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Time-Dependent Density-Functional Description of the La State In Time-Dependent Density-Functional 1 Description of the La State in Polycyclic Aromatic Hydrocarbons THESIS Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Ryan Matthew Richard, B.S. Department of Chemistry ***** The Ohio State University 2011 Thesis Committee: John M. Herbert, Adviser Sherwin J. Singer c Copyright by Ryan Matthew Richard 2011 Abstract The electronic spectrum of alternant polycyclic aromatic hydrocarbons (PAHs) in- 1 1 cludes two singlet excited states that are often denoted La and Lb. Time-dependent 1 density functional theory (TD-DFT) afford reasonable excitation energies for the Lb 1 state in such molecules, but often severely underestimates La excitation energies, and 1 fails to reproduce observed trends in the La excitation energy as a function of molec- ular size. Here, we examine the performance of long-range-corrected (LRC) density 1 1 functionals for the La and Lb states of various PAHs. With an appropriate choice for the Coulomb attenuation parameter, we find that LRC functionals avoid the severe 1 underestimation of the La excitation energies that afflicts other TD-DFT approaches, 1 while errors in the Lb excitation energies are less sensitive to this parameter. This 1 suggests that the La states of certain PAHs exhibit some sort of charge-separated character, consistent with the description of this state within valence-bond theory, but such character proves difficult to identify a priori. We conclude that TD-DFT calculations in medium-size, conjugated organic molecules may involve significant but hard-to-detect errors. Comparison of LRC and non-LRC results is recommended as a qualitative diagnostic. ii Acknowledgments RMR wishes to express his sincerest gratitude to JMH for his patience, insights, and guidance in preparing this thesis and the manuscript from which it is derived. RMR also wishes to thank The Ohio State University. iii Vita 2004 . .Parma Senior High School 2008 . .B.S. Chemistry, Cleveland State Uni- versity 2008 to present . Graduate Teaching Associate, The Ohio State University Publications Richard, Ryan M.; Ball, David W. B3LYP calculations on the thermodynamic proper- ties of a series of nitroxycubanes having the formula C8H8-x(NO3) (x=1-8). Journal of Hazardous Materials (2009), 164, 1595-1600. Richard, Ryan M.; Ball, David W. Density functional calculations on the thermo- dynamic properties of a series of nitrosocubanes having the formula C8H8-x(NO)x (x=1-8). Journal of Hazardous Materials (2009), 164, 1552-1555. Richard, Ryan M.; Ball, David W. Ab intio calculations on the thermodynamic prop- erties of azaborospiropentanes. Journal of Molecular Modeling (2008), 14, 871-878. Richard, Ryan M.; Ball, David W. Enthalpies of formation of nitrobuckminster- fullerenes: Extrapolation to C60(NO2)60. Journal of Molecular Structure: THEOCHEM (2008), 858, 85-87. Richard, Ryan M.; Ball, David W. G2, G3, and complete basis set calculations on the thermodynamic properties of triazane. Journal of Molecular Modeling (2008), 14, 29-37. Richard, Ryan M.; Ball, David W. G2, G3, and complete basis set calculations of the thermodynamic properties of cis- and trans-triazene. Journal of Molecular Modeling (2008), 14, 21-27. iv Richard, Ryan M.; Ball, David W. Ab initio calculations on the thermodynamic prop- erties of azaspiropentanes. Journal of Physical Chemistry A (2008), 112, 2618-2627. Richard, Ryan M.; Ball, David W. Ab initio calculations on the thermodynamic prop- erties of spiropentane and its boron-containing derivatives. Journal of Molecular Structure: THEOCHEM (2008), 851, 284-293. Richard, Ryan M.; Ball, David W. G2, G3, and complete basis set calculations of the thermodynmaic properties of aminoborane, diaminoborane, and triaminoborane. Journal of Molecular Structure: THEOCHEM (2007), 823, 6-15. Richard, Ryan M.; Ball, David W. G2, G3, and complete basis set calculations of the thermodynamic properties of a small beryllium molecules. Journal of Undergraduate Chemistry Research (2007), 6, 129-138. Richard, Ryan M.; Ball, David W. B3LYP, G2, G3, and complege basis set calcula- tions of the thermodynamic properties of small cyclic and chain hydroboranes. Journal of Molecular Structure: THEOCHEM (2007), 814, 91-98. Richard, Ryan M.; Ball, David W. G2, G3, and complete basis set calculations of op- timized geometries, vibrational frequencies, and thermodynamic properties of azatri- boretidine and triazaboretidine. Journal of Molecular Structure: THEOCHEM (2007), 806, 165-170. Richard, Ryan M.; Ball, David W. Optimized geometries, vibrational frequencies, and thermochemical properties of mixed boron- and nitrogen-containing three-membered rings. Journal of Molecular Structure: THEOCHEM (2007), 806, 113-120. Richard, Ryan M.; Ball, David W. G2, G3, and complete basis set calculations of the thermodynamic properties of boron-containing rings: cyclo-CH2BHNH, 1,2-, and 1,3-cyclo-C2H4BHNH. Journal of Molecular Structure: THEOCHEM (2006), 776, 89-96. v Fields of Study Major Field: Chemistry vi Table of Contents ABSTRACT . ii ACKNOWLEDGMENTS . iii VITA ........................................ iv LIST OF FIGURES . ix LIST OF TABLES . xii CHAPTER PAGE 1 Background . 1 2 Methods . 4 3 Results . 8 3.1 Linear-condensed acenes . 8 3.2 Non-linear Polyaromatic Hydrocarbons . 14 4 Analysis and Discussion . 20 4.1 Valence-bond considerations . 20 4.2 Difference densities . 23 4.3 Tozer's CT metric . 26 4.4 Atomic partition of particle/hole densities . 29 vii 5 Summary and Conclusion . 34 BIBLIOGRAPHY . 38 APPENDICES A Supplemental Information . 43 viii List of Figures FIGURE PAGE 1 1 1 Transition densities for (a) the La state and (b) the Lb state of naph- thalene, computed at the TD-B3LYP level. The isosurface in either plot encapsulates 90% of the transition density. 3 1 2 TD-DFT errors in the vertical excitation energies for the La state, expressed in wavelength units. Panel (a) illustrates the divergence of the TD-B3LYP and TD-BP86 excitation energies as a function of n, while panel (b) shows a close-up view of the errors engendered by several different LRC functionals. 9 1 3 TD-DFT errors in the vertical excitation energies for the La state, expressed in energy units. 11 1 4 TD-DFT errors in the vertical excitation energies for the Lb state, expressed in (a) wavelength units and (b) energy units. 12 5 Clar-type resonance structures54, 55 of the nonlinear PAHs considered in this work, along with the numbering scheme that is used to refer to them in the text and figures. 15 1 6 Errors (theory minus experiment) in La excitation energies for the PAHs depicted in Fig. 5. Dashed horizontal lines represent the average error for each method. Experimental benchmarks are band maxima in non-polar solvents. The solvent correction suggested in Ref.50 would reduce the TD-LRC-DFT errors by 0.11 eV and would make the TD- B3LYP errors more negative by 0.11 eV. 16 ix 1 7 Errors (theory minus experiment) in Lb excitation energies for the PAHs depicted in Fig. 5. Dashed horizontal lines represent the average error for each method. Experimental benchmarks are band maxima in non-polar solvents, and omissions from the data set in Fig. 5 correspond 1 to molecules for which no experimental value for the Lb excitation energy is available. The solvent correction suggested in Ref.50 would reduce all of the errors by 0.03 eV. 18 1 8 NTOs for the La state of naphthalene, computed at the TD-B3LYP/ cc-pVTZ level. 21 9 NTOs for two representative PAHs: (a) benzo[e]pyrene [7] and (b) di- benz[a; c]anthracene [11]. The structure of each molecule is also shown. The NTOs shown in (a) accounts for 93% of the transition density and those in (b) account for 84% of the transition density. Each NTO was computed at the TD-B3LYP/cc-pVTZ level. 22 1 1 10 Difference densities, ∆ρ, for the La and Lb states of the linear acene sequence computed using two different TD-DFT methods. In each plot, the red/blue isosurfaces encapsulate 95% of the positive/negative part of ∆ρ. The difference between the two difference densities, ∆∆ρ, is also plotted, using the same isocontour that is used to plot ∆ρ at the TD-LRC-!PBE level. 24 1 1 11 Difference densities, ∆ρ, for the La and Lb states of PAHs [7] and [11] computed using two different TD-DFT methods. The red/blue isosurfaces encapsulate 95% of the positive/negative part of ∆ρ. The difference between the two difference densities, ∆∆ρ, is also plotted, using the same isocontour as the TD-LRC-!PBE plots. 25 12 Errors in TD-DFT excitation energies for the nonlinear PAHs, plotted as a function of the CT metric, Λ. The LRC-!PBE and LRC-!PBEh functionals afford similar results, so only the latter is shown here. 28 x 13 Differences between excited-state and ground-state Mulliken charges [∆QA, from Eq. (4.8)] for the carbon atoms in hexacene, computed at the TD-B3LYP/6-31G* level. Only the symmetry-unique carbon 1 atoms have been labeled, with S0 ! La charge differences on the left 1 side and S0 ! Lb charge differences on the right side. 31 14 Differences between excited-state and ground-state Mulliken charges for the carbon atoms in PAHs [7], [10], and [11]. Charges were com- puted at the TD-B3LYP/6-31G* level and are listed for the symmetry- 1 unique carbon only; blue labels (left side) correspond to La and red 1 labels (right side) correspond to Lb. .................. 32 15 Vertical excitation energy for the La state . 50 16 Vertical excitation energy for the La state in eV . 51 17 Absolute errors for the La state . 52 18 Absolute errors for the La state in eV . 53 19 Vertical excitation energy for the Lb state . 54 20 Vertical excitation energy for the Lb state in eV .
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