Electronic and Optical Properties of the Narrowest Armchair Graphene Nanoribbons Studied by Density Functional Methods
Chia-Nan Yeh,A,CPei-Yin Lee,A,Cand Jeng-Da ChaiA,B,D
ADepartment of Physics, National Taiwan University, Taipei 10617, Taiwan.
BCenter for Theoretical Sciences and Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan.
CThese authors contributed equally to this work.
DCorresponding author. Email: [email protected]
In the present study, a series of planar poly(p-phenylene) (PPP) oligomers with n phenyl rings (n¼ 1–20), designated as n-PP, are taken as finite-size models of the narrowest armchair graphene nanoribbons with hydrogen passivation. The singlet-triplet energy gap, vertical ionization potential, vertical electron affinity, fundamental gap, optical gap, and exciton binding energy of n-PP are calculated using Kohn-Sham density functional theory and time-dependent density functional theory with various exchange-correlation density functionals. The ground state of n-PP is shown to be singlet for all the chain lengths studied. In contrast to the lowest singlet state (i.e., the ground state) of n-PP, the lowest triplet state of n-PP and the ground states of the cation and anion of n-PP are found to exhibit some multi-reference character. Overall, the electronic and optical properties of n-PP obtained from the vB97 and vB97X functionals are in excellent agreement with the available experimental data.
Manuscript received: 25 March 2016.
Manuscript accepted: 5 May 2016.
Published online: 26 May 2016.
Introduction
In recent years, graphene, a single layer of carbon atoms tightly packed into a honeycomb lattice, has received considerable attention due to its remarkable properties and technological applications.[1–9]Graphene exhibits high carrier mobility and long spin diffusion length, giving promises for graphene-based electronics and spintronics. However, as graphene has a van- ishing band gap, it cannot be directly adopted for transistor applications. Accordingly, developing methods to open a band gap in graphene is necessary for its practical applications.
To generate a non-vanishing and tunable band gap in graphene, the charge carriers can be confined to quasi-one- dimensional systems, such as graphene nanoribbons (GNRs), long and narrow graphene strips. Consequently, several experi- mental techniques have been developed for synthesizing GNRs.[10–13] Because of their fascinating electronic, optical, and magnetic properties, GNRs have recently gained increasing interest.[14–45]However, the electronic and optical properties of GNRs can be very sensitive to their width, length, edge shape (zigzag, armchair, or chiral), and edge termination. To properly design GNR-based nanodevices, a thorough understanding of the related parameters governing the electronic and optical properties of GNRs is of fundamental and practical significance.
While there has been a growing interest in GNRs, the study of the electronic and optical properties of long-chain GNRs remains very challenging. From the experimental perspectives, the difficulties in the synthesis of long-chain GNRs and their instability following isolation have been attributed to their
radical character. Accordingly, there have been very few reports of experimental data for the properties of long-chain GNRs. From the theoretical perspective, as GNRs belong to p-conjugated systems, they may exhibit multi-reference character in certain circumstances, where conventional single-reference methods may be inadequate. For instance, zigzag GNRs (ZGNRs), which are GNRs with zigzag-shaped edges on both sides, have been exten- sively studied, and long-chain ZGNRs have been found to exhibit polyradical character in their ground states, where the active orbitals are mainly localized at the zigzag edges.[26,29,35–38,42–44]
In contrast to ZGNRs, armchair GNRs (AGNRs), which are GNRs with armchair-shaped edges on both sides, are expected to possess relatively large fundamental gaps. However, the properties of long-chain AGNRs have not been extensively studied, relative to those of long-chain ZGNRs. We believe that a comprehensive understanding of the properties of AGNRs is also essential for the optimal design of GNR-based nanodevices.
For a theoretical study of the electronic and optical properties of AGNRs, density functional methods, such as Kohn-Sham density functional theory (KS-DFT).[46,47] (for ground-state properties) and time-dependent density functional theory (TDDFT)[48](for excited-state properties), are ideal, due to their computational efficiency and reasonable accuracy for large systems.[49–52]
Therefore, in this work, we adopt KS-DFT and TDDFT with various exchange-correlation (XC) density functionals to study the electronic and optical properties of the narrowest AGNRs (NAGNRs) with different lengths. The rest of this paper is Aust. J. Chem. 2016, 69, 960–968
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organized as follows. In the next section, we describe our model systems and computational details. The calculated electronic and optical properties are then compared with the available experimental data and those obtained from high-level ab initio methods. Our conclusions are presented in the last section.
Model Systems and Computational Details
As illustrated in Fig. 1, we adopt a series of planar poly (p-phenylene) (PPP) oligomers with n phenyl rings, designated as n-PP, as finite-size models of the NAGNRs with hydrogen passivation. Note that the number of electrons in n-PP (C6nH4nþ 2) is 40nþ 2, which rapidly increases with an increase in n. Therefore, efficient methods, such as KS-DFT and TDDFT, are highly desirable for the study of long-chain n-PP.
For the KS-DFT and TDDFT calculations, we adopt seven XC density functionals, which can be categorized into three different types of density functionals, such as semilocal functionals,[53] global hybrid functionals,[54] and long-range corrected (LC) hybrid functionals:[55–65]
# semilocal functionals: LDA,[66,67]PBE,[68]and BLYP[69,70]
# global hybrid functionals: PBE0[71]and B3LYP[72,73]
# LC hybrid functionals: vB97[60]and vB97X[60]
for the study of various electronic and optical properties of n-PP (up to n¼ 20), involving
# singlet-triplet energy gap
# vertical ionization potential
# vertical electron affinity
# fundamental gap
# optical gap
# exciton binding energy
Note that vB97 and vB97X have been recently shown to provide excellent performance for a wide range of applications, especially for those closely related to frontier orbital energies.[74,75]
To estimate the electronic and optical properties of n-PP at the polymer limit (n! 1), a fitting function of the form ða þ b=nÞ is adopted for the extrapolation of the calculated and experimental data. Note that this fitting function has been previously adopted to estimate the vertical ionization potential and optical gap of PPP.[76,77]
In addition, the expectation value of the total spin-squared operator h^S2i is adopted as a measure of the degree of spin contamination in KS-DFT. For a system with strong multi- reference character, the value ofh^S2i obtained from KS-DFT with conventional (semilocal, global hybrid, and LC hybrid) density functionals can be significantly different (e.g., more than 10%
difference)[78]from the exact value SðS þ 1Þ, where S can be 0 (singlet), 1/2 (doublet), 1 (triplet), 3/2 (quartet), and so on. For such a system, KS-DFT employing conventional density func- tionals can yield unreliable results. To properly describe strong static correlation in such a system, it may be essential to adopt
multi-reference methods[26,36,38,43] for small-sized systems or thermally-assisted-occupation density functional theory (TAO- DFT)[35,42,44]for medium- to large-sized systems.
All calculations are performed with a development version of Q-Chem 4.0.[79]Results are computed using the 6-31G(d) basis set with the fine grid EML(75,302), consisting of 75 Euler- Maclaurin radial grid points[80]and 302 Lebedev angular grid points.[81]
As there may be more than one way of calculating the electronic and optical properties using KS-DFT and TDDFT, respectively, we briefly describe how these properties are computed as follows.
Singlet-Triplet Energy Gap
The singlet-triplet energy gap (EST) of a neutral molecule is defined as
EST¼ ET& ES; ð1Þ the energy difference between the lowest triplet (T) and singlet (S) states, calculated at the respective optimized geometries.
Vertical Ionization Potential
The vertical ionization potential (IP) of a neutral molecule is defined as
IPð1Þ ¼ EtotalðcationÞ & EtotalðneutralÞ; ð2Þ the energy difference between the cationic and neutral states, calculated at the ground-state geometry of the neutral molecule.
For the exact KS-DFT, the vertical IP of a neutral molecule is identical to the minus HOMO (highest occupied molecular orbital) energy of the neutral molecule[82–87],
IPð2Þ ¼ &2HOMOðneutralÞ: ð3Þ Accordingly, IP(2) is identical to IP(1) for the exact KS-DFT.
For KS-DFT employing approximate XC density functionals, the calculated IP(1) and IP(2) values may differ, reflecting the accuracy of the calculated total energies and HOMO energies, respectively.
Vertical Electron Affinity
The vertical electron affinity (EA) of a neutral molecule is defined as
EAð1Þ ¼ EtotalðneutralÞ & EtotalðanionÞ; ð4Þ the energy difference between the neutral and anionic states, calculated at the ground-state geometry of the neutral molecule.
By comparingEqn 2withEqn 4, the vertical EA of a neutral molecule is identical to the vertical IP of the anion, which is, for the exact KS-DFT, the minus HOMO energy of the anion, calculated at the ground-state geometry of the neutral molecule, EAð2Þ ¼ &2HOMOðanionÞ: ð5Þ In addition, the vertical EA of a neutral molecule is conven- tionally approximated by the minus LUMO (lowest unoccupied molecular orbital) energy of the neutral molecule,
EAð3Þ ¼ &2LUMOðneutralÞ: ð6Þ
n
Fig. 1. Structure of n-PP (C6nH4nþ 2), consisting of n benzene rings.
However, even for the exact KS-DFT, there is a distinct difference between EA(3) and EA(2), due to the derivative discontinuity (Dxc) of the XC density functional:[83,87–97]
EAð3Þ & EAð2Þ ¼ 2HOMOðanionÞ & 2LUMOðneutralÞ ¼ Dxc. Global and LC hybrid functionals, which belong to the generalized Kohn-Sham (GKS) method[98](not pure KS-DFT), effectively incorporate a fraction of Dxc of the XC density functional in KS-DFT. A recent study has shown that the difference between 2HOMOðanionÞ and 2LUMOðneutralÞ is small for LC hybrid functionals.[99]Hence, for LC hybrid functionals, EA(3) should be close to EA(2), the true vertical EA.
Fundamental Gap
The fundamental gap (Eg) of a neutral molecule is defined as Eg¼ IP & EA, the difference between the vertical IP and EA of the neutral molecule. Since there are various ways of computing the vertical IP and EA in KS-DFT, we adopt the following three ways of calculating Eg:
Egð1Þ ¼ IPð1Þ & EAð1Þ ¼ EtotalðcationÞ þ EtotalðanionÞ
& 2EtotalðneutralÞ ð7Þ
Egð2Þ ¼ IPð2Þ & EAð2Þ ¼ 2HOMOðanionÞ & 2HOMOðneutralÞ ð8Þ Egð3Þ ¼ IPð2Þ & EAð3Þ ¼ 2LUMOðneutralÞ & 2HOMOðneutralÞ ð9Þ Note that Egð3Þ is the HOMO-LUMO gap in KS-DFT or the Kohn-Sham (KS) gap. For the exact KS-DFT, while both Egð1Þ and Egð2Þ yield the exact fundamental gap, there is a distinct difference between Egð2Þ and Egð3Þ, due to the Dxcof the XC density functional: Egð2Þ & Egð3Þ ¼ EAð3Þ & EAð2Þ ¼ Dxc. However, for LC hybrid functionals, as EA(3) is expected to be close to EA(2), Egð3Þ should be close to Egð2Þ, the true fundamental gap.[99]
Optical Gap
The optical gap (Eopt) of a neutral molecule is defined as Eopt¼ Eexcitedtotal ðneutralÞ & EtotalðneutralÞ; ð10Þ the energy difference between the lowest dipole-allowed excited state and the ground state, calculated at the ground-state geo- metry of the neutral molecule. Since Eoptis an excited-state property, it cannot be directly obtained with KS-DFT. For consistency with the ground-state calculations, TDDFT (with the same density functionals for the ground-state calculations) is adopted to compute Eopt.
Exciton Binding Energy
The exciton binding energy (Eb) of a neutral molecule is defined as Eb¼ Eg& Eopt, the difference between the fundamental and optical gaps. A system with small Eboften possesses high charge separation efficiency, and hence is favourable for photovoltaic applications, while the opposite may be desirable for light- emitting devices. Therefore, it is important to study the Eb values of n-PPs for understanding their potential applications. In this work, we adopt the following three ways of calculating Eb: Ebð1Þ ¼ Egð1Þ & Eopt ð11Þ Ebð2Þ ¼ Egð2Þ & Eopt ð12Þ
Ebð3Þ ¼ Egð3Þ & Eopt ð13Þ For the exact KS-DFT and TDDFT, Ebð1Þ and Ebð2Þ yield the exact exciton binding energy, while Ebð3Þ deviates from the exact exciton binding energy by an amount of Dxc. For LC hybrid functionals, as Ebð3Þ should be close to Egð2Þ, Ebð3Þ is expected to be close to Ebð2Þ, the true exciton binding energy.[75]
Results and Discussion
Fig. 2shows the singlet-triplet energy gap (EST) of n-PP as a function of the chain length, calculated using KS-DFT with various XC density functionals.[100]The results are compared with the available experimental EST data.[101–103]Overall, the calculated EST curves decrease with increasing chain length, showing consistency with the experimental data. The ground state of n-PP remains singlet for all the chain lengths studied.
Based on the calculated values ofh^S2i,[100]the lowest singlet state (i.e., the ground state) of n-PP exhibits single-reference character (i.e., has no spin contamination andh^S2i ¼ 0:0000), while the lowest triplet state of n-PP possesses some multi- reference character (i.e.,h^S2i>2:0), where the degree of spin contamination increases with the fraction of Hartree-Fock (HF) exchange adopted in a density functional.[99,104–108]For vB97 and vB97X, the lowest triplet state of long-chain n-PP is slightly spin-contaminated, partially degrading the accuracy of vB97 and vB97X for EST. Besides, the unphysical oscillations in the EST
curves obtained from vB97 and vB97X are found to be closely related to the degree of spin contamination.[100]
The ESTvalue of n-PP at the polymer limit (n! 1) is shown inTable 1. The extrapolated ESTvalue is 1.67 eV for LDA and PBE, 1.65 eV for BLYP, 2.15 eV for PBE0, 2.08 eV for B3LYP, 2.56 eV for vB97, 2.58 eV for vB97X, and 2.05 eV for the experimental ESTdata. Based on the calculated and extrapolated results, the global hybrid functionals (PBE0 and B3LYP) slightly outperform the semilocal functionals (LDA, PBE, and BLYP) and LC hybrid functionals (vB97 and vB97X).
One may wonder why the narrowest AGNRs, i.e., n-PPs, possess relatively stable singlet ground states (i.e., with non- radical character), when compared to the narrowest ZGNRs, i.e., n-acenes (acenes containing n linearly fused benzene rings), which have been shown to possess much less stable singlet ground states (i.e., with much smaller EST values), and exhibit increasing
20 15
10 n 5
1 4
3
Singlet-triplet energy gap [eV] 2
LDA
ωB97 ωB97X Expt PBE
BLYP PBE0
B3LYP
Fig. 2. Singlet-triplet energy gap (EST) of n-PP as a function of the chain length, calculated using KS-DFT with various density functionals. Here EST
is calculated usingEqn 1. For comparison, the experimental data[101–103]are taken from the literature.
polyradical character with increasing chain length.[26,29,35–38,42–44]
We expect that the geometrical arrangements of the aromatic rings in n-PP and n-acene should be responsible for the stability of these molecules.[32]For n' 3, n-PP and n-acene are polycyclic aro- matic hydrocarbons (PAHs), molecules containing three or more aromatic rings made of carbon and hydrogen atoms only. Based on Clar’s rule, the Kekule´ structure with the largest number of disjoint aromatic sextets is the most important structure for the stability of PAHs.[109]As illustrated inFig. 3, the aromatic rings of n-PP are connected with each other by a single carbon-carbon bond, isolating each aromatic ring as if n-PP is just the combination of isolated benzenes. Therefore, there are n aromatic sextets in the Kekule´ structure of n-PP. By contrast, there is only one aromatic sextet in the Kekule´ structure of n-acene. Therefore, for a given number of aromatic rings n (n' 3), n-PP is always more stable than n-acene. This suggests that the geometrical arrange- ment of the aromatic rings of PAHs should be responsible for the properties of PAHs. This argument is consistent with the results of other studies.[39–41]
At the ground-state geometry of n-PP, the IP (Fig. 4), EA (Fig. 5), Eg(Fig. 6), Eopt(Fig. 7), and Eb(Fig. 8) of n-PP as a function of the chain length, are calculated using KS-DFT and TDDFT with various XC density functionals.[100]
As shown inFig. 4a, relative to the experimental IP values,[76]
the IP(1) values calculated using vB97 and vB97X are more accurate than those calculated using the other functionals. The ground state of cationic n-PP exhibits some spin contamination
(h^S2i>0:75), where the degree of spin contamination is vanish- ingly small for the semilocal functionals and global hybrid functionals, but is noticeable for the LC hybrid functionals.[100]
The results are consistent with the argument that the larger the fraction of HF exchange adopted in a density functional, the easier the resulting KS determinant becomes spin-contaminated for multi-reference systems.[99,104–108]
The calculated IP(2) values (Fig. 4b) are more sensitive to the choice of the XC functional than the calculated IP(1) values. The semilocal functionals and global hybrid functionals severely underestimate the IP(2) values due to the incorrect asymptotic behaviour of the associated XC potentials. By contrast, owing to the correctð&1=rÞ asymptote of the underlying XC potentials and the fact that the ground state of n-PP exhibits single-reference character (where vB97 and vB97X are expected to perform
Table 1. Singlet-triplet energy gap (EST), vertical ionization potential (IP), vertical electron affinity (EA), fundamental gap (Eg), optical gap (Eopt), and exciton binding energy (Eb) (in eV) of n-PP at the polymer limit (n -N)
The extrapolated values are obtained by non-linear least-squares fittings of the corresponding properties of 1- to 20-PP, calculated using KS-DFT and TDDFT with various density functionals, where a fitting function of the formða þ b=nÞ is adopted. For the extrapolated experimental values, only those with more than two data points are calculated (using the same fitting function).[100]For each extrapolated value, the coefficient of determination R2, which is a statistical
measure of the goodness-of-fit (R2¼ 1 for a perfect fit), is given in parenthesis. For Eopt, the extrapolated SAC-CI value is 3.26 eV (R2¼ 0:9757)[77]
LDA PBE BLYP PBE0 B3LYP vB97 vB97X Expt
EST 1.67 1.67 1.65 2.15 2.08 2.56 2.58 2.05
(0.9967) (0.9954) (0.9952) (0.9714) (0.9755) (0.9289) (0.9328) (0.9871)
IP(1) 5.33 5.19 4.92 5.77 5.55 6.95 6.84 7.33
(0.9619) (0.9598) (0.9604) (0.9747) (0.9717) (0.9762) (0.9797) (0.9930)
IP(2) 4.67 4.50 4.23 5.25 5.00 7.29 7.12 7.33
(0.9973) (0.9970) (0.9970) (0.9974) (0.9975) (0.9972) (0.9966) (0.9930)
EA(1) 2.18 2.01 1.74 1.55 1.54 0.44 0.48 1.15
(0.9623) (0.9618) (0.9614) (0.9763) (0.9727) (0.9893) (0.9908) (0.9804)
EA(2) 1.49 1.34 1.07 1.05 1.00 0.96 0.90 1.15
(0.9430) (0.9415) (0.9406) (0.9583) (0.9520) (0.9835) (0.9847) (0.9804)
EA(3) 2.90 2.72 2.45 2.07 2.11 0.12 0.22 1.15
(0.9980) (0.9974) (0.9982) (0.9966) (0.9972) (0.9974) (0.9965) (0.9804)
Egð1Þ 3.17 3.18 3.17 4.22 4.01 6.51 6.36 6.24
(0.9621) (0.9608) (0.9609) (0.9755) (0.9722) (0.9844) (0.9866) (0.9882)
Egð2Þ 3.17 3.16 3.16 4.21 3.99 6.33 6.22 6.24
(0.9595) (0.9597) (0.9592) (0.9753) (0.9703) (0.9914) (0.9914) (0.9882)
Egð3Þ 1.77 1.78 1.77 3.18 2.89 7.17 6.90 6.24
(0.9981) (0.9979) (0.9981) (0.9975) (0.9978) (0.9978) (0.9970) (0.9882)
Eopt 1.73 1.73 1.73 2.56 2.42 3.52 3.44 3.57
(0.9992) (0.9994) (0.9995) (0.9976) (0.9981) (0.9896) (0.9902) (0.9944)
Ebð1Þ 1.46 1.45 1.45 1.66 1.59 2.98 2.91 2.63
(0.8033) (0.7852) (0.7875) (0.7914) (0.7921) (0.8118) (0.8464) (0.9748)
Ebð2Þ 1.44 1.43 1.43 1.64 1.57 2.80 2.77 2.63
(0.7844) (0.7728) (0.7700) (0.7872) (0.7763) (0.8893) (0.8861) (0.9748)
Ebð3Þ 0.04 0.04 0.05 0.62 0.47 3.64 3.45 2.63
(0.9924) (0.9904) (0.9909) (0.7620) (0.8848) (0.7947) (0.7159) (0.9748)
(a)
(b)
Fig. 3. Kekule´ structures of (a) 6-acene and (b) 6-PP. Here Clar’s aromatic sextets are marked with circles.[109]
reasonably well[74,75]), vB97 and vB97X yield extremely accu- rate IP(2) values. As shown inTable 1, the IP(2) value of n-PP at the polymer limit is 7.29 eV for vB97 and 7.12 eV for vB97X, which are in excellent agreement with the extrapolated experi- mental IP value (7.33 eV). From the calculated and extrapolated IP values, the IP(2) values obtained from vB97 and vB97X are reliably accurate.
For the calculated EA(1) values (Fig. 5a), due to the slight spin contamination (h^S2i>0:75) in the ground state of anionic n- PP, the LC hybrid functionals are slightly less accurate than the other functionals.[100] By contrast, as shown in Fig. 5b, the EA(2) values of short-chain n-PP (n( 4) calculated using vB97 and vB97X match very well with the experimental data, Expt1 (vertical EA)[110,111]and Expt2 (adiabatic EA),[112,113]which can be attributed to the correctð&1=rÞ asymptote of their XC potentials, while the other functionals significantly underesti- mate the EA(2) values, due to the incorrect XC potential
asymptotes. However, the accuracy of vB97 and vB97X is slightly degraded for longer-chain n-PP, as the ground state of anionic n-PP becomes slightly spin-contaminated. For the EA (3) values (Fig. 5c), the global and LC hybrid functionals perform comparably, outperforming the semilocal functionals.
FromTable 1, the EA(2) value of n-PP at the polymer limit is 0.96 eV for vB97 and 0.90 eV for vB97X, which are in good agreement with the extrapolated experimental EA value (1.15 eV). Based on the calculated and extrapolated EA values, the EA(2) values obtained from vB97 and vB97X are reason- ably accurate.
For the Egð1Þ values (Fig. 6a), while vB97 and vB97X slightly underestimate both the IP(1) and EA(1) values, they accurately predict Egð1Þ, possibly due to the cancellation of errors. For the Egð2Þ values (Fig. 6b), as vB97 and vB97X accurately predict both the IP(2) and EA(2) values, they accu- rately predict Egð2Þ. By contrast, the semilocal functionals and
9 8 7 6 5
4 1 5 10 15 20
1 5 10
n
IP(1) (b) IP(2)
(a)
LDA
PBE PBE0 ωB97
ωB97X
BLYP B3LYP Expt
n
15 20
9 8 7 6 5
Ionization potential [eV] 4 Ionization potential [eV]
Fig. 4. Vertical ionization potential (IP) for the lowest singlet state of n-PP as a function of the chain length, calculated using KS-DFT with various density functionals. Here IP(1) and IP(2) are calculated usingEqns 2and3, respectively. For comparison, the experimental data[76]are taken from the literature.
2 1 0
!1
Electron affinity [eV]
!2
!3
!4
!5
2 1 0
!1
Electron affinity [eV]
!2
!3
!4
!5
2 1 0
!1
Electron affinity [eV]
!2
!3
!4
!5
1 5 10
n 15 20
1 5 10
EA(3) EA(1) (a)
(c)
(b)
EA(2)
n
15 20
1 5 10
LDA wB97
wB97X Expt1 Expt2 PBE
BLYP PBE0 B3LYP
n 15 20
Fig. 5. Vertical electron affinity (EA) for the lowest singlet state of n-PP as a function of the chain length, calculated using KS-DFT with various density functionals. Here EA(1), EA(2), and EA(3) are calculated usingEqns 4–6, respectively. For comparison, the experimental data: Expt1 (vertical EA)[110,111]and Expt2 (adiabatic EA),[112,113]are taken from the literature.
global hybrid functionals severely underestimate both the Egð1Þ and Egð2Þ values. For the Egð3Þ values (Fig. 6c), vB97 and vB97X slightly overestimate Egð3Þ, whereas the other func- tionals significantly underestimate Egð3Þ. From Table 1, the Egð1Þ value of n-PP at the polymer limit is 6.51 eV for vB97 and 6.36 eV for vB97X, and the Egð2Þ value of n-PP at the polymer limit is 6.33 eV for vB97 and 6.22 eV for vB97X, which are in excellent agreement with the extrapolated experimental Eg value (6.24 eV). According to the calculated and extrapolated Eg values, the Egð1Þ and Egð2Þ values calculated using vB97 and vB97X are reliably accurate.
The Eoptof n-PP is found to be the singlet-singlet (S0! S1) gap for each case studied. As shown inFig. 7, the Eoptvalues calculated using vB97 and vB97X are in excellent agreement with the experimental data[77,114] and those obtained with the highly accurate symmetry-adapted-cluster configuration- interaction (SAC-CI) method.[77]Relative to the experimental
data, vB97 and vB97X perform slightly better than the SAC-CI method. The other functionals severely underestimate the Eopt value of long-chain n-PP. FromTable 1, the Eoptvalue of n-PP at the polymer limit is 3.52 eV for vB97 and 3.44 eV for vB97X, which are in excellent agreement with the extrapolated experi- mental Eoptvalue (3.57 eV) and the extrapolated SAC-CI value (3.26 eV). Based on the calculated and extrapolated Eoptvalues, the Eoptvalues calculated using vB97 and vB97X are reliably accurate.
As shown inFig. 8aandFig. 8b, the Ebð1Þ and Ebð2Þ values calculated using vB97 and vB97X decrease monotonically with the increase of n, and quickly approach some constants at about n¼ 5. By contrast, the Ebð1Þ and Ebð2Þ values calculated using the other functionals decrease more slowly, and approach some constants at the larger values of n. For the Ebð3Þ values (Fig. 8c), while vB97 and vB97X slightly overestimate Ebð3Þ, the other functionals significantly underestimate Ebð3Þ. Note that the Ebð3Þ values obtained from the semilocal functionals are unphysically negative. FromTable 1, the Ebð1Þ value of n-PP at the polymer limit is 2.98 eV for vB97 and 2.91 eV for vB97X, and the Ebð2Þ value of n-PP at the polymer limit is 2.80 eV for vB97 and 2.77 eV for vB97X, which are in excel- lent agreement with the extrapolated experimental Eb value (2.63 eV). From the calculated and extrapolated Eb values, the Ebð1Þ and Ebð2Þ values obtained from vB97 and vB97X are reliably accurate.
Conclusions
In conclusion, we have studied the electronic and optical properties (i.e., the singlet-triplet energy gaps, vertical ioniza- tion potentials, vertical electron affinities, fundamental gaps, optical gaps, and exciton binding energies) of NAGNRs with different lengths, using KS-DFT and TDDFT with various XC density functionals. The ground states of NAGNRs have been shown to remain singlet for all the lengths studied. With the increase of the NAGNR length, the singlet-triplet energy gaps,
11
(a) (b)
(c) 9 7 5 3
1 1 5 10 15 20
n Eg(1)
Eg(3)
Eg(2)
1 5 10 15 20
n
1 5 10 15 20
n
Fundamental gap [eV]
11 9 7 5 3 Fundamental gap [eV] 1
11 9 7 5 3 Fundamental gap [eV] 1
LDA
Expt1 Expt2 PBE
BLYP PBE0 B3LYP
wB97wB97X
Fig. 6. Fundamental gap (Eg) for the lowest singlet state of n-PP as a function of the chain length, calculated using KS-DFT with various density functionals. Here Egð1Þ, Egð2Þ, and Egð3Þ are calculated usingEqns 7–9, respectively. For comparison, the experimental data: Expt1 (vertical IP – vertical EA)[76,110,111]
and Expt2 (vertical IP – adiabatic EA)[76,112,113]
, are taken from the literature.
7
6
5
4
3
2
1 5 10 15 20
n
SAC-Cl Expt PBE
BLYP PBE0 B3LYP
Optical gap [eV]
LDA wB97
wB97X
Fig. 7. Optical gap (Eopt) for the lowest singlet state of n-PP as a function of the chain length, calculated using TDDFT with various density functionals.
Here Eoptis calculated usingEqn 10. For comparison, the experimental data[77,114]and SAC-CI data[77]are taken from the literature.
vertical ionization potentials, fundamental gaps, optical gaps, and exciton binding energies decrease monotonically, whereas the vertical electron affinities increase monotonically. While the neutral NAGNRs possess stable single-reference singlet ground states, the lowest triplet states and the ground states of the cationic and anionic NAGNRs exhibit some multi-reference character. Nevertheless, as the degree of spin contamination for each case is not very severe, it seems unnecessary to employ computationally expensive multi-reference methods in this study.
Overall, the electronic and optical properties calculated using the vB97 and vB97X functionals are in excellent agreement with the available experimental data, with the effective conjugation length of NAGNR being estimated to be close to 10 benzene rings. While the electronic and optical properties of NAGNRs have been shown to be controllable with the adequate choice of NAGNR length, how these properties vary with different widths, edge types, and edge terminations remain unanswered. We plan to address some of these questions in the near future.
Supplementary Material
Full details of the calculations including the singlet-triplet energy gap, vertical ionization potential, vertical electron affinity, fundamental gap, optical gap, and exciton binding energy of n-PP (n¼ 1–20) are available on the Journal’s website.
Acknowledgements
This work was supported by the Ministry of Science and Technology of Taiwan (Grant No. MOST104-2628-M-002-011-MY3), National Taiwan University (Grant No. NTU-CDP-105R7818), the Center for Quantum Science and Engineering at NTU (Subproject Nos.: NTU-ERP-105R891401 and NTU-ERP-105R891403), and the National Center for Theoretical Sciences of Taiwan.
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