Subscriber access provided by FRITZ HABER INST DER MPI Article High-Throughput Investigation of the Geometry and Electronic Structures of Gas-Phase and Crystalline Polycyclic Aromatic Hydrocarbons Bohdan Schatschneider, Stephen Monaco, Jian-Jie Liang, and Alexandre Tkatchenko J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp5064462 • Publication Date (Web): 07 Aug 2014 Downloaded from http://pubs.acs.org on August 12, 2014
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The Journal of Physical Chemistry C is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties. Page 1 of 36 The Journal of Physical Chemistry
1 2 3 High-Throughput Investigation of the Geometry and Electronic Structures of Gas-Phase 4 5 and Crystalline Polycyclic Aromatic Hydrocarbons 6 7 Bohdan Schatschneider a*, Stephen Monaco a, Jian Jie Liang b, and Alexandre Tkatchenko c 8 9 a The Pennsylvania State University, Fayette-The Eberly Campus 10 b 11 Accelrys Inc., 5005 Wateridge Vista Drive, San Diego, CA 92121 USA c 12 Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany 13 14 15 Abstract : The quest for cheap, light, flexible materials for use in electronic applications has 16 resulted in the exploration of soft organic materials as possible candidates, and several polycyclic 17 18 aromatic hydrocarbons (PAH) have been shown to be versatile (semi) conductors. In this 19 20 investigation, dispersion inclusive density functional theory is used to explore all of the current 21 22 crystalline PAHs within the Cambridge Structure Database (CSD) from both structural and 23 24 electronic standpoints. Agreement is achieved between the experimental and calculated 25 crystalline structures as well as the electronic properties: Specifically, variation between the 26 27 mass densities, unit cell parameters and intermolecular close contact fractions were within +5%, 28 29 ±2%, and ±1% of experiment, respectively. It is found that a simple addition of a ~1 eV constant 30 31 to the DFT PBE gaps provides good agreement with the experimental optical gaps of both gas 32 phase (within ±2.6%) and crystalline (within ±3.5%) PAHs. Structural and electronic analysis 33 34 revealed several correlations/trends; where ultimately, limits in the band gaps as a function of 35 36 structure are established. Finally, analysis of the difference between band gaps of the isolated 37 38 molecules and crystals ( ) demonstrates that can be captured qualitatively 39 40 by PBE and PBE0 functionals, yet significant quantitative deviations remain between these 41 functionals and experiment. 42 43 44 45 46 47 48 Keywords : Dispersion Corrected Density Functional Theory (DFT D), Organic Molecular 49 Crystals, Band Gap, Organic Semiconductors. 50 51 52 53 54 *Corresponding Author: Email: [email protected] Phone: 1 (724) 430 4257 55 56 57 58 59 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 2 of 36
1 2 3 1. Introduction : The quest for inexpensive, flexible, light weight materials for use in 4 5 electronic applications has led researchers to explore what useful arrays can be constructed 6 7 through organic means, and the ability to predict useful properties within organic electronic 8 9 materials is quickly becoming an essential part of organo electronic product design. Polycyclic 10 11 aromatic hydrocarbons (PAH) make up a group of organic molecular crystals (OMC) which have 12 shown promise for use in electronics and electro optics 1, 2 . Despite their versatility and 13 14 abundance, theoretical exploration of the electronic properties of most crystalline PAHs remains 15 16 uncharted. Here we use dispersion inclusive density functional theory (DFT) to shed light onto 17 18 the structural and electronic trends occurring within this promising group of materials. 19 The topology of individual PAH molecules can be classified as: 1) linear/nanoribbons 20 21 (comprised of homologous groups of oligoacenes, phenacenes, and oligorylenes) 2) circular 22 23 flakes/discs (K region PAHs and circumacenes) and 3) triangular. Within each topology two 24 25 edge/periphery types exist: 1) zig zag and 2) arm chair. In general, it is known that in a given 26 topology of PAHs (as well as nanoflakes) that the size of the band gap (E ) decreases 3 5 as the 27 g 28 number of aromatic rings (carbon atoms) increases, where arm chair edge PAHs have larger 29 30 band gaps and cohesive energies (enthalpies of formation) than their zig zag counter parts. 31 5 32 Similar trends can also be found in fully benzenoid PAHs . It is then understood that the 33 34 ionization energy (I) decreases and the electron affinity (A) increases with increasing number of 35 rings in a homologous class 4 (this follows from the above E trend as the fundamental/transport 36 g 6 37 gap is I A) . It is also recognized that as the number of edge atoms (N s) increase with respect to 38 3 39 the total number of atoms (N), the cohesive energy decreases . 40 41 Traditionally, PAHs can be either heterocyclic or only carbon containing, and assemble 42 in molecular crystalline arrays under ambient conditions. Those PAHs containing only hydrogen 43 44 and aromatic carbon can be classified into five crystalline motifs, definable by the π⁰ parameter 45 46 (a product of the inter planar angle between molecular components and the fraction of C···C 47 7 48 intermolecular close contacts ). The five motifs: a) herringbone (HB) b) sandwich herringbone 49 50 (SHB) c) beta herringbone (β HB) d) gamma (γ) and e) beta (β) are depicted in Figures 1a-1e 51 using structures which exemplify the characteristics of each motif 8,9. 52 53 Each motif has characteristic amounts of C···C (π···π), C···H, and H···H intermolecular 54 55 close contact interactions7 10 . It is known that the HB and β HB structures are heavily dependent 56 57 on C···H close contact interactions for crystal stabilization as calculated via Hirshfeld surface 58 59 60 ACS Paragon Plus Environment Page 3 of 36 The Journal of Physical Chemistry
1 2 3 analysis 7, but get the least amount of crystal stabilization from C···C close contacts on average 4 5 when compared to other motifs. The SHB structures are dependent on C···H interactions as 6 7 well, but become increasingly dependent on C···C interactions compared to the HB and β HB 8 9 motifs as half of each molecular component is involved in π···π stacking interactions. The β and 10 11 γ motifs are the most dependent on C···C interactions as the large molecular components have 12 low Ns/N ratios (compared to other motifs), providing ample π orbitals for stacking interactions. 13 14 15 16 17 18 19 20 21 22 23 24 (c) 25 (a) (b) 26 27 28 29 30 31
32 33 (e) 34 (d) 35 36 37 Figure 1. PAH motifs. a) HB anthracene [ANTCEN]. b) SHB – Quaterylene [QUATER10]. c) β HB – 1, 2, 3, 4, 38 tetraphenylbenzene [FOVVOB]. d) γ – coronene [CORONE01]. e) β – anthra[2,1,9,8 hjkl ]benzo[de]naphtha 39 [2,1,8,7 stuv ]pentacene [BOXGAW]. 40 41 42 Despite the extensive knowledge of gas phase PAH band gap trends, little is known about 43 44 the band gap vs. structural/motif dependence in crystalline PAHs. We previously investigated 45 11, 12 46 the oligoacenes (a prototypical group of PAHs) in great detail using dispersion inclusive 47 density functional theory. In those studies (as well as others 13, 14 ) we benchmarked the 48 49 Tkatchenko Scheffler dispersion energy method (PBE+vdW) for structural and properties 50 51 predictions of molecular crystals with a specific aim on the larger PAH family. The oligoacene 52 53 investigations mentioned above demonstrated that PBE+vdW can provide excellent structural 54 55 agreement with experiment, as well as accurately model the pressure induced structural changes 56 of naphthalene, anthracene, and pentacene. PBE+vdW was also used to reproduce the pressure 57 58 59 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 4 of 36
1 2 3 induced phase transition of tetracene along with the pressure induced changes to the HOMO 4 5 LUMO band gaps of tetracene and pentacene. With the aforementioned success in mind, we set 6 7 our sights on investigating the structural and electronic trends of all 91 PAHs currently in the 8 9 Cambridge Structural Database (CSD) in order to gain further chemical insight into this unique 10 11 class of crystals. 12 13 14 15 2. Methodology : All PAHs investigated in this study are listed in Table S1 of the 16 Supporting Information (SI). Experimental x ray structures were acquired as referenced in the 17 18 CSD (reference codes have been provided). Structure searches were conducted using CSD 19 20 version 5.32 via Conquest version 1.3 with the following search criteria selected: 3D coordinates 21 22 determined, not disordered, no errors, not polymeric, no ions, no powder structures and only 23 organics. All selected PAH structures contain only hydrogen and aromatic carbon atoms. 24 25 Structures were limited to carbon and hydrogen atoms as other elements can have significant 26 15 27 effects on the electronic and structural properties. Crystal Explorer 3.0 was used to generate 28 29 the Hirshfeld surfaces and corresponding fingerprint plots. 30 Density functional theory with dispersion interactions was utilized in the CASTEP 31 32 program for the optimization of the PAH crystal structures using the Perdew Burke Ernzerhof 33 34 exchange correlation functional (PBE) 16 with the Tkatchenko Scheffler dispersion energy 35 17 36 method (+vdW) . Where, the cost of the TS vdW energy is negligible compared to the 37 38 underlying DFT calculation. The convergence criteria and methods used for calculating all 39 crystalline PAH structures and band gaps were as described in previous investigations of 40 41 11 oligoacenes . Isolated molecular band gaps ( ) were also calculated in CASTEP using the 42 43 same convergence criteria as used for the crystals except the were calculated on the 44 45 optimized molecules using only Gamma point. Molecules were placed in a periodic cell with a 46 47 minimum of 10 Å to the cell boundary before geometry optimization. 48 Band gaps were obtained without dispersion correction as the fully self consistent 49 50 implementation of the PBE+vdW method (i.e. where the potential due to vdW interactions enters 51 52 self consistently in the solution of the Kohn Sham equations) leads to negligible modifications of 53 54 electronic properties for molecular crystals. For example, for benzene and naphthalene crystals, 55 56 the band gap modifications amount to only a few tenths of meV. 57 58 59 60 ACS Paragon Plus Environment Page 5 of 36 The Journal of Physical Chemistry
1 2 3 The long range van der Waals energy was determined from the TS method; i.e ., the 4 5 difference between the PBE total energy and the dispersion corrected total energy. The entire set 6 7 of calculations and corresponding analyses were automated through PERL scripting. Problem 8 9 structures (outliers) were addressed separately and recalculated 10 11 12 3. Results and Discussion : 13 14 3.1 Structural Analysis . The first step in establishing a computational method for use in 15 18, 19 16 a high throughput scenario is to establish its reliability for calculating structural properties . 17 18 Good agreement between the experimental and calculated crystal structures is demonstrated in 19 Figures 2-4. The percent variation between the experimental and calculated unit cell parameters 20 21 is shown in Figure 2 , where the majority of the calculated unit cell parameters are within ±2% of 22 23 experiment. 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 Figure 2. Histograms of percent variation between PBE+vdW and experimental unit cell parameters for 91 PAHs. 39 Negative numbers indicate the calculated unit cell parameters are smaller in the given cell dimension. 40 41 Comparison of the experimental and calculated densities is provided in Figure 3 . Nearly 42 43 all of the geometry optimized structures (run at 0 K) are denser than experiment. One reason for 44 the difference is that most of the PAH x ray studies were conducted at room temperature, leading 45 46 to lower densities than those calculated at 0 K (see SI Table S2 ). Despite the temperature 47 48 difference between calculation and experiment, ~77% of the calculated densities are within +5% 49 50 of experiment (indicated by green lines in Figure 3 ). When the x ray structures were determined 51 at lower temperatures (< 273 K), variation decreases to within +2.3% on average. Some 52 53 calculated structures corresponding to room temperature X ray structures were more than +5% as 54 55 dense, however no obvious phase/motif transitions were observed via analysis of the space group 56 7 57 or the π⁰ parameter for these structures. 58 59 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 6 of 36
1 2 3 1.7 4 5 1.6 6 7 1.5 8 9 1.4 10 11 1.3 12 13 1.2 14 15 Calculated Density (g/cm3) Density Calculated 1.1 16 Low Temp Room Temp 17 1 18 1 1.1 1.2 1.3 1.4 1.5 1.6 19 Experimental Density (g/cm3) 20 21 Figure 3. Comparison of calculated and experimental densities for 91 PAHs. Black line represents exact 22 agreement between experiment and calculation. Green dotted lines represent ±5% variation. Red balls are 23 structures obtained at room temperature and blue balls are structures obtained below room temperature. 24 25 Calculated structures more than +5% from experiment have structure numbers from Table S2 in the SI of: 26 2 5, 7, 16 18, 27, 30, 34, 35, 39, 42, 43, 45, 57, 63, 77, 82, 87 27 28 To assess the geometry within the unit cell, a quantitative, whole molecule comparison of 29 30 the intermolecular close contacts was performed from the breakdown of fingerprint plots 31 constructed via the deconvolution of the Hirshfeld surface. Fingerprint plots provide two 32 33 dimensional depictions of the Hirshfeld surface plotted as a function of distance between the 34 35 interior and exterior nuclei and a point on the Hirshfeld surface. Fingerprint plots can be 36 37 decomposed according to select atom atom contacts, providing a quantitative measure of the 38 39 intermolecular contact fractions occurring at the surface. For a full description of Hirshfeld 40 surface analysis and its possible applications for understanding intermolecular interactions within 41 42 molecular crystals we refer the reader to our previous work 7, 11, 13 and several encompassing 43 10, 15, 20 44 reviews/technical discussions . 45 46 Figure 4 shows that agreement is achieved between the calculated and experimental 47 intermolecular close contact fractions when using PBE+vdW for structure prediction. The 48 49 slightly increased C···H and decreased H···H close contacts in the calculated structures as 50 51 compared to that of experiment in Figure 4 are a reflection of the increased densities in the 52 53 calculated structures (resulting in more efficiently packed arrays). This observation is consistent 54 55 with the fact that lowering the temperature of PAHs will have similar effects as applying 56 pressure; i.e. , increasing the densities and rearranging the intermolecular close contacts. More 57 58 59 60 ACS Paragon Plus Environment Page 7 of 36 The Journal of Physical Chemistry
1 2 3 specifically, when PAHs (oligo( para-phenylenes) 21 , oligoacenes 11, 22, 23 and heterocyclic 13 4 5 structures) are placed under pressure, the inter planar angle increases to increase the π···π 6 7 interactions. The increased interplanar angle/π···π interactions has been shown to increase the 8 7, 13 9 C···C and C···H interactions while decreasing the H···H contacts . These same effects also 10 11 surface when the temperature of PAHs is decreased (at ambient pressure). 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 Figure 4. Comparison of intermolecular close contact fractions between calculated and experimental 45 structures as derived via Hirshfeld surface analysis. Blue line represents exact agreement between 46 experiment and DFT. Values for individual PAHs are available in SI Table S3 . 47 48 49 There is one notable outlier in Figure 4 , namely DBPERY of the γ motif. In the case of 50 DBPERY, there is compression of the unit cell parameters ( a, b, & c), a +4.1% change in 51 52 density, and a small alteration (~0.4°) of the interplanar angle (θ) with respect to experiment. 53 54 These structural changes cause the outlying intermolecular interaction behavior, yet no motif or 55 56 phase transition was observed. 57 58 59 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 8 of 36
1 2 3 Comparison of the C···C intermolecular close contacts (π···π stacking) and the density of 4 5 the experimental structures extracted from the CSD shows that a direct relationship exists for 6 7 PAHs (see Figure 5a). The average densities of each motif in Figure 5a proceed from the least 8 9 dense HB/β HB structures, to SHB structures, followed by the densest, γ and β structures. A 10 ± 11 linear best fit of the data in Figure 5a shows that ~90% of PAH mass densities are within 5% of 12 the line. Extrapolation of the best fit line to 100% C···C interaction (as in the case of graphite) 13 14 results in a density of 2.154 g/ml. This density is in agreement with that of graphite flakes (2.144 15 24 25 16 g/ml) , and within ~5% of highly orientated pyrolytic graphite (HOPG (2.266 g/ml) ). 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 (a) (b) 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 (c) 48 49 Figure 5. (a) Density as a function of C···C intermolecular close contact fractions from experimental PAH 50 structures (as extracted from CSD). Blue line is a linear best fit to the data ( ρ = 0.0091 g/ml (C···C%) + 51 1.2435 g/ml) and the green lines indicate ±5% from the best fit line. (b) van der Waals energy per atom as a 52 function of density. Values obtained from optimized structures. Blue line is a linear best fit to the data 53 (EvdW = 0.072 eV·ml/g ( ρ) + 0.0234 eV). (c) van der Waals energy per atom as a function of the C···C 54 intermolecular close contact fractions. Values obtained from optimized structures. Blue line is a linear best 55 fit to the data ( EvdW = 0.000661 eV·(C···C%) 0.0694 eV). 56 57 58 59 60 ACS Paragon Plus Environment Page 9 of 36 The Journal of Physical Chemistry
1 2 3 When the long range vdW dispersion energy per atom (including contributions from both 4 5 hydrogen and carbon) is plotted as a function of density for the optimized structures, it can be 6 7 seen that the vdW contribution to the crystal stabilization increases as a function of the density 8 9 (see Figure 5b ). Also, the average vdW energy per atom (E vdW ) increases similarly to the 10 11 density with respect to motif in Figure 5b ; i.e., the HB/β HB structures have the least EvdW , with 12 the SHB structures having more, followed by the γ and β structures (which are densest) 13 14 containing the highest EvdW contributions to crystal stability. A linear fit of the data in Figure 5b 15 16 shows that 98% of the data is within ±5% of the line. When the best fit line is extrapolated to a 17 18 density of 2.266 g/ml (that of HOPG), the vdW energy per atom is 0.14 eV/atom: When HOPG 19 is geometry optimized using PBE+vdW under the same convergence criteria, E is also 0.14 20 vdW 21 eV/atom. 22 23 The relationship between the C···C intermolecular close contacts and the EvdW as 24 25 calculated from the geometry optimized structures is shown in Figure 5c . Here, a direct 26 27 relationship is observed between the C···C interactions and the average EvdW . Extrapolation of 28 the best fit line in Figure 5c to 100% C···C% results in a E vdW of 0.136 eV/atom; the same EvdW 29 30 as calculated from PBE+vdW and the best fit line equation of Figure 5b when the density is 31 32 2.21 g/ml (ρ of graphite). It is uncanny that the linear fits in each of these figures, when 33 34 extrapolated, results in values associated with an infinite PAH array, namely graphite. In 35 addition, while the entire stabilization of the crystalline structure due to long range vdW 36 37 contributions does not come from just the C···C interactions, an increase in the C···C contacts 38 39 (or π···π interactions) can have up to a ~30% stabilizing effect on PAH matrices when 40 41 comparing say, HB structures with those of γ and β arrangements. 42 To sum up the trends present in Figure 5, in can be said that the densest PAHs will have 43 44 the largest vdW stabilization energies and most prevalent amount of C···C contacts (or π···π 45 46 interactions). 47 48 49 50 3.2.1 Electronic Properties . Previously, we calculated the structures and fundamental 51 11 52 band gaps ( ) of crystalline oligoacenes, from naphthalene (2A) to pentacene (5A) . In that 53 27, 28 54 study, and others , it was shown that the Kohn Sham gap ( ) could correctly predict the 55 56 trend of decreasing with increasing number of rings, but that quantitative prediction of 57 58 (I A) using KS orbitals led to severe underestimations due to the inherent physical differences 59 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 10 of 36
1 2 3 4 existing between and . While should not be used to predict , it’s been shown 5 6 to be an excellent tool for the prediction of the optical gap ( ) – expanded upon below. In 7 8 order to gain insight into the electronic properties of the entire class of PAHs in a high 9 10 throughput manner, ’s of gas/solution phase and crystalline PAHs were explored via 11 12 prediction with . 13 14 15 Structure Mol Mol Mol CSD Code Motif 16 Number 17 4 18 QUATER10 1 SHB 0.93 1.86 1.88 19 PENCEN* 2 HB 1.14 2.31 2.31 4, 28 20 POBPIG 3 HB 1.47 2.47 37 21 38 22 QQQCIG04 4 HB 1.48 2.51 23 CORXAI10 5 β 1.65 2.68 39 24 TETCEN01* 6 HB 1.61 2.91 2.71 4, 29, 34 25 PERLEN07 7 SHB 1.86 3.14 2.82 4, 5 26 4 27 OVALEN01 8 γ 1.92 3.09 2.88 28 TBZPER 9 β 1.91 2.92 40 29 DBPERY 10 γ 1.96 3.01 41 30 40 31 TBZPYR 11 β 2.16 3.23 32 DBZCOR 12 γ 2.30 3.27 41 33 HBZCOR 13 γ 2.45 3.76 3.35 5, 4 1 34 BNPERY 14 SHB 2.33 3.35 41 35 29 36 TEBNAP 15 β -HB 2.61 3.43 37 ANTCEN* 16 HB 2.32 3.75 3.45 4 38 WOQPAT 17 γ 2.56 3.92 3.46 5 39 29 40 BEANTR 18 HB 2.48 3.53 41 DBNTHRO02 19 HB 2.61 3.57 29 42 CEQGEL 20 SHB 2.74 3.65 40 43 4 44 CORONE01 21 γ 2.85 4.22 3.74 4, 5 45 ZZZOYC01 22 HB 2.92 4.43 3.80 46 PYRENE02 23 SHB 2.61 4.04 3.83 40 47 29 BZPHAN 24 HB 2.90 3.84 48 4 49 CRYSEN01 25 HB 2.94 4.45 3.89 50 CORANN12* 26 HB 3.01 4.60 4.13 5 51 PHENAN 27 HB 3.34 4.93 4.24 4, 29 52 5, 29 53 TRIPHE12 28 HB 3.53 5.10 4.36 4 54 NAPHTA04* 29 HB 3.37 5.07 4.45 55 56 57 58 59 60 ACS Paragon Plus Environment Page 11 of 36 The Journal of Physical Chemistry
1 2 3 Table 1 . Predicted and experimental gas/solution phase optical gaps of select PAHs. Eg in units of eV. 4 All PBE/PBE0 gaps were calculated from PBE+vdW geometry optimized structures. See supplement 5 Table S1 for molecule names. *Indicates gas phase 6 7 8 Experimental optical gaps ( ) of the PAHs were obtained from the p bands of their 9 10 optical spectra (standard practice for PAHs).5, 29 For solution spectra, solvent constants were 11 12 29 added to account for solvent stabilization effects. It is noted that acquisition of the of 13 14 crystalline PAHs is technically challenging as one must obtain the optical excitation of extremely 15 16 thin crystals and/or films due to the materials’ large extinction coefficients. Furthermore, 17 18 intermolecular coupling can be large enough in PAHs that exciton and charge transfer states can 19 dominate the spectral features, in addition to polarization and temperature effects. 30 32 For these 20 21 reasons, Eg benchmarking of crystalline PAHs can be difficult with predictive methods and may 22 23 be why previous large scale PAH investigations 3, 4, 33 36 were only performed on gas phase 24 25 molecules. 26 27 5.49 4.5 28 29 4.99 4 30 31 32 4.49 3.5 33 34 3.99 3 35 36 3.49 2.5 37 38 2.99 2 39 40 Gas Phase PBE Gap (eV) 41 2.49 1.5 42 EXP 43 1.99 PBE0 1 44 PBE 45 Gas/Soluition Phase Optical andPBE0 gaps (eV) 1.49 0.5 46 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 47 48 Structure Number 49 50 Figure 6 . Comparison of predicted and experimental gas/solution phase optical gaps for select PAHs. 51 and are on the primary axis. are as referenced in Table 1 (green triangles are gas phase and 52 grey diamonds are solution phase). are plotted on the secondary vertical axis (on the right), and 53 aligned with experimental values (see details in the text) by shifting the zero crossing of the secondary 54 vertical axis by 0.99 eV relative to that of the primary axis (same scale as the primary axis). Structure 55 reference numbers correlate with Table 1 . 56 57 58 59 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 12 of 36
1 2 3 4 The is defined as the energy needed for electronic transition of one electron from 1) 5 6 the HOMO to LUMO energy levels in molecules and/or 2) the valence to conduction bands in 7 solids where little distance exists between the electron hole pair (bound excitonic state). While 8 9 4, 63, 64 optical spectroscopy is typically used to establish , recent work shows that the 10 11 calculation of (from either exact or approximated (GGA, LDA) models) can provide reliable 12 13 physical and quantitative representation of this phenomena. While – the difference between 14 15 the highest occupied one electron state and the lowest unoccupied one for molecules in the gas 16 17 phase has been shown to reliably predict , application to extended solid networks has met 18 19 with significant scrutiny. However, it is expected that in molecular solids bound by weak 20 21 intermolecular interactions (such as PAHs) that could predict as in the “normal” 22 23 molecular situation because strongly bound Frenkel excitons remain localized within the excited 24 63 25 molecule. 26 27 Structure Crystal Crystal Crystal Other 28 CSD Code Motif 29 Number Values 30 QUATER10 1 SHB 0.63 1.60 42 1.7 43
31 PENCEN 2 HB 0.90 2.03 1.85 28 1.8 32 POPIG 3 HB 1.24 2.25 44 2.28 45 33 QQQCIG14 4 HB 1.18 2.3 8 2.3146 34 35 QQQCIG04 7 HB 1.04 2.27 2.32 48 36 QQQCIG13 5 HB 1.36 2.65 2.36 46 37 TETCEN01 6 HB 1.37 2.66 2.38 30,47 38 39 PERLEN07 8 SHB 1.45 2.52 2.65 49 2.61 50 2.67 51 40 HBZCOR 9 γ 1.65 2.80 2.8052 41 BEANTR 10 HB 2.10 3.514 3.14 53 42 ANTCEN 11 HB 1.93 3.33 3.16 31 43 44 PYRENE02 12 SHB 2.16 3.30 54
45 CORONE01 13 γ 2.27 3.52 3.35 55 46 TERPHEN02 14 HB 2.65 3.7356 47 48 BIPHEN 15 HB 3.23 4.826 4.18 57
49 50 Table 2. Predicted and experimental optical gaps of select crystalline PAHs. Eg in units of eV. All 51 PBE/PBE0 gaps were calculated from PBE+vdW geometry optimized structures. See supplement Table 52 S1 for structure names. 53 54 55 Regardless of the aforementioned difficulties and successes, Figures 6 & 7 and Tables 1 56 57 & 2 show that calculated using PBE ( ) does an excellent job of predicting the 58 59 60 ACS Paragon Plus Environment Page 13 of 36 The Journal of Physical Chemistry
1 2 3 relative optical gaps of PAHs in both the gas and crystalline phases. Interestingly, it appears that 4 5 for PAHs, an accurate value of the may be obtained by simply adding a constant of ~ 1 eV 6 7 (here on called, ξ ) to regardless of phase. Specifically, ξ is 0.99eV for solution/gas 8 PBE PBE 9 phase comparison while ξ PBE is 1.05eV for the crystalline phase. The addition of ξPBE results in 10 11 calculated gaps that differ from by only ±2.6% on average for the isolated molecules and 12 13 ±3.5% on average for the crystalline ensembles. The single parameter fit to the PBE gap is a 14 15 welcomed finding as calculations utilizing PBE are an order of magnitude cheaper compared 16 11 17 to calculations using more “precise” methods such as hybrid functionals like PBE0 , 18 19 HSE03 11 , and B3LYP 4 or quasiparticle corrections 28 and time dependent methods 4. 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Figure 7. Comparison of predicted and experimental optical gaps of crystalline PAHs. Experimental and 41 PBE0 gaps are on the primary axis. Experimental gaps are as referenced in Table 2. PBE gaps are plotted 42 on the secondary vertical axis (on the right), and aligned with experimental values by shifting the zero 43 crossing of the secondary vertical axis by 1.05 eV relative to that of the primary axis (same scale as the 44 primary axis). Structure reference numbers correlate with Table 2. 45 46 It is also interesting to note that the PBE method can distinguish between the optical gaps 47 48 of different polymorphs. This can be seen in Table 2 and Figure 7 for the case of monoclinic 49 50 (QQQCIG13), triclinic (QQQCIG14), and orthorhombic (QQQCIG04) rubrene. Here the subtle 51 52 differences in molecular packing lead to differences in the optical spectra of each polymorph 53 which are captured by PBE. 54 55 Calculation of using the hybrid PBE0 functional ) resulted in values that 56 57 agreed reasonably with experimental for the gas phase (within +12% on average), while 58 59 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 14 of 36
1 2 3 better agreement was found to exist for the crystalline band gaps (within ±5.6% on average). 4 5 Though the PBE0 calculation are parameter free, they come at significant additional cost to that 6 7 of PBE and result in larger errors when compared to the adjusted PBE gaps. Furthermore, the 8 9 incorporation of Hartree Fock orbitals into the PBE0 band gap calculation changes the physical 10 11 meaning of the number compared to PBE; i.e ., its known that hybrid functionals are a better 12 63 13 assessment of as opposed to , that is why is (nearly) always larger than 14 15 in Figures 6 & 7 . The reason that is closer to in the crystalline case ( Figure 7 ) than 16 17 in the gas phase is simply because the and have nearly the same value in crystals as a 18 28, 63 19 result of exciton delocalization and dielectric screening effects. 20 5 21 22 4.5 23 4 24 25 3.5 26 3 27 28 2.5 29 2 30 31 1.5 32
PBE Gap From Optimized Structure (eV) Structure Optimized GapFrom PBE 1 33 34 0.5 35 36 0 0 1 2 3 4 5 37 PBE Gap From Experimental Structure (eV) 38 39 Figure 8. DFT PBE gaps from the geometry optimized and the experimental structures. Gaps were 40 calculated on PBE+vdW geometry optimized structures and experimental structures as extracted from the 41 CSD. Black line represents exact agreement between experiment and DFT while the green dotted lines 42 represent ±5% variation. 43 44 A comparison of two sets of calculated gaps is shown in Figure 8: one set based on the 45 46 experimental structures, and the other on the geometry optimized structures. It can be seen that 47 48 the experimental room temperature structures produce larger gaps than the geometry optimized 49 50 structures ~96% of the time, but that the values from both sets are within ±5% on average. This 51 52 difference occurs because the calculated structures are optimized at 0K while the experimental 53 structures were all obtained at temperatures above 100K. This difference in thermal conditions 54 55 results in variations in the densities (ρ), and consequently intermolecular interactions. 56 57 Ultimately, the denser PBE+vdW optimized structures have smaller band gaps than the high 58 59 60 ACS Paragon Plus Environment Page 15 of 36 The Journal of Physical Chemistry
1 2 3 temperature experimental counterparts (the band gap density dependence is expanded upon in 4 5 section 3.3 where we show that increased ρ results in smaller Eg’s). 6 7 8 3.3 Structure and Band Gap. It is useful to have properties trends and limits to guide 9 10 the design of new materials, and in this section we demonstrate that limits exist in the possible 11 12 optical gaps of PAH crystals as a function of the crystalline structure. The relationship between 13 the intermolecular close contact fractions as obtained via Hirshfeld surface analysis and the 14 15 predicted is shown in Figure 9. The C···C interactions are shown to have an inverse 16 17 18 relationship with the maximum possible in Figure 9a. It is interesting to note that if the 19 optical gaps and C···C% of the two PAH extrema, namely benzene and graphite, are used to 20 21 create a boundary line, all PAH optical gaps will exist below this limit (see Figure 9a). In 22 23 contrast to the inverse relationship of Figure 9a, Figure 9b shows a direct relationship between 24 25 the H···H contacts and the maximum value. When a boundary line is drawn as a 26 27 function of H···H% using benzene and graphite as extrema (as done in Figure 9a), 99% of all 28 29 PAH optical gaps will exist below the limiting function. 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
46 Figure 9. Comparison of predicted optical gaps and intermolecular close contact fractions for geometry 47 optimized structures. in both a and b were calculated on PBE+vdW geometry optimized structures and 48 49 adjusted with ξPBE to give the predicted . a) C···C close contact fractions vs. . Blue line represents a 50 boundary as a function of C···C%. and C···C% of BEZEN (5.342 eV, 0%) and graphite (0.04eV 51 58 , 100%) were used for boundary construction. b) H···H close contact fractions vs. . Optical gaps and 52 H···H% of BEZEN (5.342 eV, 64.6%) and graphite (0.04eV 58 , 0%) were used for boundary construction. 53 54 The inverse correlation of Figure 9a occurs because the C···C contacts are associated 55 56 with π···π interactions and more efficient packing within crystalline PAHs. Therefore, an 57 58 59 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 16 of 36
1 2 3 increase in the C···C contacts would increase the electrodynamic interaction of the π electrons 4 5 through increased overlap of neighboring molecules .59 The positive correlation of Figure 9b can 6 7 be explained by the fact that higher percentages of H···H contacts in PAHs are associated with 8 9 inefficiencies in the molecular packing. This leads to less dense structures with weaker 10 11 interactions between π orbitals, and therefore larger gaps. 12 It has been shown both experimentally 60 62 and theoretically 11 that the more dense a 13 14 PAH crystal becomes (through high pressure), the smaller Eg becomes. This trend of decreasing 15 16 Eg with increasing density also holds for ambient pressure PAH crystals. The blue line in Figure 17 18 10a exemplifies the fact that on average, the higher the density in the ambient pressure 19 20 structures, the smaller the maximum possible will be. It is therefore no coincidence that the 21 22 β and γ structures, which are among the densest on average, possess the smallest average 23 maximum gaps. 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 Figure 10 . Predicted optical gaps as a function of density and vdW energy for PAHs. in both a and b 40 were calculated on PBE+vdW geometry optimized structures and adjusted with ξPBE to give the predicted 41 . (a) Comparison of calculated densities and . Blue line represents the vs. ρ limit. (b) vdW 42 energy as a function of . Blue line represents the vs vdW energy per atom limit 43 .
44 45 When the predicted optical gaps and densities of benzene and HOPG are taken as limiting 46 47 cases, there is an upper limit or ceiling which dictates how large can be for a PAH at a 48 49 given density (ρ); the ceiling of the optical gap ( ) as a function of ρ is represented by a blue 50 line in Figure 10a, and Equation 1 : 51 52 53 (ρ) = ( 4.62 eV·mL/g) ρ + 10.50 eV (1) 54 55 56 57 58 59 60 ACS Paragon Plus Environment Page 17 of 36 The Journal of Physical Chemistry
1 2 3 Similarly, when E is plotted as a function of the predicted optical gap, a similar 4 vdW 5 scenario can be observed (see Figure 10b). For this limiting function, the optical gap and E 6 vdW 7 for graphite and benzene are used as limiting cases. The line describing this limit is expressed 8 9 as: 10 11 12 (E vdW )= (68.28) EvdW + 9.60 eV (2) 13 14 In Figures 9 & 10 , a lower limit also appears to be present in the band gaps of stable 15 16 PAH molecular crystals. Though it is known that the band gap of PAHs can be lowered below 17 2 18 1.5 eV through the substitution of hydrogen with halogens, chalcogens and other “R groups” , 19 20 as well as expansion of the system via the addition of more rings (often leading to unstable 21 configurations), all reported stable monomolecular PAH crystals appear to push down against a 22 23 1.5 eV lower limit. It is expected that larger discotic PAHs could be stable in the crystalline state 24 25 and produce band gaps below 1.5 eV, but at present crystal structures of these materials have not 26 27 been reported. Therefore, if the density, C···C%, or vdW energy of a PAH is known, one can 28 29 expect the band gap to be above 1.5 eV, and at or below the ceiling of that variable. 30 31 32 3.4 Comparison of Crystalline and Molecular Optical Gaps : It is known that the 33 34 energy needed for a system to excite an electron is reduced when going from the gas/solution 35 36 state to the crystal because of stabilizing interactions within the crystal. This can be 37 38 demonstrated by the fact that the of isolated molecules is always larger than those of the 39 40 molecules in the crystal as shown in Figure 11 . The difference between the crystalline (xtal) and 41 42 molecular (mol) band gaps ( ) can help describe the intermolecular interactions 43 present in the crystal: 44 45 = (3) 46 47 48 Comparison of the experimentally available and those calculated with PBE 49 50 are presented in Figure 12 . While agreement between and is good when adjusted 51 52 with ξ PBE (as shown in Figures 6 & 7 ) the numbers do vary slightly from experiment. Since 53 54 depends on both molecular and crystal values, if one or both are slightly off, 55 56 then the can deviate significantly from experiment (this is because variations between 57 58 the and is of the same order of magnitude as ). So, while PBE produced 59 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 18 of 36
1 2 3 several good correlations with experiment in Figure 12 (namely between structures 4 5 3,4,8 & 10), significant deviations exist for the remaining structures. 6 7 8 9 4 0 10 -0.1
3.5 ) 11 -0.2 12
3 ) (eV -0.3 13 g 14 2.5 -0.4 15 -0.5 2 16 ) - (Mol E
g -0.6
17 PBE (eV) Gap 1.5 -0.7 18
(Xtal E -0.8 EXP 19 1 Xtal Eg -0.9 PBE0 20 Mol Eg PBE 0.5 21 -1 22 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 Structure Number 23 Stucture Number 24 25 Figure 11 . Comparison of and as calculated Figure 12 . Comparison of experimental and calculated 26 with PBE. Values from Tables 1 & 2. for select PAHs. Values from Tables 1 & 2. 27 Stuctures: 1) QUATER10 2) PENCEN 3) QQQCIG04 4) Structures: 1) QQQCIG04 2) PERLEN07 3) POPIG 4) QUATER10 28 QQQCIG14 5) POPIG 6) QQQCIG13 7) TETCEN01 8) PERLEN07 5) ANTCEN 6) TETCEN01 7) HBZCOR 8) BEANTR 9) CORONE01 29 9) HBZCOR 10) ANTCEN 11) BENATHR 12) PYRENE02 13) 10) PYRENE02 11) PENCEN CORONE01 14) TERPHE02 15) BIPHEN 30 31 32 In order to obtain a more quantitative prediction of , we investigated whether 33 34 higher level methods such as PBE0 could alleviate the error. Figure 12 shows that the use of a 35 36 hybrid functional actually increases the error associated with calculating . In the case 37 38 of PBE0, the increased error is due to the fact that is normally predicted to be ~12% larger 39 40 than the experimental gap while the error associated with is smaller (5.6%). Therefore, 41 42 when the difference between the two is taken, is typically overestimated with respect 43 44 to PBE values. To correctly predict , it will be necessary to employ higher level 45 methods such as GW. 46 47 48 49 4. Conclusion. Dispersion inclusive DFT was used to model the structural and electronic 50 51 properties of all PAHs available in the CSD. It was found that addition of a ~ 1 eV constant to 52 the DFT PBE gap provided good agreement with experiment optical gaps. Hirshfeld surface 53 54 analysis revealed that relationships exist between the density, intermolecular cohesive energy, 55 56 and the relative fractions of C···C intermolecular contacts for the structures. Relationships 57 58 59 60 ACS Paragon Plus Environment Page 19 of 36 The Journal of Physical Chemistry
1 2 3 between the close contact fractions and the optical gaps were also established; i.e ., limits in the 4 5 maximum optical gaps of PAH crystals were established as a function of C···C close contact 6 7 fractions. Similarly, it was demonstrated that a limit between the maximum optical gap and the 8 9 intermolecular cohesive energy exists. An inverse correlation was found to exist between the 10 11 density and maximum optical gap in these organic molecular crystals (OMC), where the beta and 12 gamma motifs provide the smallest average optical gaps. A 1.5 eV minimum optical gap 13 14 boundary was established for all stable PAH crystal structures currently available in the CSD. It 15 16 was also shown that prediction of needs a higher level theory than semi local or 17 18 hybrid functionals. 19 20 21 5. Supporting Information: List of CSD Refcodes and IUPAC structure names. List 22 23 of calculated and experimental: 1) unit cell parameters 2) Hirshfeld surface close contact 24 25 fractions 3) π ⁰ parameter 4) interplanar angle for all structures and 5) molecular and crystalline 26 27 . This material is available free of charge via the Internet at http://pubs.acs.org . 28 29 30 6. Acknowledgements: This work was supported in part by the Eberly Science Fund 31 32 and the NSF DMR 1410736. 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 20 of 36
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1 2 3 Table of Contents Graphic 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment Page 25 of 36 The Journal of Physical Chemistry
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Figure 1. PAH motifs. a) HB anthracene [ANTCEN]. b) SHB – Quaterylene [QUATER10]. c) β HB – 1, 2, 3, 26 4, tetraphenylbenzene [FOVVOB]. d) γ – coronene [CORONE01]. e) β – anthra[2,1,9,8 27 hjkl]benzo[de]naphtha [2,1,8,7 stuv]pentacene [BOXGAW]. 28 196x103mm (96 x 96 DPI) 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 26 of 36
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Figure 2. Histograms of percent variation between PBE+vdW and experimental unit cell parameters for 91 18 PAHs. Negative numbers indicate the calculated unit cell parameters are smaller in the given cell dimension. 19 270x75mm (96 x 96 DPI) 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment Page 27 of 36 The Journal of Physical Chemistry
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Figure 3. Comparison of calculated and experimental densities for 91 PAHs. Black line represents exact 31 agreement between experiment and calculation. Green dotted lines represent ±5% variation. Red balls are 32 structures obtained at room temperature and blue balls are structures obtained below room temperature. 33 Calculated structures more than +5% from experiment have structure numbers from Table S2 in the SI of: 34 2-5, 7, 16-18, 27, 30, 34, 35, 39, 42, 43, 45, 57, 63, 77, 82, 87 245x168mm (96 x 96 DPI) 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 28 of 36
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Figure 4. Comparison of intermolecular close contact fractions between calculated and experimental 32 structures as derived via Hirshfeld surface analysis. Blue line represents exact agreement between 33 experiment and DFT. Values for individual PAHs are available in SI Table S3. 34 221x158mm (96 x 96 DPI) 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment Page 29 of 36 The Journal of Physical Chemistry
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Figure 5. (a) Density as a function of C•••C intermolecular close contact fractions from experimental PAH 33 structures (as extracted from CSD). Blue line is a linear best fit to the data (ρ = 0.0091 g/ml (C•••C%) + 34 1.2435 g/ml) and the gree n lines indicate ±5% from the best fit line. (b) van der Waals energy per atom as a function of density. Values obtained from optimized structures. Blue line is a linear best fit to the data 35 (EvdW = 0.072 eV•ml/g (ρ) + 0.0234 eV). (c) van der Waals energy per atom as a function of the C•••C 36 intermolecular close contact fractions. Values obtained from optimized structures. Blue line is a linear best fit 37 to the data (EvdW = 0.000661 eV•(C•••C%) 0.0694 eV). 38 211x156mm (96 x 96 DPI) 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 30 of 36
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Figure 6. Comparison of predicted and experimental gas/solution phase optical band gaps for select PAHs. E_g^optand E_g^PBE0are on the primary axis. E_g^opt are as referenced in Table 1 (green 29 triangles are gas phase and grey diamonds are solution phase). E_g^PBE are plotted on the secondary 30 vertical axis (on the right), and aligned with experimental values (see details in the text) by shifting the zero 31 crossing of the secondary vertical axis by 0.99 eV relative to that of the primary axis (same scale as the 32 primary axis). Structure reference numbers correlate with Table 1. 33 194x118mm (96 x 96 DPI) 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment Page 31 of 36 The Journal of Physical Chemistry
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Figure 7. Comparison of predicted and experimental optical gaps of crystalline PAHs. Experimental and 29 PBE0 gaps are on the primary axis. Experimental gaps are as referenced in Table 2. PBE gaps are plotted 30 on the secondary vertical axis (on the right), and aligned with experimental values by shifting the zero 31 crossing of the secondary vertical axis by 1.05 eV relative to that of the primary axis (same scale as the 32 primary axis). Structure reference numbers correlate with Table 2. 33 212x134mm (96 x 96 DPI) 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 32 of 36
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Figure 8. Band gaps from the geometry optimized and the experimental structures. Band gaps were 33 calculated using PBE on PBE+vdW geometry optimized structures and experimental structures as extracted 34 from the CSD. Black line represents exact agreement between experiment and DFT while the green dotted 35 lines represent ±5% variation. 36 150x114mm (96 x 96 DPI) 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment Page 33 of 36 The Journal of Physical Chemistry
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Figure 9. Comparison of predicted optical gaps and intermolecular close contact fractions for geometry 20 optimized structures. E_g^PBEin both a and b were calculated on PBE+vdW geometry optimized structures 21 and adjusted with ξPBE to give the predicted E_g^opt. a) C•••C close contact fractions vs. E_g^opt. Blue line represents a E_g^opt boundary as a function of C•••C%. E_g^opt and C•••C% of BEZEN (5.342 eV, 22 0%) and graphite (0.04eV 58, 100%) were used for boundary construction. b) H•••H close contact 23 fractions vs. E_g^opt. Optical gaps and H•••H% of BEZEN (5.342 eV, 64.6%) and graphite (0.04eV 58, 24 0%) were used for boundary construction. 25 248x83mm (96 x 96 DPI) 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 34 of 36
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Figure 10. Predicted optical gaps as a function of density and vdW energy for PAHs. E_g^PBEin both a and 20 b were calculated on PBE+vdW geometry optimized structures and adjusted with ξPBE to give the predicted E_g^opt. (a) Comparison of calculated densities and E_g^opt. Blue line represents the 21 E_g^optvs. ρ limit. (b) vdW energy as a function of E_g^opt. Blue line represents the E_g^optvs vdW 22 energy per atom limit. 23 253x84mm (96 x 96 DPI) 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment Page 35 of 36 The Journal of Physical Chemistry
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 Figure 11. Comparison of E_g^Xtal and 〖 E〗_g^Mol as calculated with PBE. Values from Tables 1 & 2. 28 Stuctures: 1) QUATER10 2) PENCEN 3) QQQCIG04 4) QQQCIG14 5) POPIG 6) QQQCIG13 7) TETCEN01 8) 29 PERLEN07 9) HBZCOR 10) ANTCEN 11) BENATHR 12) PYRENE02 13) CORONE01 14) TERPHE02 15) BIPHEN 30 31 104x79mm (96 x 96 DPI) 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment The Journal of Physical Chemistry Page 36 of 36
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Figure 12. Comparison of experimental and calculated 〖∆E 〗_g^(Xtal Mol) for select PAHs. Values from 29 Tables 1 & 2. 30 Structures: 1) QQQCIG04 2) PERLEN07 3) POPIG 4) QUATER10 5) ANTCEN 6) TETCEN01 7) HBZCOR 8) 31 BEANTR 9) CORONE01 10) PYRENE02 11) PENCEN
32 102x81mm (96 x 96 DPI) 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment