screening parameter on the performance of screened hybrid
functionals, J. Chem. Phys. 125 (2006) 224106–224110.
[74] T.M. Henderson, A.F. Izmaylov, G. Scalmani, G.E. Scuseria, Can short-range
hybrids describe long-range-dependent properties?, J Chem. Phys. 131 (2009)
44108–44116.
[75] C. Adamo, V. Barone, Exchange functionals with improved long-range behavior
and adiabatic connection methods without adjustable parameters: the mPW
and mPW1PW models, J. Chem. Phys. 108 (1998) 664–675.
[76] C. Adamo, V. Barone, Toward reliable adiabatic connection models free from
adjustable parameters, Chem. Phys. Lett. 274 (1997) 242–250.
[77] C. Adamo, V. Barone, Toward reliable density functional methods without
adjustable parameters: the PBE0 model, J. Chem. Phys. 110 (1999) 6158–6169.
[78] A.D. Becke, Density-functional thermochemistry. V. Systematic optimization of
exchange-correlation functionals, J. Chem. Phys. 107 (1997) 8554–8560.
[79] H.L. Schmider, A.D. Becke, Optimized density functionals from the extended
G2 test set, J. Chem. Phys. 108 (1998) 9624–9631.
[80] Y. Zhao, N.E. Schultz, D.G. Truhlar, Exchange-correlation functional with broad
accuracy for metallic and nonmetallic compounds, kinetics, and noncovalent
interactions, J. Chem. Phys. 123 (2005) 194101–194118.
[81] S.H. Vosko, L. Wilk, M. Nusair, Accurate spin-dependent electron liquid
correlation energies for local spin density calculations: a critical analysis, Can.
J. Phys. 58 (1980) 1200–1211.
[82] Y. Zhao, D.G. Truhlar, Comparative DFT study of van der Waals complexes:
rare-gas dimers, alkaline-earth dimers, zinc dimer, and zinc-rare-gas dimers, J.
Phys. Chem. A 110 (2006) 5121–5129.
[83] A.D. Becke, Density-functional exchange-energy approximation with correct
asymptotic behavior, Phys. Rev. A 38 (1988) 3098.
[84] T. Yanai, D.P. Tew, N.C. Handy, A new hybrid exchange-correlation functional
using the Coulomb-attenuating method (CAM-B3LYP), Chem. Phys. Lett. 393
(2004) 51–57.
[85] Y. Tawada, T. Tsuneda, S. Yanagisawa, T. Yanai, K. Hirao, A long-rangecorrected time-dependent density functional theory, J. Chem. Phys. 120 (2004)
8425–8433.
[86] O.A. Vydrov, J. Heyd, A.V. Krukau, G.E. Scuseria, Importance of short-range
versus long-range Hartree–Fock exchange for the performance of hybrid
density functionals, J. Chem. Phys. 125 (2006) 74106.
[87] O.A. Vydrov, G.E. Scuseria, Assessment of a long-range corrected hybrid
functional, J. Chem. Phys. 125 (2006) 234109.
[88] O.A. Vydrov, G.E. Scuseria, J.P. Perdew, Tests of functionals for systems with
fractional electron number, J. Chem. Phys. 126 (2007) 154109.
[89] J.D. Chai, M. Head-Gordon, Long-range corrected hybrid density functionals
with damped atom–atom dispersion corrections, Phys. Chem. Chem. Phys. 10
(2008) 6615–6620.
[90] J.D. Chai, M. Head-Gordon, Systematic optimization of long-range corrected
hybrid density functionals, J. Chem. Phys. 128 (2008) 84106–84115.


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