Theoretical Studies of the Nonlinear Optical Properties of Organic Materials Citation Lu, Daqi (2000) Theoretical Studies of the Nonlinear Optical Properties of Organic Materials. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/myx9-7278. https://resolver.caltech.edu/CaltechTHESIS:10012010-103544088 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Applied Physics) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Applied Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Goddard, William A., III Thesis Committee: Goddard, William A., III (chair) Johnson, William Lewis (co-chair) Kornfield, Julia A. Defense Date: 19 November 1999 Record Number: CaltechTHESIS:10012010-103544088 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:10012010-103544088 DOI: 10.7907/myx9-7278 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 6085 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 01 Oct 2010 18:09 Last Modified: 14 Jul 2023 20:00 Thesis Files Preview PDF - Final Version See Usage Policy. 40MB Repository Staff Only: item control page

Theoretical Studies of the Nonlinear Optical Properties of Organic Materials Thesis by Daqi Lu In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2000 (Submitted Nov 19, 1999) 11 © 2000 Daqi Lu All Rights Reserved 111 To my parents IV Acknowledgements I would like to thank the following people: 1. Mentor Bill Goddard for his guidance and support. 2. Prof. Guanhua Chen, Prof. Seth Marder and Prof. Jason Perry for the cooperation on this project. 3. Group members, Guanghua Gao, Jinsong Hu, Xinlei Hua, Rick Muller, Yuejin Guo, Ersan Demiralp, Siddharth Dasgupta, Mario Blanco, Tahir Cagin, Matt Carlson, Derek Debe,Cecco Faglioni, Darryl Willick, and Yongchun Tang for their help. 4. IQC coworkers, Eric Wu, Derek Xu, Bill Larsen, C. J. Stoltman, John Busby, Li Ping, Scott Louie, and Connie Chan for going through the frustration and happiness of an Internet startup with me. 5. My wife, Winnie Wang, for her patience and understanding. v Contents Acknowledgements iv 1 Introduction 1 2 1.1 Valence-Bond Charge-Transfer Exciton (VB-CT-E) Mode1 4 3 1. 2 Valence-Bond Charge-Transfer (VB-CT) Mode15 ..... 3 1.3 Valence-Bond Charge-Transfer Solvent (VB-CT-S) Mode16 3 1.4 Valence-Bond Charge-Transfer Temperature (VB-CT-T) Mode17 4 1.5 Saturation of the Second Hyperpolarizability for Polyacetylenes 8 4 1.6 ab initio Predictions of Large Hyperpolarizabilities Push-Pull Polymers 9 5 1. 7 References................................. 5 The Valence-Bond Charge-Transfer-Exciton (VB - CT E) Model for Predicting Nonlinear Optical Properties of Polymeric Materials 9 2.1 Introduction............................... 9 2.2 The Valence-Bond Charge-Transfer-Exciton (VB-CTE) View of Poly- 2.3 mer Excited States . . . 10 2.2.1 The Hamiltonian 10 2.2.2 Dipole Moment . 12 2.2.3 Neglect of Coulomb Interactions. 14 Saturation Length. . . . . . . . . . . . . 16 2.3.1 Empirical Formula for Saturation Lengths 2.3.2 Effects of Coulomb Interation and Frequency of Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.4 16 18 Calculations of the Saturation Lengths for Oligothiophenes and Polyacetylene . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4.1 Oliothiophenes, Comparison to Experiment. 21 2.4.2 Polyacetylene . . . . . . . . . . . . . . . . . 25 VI 2.5 Discussion . . . . . . . . . . . 26 2.5.1 Comparision with Experiment 26 2.5.2 Predictions for Other Materials, Design Considerations 26 2.5.3 Comparisons to Other Theory 29 2.6 Summary 29 2.7 References 30 3 The Valence-Bond Charge-Transfer Model (VB-CT) for Nonlinear 4 Optical Properties of Charge-Transfer Organic Molecules 32 3.1 Introduction. 32 3.2 Theory.... 33 3.2.1 The VB-CT Model 33 3.2.2 Inclusion of the Bond Length Alternation (BLA) Coordinate 35 3.2.3 Application of an Electric Field 36 3.2.4 Polarizabilities 38 ......... 3.3 Predictions of /-l, C¥ , /3, and '"Y From VB-CT Theory 39 3.4 Discussion .. . . . . . . . . . . . . . . . . . 45 3.4.1 Comparison with AMI Calculations. 45 3.4.2 Comparison with Two-Level Models. 45 3.4.3 Additional Excited States . . . . . 46 3.4.4 Applications to Various Molecules. 47 3.5 Summary 48 3.6 References 49 The Valence-Bond Charge-Transfer Solvation Model (VB-CT-S) Nonlinear Optical Properties of Organic Molecules in Polar Solvents 51 4.1 Introduction..... 51 4.2 The VB-CT-S Model 52 4.2.1 No Solvent. . 52 4.2.2 Solvation Effects 53 Comparison with Experiment 57 4.3 VB 4.4 5 6 7 4.3.1 Application to 1,1 dicyano, 6-(di-butyl amine) hexatriene,(l) 57 4.3.2 Frequency Dependent Correction for Hyperpolarizations 60 Discussion. .. . . . . . . . . 60 4.4.1 Summary of VB-CT-S 60 4.4.2 Estimation of Molecular Based Properties 61 4.4.3 Estimation of Solvent Properties 61 4.4.4 Estimation of Solvation Energy 62 4.5 Summary 62 4.6 References 63 The Temperature Dependence For Nonlinear Optical Properties of Donor Acceptor Push-Pull Organic Molecules 65 5.1 Introduction 65 5.2 Theory . . . 65 5.3 Application to 1,1 dicyano, 6-(di-butyl amine) hexatriene 68 5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.4.1 Electronic and Vibrational Hyperpolarizabilities 70 5.4.2 Macroscopic Hyperpolarizabilities 70 5.5 Summary 72 5.6 References 73 Saturation of the Second Hyperpolarizability for Polyacetylenes 75 6.1 Introduction 75 6.2 Calculations 75 6.3 Discussion 78 6.4 Summary 80 6.5 References 82 ab initio Predictions of Large Hyperpolarizability Push-Pull Polymers: Julolidinyl-n-isoxazolone and Julolidinyl-n-N, N'-diethylthiobarbituric VIII 84 acid 7.1 Introduction 84 7.2 Results . . . 84 7.3 Calculational Details 90 7.3.1 Basis Sets 90 7.3.2 Geometry 90 7.3.3 Hyperpolarizabilities 91 7.4 References . . . . . . . . . . 92 A Evaluation of Matrix Elements 94 B Application of Perturbation Theory 97 C The Evaluation of t and V 100 D Evaluation of the Temperature Effect 102 IX List of Figures 1.1 Molecule with high (3 . . . . . 2 1.2 Donor acceptor type molecule 2 2.1 Molecule with good NLO properties. 9 2.2 Polymer oligothiophene. . . . . . . . 10 2.3 Dependence of saturation behavior on TJ = t/V for (a) polarizability and (b) hyperpolarizability. . . . . . . . . . . . . . . . . . . . . . .. 2.4 17 The saturation behavior of polyacetylene for (a) polarizability and (b) hyperpolarizability. The solid line (NoQ) uses sum-over-state or perturbation without Coulomb interaction. The dashed line (Sum-Q) uses sum-over-state with Coulomb interaction . . . . . . . . . . . . . . .. 2.5 20 Effect offrequency on the saturation behavior of (a) polarizability and (b) hyperpolarizability. The resonance frequency of a is 2.8 eV (corresponding to w/V = 0.49) whereas the resonance frequency of I is 1.4 eV (corresponding to w/V = 0.24). . . . . . . . . . . . . . . . . . .. 2.6 22 Comparison of theory and experiment for saturation behavior of oliogthiophene. (a) polarizability (w =1.95 eV) and (b) hyperpolarizability (w =1.16 eV). Dashed line with error bar uses experiment data. The solid line uses sum over states with Coulomb interaction. 24 2.7 Repeating units for polymers of Table 2.3 . 28 3.1 Donor acceptor type molecule . . . . . . . 32 x 3.2 Energies of the ground (Egr) and excited (Eex) states as a function of bond length alternation (BLA), q. Vo is the adiabatic difference between the pure VB and CT states and qopt is the optimum BLA coordinate, qopt. These calculations used k and t from (3.17). (a) Vo = leV leads to qopt = -0.069A, (b) Vo = 0 leads to qopt = 0, (c) Vo = -leV leads to qopt = +0.069A. 3.3 37 The relation between hyperpolarizability and charge transfer character (1). The right-hand side figures are the comparision between VB-CT caculation (solid line) and AMI caculation (squares) (reference 9). (a) Polarizability, a, (b) hyperpolarizability, (3, (c) second hyperpolarizability, {, and (d) third hyperpolarizability, O. . . . . . . . . . . . .. 3.4 40 The relation between the dipole moment Pz (in units of MeT) and V (the VB-CT excitation energy). (a) polarizability, a, (b) hyperpolarizability, {3, (c) second hyperpolarizability, {, and (d) third hyperpolarizability, 6 obtained from the derivatives of Pz with respect to -V as shown by equations (3.30)-(3.33). 42 3.5 Valence bond state of the bridge .. 46 3.6 Charge transfer state of the bridge 47 3.7 Molecule with f = 0.5 . . . . . . . 47 3.8 Molecule with the positive maximum in {3 48 3.9 Molecule with the positive maximum in { 48 4.1 Molecule 1,1 dicyano, 6-(di-butyl amine) hexatriene 51 4.2 The Marcus solvation model. rD and r A are the radii of the donor and acceptor, respectively. RDA is the distance between the donor and acceptor. 54 Xl 4.3 The predicted dependence of pol ariz abilities on solvent polarity (expressed in terms of the static dielectric constant E). (a) Polarizability, a, (b) Hyperpolarizability, fJ, (c) Second hyperpolarizability, 'Y, (d) Third hyperpolarizability, r5zzzzz. The values plotted are the static averaged values. The parameters used are: Va = 0.833 eV, SF = 0.0685A-1, t = 1.184 eV, RDA = 7.30A, Q = 0.738, k =33.55 eV/A2. For 'Y in (c) a comparison is made between theory (solid line) and experiment (dots). Here the experimental results were corrected to static values using (4.39). 5.1 ........................... 58 Temperature dependence of the polarizability (a), and hyperpolarizabilities (f3, and 'Y) for 1,1 dicyano,6-(di-butyl amine) hexatriene (1). The dash lines (analytical) are from the formula (5.31 - 5.33). The solid lines are from the numerical evaluation of (5.17). The crosses are the vibrational contributions. The triangles are the electronic contributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.1 71 Recent results on the saturation of 'Y/N for polyacetylenes (PA). The open squares are from the experient (reference 8) on a substituted, imperfect PA. The open circles are from semiempirical theory (reference 9).The filled squares are from the current calculation. . . . . . . . .. 6.2 76 (a) Pol ariz ability of polyacetylenes chains from ab initi 0 calculations (J aguar /NLO). The filled squares are from the current ab initio HF calculations. The open circles are for VB-CT using t = 0.83e V and V = 1.0eV. The solid line is a fit to an equation like (6.3). The parameters are a oo = 22.6 x 1O-24 esu , Aa = 0.249, Ba = 0.84 , and Ca = 0.08. (b) Second hyperpolarizability of polyacetylenes chains from ab initio calculations (Jaguar/NLO). The line is a fit to equation (6.3). The parameters are "(00 = 380 X 10- 36 esu, A, = 0.073, B, = 11.4, and C, = 0.182. 6.3 ........... 77 Molecule used in experiment (reference 8) 78 Xll 6.4 Second hyperpolarizability of polyacetylenes chains. The filled squares are from the current ab initio HF calculations while previous ab initio results (reference 14) are shown with open squares. The open circles are for VB-CT using t = 0.83eV and V = 1.0eV. Direct experimental results (reference 16) are shown with filled triangles. . 79 6.5 Solvent used in experiment (reference 19) . . . . . . . 81 6.6 Effect of disorder in second hyperpolarizability based on VB-CT with t = 0.83eV and V = 1.0eV. Filled circles indicates the ordered polyene while open circles are with a random disorder of Vi = V ± O.4e V. 7.1 (a) Julolidinyl-n-isoxazolone and (b) Julolidinyl-n-N, N'-diethylthiobarbituric acid 7.2 .................. . ................. 85 The dependence of hyper pol arizability wf3 (0) for Jololidine-n-isoxazolone on N = n + 1, the number of double bonds in the linker. . . . . . .. 7.3 81 86 The dependence of CY xx , f3xxx, and 'Yxxxx on N = n + 1, the number of double bonds in the linker. The solid lines are the least squares fit to the calculated values. The units are 10- 23 esu for CY xx , 10- 30 esu for f3xxx, and 10- 37 esu for 'Yxxxx. A.1 The MO of butadiene. 89 The wavefunction of LUMO is 0.602Xl + 0.372X2 - 0.372X3 - 0.602X4. The wavefunction of HOMO is 0.602Xl ~ 0.372X2 - 0.372X3 + 0.602X4, where Xi are the four atomic orbitals.. 95 Xlll List of Tables 2.1 Theoretical properties of polyacetylene as a function of polymer length, N. Q indicates the use of (2.2) whereas NoQ indicates the use of (2.16). All cases use zero frequency. The monomer is butadiene. Using V = 2.85eV. b (a) = ~(azz). C ('Y) = ~(-yzzzz). 2.2 a ......... 18 Experimental and theoretical properties of oligothiophenes as a function of polymer length, N. NoQ indicates the use of (2.16) whereas Q indicates the use of (2.2). w = 0 indicates the static case whereas w =j:. 0 indicates finite frequency. w = 1.95eV for (a(-w,w)) and w = 1.16eV for (-y(-2w;w,w, 0)). aS ee reference 4. bUsing V = 4.0geV,t = 0.83eV anda=V2. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 2.3 25 Predicted saturated values polarizability (a), hyperpolarizability b), and saturation lengths (La, Ly) of polymers. Based on equation (2.19) and (2.21) with w = o. aCalculated values from reference 15. bExperimental values from reference 15. cCalculated using (2.28) or (2.29). dUsing a = V2. eYalues at saturation using (2.19) and (2.21) and w = O. An approximate correction for finite frequency is to multiply a( -w, w) by Egj(Eg - w) and to multiply 'Y(-3w;w,w,w) by E g3j(Eg - w)(Eg2w)(Eg - 3w). 'See Figure 2.7. gUsing (2.25) and (2.26) unless otherwise noted. hDerived in text rather than from B. iThe entries in the table are in terms of butadiene monomers. The number of double bonds is 14 for a and 24 for ,. 27 xiv 7.1 Hyperpolarizabilities for the Julolidinyl-n-isoxazolone class and Julolidinyl- n-N, N'-diethylthiobarbituric acid class. Here /-l is the dipole moment, ,8(0) is the static hyperpolarizability, and () is the angle between j1 and $. The units are 10- 48 esu for J-l" ,8(0), 10- 18 esu for /-l, and 10-30 esu for ,8(0). The experiments used CHCl3 solvent whereas theory considered 87 molecules in the gas phase. 7.2 Dipole moment /-l, polarizability O!, hyperpolarizabilities j3 and, for Julolidinyl-n-isoxazolone class and Julolidinyl-n-N, N'-diethylthiobarbituric acid class. The units are 10- 18 esu for /-l, 10- 24 esu for O!, 10- 30 esu for ,8, and 10- 36 esu for ,.a Molecule in Figure 7.l.a with n Molecule in Figure 7.1.b with n = 6. 7.3 = 6. b ................ 88 Hyperpolarizabilities of molecules in Figure 7.1.a with n = 0,1,2. no solvent indicates a standard gas phase calculation using Jaguar, solvent indicates Jaguar/Solvate calculation using chloroform, the solvent used in the experiments. The units are 10- 24 esu for O!, 10- 30 esu for ,8, and 10- 36 esu for J. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7.4 89 Hyperpolarizabilities of molecules in Figure 7.l.a with n = 1 with different basis sets, 6-31G, 6-31G** (including polarization functions), and 6-31 G++ (including diffuse functions). The units are 10- 24 esu for O!, 10- 30 esu for ,8, and 10- 36 esu for ,. 90 1 Chapter 1 Introduction There is a great deal of interest in developing organic nonlinear optical (NLO) materials 1 for use in 1. optical processing of data/images, 2. optical storage of data/images 3. optically based telecommunications, and 4. optically based computers. The important properties for these applications are the hyperpolarizabilities, which describe how the dipole moment (/1) depends on an applied electric field (£), (1.1) Here ct, {3, and I are the polarizability, first hyperpolarizability, and second hyperpolarizability, respectively. The properties of most interests are {3 and I which control 1. frequency doubling (better focus, more data), 2. changes in refractive index (electro-optical switches for telecommunications), and 3. frequency mixing. Currently LiNb0 3 is the material of choice for these applications 1 . However, polymers would provide great advantages in ease of processing and for tailoring the properties to match precise requirements. Recent advances in developing new high {3, I organic materials 2 - 3 include the development of such materials such as polymer in Figure 1.1. The general pattern 2 o - Figure 1.1: Molecule with high {3 Figure 1.2: Donor acceptor type molecule for good NLO materials are donor acceptor type molecules with a conjugated chain in between (Figure 1.2). A typical approach for predicting polarizabilities involves summing over intermediate states formed from molecular orbitals. Thus for a laser frequency w, the polarizability and hyperpolarizabilities have the form (2.31) - (2.33). Given a good description of the excited states, this sum-over-states approach can be used to predict accurate values of CY, {3, and 'Y. However,there are two problems: 1. It rapidly becomes complicated and expensive as system size increases. 2. There is no obvious relationship between cy,{3,'Y or between these properties and other properties of the system. We have developed a new approach (denoted VB-CT) for predicting NLO properties. There are four models in this approach to describe the relationship between the NLO properties and the linker (VB-CT-E), the structure (VB-CT), the solvent (VB-CT-S), and the temperature (VB-CT-T). For the past few years, this model has gained vast popularity and has become the standard model for predicting the NLO properties of conjugated system. A partial list of the papers that reference VB-CT model is in reference 10-60. 3 In order to accurately predict the NLO properties, we also did ab initio calculations on long polyacetylene chain 8 and polymers9 with the best NLO properties at both Hartree Fock and DFT level. The results agree well with the experiment. 1.1 Valence-Bond Charge-Transfer Exciton (VBCT-E) Model4 In Chapter 2, we develop a theory to predict the polarizability (a), second hyperpolarizability b), and the saturation lengths for nine polymeric materials. The theory (VB-CT-E) is based on a valence-bond view on the ground and excited states and considers the excited states as charge transfer excitons. It involves just two parameters which can be extracted from simple molecular orbital calculations and/or from experimental values of bandgap and bandwidth. For the one system (oligothiophenes) with experimental data on saturation length, the predicted L'Y and La are in good agreement with experiment. 1.2 Valence-Bond Charge-Transfer (VB-CT) Mode15 In Chapter 3, we describe the nonlinear optical properties of charge-transfer organic materials in the framework of a simple valence-bond charge-transfer model. This model leads to analytic formulas for the absorption frequency, hyperpolarizabilities, and bond length alternation, all of which are described in terms of two parameters, V, and t related to the bandgap, and bandwidth. This model provides a clear physical picture for the dependence of the hyperpolarizabilities on the structure of chargetransfer molecules and leads to good agreement with the trends predicted by the AMI calculations. 1.3 Valence-Bond Charge-Transfer Solvent (VBCT-S) Model 6 In Charpter 4, we develop a model to precit the solvation effect on the NLO properties of charge transfer organic materials such as 1,1 dicyano,6-(di-butyl amine) hexatriene. 4 This model is based on the VB-CT framework, using a continuum description of the solvent. The resulting model leads to analytic formulas for the absorption frequency, the hyperpolarizabilities, and the bond length alternation with only one solvent dependent parameter E (the dielectric constant of the solution). The theory involves just four solvent-independent parameters, VO, t, SF, and Q, which are related to the bandgap, bandwidth, geometry, and dipole moment of the CT molecule. The results are in good agreement with experiment. 1.4 Valence-Bond Charge-Transfer Temperature (VBCT-T) Model 7 In Chapter 5, we predict the temperature effect on the NLO properties of the charge transfer organic materials. This model is based on the valence-bond charge-transfer (VB-CT) framework, but considers the effect of finite temperatures on the hyperpolarizabilities. This leads to analytic formulae for the hyperpolarizabilities as the function of temperature. Both electronic and vibrational contributions to the hyperpolarizabilities are discussed in this model. 1.5 Saturation of the Second Hyperpolarizability for Polyacetylenes 8 Polyacetylene (PA) polymers - (-c H = C H - ) N - lead to large second hyperpolarizabilities (ry) that increase with N. For small N the increase is quite rapid, eventually saturating (becoming linear in N) for N 2: N;at. Both experimental and theoretical attempts have been made to estimate N;at. In 1994 this led to two Science papers, one obtaining N;at = 20 (semiempirical theory) and the other obtaining N;at = 125 (experiment, but on a substituted disordered form of PA)! In Chapter 6, we did ab initio calculations up through N = 49 (C98HlOO) and show that N;at = 45 ± 5. The much larger experimental value, N;at = 125, is explained in terms of the structural defects and disorder introduced by the synthetic method. 5 1.6 ab initio Predictions of Large Hyperpolarizabilities Push-Pull Polymers 9 Recently significant advances have been made in engineering push-pull organic chrDmophores to have very large hyperpolarizabilities ((3), leading to materials with J1 . (3 as high as 15,000 10- 48 esu. Such developments have been slow and costly because of difficulties in synthesis, purification, and measurement. As an alternative we have developed a new Quantum Mechanical program (JaguarjNLO) which provides predictions of (3 for such molecules far faster than previously possible. In Chapter 7, We have applied JaguarjNLO to predicting 0:, (3, and 'Y for the high (3 push-pull organics and find excellent agreement with experiment. In summary, VB-CT model provides a qualititive framework, and ab initio calculation provides an accurate prediction for the NLO properties of organic materials. Combined together, these methods can be used as an effective tool for developing new nonlinear optical materials. 1.7 References 1. Prasad, P. N.; Williams, D. J. Eds, Introduction to Nonlinear Optical Effects in Molecules and Polymers; Wiley: New York, 1991. 2. Marder, S. R. et al., Science 263,511(1994). 3. Marder, S. R.; Torruellas, W. E.; Blancharddesce, M.; Ricci, V. et al., Science 1997, 276, 1233. 4. Lu, D.; Chen, G.; Goddard, W. A. III, J. Chern. Phys. 101, 4920 (1994). 5. 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Zyss, J.; Nicoud, J., Curro Opin. Solid. St. M 1, 533 (1996) 9 Chapter 2 The Valence-Bond Charge-Transfer-Exciton (VB - GTE) Model for Predicting Nonlinear Optical Properties of Polymeric Materials 2.1 Introduction Nonlinear optical (NLO) properties are important in numerous applications from lasers to optical switches and electronics. 1 Some of the best NLO properties are displayed by organic materials, for example charge-transfer type organic molecules where contributions to NLO come mostly from charge resonance between the donor and acceptor. 2 However, for molecules with a long polylene bridge, the bridge can also contribute significantly, leading to even better nonlinear optical properties. Thus S. Marder and coworkers 3 have shown that crystal (Figure 2.1) lead to 'Y = 10- 34 esu. Although NLO properties improve with increasing length of the polymer bridge, experiments 4 indicate that it saturates near 7 units for oligothiophenes (Figure 2.2). In order to design and optimize such materials, it is of interest to predict the NLO properties as a function of donor and acceptor, as a function of the nature and length of the polymer bridge, and as a function of solvent. There have been many recent theoretica1 5 - 8 and experimenta13 studies directed at understanding and Figure 2.1: Molecule with good NLO properties 10 s ---<f s s '>--- Figure 2.2: Polymer oligothiophene optimizing the NLO properties of organic compounds. The conventional method of calculating hyperpolarizabilities consists of calculating the occupied and excited molecular orbitals of a molecule and summing over the excited states,9 equations (2.31) to (2.33). These methods work well for small molecules; however, for large and complex molecules, such methods rapidly become extremely tedious and compute intensive. We develop here an alternate approach based on a valence-bond view of the bonding and a charge-transfer-exciton view of the excited states. This method allows very rapid calculations and leads to results in good agreement with experiment. With an additional approximation it leads to analytical formulae that provide physical insight into the factors determining the magnitude of a and 'Y and the saturation length. 2.2 The Valence-Bond Charge-Transfer-Exciton (VBCTE) View of Polymer Excited States 2.2.1 The Hamiltonian As a prototype we will use a rigid-chain polymer with N monomers and take N to be odd for convenience. We assume that the most important states of the polymer can be discribed with a valence-bond (VB) description using only the highest-occupied molecular orbital (HOMO) and the lowest-unoccupied molecular orbital (LUMO) of each monomer unit. In the ground state the HOMO is doubly occupied and the LUMO is empty. Now consider the excitation of a monomer (site p) near the middle of a long polymer. The occupied LUMO may be considered as on the same monomer 11 (leading to a covalent excitation energy flEe), or on adjacent monomers (sites p± 1), or on sites farther away (sites p ± 2, p ± 3, etc.). We let <ppq denote the many electron charge-transfer-exciton wavefunction in which an electron at site p is excited to a monomer q units the right of p leaving a hole at site p. Considering a fixed site p for the hole, the Hamiltonian matrix between various <ppq becomes -t V-2 -t -t V-I at at 0 -at -at VI -t -t ~ H(O) = (2.1) -t where matrix elements are kept only between adjacent sites. The diagonal matrix elements Vq are the energy required to remove one electron from the LUMO of one monomer and place it in the HOMO of the monomer q units away. They become (2.2) where I P is the ionization potential of the monomer, EA is the electron affinity of the monomer, Ro is the distance between the nearest monomer centers, and q is the number of monomers between the donor and acceptor states (q = ±1, ±2, etc.). The transfer matrix elements are if Iql > 1 (2.3) 12 if q= 1 (2.4) if q = -1 (2.5) where the opposite sign of the matrix elements for q = ±1 arise from the antisymmetric LUMO and HOMO of butadiene (see Appendix A and Fig. A.I). We have ignored the covalent excitation at q = 0 because it does not contribute to changes in the dipole moment. The evaluation of the matrix element t and constant a is discussed in Appendix A where we find that for the butadiene unit of thiophene t = 0.36;3 (2.6) a=V2 (2.7) where ;3 is the resonance integral between the adjacent Sp2 carbon orbitals. 2.2.2 Dipole Moment Writing the dipole operator as j.1> = '£1 q1R1 where ql is the charge on monomer l,the Hamiltonian in an electric field £ becomes H=HO-j.1>'£' (2.8) In order to examine the polarizability and hyperpolarizability, we need matrix elements of the dipole operators. (2.9) where a is the component (x, y, or z). Assuming that only the component along the polymer axis (say z) is important, this leads to 13 I{j,z = (4)pil LQl R l l4>pj) = c5ij(-e~) (2.10) l and the dipole moment matrix becomes 0 -2eRo -eRo Pz = (2.11) 0 eRo 0 2eRo For a finite electric field £, the energy of the ground state becomes (2.12) leading to the static polarizability and hyperpolarizability (2.13) (2.14) (2.15) The above analysis is for a fixed site p. The total values are approximately N times as large. 14 2.2.3 Neglect of Coulomb Interactions Although the Coulomb correction in (2.2) may be large, we will first consider the simple approximation where these terms are ignored, leading to VqNoQ = V = IP - EA. (2.16) The energy of ground state \0) is calculated by using the following perturbation formulas l l . The Hamiltonian in an electric field H = HO - P . £ can be written as H = Ho + W where Ho V V Ho= o (2.17) V V is the unperturbed Hamiltonian matrix and 15 -t w= 2eRo&z -t -t eRo&z at at o -at -at -eRo&z -t -t -2eflo&z-t (2.18) is considered as the perturbation Hamiltonian matrix. Using the standard perturbation formulas (Appendix B) leads to (2.19) (2.20) f3zzz = 0 /zzzz = 8a2e4 R 4 fJ2 V3 0 G(fJ) (2.21) where 00 A(fJ) = L A2ifJ2i (2.22) i=O 00 G(fJ) = L G2i fJ2i (2.23) i=O and t V fJ = - (2.24) 16 is the dimensionless energy dominating the saturation length. As shown in Appendix C, t is related to the bandwidth (B) by (2.25) B~4t while V is related to the energy gap (Eg) by (2.26) Eg ~ V - 2t. Thus t !i TJ=V= E ~!i 9 (2.27) 2 We will see that a large t leads to a larger saturation length while large V or bandgap leads to a smaller TJ and a small saturation length. For Eg = 0, (2.27) comes to TJ = ~. Thus the range of TJ is 0 to ~. 2.3 Saturation Length 2.3.1 Empirical Formula for Saturation Lengths As shown in Appendix A, a I"V y2, so the polynomials in (2.22) and (2.23) are functions of only one parameter TJ = fr . Consequently, the saturation length is a function only of TJ = fr. The length dependence of a and , are shown for various values of TJ in Figure 2.3 (scaled by the limiting values). We will define the the saturation length La and L-y as the length for which a and " respectively, attain 95% of the limiting value. To a good approximation we find (Figure 2.3.b) that (2.28) for TJ = 0.1 to 0.4. Similarly the saturation length for polarizability is approximated by (2.29) 17 (a) 1.2 Polarizability (a) 1 Z 0.8 'D 0.6 ........ ij - - - -= :-::::-- -- /' / Q) r-1 rQ ---t/V=O .1 t/V=0.2 ---- t/V=0.3 --------- t/V=O. 4 0.4 0 / / (f) 0.2 0 -0.2 0 2 4 10 8 6 12 Number of monomer units (b) 1.2 Hyperpolarizability 1 Z ........ 0.8 'd 0.6 ./ / / Q) r-1 rQ 0 / / / / (f) 0.2 - ----- -- / / / 0.4 ~ /' / ?- -- ( y) / t/V=O.l - - t/V=0.2 - ---t/V=0.3 --------- t/V=O. 4 0 -0. 0 5 20 10 15 Number of monomer units 25 Figure 2_3: Dependence of saturation behavior on TJ = t/V for (a) polarizability and (b) hyperpolarizability 18 as shown in Figure 2.3.a. Combining (2.28) and (2.29) leads to (2.30) L'Y rv 1.9L a - 0.9. N Excitation (a)b (10-23 esu ) Energya (eV) Q NoQ 3 5 7 9 11 13 15 17 2.47 2.00 1.86 1.82 1.81 1.80 1.80 1.80 3.20 8.39 12.98 16.97 20.79 24.58 28.36 32.15 8.38 22.83 35.62 46.61 57.09 67.48 77.87 88.25 hY(10-34 esu ) Q NoQ 3.10 30.51 84.90 136.78 177.73 212.91 246.28 279.23 5.56 75.07 236.01 385.49 498.88 596.08 688.98 781.02 Table 2.1: Theoretical properties of polyacetylene as a function of polymer length, N. Q indicates the use of (2.2) whereas NoQ indicates the use of (2.16). All cases use zero frequency. The monomer is butadiene. a Using V = 2.85eV. b (a) = ~(azz). C (ry) = t(')'zzzz). 2.3.2 Effects of Coulomb Interation and Frequency of Electromagnetic Fields The results in Section 2.2.3 and Section 2.3.1 are approximate since the Coulomb interaction in (2.2) is ignored. Including the Coulomb interactions between electrons and holes [using (2.2) instead of (2.16)] prevent us from obtaining analytical results as in Section 2.2.3. Also for frequency dependence polarizabilities we again cannot get analytical formulas. So we use sum-over-state approach in which the polarizability and hyperpolarizability have the form (2.31) 19 (2.32) (2.33) Here (i) 9 indicates the ground state and L' indicates that 9 is excluded from the sum. (ii) ril indicates the dipole matrix element for component i between states land k and r11 = r11 - r~g. (iii) Wo- = Li Wi· (iv) K( -Wo-; WI, W2, W3) is a numerical factor determined by the nature of the nonlinear optical process9 . (v) Lo-;1,2,3 denotes the average of all terms generated by permuting a,Wl,W2,W3· The excitation energy are obtained by diagonalizing the Hamiltonian matrix (2.1) and the dipole matrix elements can be obtained by transformation of matrix (2.11). The resulting values of a and r are listed in Table 2.1 and 2.2 for polyacetylene and oligothiophene respectively. Tables 2.1 an 2.2 show that Coulomb interactions do affect the magnitude of a and f. Thus including Coulomb interactions leads to a zz 2 to 3 times larger and to rzzzz, 2 to 3 times larger. However, as shown in Figure 2.4, the saturation length is nearly identical. The reason for this is that at the point 20 (a) 1.2 ro=O, ------- Polarizability (a), Polyacetylene 1 ~ 0.8 0.6 I 0.4 ---NoQ - - Sum-Q 0.2 o _0.2L-~-~-L-~--L-~-L-~-~--~~--~ o Z 0.8 '0 0.6 ?(]) 10 6 8 4 Number of Monomer Units 12 Hyperpolarizability (y), m=O, Polyacetylene 1 '-... 2 rl rU U 0.4 / ---NoQ - - Sum-Q Cf) 0.2 o o 2 12 14 10 Number of monomer units 4 6 8 16 Figure 2.4: The saturation behavior of polyacetylene for (a) polarizability and (b) hyperpolarizability. The solid line (NoQ) uses sum-over-state or perturbation without Coulomb interaction. The dashed line (Sum-Q) uses sum-over-state with Coulomb interaction 21 where the separation between the electron and hole is approximately the saturation length, the Coulomb interaction is much less than V and thereby has negligible affect on exciton size. The frequency also affects the magnitude and saturation length of a and I. When the frequency is near resonance, the magnitude and saturation length change a great deal. Thus from Table 2.2, the magnitude of IZZZZ increases by a factor of 10. This is because w = 1.16eV is near the resonance frequency of w = 1.4eV. The saturation length also increases by 2 units (Figure 2.5.b). However, for frequencies far from resonance, the change in saturation length is negligible (Figure 2.5), and the change in magnitude is about one order of magnitude. 2.4 Calculations of the Saturation Lengths for Oligothiophenes and Polyacetylene For oligothiophenes, the saturation length of a and 1 with chain length has recently been observed experimentally4. We will use the above theory to predict the saturation properties of polyacetylene and oliogthiophenes. 2.4.1 Oliothiophenes, Comparison to Experiment To estimate the NLO properties of oligothiophenes, we evaluate a and t by ignoring the sulfur and using Huckel theory (MO's for butadiene in Figure A.1). The analysis in Appendix A comes to t = 0.36jJ. (2.34) A fit to the band states of polyacetylene 12 leads to jJ = 2.4(1.0 ± O.7c5)eV (2.35) where 8 is the bond length alternation (half the difference in bond length between double and single bonds). Butadiene leads to 28 = (1.47 - 1.34) = 0.13A, and hence we take 22 (a)l .2 Polarizability 1 Z (a) --~~~~~~~~~-- 0.8 '-- o. 6 OJ r-I o . 4 t5 'D - - - W/V = O. 0 W/V=0.2 W/V = 0.4 --------- W/V=O. 6 rD 0 (fl 'I 0.2 " 0 -0. 0 10 12 4 6 8 Number of monomer units 2 14 (b) 1 .2 Hyperpolarizability (y) 1 ~~~--~~~~- 0.8 Z '-?- 0.6 '0 0.4 OJ rD 0 - - - w/V=O . 0 - w/V=O. 1 - - .- - w/V=O. 2 --------- w/V=O. 3 / r-I 0.2 I (fl ---- ::=-: - - - - - 0 -- -0. 0 2 14 10 12 8 Number of monomer units 4 6 16 Figure 2.5: Effect of frequency on the saturation behavior of (a) polarizability and (b) hyperpolarizability. The resonance frequency of a is 2.8 eV (corresponding to w IV = 0.49) whereas the resonance frequency of I is 1.4 eV (corresponding to wIV = 0.24). 23 0= 0.065A. (2.36) With (2.6) and (2.34), (2.35) leads to t = 0.83eV. (2.37) We solved for the eigenvalues of (2.1) as a function of chain length (see Table 2.2) and find that v = 4.0geV (2.38) leads not only to the experimental energy gap4 Eg = 2.77eV for saturation, but also an excellent fit with the observed transitions for shorter polymers. From Appendix C, v ~ Eg + 2t (2.39) which would lead to V = 2.77 + 1.67 = 4.44eV, in reasonable agreement with the more exact calculation of 4.09 eV. Equations (2.37) and (2.38) lead to tlV = 0.20, suggesting saturation lengths of La = 5 monomers and L, = 8.6 monomers, which are in excellent agreement with the experiment La rv 5 and L, rv 84. Using the above parameters and including Coulomb interaction leads to the a and 'Y in Figure 2.6 (using experimental 4 frequencies). The results are in excellent agreement with experiment except for N = 7. Experimently alN and 'YIN are greater than the values for larger N whereas experiment and caculations show that alNand 'YIN increase monotonically with N. Consequently, we suggest there may be an experimental artifact at N = 7. The w = 0 values for a and 'Y calculated from formulas (2.22) and (2.23) and from the sum-over-state (2.31) and (2.33) are listed for comparison in Table 2.2. The calculated 'Y at w = 1.16eV is off from the experiment by 6% for N = 11 and by 13% for N = 9. However, the caculated a at w = 1.95eV is about 3 to 4 times the observed values. 24 (a) Polarizability (a (-~,O)) ), 0)=1.95 eV 1.5 Oligothiophenes .. 1-..... . z 'B 1 \j OJ rl rD u 0.5 - - - Sum-Q ........ ·Experiment U) o o 2 10 4 8 6 Number of monomer units 12 (b) 1.2 Hyperpo1arizabili ty (y (-20),0),0),0) ) , 0)=1 .16eV, Olig.1~~~oPhen_e_s_ _ _ __ ./ 0.8 r 0.4 ·} ......1 - - - Sum-Q ......... Experiment o o 2 4 10 12 14 6 8 Number of monomer units 16 Figure 2.6: Comparison of theory and experiment for saturation behavior of oliogthiophene. (a) polarizability (w =1.95 eV) and (b) hyperpolarizability (w =1.16 eV). Dashed line with error bar uses experiment data. The solid line uses sum over states with Coulomb interaction. 25 N 3 5 7 9 11 Excitation Energies (eV) Calc b Expt 3.52 3.67 3.01 2.99 2.84 2.81 2.79 2.78 2.77 2.77 (CYzz(10-23esu) Expt a Calculation w =0 w = 1.95 3.90 1.1 2.71 2.6 6.36 10.93 9.35 17.13 6.6 22.45 12.09 14.78 27.51 7.0 h'zzzz(10-34 esu) Expt a Calculation w=O w = 1.16 9.9 2.10 5.39 107 60.3 12.8 188.4 360 26.5 321.78 37.11 370 460 429.2 45.98 Table 2.2: Experimental and theoretical properties of oligothiophenes as a function of polymer length, N. NoQ indicates the use of (2.16) whereas Q indicates the use of (2.2). w = 0 indicates the static case whereas w =1= 0 indicates finite frequency. w = 1.95eV for (a(-w,w) and w = 1.16eV for (1'(-2w;w,w, 0). aS ee reference 4. bUsing V = 4.0geV, t = 0.83eV and a = 0. 2.4.2 Polyacetylene Next we consider the case of a simple trans polyene (polyacetylene). We still use butadiene as the monomer comes to t = 0.83eV (2.40) (2.41) just as for oligothiophenes. We solved for the eigenvalues of matrix (2.1) as a function of chain length and chose V so that excitation energy for large N equals the experimental bandgap, Eg = 1.8eV. This leads to v = 2.85eV. (2.42) In comparison use of (2.39) with the observed gap leads to V = 3.47eV, in fair agreement with the calculated value. Equations (2.40) and (2.42) lead to t/V = 0.29 which from (2.28) and (2.29) suggest La ~ 6.8 and L, ~ 12. That means that saturation length of a is about 14 26 double bonds and the saturation length for '"Y is about 24 double bonds. This result is in agreement with other theoretical results 14 (21 double bonds). The saturation length for '"Y of oligothiphenes is 16 double bonds, considerably shorter than that of polyacetylene. This results primarily from the higher value of V and hence the bandgap arise from the the sulfur in oligiothiophenes. 2.5 Discussion 2.5.1 Comparision with Experiment Considering the simplicity of this VB - CT E model, the predicted results are excellent. Important approximations which may cause discrepancies are: 1. This model is derived for the long chain limit and should hold for short chain length. 2. The hole is assumed fixed. 3. For the nearest neighbors interaction, the expressions for Coulomb interaction should be corrected for sheilding. 4. The covalent (same site) exciton is neglected. 5. The correlation between electron-hole pair is neglected. 2.5.2 Predictions for Other Materials, Design Considerations The analytical results (2.19)-(2.27) (obtained for a static external field with neglect of Coulomb interactions) indicate that the magnitudes of a and '"Y for this class of polymers are determined by two parameters: t - the monomer hopping matrix element (one quarter the bandwidth, B) V - the exciton excitation energy (~ Eg + 4t ~ Eg + B/2) 27 Polymer! a b c d e f g h J Ba (eV) 3.9 3.5 2.8 3.8 2.5 3.3 2.7 1.3 Eb 9 (eV) 2.1 3.4 3 3.2 2.2 3.3U 5.4 a 3.1 a 2.77 1.8 t g (eV) 1.0 0.9 0.7 0.95 0.6 0.8 0.7 0.3 0.83 h 0.83 h vg (eV) TJ 4.1 5.1 4.4 5.1 3.4 5.0 6.8 3.8 4.09 2.85 0.24 0.17 0.16 0.19 0.18 0.16 0.10 0.08 0.20 0.29 La C L C 'Y a zz d,ejN (l0-23 esu ) 5.8 4.4 4.2 4.8 4.6 4.2 3.0 2.6 5.0 10.1 7.5 7.1 8.2 7.8 7.1 4.8 4.0 9.0 12.0i 2.41 1.13 2.49 1.13 3.88 0.40 0.083 1.09 1.77 5.65 6.8 i ,zzzz d,e j N (l0- 34 esu) 11.30 2.54 15.93 2.55 44.90 0.51 0.020 3.45 6.87 82.06 Table 2.3: Predicted saturated values polarizability (a), hyperpolarizability (,), and saturation lengths (La, L'Y) of polymers. Based on equation (2.19) and (2.21) with w = O. aCa1culated values from reference 15. bExperimental values from reference 15. cCa1culated using (2.28) or (2.29). dUsing a = Vi, eValues at saturation using (2.19) and (2.21) and w = O. An approximate correction for finite frequency is to multiply a( -w, w) by Egj(Eg - w) and to multiply ,( -3w; w, w, w) by E/ j(Eg w)(Eg - 2w)(Eg - 3w). !See Figure 2.7. gUsing (2.25) and (2.26) unless otherwise noted. hDerived in text rather than from B. iThe entries in the table are in terms of butadiene monomers. The number of double bonds is 14 for a and 24 for ,. In addition the monomer length Ro plays an obvious role. a ~ 1/Vi is nearly constant and hence has little effect. Thus we hope that these relations will provide experimental with insights in designing and developing new materials. The saturation length depends only on one parameter, TJ = tjV. Inclusion of Coulomb interactions and finite frequency (far from resonance) leads to little change in the saturation length. These results lead to the simple empirical formulae (2.31) and (2.32) for relating saturation length to TJ. Thus to design new materials with longer saturation length and with larger magnitudes for a and " one should increase t and/or decrease V. To predict a and, for other materials we need only estimate t and V and substitute into (2.19) and (2.21). The value of t can be estimated from the bandwidth of the conduction band, B ~ 4t. Given t, the value for V can be estimated from 28 b. Polyparaphenylene a. Polydiacetylene R I +c/l I R c. Polyparaphenylene vinylene '1 CH e. Poly thiophene vinylene d. Polyprrople w-t I H f. Polyvinylene sulfide C I -CH g. Polymethineimine i. Polyacetylene (trans) h. Polybenzothiophene j. Poly thiophene Figure 2.7: Repeating units for polymers of Table 2.3 29 the energy gap (between the conduction and valence bands) using equation (2.26), V = Eg + 2t. These values of Band Eg can be obtained either from experiment or theory. To illustrate the use of this approach we predict in Table 2.3 the values of a, " and the saturation lengths (Lo:, L'Y) for a number of polymers (see Figure 2.7) based on published values is of the bandwidth (B) and bandgap (Eg). 2.5.3 Comparisons to Other Theory Another recent approach to predicting NLO properties is by Mukamel and coworkers 14 who studied polyacetylene chains without donor acceptor groups. They used the Pariser-Parr-Pople (PPP) Hamiltonian and solved the equations of motion for pairs of electrons and holes by assuming pairs of different types are decoupled. This approach greatly simplifies the general calculation over a standard sum over molecular orbital states approach by allowing the motion of electron-hole, electron-electron and holeholeairs. VB-CTE model further simplifies the calculation by considering only the electron-hole pairs (excitons) since the electron-electron and hole-hole pairs are much higher in energy. 2.6 Summary By focusing solely on electron-hole pairs, we constructed a simple Hamiltonian for a conjugated polymer chain and solved for the properties using perturbation theory. Analytical formulae are derived for a and, in terms of just two parameters: V and t plus the overall monomer length, Ro. These formulae predict the experimental values of a and, within an order of a magnitude and show that a and, can be increased by (i) decreasing the cost of creating an exciton (V) and (ii) once it is created by increasing the energy gain (t) of delocalizing it over many repeat units. We find that the saturation length is determined by only single parameter: the ratio of t (the hopping matrix element between nearest monomers) and V (the exciton excitation energy). These two parameters are obtained easily from experiment or theory. In addition we obtained the simple empirical formula (2.31) and (2.32) for 30 predicting the saturation lengths. These results should be valuable for designing new materials and for interpreting experiments. 2.7 References 1. Prasad, P. N.; Williams, D. J. Eds., Introduction to Nonlinear Optical Effects in Molecules and Polymers, Wiley, New York, 1991. 2. Marder, S. R; Beratan, D. N.; Cheng, L. T., Science 252, 103 (1991). 3. Cheng, L. T.; Tam, W., Stevenson, S. H.; Meredith, G. R; Rikken, G.; Marder, S. R, J. Phys. Chem. 95, 10631 (1991). 4. Thinpont, H.; Rikken, G. R; Meijer, E. W.; ten Hoeve, W.; Wynberg, H., Phys. Rev. Lett. 65, 2141 (1990). 5. Beratan, D. N.; Onuchic, J. N.; Perry, J. W., J. Phys. Chem. 91, 2696 (1987). 6. Shuai, Z.; Bredas, J. L., Phys. Rev. B 44, 5962 (1991). 7. Heflin, J. R; Wong, K. Y.; Zamani-khamiri, 0.; Garito, A. F., Phys. Rev. B 38, 1573 (1988). 8. Greene, B. I.; Orenstein, J.; Schmitt-Rink, S., Science 247, 679 (1990). 9. Orr, B. J.; Ward, J. F., Mol. Phys. 20, 513 (1971). 10. Pople, J. A.; Walmsley, S. H., Faraday Soc. and Cont. 15, 441 (1962). 11. Corson, E. M., Perturbation methods in the quantum mechanics of n-electron systems,Hafner Publishing Company, New York. 12. Tavan, P.; Schulten, K., Phys. Rev. B 36, 4337 (1987). 13. Fann, W. S.; Benson, S.; Madey, J. M. J.; Etemad, S.; Baker, G. L.; Kajzar, F., Phys. Rev. Lett. 62, 1492 (1989). 14. Mukamel, S.; Wang, H. X., Phys. Rev. Lett. 69, 65 (1992). 31 15. Skotheim, T. A. Eds., Handbook of Conducting Polymers,Vol 2. Marcel Dekker, Inc., p 859. 16. Mathematica, Version 2.0, Wolfram Research, Inc., Champaign, Illinois. 17. Richardson, C. H., An Introduction to the Caculus of Finite Differences, D.van Nostrand Company, Inc. 1954, p 108. 32 Chapter 3 The Valence-Bond Charge-Transfer Model (VB-CT) for Nonlinear Optical Properties of Charge-Transfer Organic Molecules 3.1 Introduction Nonlinear optical (NLO) materials are playing an increasingly important role for a wide range of applications, including laser technology, telecommunications, data storage, and optical switches l - 4 . Exemplary NLO properties are exhibited by conjugated organic molecules with terminal electron donor and acceptor groups (Figure 3.1). Such molecules possess a low energy charge transfer (CT) state and exhibit large second and third order nonlinearities 5 - 7 . Marder, Perry, and coworkers 7 have studied the NLO properties of a series of such materials and have shown strong correlation between hyperpolarizabilities and bond length alternation (BLA) in the conjugated bridge. They used solvents of varying polarity to modify the BLA for the conjugated organic molecules and measured the corresponding hyperpolarizabilities. Gorman and Marder7e carried out finite-field semiempirical molecular orbital calculations (at the AMI level) to obtain numerical relationships between the hyperpolarizabilities and the BLA, with results consistent with experimental trends. However, there is not yet a simple analytical model to explain the relationship between BLA and the various order polarizabilities. <I>VB <I>er Figure 3.1: Donor acceptor type molecule 33 Here we start with the valence-bond charge-transfer model (VB-CT) previously used to describe the hyperpolarizabilities of polyenes 8 and extend it to systems terminated with donor and acceptor units. This VB-CT model is used to describe the polarizability and hyperpolarizabilities of charge-transfer type molecules. We derive analytic formulea that explain the relation of the polarizability and hyperpolarizabilities to BLA. Because of its basis in classical resonance theory, this VB-CT model provides insight into the factors determining CY, j3, and ,. The basic theory is developed in Section 3.2 and predictions of hyperpolarizabilities are given in Section 3.3. Section 3.4 compares the VB-CT results to quantum chemical calculations (AMI) and shows how to use the theory in interpreting experimental results and predicting properties. 3.2 Theory 3.2.1 The VB-CT Model We consider a molecule of the form in Figure 3.1 where WVB is the normal valencebond configuration (no charge transfer between donor and acceptor). The second state WCT is obtained by moving an electron from donor to acceptor, leading to the alternative VB description of the intervening polyene unit. Taking the energy of WVB as the reference, the Hamiltonian matrix becomes Ho = (EVB -t where -t -t) (0 -t), ECT -t (3.1) V (wvBlllIWCT) is the many-body charge transfer matrix element (t is positive), V = ECT - E vB , (3.2) and we assume that (WvBIWCT) = O. V is determined by the nature of the donor and acceptor molecules, or the topology of the conjugated linker (which determines the change in aromaticity), and the effect of solvent polarity. Solving for the eigenvalues 34 of (3.1) leads to the energies 1 1 2 2 = -V - _v'V2 +4t2 (3.3) 1 E ex = -V + -12 v'V2 + 4t2' 2 (3.4) E gr ' where gr and ex indicate the ground and excited states. The energy gap is (3.5) which can be related to the wavelength for the maximum in the absorption spectrum he Amax = E· (3.6) 9 Denoting the fraction of the charge-transfer configuration in the ground state as f, we write the ground state eigenfuntion as (3.7) From (3.1) and (3.3) we obtain (3.8) so that (3.9) Also f = 8Egr 8V· (3.10) 35 3.2.2 Inclusion of the Bond Length Alternation (BLA) Coordinate Since \lI CT and \lIVB involve alternate resonant descriptions of the intervening polyene unit, the increase of f from 0 to 1 will change each double bond (R = 1.331\) of the polyene to a single bond (R = 1.45A) and vice versa. [These distances are based on the experimental average bond lengths trans-1,3,5,7-octatetraene9 .] Thus the bond length alternation (BLA) coordinate changes from q = -O.12A to q = +O.12A as the CT fraction f goes from 0 to 1, leading to a one-to-one relationship. We will describe the bond length distortion as a vibrational coordinate by associating potentials (3.11) and (3.12) with the VB and CT states. The vibrational contributions to the Hamiltonian (3.1) are (3.11) (3.12) where q~B and q~T are equilibrium positions and Yo is the adiabatic energy difference for these two states. For f = 0 the equilibrium BLA coordinate is q~B = -O.12A and for f = 1 it is q~T = +O.12A. Thus V and the ground state energy Egr are replaced by (3.13) and (3.14) The equilibrium coordinate of \lI gn qopt, is obtained by solving dE gr dq=O. This leads to qopt 0) 1(0 0) V ="21(0 qVB + qCT +"2 qVB - qCT JV2 + 4t 2 (3.15) 36 = q~ B - f (q~ B - q~T ) = -0.12 + 0.24f. (3.16) Thus f and qopt are linearly related to each other. We should note that the V in (3.13) is the vertical energy difference (3.2) for a particular value, q, whereas Va is the adiabatic energy difference with each surface at its minimum. Eqn (3.13) with q = qopt and eqn (3.16) leads to a nonlinear equation which we solve iteratively for qopt· Figure 3.2 illustrates the variation of the ground state potential surface and qopt for Va = 1.0,0.0, and -1.0eV using t = 1.1eV (3.17) [derived from the experimental data on molecule in Figure 3.1, see discussion in Section 3.4.4 land k = 33.55eV/ A2 = 773.7keal/mozA 2 = 5.38mdyn/ em (3.18) (from UFF 10 ) for the force constant in (3.11) and (3.12). Vo is determined by the nature of the donor, the acceptor, the conjugated linker, the solvent, etc. For Va = leV (Figure 3.2.a), we obtain qopt = -0.069A. For Va = 0 (degenerate VB and CT), we obtain qopt = 0 (Figure 3.2.b). Further stabilization of CT to Vo = -leV (Figure 3.2.c) reverses the BLA to qopt = +0.069A. 3.2.3 Application of an Electric Field For conjugated donor acceptor systems in Figure 3.1, the polarizability and hyperpolarizabilities are dominated by the z component (along the chain axis), and we will ignore all other components. Assuming that only 'l1 CT contributes to the dipole moment, we write 37 Vo (a) 5 :;- 4 e> Q) >. 3 W 2 1 eV Eex ~ Ect c . / Evb (ij :g 1 .ill 0 a. 0 -1 ~~~~~~~~~-L~~~~~~~~~~~~~~~~ -0.4 (b) :;~ >. e> Q) c -0.3 -0.2 4 -0.1 Vo :g .ill ~ 0.1 0.2 0.3 0.4 = 0 eV 3 2 1 W (ij o : .. : 0 -1 -2 -0.4 (c) 4 :;- 3 >. e> Q) 2 W 1 -0.3 -0.2 -0.1 o 0.1 0.2 0.3 0.4 Vo = -1 eV ~ c (ij E 0 .ill 0 a. -1 -2 -0.4 -0.3 q, -0.2 -0.1 o 0.1 the bond length alternation 0.2 0.3 coordinate 0.4 (A) Figure 3.2: Energies of the ground (Egr) and excited (Eex) states as a function of bond length alternation (BLA), q. Vo is the adiabatic difference between the pure VB and CT states and qopt is the optimum BLA coordinate, qOft. These calculations used k and t from (3.17). (a) Va = leV leads to qopt = -0.069A, (b) Va = 0 leads to Qopt = 0, (c) Va = -leV leads to Qopt = +0.069A. 38 (3.19) MCT = QeRDA where Q is the net charge transfer for WCT. In an applied external electric field, E, as used in measurements of hyperpolarizabilities, the additional (optical) Hamiltonian is Hl = (°0 0) -MCTE (3.20) ' leading to H=Ho+Hl = EvE -t ( -t ECT - MCTE ) (3.21) Thus the relative energy V is replaced by V£ = V - MCTE. (3.22) Equations (3.1), (3.5) and (3.9) continue to apply for finite applied fields but with V replaced by V£. In particular the change in f due to the applied field is df dE 3.2.4 df dV£ dV£ dE -- 2t2 MCT (Vi + 4t 2)3/2 (3.23) Polarizabilities Given the dependence of the ground state energy on the external electric field, the dipole moment of the ground state, Pz , is obtained from (3.24) Assuming that z is the direction along the chain and that E is in this direction, the polarizability and hyperpolarizabilities can then be obtained from (3.25) 39 1d2 p f3zzz = 2 de; 1£==0, (3.26) 1 d3 Pz 6" d&3 1£==0. (3.27) 1 d4 p z 8zzzzz = 24 d&4 1£=0. (3.28) 'Yzzzz = Using (3.9), (3.14) and (3.16) we obtain (3.29) (3.30) (3.31) (3.32) 5t2P,~T V[V2 - 3t2] 8zzzzz = 24 dv,4 1£=0= E9 . P,~T d4f £ (3.33) 9 The following discussions will omit the z subscripts. 3.3 Predictions of 11, CY, /3, and r From VB-CT Theory As f increases from f = 0 to f = 1, the VB-CT model leads to a structural evolution in which the polyene double bonds for \liVE change to polyene single bonds in WeT and vice versa. Thus each bond decreases or increases by 0.12A as f goes from 0 to 1. Since there is a linear relation (3.16) between f and Qopt [the ground state bond length alternation (BLA)], and since f determines the polarizability and hyperpolarizabilities, the polarizability and all hyperpolarizabilies are determined by a single BLA parameter, qopt. This has been anticipated by Marder7 et al., who pointed out that bond length alternation is a useful parameter to examine the structure-property relationships for 40 0.2 0.4 0.6 0.8 :a 10 f = 0.5 : .......•.• t : ~ ' ..... -2 -2 (b) ~ " ........ ~ ......... : - VB-CT - V8-CT • AMI Ib) 400 200 f = 400 0.276 t 200 -200 -200 f -400 t = -400 0.724 500 -500 -500 -1000 -1000 t -1500 -1500 ", - • AM1 .' II AMI VB-CT -2000 -2000 (d) (d) 40 40 20 20 + -20 -20 -40 -40 0.2 0.4 0.6 0.8 0.2 0.4 0.6 O. B Figure 3.3: The relation between hyperpolarizability and charge transfer character (1). The right-hand side figures are the comparision between VB-CT caculation (solid line) and AMI caculation (squares) (reference 9). (a) Polarizability, O!, (b) hyperpolarizability, /3, (c) second hyperpolarizability, ,,(, and (d) third hyperpolarizability, o. 41 nonlinear optical materials. They showed that the j3 and, values can be tuned through extrema and zero-crossings by varying BLA. In addition they carried out finite-field AMI calculations 7e on molecule in Figure 3.1 and showed the relationships of a, j3 and , to qopt. Their results provide a good test of VB-CT theory. In order to illustrate the relationships as calculated using the VB-CT model, we used (3.9), (3.16), and (3.28)-(3.33) to calculate j, qopt, a, j3, " and 8 as a function of V, all with the fixed values of t and k from (3.17 - 3.18). This allowed us to obtain a, j3, and 'Y as a function of qopt, Figure 3.3. General observations from these relations are as follows: 1. a has a maximum for f = ~, which is when all bond lengths in the polyene chain are equal. 2. f3 is related to the derivative of a with respect to f, leading to a maximum at f = 0.276, a minimum at f = 0.724, and a zero value at f = ~. The maximum and minimum have the same magnitude. 3. , is related to the derivative of f3 with respect to f, leading to the largest = ~, with secondary maxima (about 1/4 the magnitude) at f = 0.173 and f = 0.827. 'Y = 0 at f = 0.276 and f = 0.724 where 1f31 is a maximum. magnitude (a minimum) at j 4. 8 is related to the derivative of 'Y with repect to f, leading to largest magnitudes at j = 0.357 (minimum) and f = 0.643 (maximum); secondary maxima in the = 0.117 (maximum) and j = 0.883 (minimum). 8 has zeroes at the places where 1,1 is a maximum (f = 0.173, f = 0.5, and magnitudes occur at j f = 0.827). These derivative relationships can be understood as follows: From (3.22), Ve is a linear function of £, leading to d ddVe d d£ = dVe d£ = -/-leT dVe . (3.34) 42 o 5 -5 -lO ~ l 0.8 0.6 0.4 0.2 o (b) 4 3 (e) 3 f3 2 l o -l -2 (d) 2 2 l o 5 o -5 -lO v Figure 3.4: The relation between the dipole moment Pz (in units of /LeT) and V (the VB-CT excitation energy). (a) polarizability, a, (b) hyperpolarizability, (3, (c) second hyperpolarizability, ,,(, and (d) third hyperpolarizability, c5 obtained from the derivatives of Pz with respect to - V as shown by equations (3.30)-(3.33). 43 The relation between the dipole moment Pz and Ve is plotted in Figure 3.4.a. We assume that the dipole moment of'llvB is zero, which corresponds to the limit with Ve = +00. The dipole moment P of the charge transfer state is J.1CT, which corresponds to the limit with Ve = -00. Taking the first, second, and third derivatives of the dipole moment with - Ve leads to a, 13, " and 6 as shown in Figure 3.4.b - 3.4.e. The relations between Ve , f, and qopt are expressed in (3.9) and (3.16). Changing variables from Ve to qopt leads to ~ = _d_dqopt = -0.12 df _d_ dYe dQopt dYe dYe dqopt (3.35) where the range of Ve , i.e., [-00,00] is mapped onto the range of qoptl [+0.12, -0.121. Thus the shapes of the curves are similiar but the signs of a and, change, as indicated by (3.29) and (3.30). These results lead to the following observations: When V = 0 (degenerate VB and CT states), we have 1 f = 2' (3.36) qopt = 0, (3.37) t = E g /2. (3.38) At this point, a is a maximum, 13 = 0, and, is a minimum (largest magnitude), and 6 = o. When IVI = Itl, we have f = 0.276 or 0.724, (3.39) qopt = ±0.0538A, (3.40) t= E J. Thus 13 takes its maximum (minimum) and, = o. (3.41) 44 When IVI = v'3ltl, we have f = 0.173 or 0.827, (3.42) = ±0.0785A, (3.43) E Jr (3.44) qopt t= Thus 11'1 is a maximum and 6 = O. When IVI = 0.595Itl, we have f = 0.357 or 0.643, (3.45) qopt = ±0.0343A, (3.46) t = 0.479Eg . (3.47) f = 0.117 or 0.883, (3.48) qopt = ±0.0919A, (3.49) t = 0.322Eg • (3.50) Thus 161 takes its largest maxima. When IVI = 2.376Itl, we have Thus 161 takes its second largest maxima. The shapes of the polarizability curves are relatively insensitive to the value t. Thus the salient factor for polarizability and hyperpolarizabilities is the bond length al ternation. 45 3.4 Discussion 3.4.1 Comparison with AMI Calculations The VB-CT results are compared with AMI calculations 7e (squares) in Figure 3.3. [The AMI calculations led to a limited bond length alternation of 0.108A, whereas the experimental value for octatetraene led to 0.12A; consequently for AMI qopt = -0.108 for f = 0 and qopt = +0.108 for f = 1.] In Figure 3.3 we used Q values of 0.69, 0.51, 0.69, 0.69 to scale the curves for a, (3, " and 6 respectively. The AMI results for (3 and, agree quite well with VB-CT theory. VB-CT has a going to zero as f --+ 0 or 1, whereas the AMI calculations lead to about half the maximum. This is probably because the current VB-CT calculations ignore the polarizability for a fixed VB or CT structure (it could have been included as an additive correction). 3.4.2 Comparison with Two-Level Models Two-level models have been used widely to understand the nonlinear optical properties of materials l l . In the absence of an external electric field, the system is described in terms of two eigenvectors corresponding to the ground and excited states. Upon applying an electric field c, the additional Hamiltonian term, -c . J.t is treated as a perturbation, where J.t is the dipole moment matrix J.t = (J-Lgg J.tge J.tge ) . J.tee (3.51) This leads, for example, to a first hyperpolarizability (3 of the form 2 (3 rv (J-Lee - J.tgg) ~e ge (3.52) where Ege is the energy gap. This expression results from the change in ground state energy due to field, but does not account explicitly for the change in structure due to the modified equilibrium position of the ground state potential. Thus, it does not 46 Figure 3.5: Valence bond state of the bridge provide a form for the higher polarizabilities, and does not provide an explicit picture for the structure/property relationships illustrated above. The VB-CT model differs from such two-level models because the nonlinear optical properties are treated as due to the dependence of the dipole moment on the "field" on VB-CT adiabatic energy difference, V, which itself depends on the mixing of VB and CT and the resultant change in structure. This VB-CT picture leads itself easily to the calculation of changes in structure and of the structure-NLO property relationships with only a few chemically meaningful parameters. This leads to a direct relationship between a, /3, ,,(, b as the structure varying between a neutral polyene-like structure and a zwitterionic structure. 3.4.3 Additional Excited States The VB-CT model assumes that all other excited states are much higher than the VB and CT states. In particular the resonant states (Figure 3.6) involving the bridge or linker must be much higher. For octatetraene (Figure 3.5) the absorption maxima is about 4 eV indicating that the resonant state of the linker in Figure 3.1 is about 4 eV above the VB state 12 . Since the donor acceptor molecules considered here have the CT state about 1 eV to 2 eV above the VB state, neglect of the resonant state should be a good approximation. When the energy of the resonance structure is close to those of VB and CT, the contributions from the linker resonant state must be included. This complicates the theory so that the results are no longer analytic. However, combining the current VB-CT theory for donor acceptor molecules with the VB-CT-E theory 8 for polymer linkers is straightforward and under development 13. 47 Figure 3.6: Charge transfer state of the bridge Figure 3.7: Molecule with f = 0.5 3.4.4 Applications to Various Molecules To illustrate the application of the VB-CT model, we will consider the nonlinear polarizabilities of several donor acceptor molecules linked by a conjugated octatetraene chain. As described in Section 3.3, t can be explicitly related to Amax for certain structural limits. Given t, one can use the expressions and the relationships given above to identify molecules that would have properties near the peak values or zero crossings of the polarizabilities. For the case of the four double bond cyanine (Figure 3.7), Amax was measured to be 516 nm in CH3 CN solvent, with I = -370 X 1O-36 esu .14 By symmetry this system has f =0.5, leading to t = E g /2, and hence t = 1.18eV. From (3.41) we predict that a molecule of the same length and linker type but with Eg = VSt would lead to I = 0 and a positive maximum in (3. Using t = 1.18eV this would occur at A = 467 nm. In fact, in the solvents C6H6 and dioxane, molecule in Figure 3.8 has Amax = 472 nm and Amax = 468 nm, respectively, and '"'( changes from 15 x 1O-36 esu to -25 X 1O-36 esu with a positive peak in J-l(3!15-16 From (3.44), we expect '"'( to have a positive maximum when Eg = V7t, or Amax = 394nm. Experimentally, molecule in Figure 3.9 is observed 15 to have a positive peak in I for solvent CH3 CN, as well as a Amax = 418nm, close to the prediction from the 48 ~N BU'" N ~ ~ c BU/ C / '" C~ N Figure 3.8: Molecule with the positive maximum in (3 Figure 3.9: Molecule with the positive maximum in 'Y VB-CT model. Thus, (55) has t = E g / V7 = 1.10eV. Thus, spectroscopic measurements of Amax, together with selected NLO data, can be utilized to predict structures and solvents needed to attain desired structure-NLO property relationships. Elsewhere 16 we will consider a more complete description of solvent effects. 3.5 Summary We presented the valence-bond charge-transfer (VB-CT) model to provide a simple means for predicting nonlinear optical properties of charge transfer type organic molecules and for explaining in simple analytic terms the relationship between these properties and structure. The absorption frequency, hyperpolarizabilities, and bond length alternation (BLA) are expressed in term of analytic formulae with a total of four independent parameters, V, t, Q, and k. Here k is a force constant appropriate for the BLA coordinate of polyene linkers and should be similar for all such materials 49 (it might depend on polyene length; we used a k based on the universal force field) 10 . Q is the ratio of the actual dipole moment of the excited CT state to that expected for perfect charge transfer. This value will depend mainly on polyene length, but will be close to the values used here (Q = 0.5 to 0.7). V and t are related to bandgap (Amax) and to bandwidth. The value of t is mainly determined by the length of the polyene spacer as well as the coupling of the donor and acceptor to the spacer. Thus, the variable most accessible to design is V which can be modified by changing the strength of the donor or the acceptor, the bridge topology, or the solvent polarity. We have shown how to estimate the polarizabilities for various materials by estimating V based on bandgap measurements. We have shown how O!, /3", and r5 are related by derivatives with respect to f and how the design of molecules with an appropriate value of f can lead to optimization of particular properties, while minimizing the others. 3.6 References 1. Prasad, P. N.; Williams, D. J. Eds, Introduction to Nonlinear Optical Effects in Molecules and Polymers (Wiley, New York, 1991). 2. Kuhn, H.; Robillard, J. Eds, Nonlinear Optical Materials (CRC Press, Inc. 1992). 3. Ashwell, G. J.; Hargreaves, R. C.; Baldwin, C. E.; Bahra, G. S.; Brown, C. R., Nature 357, 393 (1992). 4. Stegeman, G. I.; Burke, J. J.; Seaton, C. T., "Nonlinear Integrated Optics," in " Optical Engineering: Integrated Optical Components and Circuits." ed. by Huteheson, L. D.; Dekker, M., New York, 1987, Vol. 13. 5. Dulcic, A.; Flytzanis, C.; Tang, C. L.; Pepin, D.; Fetizon, M.; Hoppiliard, Y., J. Chern. Phys. 74, 1559 (1981). 50 6. Cheng, L. T.; Tam, W.; Marder, S. R; Stiegman, A. E.; Rikken, G.; Spangler, C. W., J. Phys. Chern. 95, 10643 (1991). 7. (a) Marder, S. R; Beratan, D. N.; Cheng, L. T., Science 252, 103 (1991); (b) Marder, S. R; Perry, J. W.; Bourhill, G.; Gorman, C. B.; Tiemann, B. G., Science 261, 186 (1993); (c) Marder, S. R; Gorman, C. B.; Tiemann, B. G.; Cheng, L. T., J. Am. Chern. Soc. 115, 3006 (1993); (d) Marder, S. R et al., J. Am. Chern. Soc. 115, 2524 (1993); (e) Gorman, C. B.; Marder, S. R,Proc. Natl. Acad. Sci. 90, 1129 (1993). 8. Lu, D.; Chen, G.; Goddard, W. A. III, J. Chern. Phys., 101, 4920 (1994). 9. Baughman, R H.; Kohler, B. E.; Levy, I. J.; Spangler, C. W., Synth. Met. 11, 37 (1985). 10. Rappe, A. K.; Casewit, C. J.; Colwell, K. S.; Goddard, W. A. III; Skiff, W. M., J. Am. Chern. Soc. 114, 10024 (1992). 11. Oudar, J. L., J. Chern. Phys. 67, 446 (1977). 12. Granvill, M. F.; Holtom, G. R; Kohler, B. E., J. Chern. Phys. 72,4671 (1980). 13. Lu, D.; Chen, G.; Goddard, W. A. III, work in progress. 14. Bourhill, G.; Tiemann, B. G.; Perry, J. W.; Marder, S. R, Nonlinear Optics, in press. 15. Bourhill, G.; Cheng, L. T.; Gorman, C. B.; Lee, G.; Marder, S. R; Perry, J. W.; Perry, M. J.; Tiemann, B. T., Proc. SPIE 2143 (1994). 16. Chen, G.; Lu, D.; Goddard, W. A. III, J. Chern. Phys., 101, 5860 (1994). 51 Chapter 4 The Valence-Bond Charge-Transfer Solvation Model (VB-CT-S) Nonlinear Optical Properties of Organic Molecules in Polar Solvents 4.1 Introduction Lu et aU (denoted as Paper I) recently proposed the valence-bond charge-transfer (VB-CT) model to predict polarizability (a) and hyperpolarizabilities ((3, " and (5) of charge-transfer conjugated molecules. This simple model accounts for the dependence of the polarizabilities on the charge transfer energy (V) and shows that a, (3, " and <5 are all related to the bond-length alternation (BLA), which is in turn related to the fraction, f, of the wavefunction having CT character. This leads to a derivative relationship among a, (3, , and <5. In Section 4.2 we start with the VBCT model and employ the Marcus solvation mode1 2 to predict how solvation affects the polarizabilities of molecule in Figure 4.1. This leads to the valence-bond chargetransfer-solvation (VB-CT-S) model in which the solvent is described as a continuous medium with dielectric constant E and the donor and acceptor groups are represented by two spheres of radius rD and r A, respectively (see Figure 4.2). VB-CT-S describes how the absorption edge, BLA, and hyperpolarizabilities are related to the solvent properties. In Section 4.3 VB-CT-S is employed to interpret and explain recent experimental Figure 4.1: Molecule 1,1 dicyano, 6-(di-butyl amine) hexatriene 52 observations by Marder, Perry, and coworkers 3 on the effect of solvation on the second hyperpolarizability, '"Y. We also predict the values of a, /3, and () as a function of solvent polarity. 4.2 The VB-CT-S Model 4.2.1 No Solvent The VB-CT model of Paper I assumes that the wavefunction of the molecule and all properties can be described as a linear combination Wgr = R WVB + If WCT (4.1) of the two valence bond configurations in Figure 3.1, WVB and WCT. Here WVB is the wavefunction for the valence bond ground state (no charge transfer from donor to acceptor) while WCT describes the state in which an electron is moved from donor to acceptor while readjusting the other bonds. The optimum resonance fraction, f, in (4.1) is determined by the relative energy of WVB and WCT, the coupling between them, the change in the dipole moments, and the solvent polarity. Without solvent the Hamiltonian is (4.2) where WCT and WVB are assumed to be orthogonal (4.3) Here (4.4) 53 is the charge transfer matrix element (t is positive), and (4.5) is the difference in energy between WCT and WVB. This leads to a bandgap of (4.6) Introducing the bond length alternation coordinate, qopt, V is replaced by V = Vo + ~k [ (q - q~T ) 2- (q - q~ B) 2] , (4.7) and the bond length alternation can be written as qopt = (1 - 1) q~B + fq~T = q~ B - f (q~ B - q~T ) (4.8) where q~B = -O.12A and q~T = +O.12A. 4.2.2 Solvation Effects Placing a CT molecule into a polar solvent leads to reorientation of both the solvent and solute molecules. This changes the relative energy of WVB and WCT, (4.5), which through (4.2) changes the optimum fraction, f, of CT character in the ground state (4.1). Assuming that only CT contributes, the dipole moment of the ground state becomes J1 = f J1CT = fQeR DA (4.9) J1CT = QeRDA. (4.10) where As shown in Chapter 3, the fraction of CT character in the ground state is 54 Acceptor Donor Figure 4.2: The Marcus solvation model. TD and T A are the radii of the donor and acceptor, respectively. RDA is the distance between the donor and acceptor. y'V2 + 4t 2 - V f = 2y'V2 + 4t2 1 2 JE; - 4t2 2Eg (4.11) In the VB-CT-8 model we assume that electronic states other than Wv Band WCT have much higher energies and that t (the coupling between WVB and WCT) is independent of solvent. Thus quantitative evaluation of the solvation effects requires only the change in relative energy, (4.5), Vs = V +~Vs (4.12) due to the presence of solvent. Here V is the energy difference between 'livE and WCT without solvent and ~ Vs is the change in the relative energy caused by the addition of solvent. Including BLA the final V is obtained from (4.12) using Vs in place of V. To evaluate ~ Vs we approximate 2 the donor and acceptor by two spheres of radius TD and T A with charges distributed symmetrically as in Figure 4.2. For charges of Ze and - Z e on the donor and accepter atoms, the electric displacement field D is 55 (4.13) where x s , Xd, Xa are the displacement vectors for solvent, donor and acceptor, respectively. Assuming the relaxation time of solvent molecules is much longer than the time scale of charge transfer, the induced dipole moment density of solvent is always equal to its equilibrium value P eq, which is readily expressed as (4.14) where E is the dielectric constant of the solvent, and Deq is the electric displacement field at equilibrium [i.e., Z = fQ in (15)]. For a static electric field, the energy is calculated as (4.15) The energy change upon the adding the solvent is !:1E = - -1 2Eo J 3 P eq . D dr. (4.16) Ignoring image charges, this leads to 2 !:1Vs = - - e ( 1- 1) f Q SF, 2 (4.17) 1 1 1 SF=-+---- (4.18) 47rEo - E where 2rD 2rA RDA depends only on the geometry. From (4.17) we observe that strong solvation effects arise from: 1. large E 2. small radii of donor or acceptor, and 3. large R DA . 56 Given (4.12) and (4.17) expressing V in terms of Vo, SF, and E, we can use (4.6) to determine the energy gap, E g , as a function of solvent polarity. We assume that the energy gap is given by the absorption edge Amax = he/ E g . The polarizabilities have the form (see 3.30-3.33) = 3t21-lbTV _ 3t2(I-lDA)3 rQ 3V __ ~ 8cx zz 13 E~ zzz - 4t21-l~T(V2 - t 2) "/zzzz = EJ (V2 + 4t2)5/2 - 2J.1 8V 4t2(I-lDA)414Q4 (V2 - t 2) 1 28(3zzz = (V2 + 4t2f/2 = 31-l 8V 2 2 2 2 2 2 6zzzzz = 5t J.1bT(V - 3t )V = 5t (J.1DA)5 J5Q5(V - 3t )V __ ~ 38 ,,/zzzz (V2 + 4t2)9/2 4J.1 8V . E2 (4.20) (4.21) (4.22) In solution one measures the rotationally averaged values of the polarizabilities, (4.23) I-lI3 - (I-l . (3) = Li,j (3ijj I-li' 1 "/ = (,,/) = Sb xxxx + "/yyyy + "/zzzz + 2,,/xxyy + 2"/yyzz + 2"/xxzz). (4.24) (4.25) Assuming only the z components are nonzero, (4.23)-(4.25) lead to (4.26) 13 = I-lzI3zzz = 3t21-lbTV I-l E~ _ 1 4t21-l~T(V2 - t 2)V "/ - S"/zzzz = 5E7 . (4.27) (4.28) g All quantities in (4.19)-(4.22) are defined except Q, which is the ratio of the actual solvent-free dipole moment of the CT state to the ideal value assuming that the CT state has one electron transfered from the donor to the acceptor. This can be obtained 57 by comparing to the absolute value of one of the quantities in (4.19)-(4.22) at some value of E. 4.3 Comparison with Experiment 4.3.1 Application to 1,1 dicyano, 6-(di-butyl amine) hexatriene,(l) The dots in Figure 4.3.c show the experimental values 3 of the second hyperpolarizability 'Y for molecule (1) for a variety of solvents [C6H6-n-C6H14 (E = 2.087), Dioxane (E = 2.209), CCl 4 (E = 2.238), C6H6 (E = 2.284), CH2 Cl 2 (E = 9.08), CH3 0H (E = 32.6), and CH3 CN (E = 37.5)]. We will compare the predictions of V B-CT-S theory with these experimental results. To compare VB-CT-S theory with experiment, we must evaluate six parameters: t, Vo, SF, R DA , Q, and k. Using the Universal Force Field 4 ,5 (UFF) in conjunction with Charge Equilibration theory6 to predict the charges of Figure 4.1 in vacuum, we obtain RDA = 7.30A (4.29) for molecule in Figure 4.1. Similarly UFF 4 leads to 2 2 k = 773.7kcal/A = 33.55eV/A = 5.38mdyn/cm. (4.30) The remaining parameters t, Vo, SF, and Q are each intrinsic parameters of CT molecules and can be determined directly from experiment. 3 According to Chapter 3, 'Y is zero when IVI = Itl. Experimentally3 'Y = 0 for a solvent polarity of E = 2.209, leading to Eg = 2.648eV for this polarity. Thus from IVI = It I and (7), we obtain t = Eg/V5 = 1. 184eV. (4.31 ) Vo and SF can be obtained by fitting absorption peaks in two different solvents. We 58 (a) 22.5 (b) 259 22 -VB-CT-S 21.5 ~ <1> 258 c=> on <1> 21 = ~ 'I' = ~ <:::5 I = =- 257 ~ 20.5 -VB-CT-S 20 19.5 10 256 20 30 40 50 0 10 20 E (c) 30 40 I I 50 E -50 40 I (d) I -55 20 -VB-CT-S • Experiment -60 -VB-CT-S ~ <1> c=> on <1> I = I = ~ -65 ~ -20 <-<::> -70 :?- -75 -40 -80 10 20 30 E 40 50 0 I I I I 10 20 30 40 50 E Figure 4.3: The predicted dependence of pol ariz abilities on solvent polarity (expressed in terms of the static dielectric constant E). (a) Polarizability, ct, (b) Hyperpolarizability, (3, (c) Second hyperpolarizability, 'Y, (d) Third hyperpolarizability, Ozzzzz. The values plotted are the static averaged values. The parameters used are: Vo = 0.833 eV, SF = 0.0685A -1, t = 1.184 eV, RDA = 7.30A, Q = 0.738, k =33.55 eV / A2. For 'Y in (c) a comparison is made between theory (solid line) and experiment (dots). Here the experimental results were corrected to static values using (4.39). 59 choose dioxane (El = 2.209 and 7 absorption energy Egl = 2.648eV), and CH3 CN (E2 = 37.5 and 7 Eg2 = 2.604eV). This leads to the following equations: Vo+~k [(q2 - q~T)2 - (q2 - q~B)2] - 4::0(1 - E~) hQ2 F JEgl- 2, S = 4t (4.33) where Ii depends on t and Eg [as given in (4.11)] and the BLA coordinate, qi, can be obtained from (4.7) and (4.8). Solving equations (4.32), and (4.33), leads to (4.34) and Va = 0.833eV (4.35) To separate out Q from SF, we can fit to the magnitude of"( at some E. We choose to do this for CH3 CN (E = 37.5). The experimental value 3 is "(static = -35 esu whereas the calculated value would be "( = -118 esu for Q = 1. This leads to Q4 = 0.297, (4.36) Q = 0.738. (4.37) or Substituting into (4.34) leads then to -I SF = 0.0685 A . (4.38) Given t, Va, SF, Q, R DA , and k from (4.29 - 4.38), we can calculate CY, j3, ,,(, and r5 for all solvent polarities, E. The resulting averaged values are shown in Figure 4.3. Currently only "( is available from experiment, Figure 4.3.c. We see that VB-CT-S fits reasonably well with experiment 3 despite the simplicity of this model. It will be valuable to measure the CY and {3 for this molecule in various solvents in order to 60 further test the model. 4.3.2 Frequency Dependent Correction for Hyperpolarizations In Figure 4.3.c the experimental THG results are corrected to the static values by using the following formula "IT HG _ 'Ystatic - 1 (E; E; 4 (Eg - 3w)(Eg - 2w)(Eg - w) + (Eg + 3w)(Eg + 2w)(Eg + w) 3 E39 E9 ) + + (Eg + w)(Eg + 2w)(Eg - w) (Eg + w)(Eg - 2w)(Eg - w) (4.39) where Eg is the bandgap and w is the frequency used in the experiment (w = 0.65eV). Similarly when experiments are available, the a and /3 will be corrected by using /3SHG (4.40) /3static and E2 9 4.4 Discussion 4.4.1 Summary of VB-CT-S w2 ' (4.41) The VB-CT-S model is quite simple. It involves 1. two properties (k and R DA ) which can be obtained from the force field (spec- troscopy or theory) 2. three electronic parameters (Vo, t, Q) characteristic of the isolated molecule that can be derived from theory or experiment on the isolated molecule or from experiment in solution [as illustrated in Section 4.3.1) 61 3. one solvent independent parameter, SF, which must be obtained from an experimental value of the £Y, (3, or, in a polar solvent. Given these six parameters one can predict the properties [A max , £Y, (3, " 6, and qopt] as a function of solvent polarity. 4.4.2 Estimation of Molecular Based Properties In designing new nonlinear materials, one might consider replacement of the donor, of the acceptor, or of the linker. The value for Yo should depend strongly on the ionization potential (1 P) of donor (D) and the electron affinity (EA) of acceptor (A) which might be related to the change in redox potentials for some solvent. Similarly the differential charge transfer, Q, can be estimated from 1 PD and EA A . The effect of changing the length of the linker or of replacing the polyene linker in (1) with other polymers is more difficult. In an earlier paper,8 we discussed the dependence of Amax, £Y, (3, and, on polymer chain length based on a valence-bond charge-transfer-exciton (VB-CTE) model of the polymer chain. The combination of the VB-CT-S description of donor accepter systems with the VB-CTE model for polymer linker character and length leads to a three state description that is no longer analytic. This will be considered in a later paper. 9 4.4.3 Estimation of Solvent Properties In comparing to experiment we used the static dielectric constant for the solvent. This assumes that the relaxation time of the solvent is much longer than the time scale of charge transfer (that is, of the excitation process). This is true in polar solvents where the dielectric contribution is dominated by the orientation of the polar molecule. For non-polar solvents, the contribution of the induced electronic dipole moment to the total dielectric constant is not negligible, and its response to the change of electric field may have a time scale similar to the charge transfer. In this case we should use the frequency dependent dielectric constant, Ew' This may account for the poorer fit of , to experiment 3 for the nonpolar solvents. 62 4.4.4 Estimation of Solvation Energy In estimating the effect of solvent on the energy separation Vs [equations (4.12)(4.18)]' we employed the Marcus solvent model used for predicting electron transfer rates 2. This leads to the specific form (4.18) for the geometric parameter SF in which r A and rD should be about the size of the van der Waals radii of the donor and the acceptor. Assuming that r A = rD, equation (4.38) leads to r A = 4.87A for molecule in Figure 4.2. This is a plausible value, but we may also treat SF as a characteristic shape parameter of the CT molecule. The induced dipole density of the solvent is proportional to ~. Therefore, ~Vs in (4.17) should be proportional to (1 - Dand f regardless of the structure of solvent molecule. Hence equation (4.17) may be generalized to an arbitrary geometry of the CT molecule (rather than Figure 4.2). More rigorous ways to calculate solvation effects are available. Thus Warshel et al. 10 used an explicit representation of the solvent (the SCAAS model, surface con- strained all atom solvation) to treat the effect of solvent on the empirical valence bond (EVB) description of an SN2 reaction. To treat SN 1 ionic dissociation in a liquid, Kim and Hynes l l used a nonlinear Schrodinger equation formulation. Alternatively, the solvation energy could be evaluated with the DelPhi Poisson-Boltzmann continuum solvent procedure of Honig et al. 12. The charges for such calculations could be obtained using Charge Equilibration (QEq)6. Indeed, Friesner et al. 13 have developed an interface between Jaguar14 and DelPhi 12 that does the quantum mechanics self-consistently within the self-consistent reaction field of the solvent l3 . 4.5 Summary We find that the simple V B-CT-S model provides a quantitative explanation of the dependence of hyperpolarizabilities /3 and 'Y on solvation. The theory agrees well with the experimental results by Marder, et al. 3 for the 'Y of molecule in Figure 4.1, the only system for which a hyperpolarizability has been measured for a wide range of solvents. The VB-CT-S theory employs a continuous dielectric description of the solvent and leads to analytic formula for Amax, CY, /3, and 'Y in different solvents. 63 Because of its simplicity, this theory should be of use in designing new nonlinear optical materials. 4.6 References 1. Lu, D.; Chen, G.; Perry, J. W.; Goddard, W. A. III, J.Am. Chern. Soc. 116, 10679 (1994). 2. Marcus, R. A., J. Chern. Phys. 24, 996 (1956). 3. Marder, S. R.; Perry, J. W.; Bourhill, G.; Gorman, C. B.; Tiemann, B. G., Science 261, 186 (1993). 4. Rappe, A. K.; Casewit, C. J.; Colwell, K. S.; Goddard, W. A. III; W. M. Skiff, W. M., J. Am. Chern. Soc. 114, 10024 (1992). 5. Force field calculations were carried out with POLYGRAF from Molecular Simulation Inc. (Burlington, Massachusetts). 6. Rappe, A. K.; Goddard, W. A. III, J. Phys. Chern. 95, 2260 (1991). 7. Personal correspondence with Dr. Bourhill and Dr. Perry of JPL. 8. Lu, D.; Chen, G.; Goddard, W. A. III, J. Chern. Phys., 101,4920 (1994). 9. Lu, D.; Chen, G.; Goddard, W. A. III, work in progress. 10. Warshel, A.; Weiss, R. M., J. Am. Chern. Soc. 102, 6218 (1980); Hwang, J. K.; King, G.; Creighton, S.; Warshel, A., J. Am. Chern. Soc. 110, 5297 (1988). 11. Kim, H. J.; Hynes, J. T., J. Am. Chern. Soc. 114,10508 (1992); ibid, 114,10528 (1992). 12. Jean-Charles, A.; Nicolls, A.; Sharp, K.; Honig, B.; Tempczyk, A.; Hendrickson, T. F.; Still, W. C., J.Am.Chem.Soc. 113 1454 (1991). 64 13. Tannor, D. J.; Marten, B.; Murphy, R.; Friesner, R. A.; Nicholls, A.; Honig, B.; Ringnalda, M., "Accurate First Principles Caculations of Molecular Charge Distributions and Solvation Energies from Ab Initio Quantum Mechanics and Continuum Dielectric Theory," J. Am. Chern. Soc., submitted. 14. Langlois, J. -M.; Muller, R. P.; Coley, T. R.; Goddard, W. A. III; Ringnalda, M. N.; Won, Y.; Friesner, R. A., J.Chem.Phys. 92, 7488 (1990). Greeley, B. H.; Russo, T. V.; Mainz, D. T.; Friesner, R. A.; Langois, J. -M.; Goddard, W. A. III; Donnelly, R. E. Jr.; Ringalda, M. N., "New Pseudospectral Algorithms for Electronic Structure Caculations: Length Scale Separation and Analytical Two-Electron Integral Corections," J. Am. Chern. Soc., submitted. 15. This is available as Jaguar/Solvate from Schrodinger Inc. (Pasadena, California). 65 Chapter 5 The Temperature Dependence For Nonlinear Optical Properties of Donor Acceptor Push-Pull Organic Molecules 5 .1 Introduction Extensive theoretical studies l - 13 have been done on the hyperpolarizabilities of organic materials. Most studies are for the ideal case of an isolated molecule in a vacuum at its equilibrium geometry (absolute temperature of zero). However, most measurements of hyperpolarizabilities are at room temperature and in solvent. In order to compare the theoretical calculations with the experiment, it is important to consider the temperature dependence and solvation effects on the hyperpolarizabilites of these molecules. We recently proposed the valence-bond charge-transfer (VB-CT) model 14 to predict hyperpolarizabilities of donor acceptor push-pull conjugated molecules, such as (1). In order to predict the effect of solvation on the hyperpolarizabilities of these molecules, we developed the VT-CT-S modeP5 Both studies ignored the effect of temperature. In this paper, we use the VB-CT model to consider the electronic and vibrational contributions to hyperpolarizabilities at finite temperature. This leads to analytical expressions for describing the temperature dependence of the hyperpolarizabilities. 5.2 Theory The highest hyperpolarizabilities for organic systems are obtained in push-pull donor acceptor organics which exhibit two dominant resonant structures differing by a charge transfer from the donor to the acceptor. 16 - 17 We assume as a model the typical example 1,1 dicyano,6-(di-butyl amine) hexatriene (Figure 4.1). In this system two 66 low energy vibrational modes are likely to playa role in thermal effects on nonlinear optical (NLO) properties. The vibrational mode q distorts the resonance structure of a into b in Figure 4.1 and is critical in determining the hyperpolarizability. Motion in the torsional mode t modifies the IT overlap and hence the coupling of the states in Figure 4.1. Thus motions in both q and t might modify the polarizabilities. We write the ground state energy at finite temperature as (5.1) where Et(</J) is the torsional energy and Ev(q, </J) describes the vibrational energy of the resonance system (Figure 4.1). We describe the vibrational energy of this system as in the VB-CT model, (5.2) where (5.3) and A is the force constant. However, we now allow the transition matrix element, t, between VB state and CT state to depend on the torsional angle </J, t = t(</J). (5.4) The torsional energy, Et(</J), can be expanded to a Fourier series 00 Et(</J) = - L Em cos(m</J). (5.5) m=O There are two contributions to the polarizability and hyperpolarizabilities: electronic and vibrational. From reference 14, 19, 20, and 21, the polarizabiIity and hyperpolarizabilities have the forms (5.6) 67 e _ f3 e 3t 2 J1 3 V ~5 (5.7) 4t2J14 [V2 - t 2J ~7 (5.8) - q~B)2 = 0:'e-0:' = 0:'e2t2A(q~T -------:--::---'---~3 (5.9) f3v = f3efJ = f3e6t2A(q~T - q~B)2 (5.10) 'Y = 0:' v ~3 v ee( 9V2) 'Y = 'Y 'Y = 'Y 1 + 8(V2 _ t 2) . (5.11) Here ~ = JV2 + 4t 2 and we omit the z subscripts where z is the direction along the linker. Hence, the total polarizability and hyperpolarizabilities are (5.12) (5.13) (5.14) The average value of a property p [either the polarizability (0:') or hyperpolarizabilities (f3, and 'YJ of the molecule is given by < oo p >= ~ r+ d Z Loo r d'" ("') [_ Ev(q, 4» + Et(4))] q L7r o/p 0/, q exp kT (5.15) where Z is the partition function (5.16) Since the vibrational and torsional energies are much higher than the thermal energy, the first order approximation to (5.15) and (5.16) leads to [see (A-14) of 58 Appendix A] -1-.) + [_l_CPP(qO'¢O) + _1_8 p qo, 'PO 2>' q 82 q 2E¢ < >= ( p 2p (qO'¢0)] kT 82-1-. (5.17) 'P where (5.18) and (5.19) Here qo and ¢o are the optimum vibrational coordinate and torsional angle, while tl and En are the second derivatives of t(¢) and E t (¢) evaluated at ¢o. 5.3 Application to 1,1 dicyano, 6-(di-butyl amine) hexatriene We illustrate the application of (5.17)-(5.19) by applying the theory to the molecule 1,1 dicyano,5-(di-butyl amine) hexatriene, Figure 4.1. The values of some of the parameters were fitted to the experimental data in reference 15: to = 1.184 eV, Vo = 0.833 eV, I~I = 5.39 A, and >. = 33.55 eV / A2. (5.20) Here to is the charge transfer matrix element, Yo is the difference in energy between the valence bond state and the charge transfer state, J1, is dipole moment, and>' is the force constant. Using the universal force field 18 (UFF), we expand the torsional potential in the 69 form of (5.21) where ¢>o = 0 and Vo = 0.434eV. The matrix element t can be written in the form of t = to cos ¢> (5.22) where to = 1.184eV. The matrix element V can be written in the form of 1 (q - qo )2 V = Vo + 2.).. (5.23) where the equilibrium position 15 qo = -0.0577 A. Thus, the total energy has the form E(eV) = -0.6362 + 12.16 [q + 0.0577J 2 + 0.953¢>2. (5.24) Substituting (5.6 - 5.14) and (5.24) into (5.15 - 5.16) leads to a e = (19.84 + 0.00028 T) 10-24 esu (5.25) f3e = (256.81 - 0.0173 T) 10-30 esu (5.26) "/ = (32.88 - 0.046 T) 10- 36 esu (5.27) 24 esu (5.28) f3V = (212.07 - 0.0067 T) 10- 30 esu (5.29) "tV = (254.48 - 0.079 T) 10- 36 esu (5.30) a = (25.31 + 0.0010 T) 10- 24 esu (5.31) f3 = (468.88 - 0.024 T) 10- 30 esu (5.32) a V = (5.47 + 0.00072 T) 10- 70 I = (287.36 - 0.125 T) 10- 36 esu. (5.33) Here 0: and I are the average values. Thus at T = 300 K, we obtain = 25.61 X 10-24 esu, (5.34) f3 = 461.68 X 10- 30 esu, (5.35) I = 249.86 X 10- 36 esu. (5.36) 0: In Figure 1 we compare the values from equation (5.31 - 5.33) (dash line) with numerical evaluation of equation (5.17) (solid line). We see that these two curves agree very well at temperatures up to T = 500 K. The increased discrepancy for higher temperatures may arise partly from the higher order terms. As the temperature increases from OK to 300K, 0: increases by 1%, f3 decreases by 1.7%, and 'Y decreases by 10%. Thus for a donor acceptor push-pull molecule, temperature has only a very small effect on 0:, f3 and f. 5.4 Discussion 5.4.1 Electronic and Vibrational Hyperpolarizabilities In Figure 1, we plot the contributions from electronic hyperpolarizabilities (line with cross), and vibrational hyperpolarizabilities (line with triangle). As the temperature increases from OK to 300K, o:e increases by 0.5%, o:V increases by 3%, f3e decreases by 2.1%, f3v decreases by 1.3%, Ie decreases by 50%, and IV decreases by 8%. The sensitivity of Ie to temperature occurs because molecule (1) was designed to have a large f3 e, causing Ie near the zero crossing point (see Figure 2 of reference 14). 5.4.2 Macroscopic Hyperpolarizabilities The experimentally measured hyperpolarizabilities are the average of microscopic hyperpolarizabilities over a statistical distribution of molecular orientations. For the 71 30 ~ 25 - 20 - 1 5 - 1 0 5 o o 10 0 200 30 0 400 500 600 T 6 00 .--. - 500 :::J (/l (J.) 400 o(') 6 or- - 300 - 200 - lOO - f- o o 100 2 00 300 40 0 500 600 T 400 350 r- 2 00 t- l 50 f- -.. :::J (/l (J.) <.0 (') 6or- - 100 50 o o 100 200 300 400 500 600 T Figure 5.1: Temperature dependence of the polarizability (a), and hyperpolarizabilities (f) , and ,) for 1,1 dicyano,6-(di-butyl amine) hexatriene (1). The dash lines (analytical) are from the formula (5.31 - 5.33). The solid lines are from the numerical evaluation of (5.17) . The crosses are the vibrational contributions. The triangles are the electronic contributions . 72 isotropic average, _ J.12 1 J.12 a =< a > + 3kT = 3(a xx + a yy + a zz ) + 3kT /3 = 0 J.1.f3 1 =< 1 > + 5kT 1 = S(rxxxx J.1.f3 + Iyyyy + IZZZZ + 2,xxyy + 2,xxzz + 2,yyzz) + 5kT' (5.37) Here a, /3, and 1 indicate the macroscopic hyperpolarizabilities. Considering temperature effects, the above equations become (5.38) (5.39) Here All<) A,a, and A, are the temperature coefficients in (8) while ao, f30, and 10 indicate the values at T = OK. In most cases, the modification is negligible. However, for the cases discussed in Section 5.4.1, the temperature effects can be significant. 5.5 Summary We extended the VB-CT model to treat the effects of temperature on hyperpolarizabilities. We find that temperature has a little effect on hyperpolarizabilities. The result is an analytic description which may be useful in predicting the effects of temperature and in analyzing the results from experiment. 73 5.6 References 1. Prasad, P. N.; Williams, D. J. Eds, Introduction to Nonlinear Optical Effects in Molecules and Polymers; Wiley: New York, 1991. 2. Kuhn, H.; Robillard, J. Eds, Nonlinear Optical Materials; CRC Press, Inc. 1992. 3. Perry, J. W.; Mansour, K.; Lee, I. et al., Science. 1996, 273, 1533 4. Rumi, M.; Zeribi, G.; and Mullen, K., J. Chern. Phys. 1998, 108, 685 5. Shuai, Z.; Bredas, J. L.; Saxena, A.; Bishop, A. R., J. Chern. Phys. 1998, 109, 572. 6. Maroulis, G., J. Chern. Phys. 1998, 108,5432. 7. Jacquemin, D.; Champagne, B.; Kirtman, B., J. Chern. Phys. 1997, 107, 5076. 8. Champagne, B., Chern. Phys. Lett. 1998, 287, 185. 9. Ohta, K.; Sakaguchi, T.; Hamada, K.; Fukumi, T., Chern. Phys. Lett. 1997, 274,306. 10. Norman, P.; Luo, Y.; Jonsson, D.; Agren, H., J. Chern. Phys. 1997, 106, 8788. 11. Lauderdale, W. J.; Coolidge, M. B., J. Phys. Chern. 1995, 99, 9368. 12. Matsuzawa, N.; Ata, M.; Dixon, D. A., J. Phys. Chern. 1995, 99, 7698. 13. Kirtman, B.; Luis, J. M.; Bishop, D. M., J. Chern. Phys. 1998, 108, 10008. 14. Lu, D.; Chen, G.; Perry, J.; Goddard, W. A. 111., J. Am. Chern. Soc. 1994, 116, 10679. 15. Chen, G.; Lu, D.; Goddard, W. A. 111., J. Chern. Phys. 1994, 101, 5860. 16. Marder, S. R.; Perry, J. W.; Bourhill, G.; Gorman, C. B.; Tiemann, T. G.; Mansour, K., Science. 1993, 261, 186. 74 17. Marder, S. R.; Torruellas, W. E.; Blancharddesce, M.; Ricci, V. et al., Science, 1997, 276, 1233. 18. Rappe, A. K.; Casewit, C. J.; Colwell, K. S.; Goddard, W. A. III; Skiff, W. M., J. Am. Chern. Soc. 1992, 114, 10024. 19. Kim, H.; Cho, M.; Jeon, S., J. Chern. Phys. 1997, 107, 1936. 20. Castiglioni, C.; Del Zoppo, M.; Zerbi, G. Phys. Rev. B, 1996, 53, 13319. 21. Bishop, D.; Champagne, B.; Kirtman, B., J. Chern. Phys. 1998, 109, 9987. 75 Chapter 6 Saturation of the Second Hyperpolarizability for Polyacetylenes 6 .1 Introduction It is well known 4 - 7 that for polyenes, -( -CH = CH - )N-, a and I increase rapidly with N for smaller N but saturate to increase only linearly with N above some value, N sat ([3 = 0 since the molecule is centrosymmetric). For polythiophenes (which have a polyene backbone), it was established 4- 7 that a saturates at N = 10 double bonds (a C20 polyene) whereas I saturates at N = 18 double bonds (C36 ). For the parent compound, polyacetylene (PA) , experiment IS shows that N;at > 16 and theory16 shows that N;at > 22. However, it has not been possible to study sufficiently large N to establish N;at experimentally (because of difficulties with synthesis) or theoretically (because of expense). Of two recent reports on long chain polyenes in Science, experimentS on a substituted, imperfect PA leads to N;at ~ 125 (C250 ) whereas (semiempirical) theory9 leads to N;at = 20 (C40 ), see Figure 6.1. 6.2 Calculations The largest previously ab initio theoretical calculations 16 were on N = 22 (C44), far too small to test for saturation. We have developed a new ab initio quantum mechanical program (JaguarjNLO)1l-13 which allows predictions of hyper pol ariz abilities for large molecules 10 to 20 times faster than previously possible. We report here ab initio Hartree-Fock (HF) calculations (6-31G basis set)14 on a and I with N up to 49 (C9S ). The geometry was obtained from the optimized structure15 of C22 H 24 (HF with a 6-31G basis), using the central unit to determine the geometry of all PA polymers. (This led to C = C = 1.450 A, C - C = 1.338 A and a bond angle of 124.3°.) We 76 500 400 I t-0 300 0 I I I .. . - " - t-0 -Z ?- [>. 200 r- 0 100 -0 . - " " " " I 0 0 [>. 50 0 I Ab initio theory Experiment Semiempirical Theory I I 200 100 150 Number of Double Bonds, N 250 Figure 6.1: Recent results on the saturation of ,IN for polyacetylenes (PA). The open squares are from the experient (reference 8) on a substituted, imperfect PA. The open circles are from semiempirical theory (reference 9).The filled squares are from the current calculation. tested the geometry by calculating the energy of C4s H 50 with bond lengths changed by ± 0.003 A and found that the above structure has the lowest energy. The a and, were obtained by calculating J.L(e) with three finite fields (sufficient because of the inversion symmetry). We used electric fields along the chain direction to obtain a zz and 'ZZZZ. There are two contributions to the errors. 1. Higher order terms in (1.1). By using a fourth electric field for C7s H so , we found that neglect of higher order terms leads to less than 0.01% error for a and to about 0.4% error for ,. 2. Errors in the calculation of the dipole moment (due to lack of convergence or other errors in the wavefunction). By examining the changes of a and, with convergence, we estimate that the error from this source is less than 0.01% for a and less than 3% for ,. 77 (a) 25 20 z ""- 15 ~ • 10 Jaguar Fitted 5 o (b) 10 20 30 40 Number of Double Bonds, 50 60 N 80 70 60 z ""- ~ v 50 40 30 • 20 • Jaguar ---Fitted 10 0 0 10 40 20 30 Number of Double Bonds, 50 60 N Figure 6.2: (a) Polarizability of polyacetylenes chains from ab initi 0 calculations (JaguarjNLO). The filled squares are from the current ab initio HF calculations. The open circles are for VB-CT using t = 0.83eV and V = 1.0eV. The solid line is a fit to an equation like (6.3). The parameters are 0: 00 = 22.6 x 1O-24 esu , Aa = 0.249, Ea = 0.84 , and Cu = 0.08. (b) Second hyperpolarizability of polyacetylenes chains from ab initio calculations (JaguarjNLO). The line is a fit to equation (6.3). The parameters are 1'00 = 380 X 10- 36 esu, A, = 0.073, E, = 11.4, and C, = 0.182. 78 -I 25 Figure 6.3: Molecule used in experiment (reference 8) Figure 6.2.a shows the results for a (filled squares), where we see that (6.1) Figure 6.2.b shows results for < I >= Izzzz/5 where we see that N;at = 45 ± 5 (Cgo ). (6.2) These values are obtained by fitting a/Nand 1/N to smooth curves and setting N sat to 95% of the limiting value. For large N we expect _ I _ (1+- IN = N = 100 1 exp( - AN) ) Bexp(-CN) . (6.3) The solid lines in Figure 6.2 represent the fits to such a function. 6.3 Discussion Direct experimental measurements on I for polyenes has been made by Craig et al. 18 who synthesized polymer (Figure 6.3) with N from 4 to 13. The resulting values of I (in T H F) are included in Figure 6.3 (filled triangles). The N dependence from theory and experiment agree well 2o . Previous ab initio calculations 16 (HF with 6-31G basis) considered only N up to 22. These results (open squares in Figure 6.3) are in excellent agreement with the current results (circle in Figure 6.3). 79 80 • 70 0 0 • 60 50 0 ~ 40 ~ ~ v 30 • • 0 / 20 10 0 0 ... Jaguar Fitted VB-CT Previous Calculatio Experiment o o 10 20 30 40 50 Number of Double Bonds, N 60 Figure 6.4: Second hyperpolarizability of polyacetylenes chains. The filled squares are from the current ab initio HF calculations while previous ab initio results (reference 14) are shown with open squares. The open circles are for VB-CT using t = O.83eV and V = l.OeV. Direct experimental results (reference 16) are shown with filled triangles. 80 Mukamel and coworkers9 predicted the N dependence of'Y using an electron-hole model (based on semiempirical P P P 7r electron calculations) to describe polarization. Their results (crosses in Figure 6.2) lead to saturation at N;at = 20 (C40 ). This is substantially smaller than the ab initio results (N;at = 45). The difference is probably due to approximation in the semiempirical and electron-hole descriptions. Previously5-7 we developed a simple theoretical model, valence bond-charge transfer (VB-CT), for predicting polymer length dependence of a and 'Y. This method involves two parameters (t and V) which can be fitted to either experiment or theory. [The bandgap and valence bandwidth and their dependence can be used on N.] Using t = 0.83eV and V = 1.0eV for VB-CT leads to good agreement with the ab initio results on a (as shown in Figure 1, open circles). The VB-CT results for 'Y in Figure 6.3 (open circles, scaled by a factor of 0.457) leads to N;at = 45, in excellent agreement with the ab initio calculations. 6.4 Summary Summarizing, the ab initio theoretical studies lead to N':at = 25±2 and N7at = 45±5. These theoretical results are in strong disagreement with recent experimental results 8 , which led to N;at = 125. However, these experiments did not involve pure all-trans polyenes. The synthetic method 19 led to about a 50-50% mixture of the species in Figure 6.5. We find (vide infra) that such disorder can increase N;at. In addition, there appears to be other structural defects8 ,19 in the chains that might modify the saturation length. Also the measurement of chain length was indirect 8 , leading to additional uncertainties. Given the limitations in the experiments, we believe that the current estimate of N;at = 45 ± 5 is the most reliable estimate. In order to examine the role that disorder plays in a and " we used VB-CT theory with the t and V parameters discussed above but randomly changed V (the site energy) by ±O.4eV. The result is shown in Figure 6.6 along with that for pure all-trans polyethylene. We find two effects: 81 Figure 6.5: Solvent used in experiment (reference 19) 1000 I I I I I I <> <> z '--- ~ v 800 I- 600 - 400 - 200 - <> <> <> <> <> <> - <> - <> <> • <> Ordered Disordered - <> - <> ;, 0 0 10 ~ • I • • •I • • I • • •I • • I • • 20 40 50 30 Number of Double Bonds, N 60 70 Figure 6.6: Effect of disorder in second hyperpolarizability based on VB-CT with t = O.83eV and V = 1.0eV. Filled circles indicates the ordered polyene while open circles are with a random disorder of Vi = V ± O.4eV. 1. N;at increases by a factor of about 1.4. 2. The magnitude of 'Y for a given N also increases by a factor of about 5. This suggests that deliberate inclusion of disorder may be advantageous. One might do this by including monomers with C H 3 , Cl, or F substituents. Such monomers might lead to increased solubility in various solvents, allowing synthesis of longer chains. In addition one might achieve the alignment needed for high nonlinear performance by using substituents that can be photolytically or thermally cross linked in the applied electric field. 82 6.5 References 1. For a recent survey see Chemistry and Engineering News, February 20, 1995, page 28. 2. Arizmendi, L.; Agullolope, F., MRS Bulletin, 19(3), 32-38(1994). 3. Marder, S. R et al., Science 263, 511(1994). 4. Thinpont, H.; Rikken, G. L.; Meijer, E. W.; tan Hoeve, W.; Hynberg, H., Phys. Rev. Lett. 65, 2141 (1990). 5. Chen, G.; Lu, D.; Goddard, W. A. III, J. Chern. Phys. 101, 5860 (1994). 6. Lu, D.; Chen, G.; Goddard, W. A. III, J. Chern. Phys. 101, 4920 (1994). 7. Lu, D.; Chen, G.; Perry, J. W.; Goddard, W. A. III, J. Am. Chern. Soc. 116, 10679 (1994). 8. Samuel, I. D. W.; Ledoux, I.; Dhenaut, C.; Zyss, J.; Fox, H. H.; Schrock, R R; Silbey, R J., Science 265, 1070 (1994). 9. Mukamel, S.; Tahahashi, A.; Wang, H. X.; Chen, G., Science 266, 250 (1994). 10. Hurst, G. J. B.; Dupuis, M.; Clementi, E., J. Chern. Phys. 89, 385 (1988). 11. Ringnalda, M. N.; Langlois, J.-M.; Greeley, B. H.; Murphy, R B.; Russo, T. V.; Cortis, C.; Muller, R P.; Marten, B.; Donnelly, R. E. Jr.; Mainz, D. T.; Wright, J. R.; Pollard, W. T.; Cao, Y.; Won, Y.; Miller, G. H.; Goddard, W. A. III; Friesner, R. A., Jaguar v2.01, Schrodinger, Inc., Pasadena, California, 1994. 12. Lu, D.; Marten, B.; Cao, Y.; Ringnalda, M. N.; Freisner, R. A.; Goddard, W. A. III, Chern. Phys. Lett., 242, 543 (1995). 13. Marten, B.; Cao, Y.; Friesner, R. A.; Lu, D.; Goddard, W. A. III, J. Chern. Phys., submitted. 83 14. The 6-31G basis set gives accurate predictions for such systems. For example, in reference 10 we calculated the hyperpolarizabilities of Julolidinyl-n-isoxazolone using the 6-31G basis set for n = 1 to 6. Adding polarization functions increases I uniformly by about 10%. Thus the dependence of I on N was not affected and led to excellent agreement with experiment. At the HF level the magnitude of (3 was systematically about half the experimental value. 15. Villar, H. 0.; Dupuis, M.; Watts, J. D.; Hurst, G. J. B.; Clementi, E., J. Chern. Phys. 88, 1003 (1988). 16. Kirtman, B.; Toto, J. L.; Robins, K. A.; Hasen, M., J. Chern. Phys. 102 5350 (1995). 17. Mukamel, J.; Wang, H. X., Phys. Rev. Lett. 69, 65 (1992). 18. Craig, G. S.; Cohen, R E.; Schrock, R R; Silbey, R J.; Puccetti, G.; Ledoux, I.; Zyss, J., J. Am. Chern. Soc. 115,860 (1993). 19. Fox, H. H.; Wolf, M. 0.; O'Dell, R; Lin, B. L.; Schrock, R. R; Wrighton, M. S., J. Am. Chern. Soc. 116, 2827 (1994). 20. Because of the correlation effect, the presence of solvent and end groups, and the non-zero frequency, it is difficult to compare the absolute values between the theory and experiment. Consequently, in Figure 6.4 we divided the reported experimental value of'"'( by a factor of 6. 84 Chapter 7 ab initio Predictions of Large Hyperpolarizability Push-Pull Polymers: J ulolidinyl-n-isoxazolone and Julolidinyl-n-N, N'-diethylthiobarbituric acid 7 .1 Introduction For telecommunications (electrooptic switches), optical information processing, and sensor applications, it is useful to develop organic chromophores with very large first hyperpolarizabilities, /3. A major step forward in developing such materials was made recently by Marder et al. 1 who succeeded in developing the highest /3 organics currently known, Julolidinyl-6-isoxazolone [Figure 7.1.a with n = 6] and Julolidinyl6-N, N'-diethylthiobarbituric acid [Figure 7.1.b with n = 6]. These push-pull organic materials lead to Jk . /3(0) = 13,600 x 10- 48 esu and 14,920 x 10- 48 esu, respectively. Although semiempirical and ab initio level calculations of hyperpolarizabilities of various molecules have been reported recently2-6, ab initio level calculations have not been previously reported for molecules of the size of molecules in Figure 7.1 with n = 6. We have developed a new ab initio Quantum Mechanical program (JaguarjNLO)7,8 which provides predictions of hyperpolarizabilities for such molecules far faster than previously possible and have applied it to predicting the hyperpolarizabilities for molecules in Figure 7.1.a with n = 0, 1,2, 3, and 6, for molecules in Figure 7.1.b with n= 6. 7.2 Results Hartree-Fock(HF) calculations using the 6-31G basis leads to the results in Table 7.1 and Figure 7.2. Here we see that the comparison between theory (in vacuum) and experiment (in chloroform) is good; in particular, the trend with n is excellent. 85 (b) Figure 7.1: (a) Julolidinyl-n-isoxazolone Julolidinyl-n-N, N'-diethyIthiobarbituric acid and (b) 86 \ 10 4 \ \ I \ I I I- 0 ;:l [fJ I OJ ro '"I 0 I 0 0 1000 =- - I H 0 I C!l. ::i. 0 Experiment Jaguar 0 100 \ 0 1 \ \ I I I 6 2 4 5 3 Number of Double Bonds in the Linker 7 (n+l) Figure 7.2: The dependence of hyper pol ariz ability w/3 (0) for Jololidine-n-isoxazolone on N = n + 1, the number of double bonds in the linker. The excellent agreement between theory and experiment suggests that theory can be used as an effective tool in developing new nonlinear optical materials. For Julolidinyl-6-isoxazolone the total CPU time to obtain J-l, CY, /3, J-l. /3, 'Y is 10 hours (on an HP 735 workstation). This is much faster than the processes of synthesizing and purifying the compounds and then running the experiments to measure the moments and hyperpolarizabilities. Taking x as the direction of the linker and xz as the plane of the linker leads to the components in Table 7.2. As expected, the major contribution to /3 is from /3x. For example, for molecule in Figure 7.l.a with n = 6, we calculate /3x/ /3y = 146 and /3x/ /3z = 7.2. Table 7.1 shows that the theoretical dipole moment J-l for moleclue in Figure 7.l.a changes smoothly with n (as expected). However, experiment! does not. The experimental values of J-l are obtained from the changes of the capacitance of the vessel with molecules (Figure 7.l.a) in chloroform solvent. We believe that this discrepancy is mainly experimental. The EFISH experiment! measures J-l. /3 rather than /3. Since there is no data on the relation between the J-l and /3 vectors, the experimental value of /3(0) is extracted 87 Molecule Class n 1 1 1 1 1 2 0 1 2 3 6a 6a }1' jJ(O) Exp. Jaguar 328 ± 65 919 ± 180 3000 ± 600 4753 ± 950 13600 ± 2720 14920 ± 3000 218 529 1466 3311 6960 6877 }1 Exp. Jaguar 9.5 9.1 9.0 9.8 16 10.71 12.47 15.49 16.65 17.67 15.25 jJ(O) Exp. Jaguar 34±7 100 ± 20 339 ± 70 485 ± 100 849 ± 100 31 60 117 232 442 456 () Jaguar 49 45 36 31 27 9 Table 7.1: Hyperpolarizabilities for the Julolidinyl-n-isoxazolone class and Julolidinyln-N, N'-diethylthiobarbituric acid class. Here}1 is the dipole moment, jJ(O) is the static hyperpolarizability, and () is the angle between i1 and jJ. The units are 10- 48 esu for J1" jJ(O), 10- 18 esu for }1, and 10- 30 esu for jJ(O). The experiments used CHCh solvent whereas theory considered molecules in the gas phase. from }1' jJ, assuming that }1 and jJ are parallel. However, the calculations (Table 7.1) show that the angle between }1 and jJ of Figure 7.l.a ranges from 27° to 49°, and the angle for Figure 7.l.b is 9° for n = 6. Thus the theory is useful in extracting the experimental value of jJ from }1 . jJ. Experimental values were not reported for a (polarizability) and r (second hyperpolarizability). We show in Figure 7.3 the calculated a, jJ, r as a function of number of double bonds (N) in the linker for Figure 7.l.a. The lines in the log-log plots of Figure 7.3 lead to a = a oNO.70, jJ = jJoN1. 44, and r = roN2.65. The components of a, jJ and r are listed in Table 7.2.9 The experiments were carried out for molecules in chloroform whereas the results reported above were for gas phase molecules. In order to determine the effects of solvent, we used Jaguar/Solvent 10 - 12 program which uses DelPhi 15 to evaluate the reaction field using a continuous Poisson-Boltzmann description of the solvent self consistently with using Jaguar to calculate the wavefunctions in the reaction field. Table 7.3 shows the calculated hyperpolarizabilities of molecules in Figure 7.l.a with n = 0, 1,2 in chloroform (E = 4.806). Here we see that a is the same in the gas phase and in solvent, jJ changes less than 10%, and r decreases less than 40% in solvent. 88 n /-1x /-1y /-1z (\(xx (\(yy (\(zz (\(xy (\(xz (\(yz (3xxx (3xyy (3xzz (3x (3yXX (3yyy (3yzz (3y (3zxx (3zyy (3zzz (3z Ixxxx IYYYY IZZZZ 0 1 2 3 6a 6b 9.66 -1.04 4.50 11.12 -0.35 -5.65 14.29 0.14 -5.97 15.35 -0.73 -6.42 16.71 -0.42 -5.72 15.20 0.09 1.12 59.40 23.40 36.43 -0.59 10.65 0.27 82.87 19.38 41.53 -1.69 8.66 -3.44 113.4 18.91 45.06 -2.63 5.86 -0.02 141.02 20.71 40.16 2.92 3.16 7.37 228.9 24.2 51.4 -1.8 4.0 2.9 242.6 28.0 56.7 0.8 13.4 -1.5 27.38 -1.17 2.24 28.45 -0.35 0.01 0.14 -0.20 12.17 -0.32 0.74 12.59 52.78 0.74 3.86 57.38 0.44 0.09 0.23 0.76 18.04 0.03 0.89 18.96 108.75 2.44 2.69 113.88 0.08 0.12 0.14 0.34 27.24 0.28 -0.71 26.81 206.44 14.26 14.97 229.24 13.18 0.51 0.26 13.94 31.69 -0.22 -1.63 29.84 400.48 16.39 20.91 437.77 2.78 -1.11 1.41 3.07 59.23 0.19 2.02 61.44 406.4 20.9 27.5 454.8 1.0 -0.07 -0.07 0.9 -26.7 -2.9 -4.4 -34.0 21.31 0.51 3.09 99.84 0.418 2.203 290.37 -0.80 7.22 640.70 -0.08 6.81 3736.7 15.5 38.3 4269.9 4.0 45.6 Table 7.2: Dipole moment /-1, pol ariz ability (\(, hyperpolarizabilities (3 and I for Julolidinyl-n-isoxazolone class and Julolidinyl-n-N, N'-diethylthiobarbituric acid class. The units are 10- 18 esu for /-1, 10- 24 esu for (\(, 10- 30 esu for (3, and 10- 36 esu for T a Molecule in Figure 7.1.a with n = 6. b Molecule in Figure 7.1.b with n = 6. 89 UJ Q) :j 10~ -.--I r-I -.--I ~ .~~.. fa 1000 E- . .-.- N -.--I J..j n:l r-I o 100 F- --- ---- ill 10 t----------------- _____ -----------y-------_ .. _ ... _--_.--- ------y ._.--T--..... -.--------'f---.-- ex 1 1 . . k N, Number of Double Bonds In the Lln er 10 Figure 7_3: The dependence of ax x, (3xxx, and 'Yxxxx on N = n + 1, the number of double bonds in the linker. The solid lines are the least squares fit to the calculated values. The units are 10- 23 esu for a xx , 10-30 esu for (3xxx, and 10- 37 esu for IXxxx' n a xx a zz f3x f3z 'Yxxxx 0 2 1 no solvent solvent no solvent solvent no solvent solvent 59.40 36.43 28.45 12.59 21.31 60.88 36.54 33.18 15.04 13.06 82.87 41.53 57.38 18.96 99.8 82.85 41.35 60.28 18.79 76.7 113.4 45.06 113.88 26.81 290.37 111.0 44.93 110.09 23.91 242.14 Table 7.3: Hyperpolarizabilities of molecules in Figure 7.1.a with n = 0,1,2. no solvent indicates a standard gas phase calculation using Jaguar, solvent indicates Jaguar/Solvate calculation using chloroform, the solvent used in the experiments. The units are 10- 24 esu for (x, 10- 30 esu for (3, and 10-36 esu for 'Y. 90 Basis Sets 6-31G 6-31G** a xx a zz 82.87 41.53 57.38 18.96 99.8 83.73 42.19 57.35 17.95 108.5 /3x /3z 'Yxxxx Table 7.4: Hyperpolarizabilities of molecules in Figure 7.1.a with n = 1 with different basis sets, 6-31G, 6-31G** (including polarization functions), and 6-31G++ (including diffuse functions). The units are 10- 24 esu for a, 10- 30 esu for /3, and 10- 36 esu for 'Y. 7.3 Calculational Details 7.3.1 Basis Sets All calculations in Table 7.1 and Figure 7.2 were at the Hartree-Fock(HF) level using the 6-31G basis set with Jaguar (v2.01) program8 . The results in Table 7.4 show that adding polarization functions to the basis (6-31 G**) affects the results by 1% to 8%. 7.3.2 Geometry Crystal structures have been reported 1 for molecules in Figure 7.1.a with n = 2 and 3. In estimating the structure for the other molecules, we used the geometries of the donor and acceptor from the crystal structure of n = 3. The bond lengths for the linker polyene were estimated from the structural data of n = 2,3. We assumed that the polymer linker involves a resonance of the two valence bond (VB) configurations with an average bond length of 1.385A and an average alternation of t5rn- The experiments lead to t5r2 = 0.037A and t5r3 = 0.049A, respectively, and we use 13 t5r = o.llA for n = +00. Fitting these results leads to t5rl = 0.023 A and t5r6 = 0.071 A; we assumed t5ro = 0.0 A. For molecules in Figure 7.1.b, we assumed that the donor and linker have the same geometry as for those in Figure 7.1.a. The geometry of the acceptor of molecule 91 in Figure 7.1.b was obtained by geometry optimization at the HF /3-21G level. We assumed that molecules in the gas phase and in the solid state have the same bond length. This is reasonable since the environment in a molecular crystal is far less polarized than in a polar solvent. For the solvation calculations, we also used the same geometry as for the gas phase. For a polar solvent, this could cause increased errors because of changes in the bond lengths of the linker. To estimate this effect, we used the VB-CT-S mode1 14 ,15 coupled with the experimental data. For molecule in Figure 7.1.a with n = 3, VB-CT-S estimates that 8r3 changes from 0.049A to 0.040A and f3 decreases by 7% [the parameters used were: t = 0.88eV, V = 0.7geV, Q = 0.43, rD = 3A, r A = 3A, rDA = 16A]. 7.3.3 Hyperpolarizabilities The polarizabilities were obtained by solving for the HF wavefunction in finite fields, t, and calculating the dipole moments, [1. Considering only the x components, this leads to (7.1) The 5 point calculation uses the values c = 0, ±C1, ±C2 with C2 = 2C1. This leads to (7.2) 2 1 5 Jl(O) 4 f3 = 3c? [Jl (C1) + Jl (-C1)]- 24c? [Jl (C2) + Jl (-C2)] - 4 c? + 0(c1), (7.3) ,= -6~f [Jl (C1) - (-C1)] + 121Cf (C2) - (-C2)] + O(ci). (7.4) JL [JL JL The magnitudes of the finite field were chosen to minimize the error in f3. We used C1 = 0.0024 a.u. for n = 6 , C1 = 0.0048 a.u. for n = 1,2,3, and C1 = 0.0098 a. u. for n = O. To test the accuracy, we did two extra points ±C3 with C3 = 0.0192 a.u. for molecule in Figure 7.1.a with n = 1. The f3xxx values for calculations with three points, five points, and seven points are 52.99 x 10- 30 esu, 52.78x 10- 30 esu, and 52.58 x 10- 30 esu respectively. This indicates that the numerical error in f3 is 92 less than 1%. We also tested the accuracy by calculating the hyperpolarizabilities of various molecules and comparing with the results using Gaussian 92 16 (which uses analytic methods). The agreement between these two methods is excellent. Examples include: p-nitrobenzene, f3zzz = -111 a.u.(Gaussian 92) and f3zzz = -111 a.u. (Jaguar); and orthonitroaniline, f3zzz = 189 a.u.(Gaussian 92) and f3zzz = 192 a.u. (Jaguar). Additional details of the hyperpolarizability calculations will be published elsewhere. 7 We also tested the accuracy by calculating the, of benzene. The EFISH experiment 17 yields, = 1.5 X 10-36 esu. Jaguar calculations at the HF /6-31G level lead to , = 1.1 X 10-36 esu, in reasonable agreement with experiment. 7.4 References 1. Marder, S. R et al., Science 263, 511 (1994). 2. Meyers, F.; Bn§das, J. L.; Zyss, J., J. Am. Chern. Soc. 114,2914 (1992). 3. Chopra, P.; Carlacci, L.; King, H. F.; Prasad, P. N., J. Phys. Chern. 93, 7120 (1989). 4. Clays, K.; Hendrickx, E.; Thiest, M.; Verbiest, T.; Persoons, A.; Dehu, C.; Bn§das, J. L., Science, 262, 1419 (1993). 5. Hurst, G. J. B.; Dupuis, M.; Clementi, E.,J. Chern. Phys. 89, 385 (1988). 6. Sim, F.; Chin, S.; Dupuis, M.; Rice, J. E., J. Phys. Chern. 97, 1158 (1993). 7. Marten, B.; Lu, D.; Cao, Y; Ringnalda, M. N.; Goddard, W. A. III; Friesner, R A., in preparation. 8. Ringnalda, M. N.; Langlois, J.-M.; Greeley, B. H.; Murphy, R B.; Russo, T. V.; Cortis, C.; Muller, R P.; Marten, B.; Donnelly, R E. Jr.; Mainz, D. T.; Wright, J. R; Pollard, W. T.; Cao, Y.; Won, Y.; Miller, G. H.; Goddard, W. 93 A. III; Friesner, R A., Jaguar v2.01, Schr6dinger, Inc., Pasadena, California, 1994. 9. We did not calculate the IXXyy, IXXZZ and IYYzz components because the electric fields were applied only along the chain direction. Since the dominant contribution to I is from Ixxxx, the other components can be neglected. 10. Tannor, D.; Marten, B.; Murphy, R.; Friesner, R. A.; Sitkoff, D.; Nicholls, A.; Ringnalda, M.; Goddard, W. A. III; Honig, B., J. Am. Chern. Soc. 116, 11875 (1994). 11. Marten, B.; Sitkoff, D.; Kim, K.; Murphy, R; Ringnalda, M.; Friesner, R A.; Honig, B.; in preparation. 12. Nicholls, A.; Honig, B., J Comput. Chern. 12,435-445 (1991). 13. Villar, H. 0.; Dupuis, M.; Watts, J. D.; Hurst, G. J. B.; Clement i, E., J. Chern. Phys., 88, 1003 (1987). 14. Lu, D.; Chen, G.; Perry, J.; Goddard, W. A. III, J. Am. Chern. Soc., 116, 10679 (1994). 15. Chen, G.; Lu, D.; Goddard, W. A. III, J. Chern. Phys., 101, 5860 (1994). 16. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.; Johnson, B. G.; Wong, M. W.; Foresman, J. B.; Robb, M. A.; Head-Gordon, M.; Replogle, E. S.; Gomperts, R; Andres, J. L.; Raghavachari, K.; Binkley, J. S.; Gonzalez, C.; Martin, R L.; Fox, D. J.; Defrees, D. J.; Baker, J.; Stewart, J. P.; PopIe, J. A., Gaussian, Inc., Pittsburgh, PA, 1993. 17. Karna, S. P.; Talapatra, G. B.; Prasad, P. N., J. Chem.Phys., 95,5873 (1991). 94 Appendix A Elements Evaluation of Matrix To illustrate the evaluation of (2.6) and (2.7), consider the butadiene monomer (Figure A.I), where 4 e;UMO = L C~UMOxf = 0.602xl- 0.372X~ - 0.372,Xa + 0.602X~ (A.I) i=l e: 6 OMO = L Ci~OMOxf = 0.602xl + 0.372X~ - 0.372,Xa - 0.602X~ (A.2) i=l and Xl are the four atomic orbitals on monomer p. For electron hops this leads to where 13 is the resonance integral between two adjacent Sp2 carbon centers. Thus (2.3) becomes t = 0.3613. (A.4) For excitation to adjacent sites we obtain (A.5) 95 -0.602 -0.372 0.372 0.602 0.602 LUMO 0.602 0.372 HOMO -0.602 0.372 Figure A.I: The MO of butadiene. The wavefunction of LUMO is 0.602Xl +O.372X2O.372X3 - O.602X4' The wavefunction of HOMO is O.602Xl - O.372X2 - O.372X3 + O.602X4, where Xi are the four atomic orbitals. 96 Similarly (A.6) Thus (2.5) leads to a=V2. (A.7) More generally we can write the coefficient of the outer orbital on OLUMO as C l and on eHOMO as Ch to obtain (A.8) (A.9) Thus V2Cl Ch a = (C? + C~)" (A.I0) a=V2. (A.ll) A simple approximation is Cl ~ l/VM and Ch ~ l/VM where M is the number of atoms (M = 4 for butadiene) so that a~V2. Even if Cl = 2Ch , we obtain a = O.8V2. (A.12) 97 Appendix B Theory Application of Perturbation The ground state energy EO can be decomposed as follows: (B.1) where ~o = Eo (B.2) ~l = (OIWlj) (B.3) ~2 = L 1(0IWI0)1 2 (B.4) #0 Eo - E j ~3 = L (OIWlj)(jIO; 2) (B.5) #0 n-2 ~n = L (OIWjj)(jIO; n - 1) - L ~v(OIO; n - v) j~O (B.6) v=2 Here we use the notation (010; 1) = (lIO; 1) = (lIWIO) ° (B.7) (l =I- 0) (B.8) Eo-El (010; 2) = -~ L (jIO; 1)2 2 J. (lIO; 2) = - L (lIWlj)(IO; 1) - (lIWIO)(OIWIO) j El - Eo (El - Eo)2 (B.9) (l =I- 0) (B.1O) 1 n-l (010; n) = -2 L L (j10; v)(jIO; n - v) v=l j (B.11) 98 (lIO; n) = _ L (lIWlj)(IO; n - 1) + ~ ~v(lIO; n - v) j El - Eo El - Eo v=l (l i= 0). (B.12) Substituting (B.l), (B.2), and (B.3) into (2.19)-(2.21) leads to (B.13) f3zzz = (B.14) 0 8a 2 e4 R6'fJ 2 00 2i /zzzz = V3 G 2i 'fJ (B.15) t 'fJ=- (B.16) L i=O where V Ao = 1 A2 = 11 - 8a 2 ) ~ -5 A4 = 80 - 144a2 + 60a4 ~ 32 A6 = 490 - 1488a2 + 1456a4 - 448a6 ~ -396 A8 = 2730 - 1194a2 + 19260a4 - 13440a6 + 3360a8 ~ 6608 AlO = 14322 - 82776a 2 + 189816a4 - 214720a 6 + 118800a8 - 25344a 10 ) ~ -19934 (B.17) Go = 1 G 2 = 57 - 18a2 ~ 21 G 4 = 1170 - 1104a2 + 224a 4 ~ -142 G 6 = 15403 - 27390a2 + 14760a4 - 2400a 6 ~ 463 G 8 = 157564 - 433488a 2 + 420816a 4 - 168960a6 + 23760a8 ~ 2332 GlO = 1368796 - 5233928a 2 + 7684768a 4 - 5372640a6 + 1777776a 8 - 224224a lO ~ -71860 99 G 12 = 10602592 - 52717056a 2 + 106140480a4 - 1l0176640a6 + 61850880a 8 - 17751552a 10 + 2050048a 12 ~ 1084778. (B.18) In each case we have substituted a ~ 0. The above coefficients are for long chains. If the chain length is small, the coefficients are different. For perturbation theory in which Vn is not constant, we find that the expressions for AD, Go are just functions of t, Vi, the expressions for A 2 , G2 are functions of t, VI, V2 , the expressions for A 4 , G 4 are functions of t, VI, V2 , V3 , etc. When the order of expansion increases, the coefficients in the Hamiltonian matrix involved in the expression expand gradually along the diagnal. Thus we can consider that AD, Go are contributions from charge transfer to the nearest neighbor site, A 2 , G2 are contributions from charge transfer to the second nearest neighbor sites, etc. The convergence length of the polynomial should correspond to the saturation length. In order to get the accurate values of a zz , 'Yzzzz from the polynomial, we must calculate to the N terms where 2N is the saturation length. 100 Appendix C The Evaluation of t and V We can estimate the values of t and V from the bandgap and the conduction bandwidth. The eigenvalues E of the Hamiltonian matrix (2.1) satisfy the difference equation: (C.l) When n is large, Vn approaches the constant value I P - EA. In this case (C.l) becomes a homogeneous linear difference equation 17 with the solutions (C.2) where Ai and A2 are solutions of (C.3) J( (CA) IV - EI < 2t. (C.5) A= V - E ± 2t V - E)2 - 1 2t when This leads to complex conjugate roots (oscillating bn ). These states correspond to the conduction states, leading to the bandwidth of the conduction band B = 4t. (C.6) The upper bound of the conduction band is V + 2t and the lower bound is V - 2t. When n is small, Vn is smaller than V. That will decrease the lower bound V - 2t. Because of the coupling between the ground state and excited state, the ground state will lower a little bit (- 2a~t2), and the excited state will be elevated a little bit 101 (- 2a;t2). Combining these two effects together, the bandgap is approximately Eg = V - 2t (C.7) B = 4t. (C.S) and the bandgap is 102 Appendix D Evaluation of the Temperature Effect Suppose that qo and ¢o are the optimum vibrational and torsional coordinates (at absolute zero temperature). Since the torsional energy E t (¢) and the charge transfer matrix element t(¢) are even functions of ¢ - ¢o, the Taylor expansions of Et(¢) and t(¢) lead to Et(¢) = Et(¢o) + ~Et1(¢ - ¢o? + O((¢ - ¢O)4) (D.1) t(¢) = t(¢o) + ~tl(¢ - ¢O)2 + O((¢ - ¢O)4) (D.2) The vibrational energy becomes Ev(q, ¢) = ~V(q) + ~A (q - q~B)2 - ~VV2(q) + 4t2(¢) = ~V(q) + ~A (q - q~Br - ~Vr-V-2(-q)-+-4-t-2(-¢O-)-+-4-tl-t(-¢-o)-(¢---¢o-)2-+-O-[(-¢---¢-O-)4) 1 _ 1 1 ( 0)2 - 2V(q) + 2A q - qVB 1 / 2 2 [4tlt( ¢o) (¢ - ¢O)2] - 2V V (q) + 4t ((¢o)) 1 + V2(q) + 4t 2(¢o) 2 + 0 [(¢ - ¢O)4] = ~V(q) + ~A (q - q~Br - ~JV2(q) + 4t2(¢o) _ t1t(¢o)(¢ - ¢O)2 2 2 2 JV2(q) + 4t 2(¢o) + 0 [(¢ - ¢o)4] (D.3) The first three terms describe the ground state energy without torsional contributions. Taking the Taylor expansion of these three terms about q - qo leads to (DA) 103 where qo is the equilibrium position qo = 0 0) 1 ( 0 0) V "21 (qVB + qCT +"2 qVB - qCT VV2 + 4t2(4)o) (D 5) . and (D.6) Thus (D.7) The total energy becomes (D.8) Here (D.9) The high order terms can be ignored since we assume the vibrational energy and rotational barrier are much higher than the thermal energy kT. This leads to 17r-7r d4>p(4), q)exp [- kT E] < p > = Z1 1+ -00 dq 00 104 1 = Z J+oo dq J1r d¢ [p(¢o, qo) + 8q8p (q - qo) + 8¢ 8p (¢ - ¢o)+ -1r -00 [E ] 2 2 2 1 8 p - qO) 2+ -8 (p q - qO)(¢ - ¢O) + --(¢ 18 p --(q - ¢O) 2] exp - - . 2 2 28 q 8q8¢ 28 ¢ kT (D.lO) Evaluating each term in the above equation leads to J+oo dq J1r d¢p( ¢o, qo)exp [- kTE] = p( ¢o, qo) (D.ll) j+oo dq j1r d¢[-(q 8p - qo) + -(¢ 8p - ¢o))exp [ - E] - = 0 (D.12) Z1 -1 Z -00 -1 Z ~ Z /+00 d -00 /1r '" -1r -00 8¢ 8q -1r kT J+oo dq J1r d¢-(q 8p - qo)(¢ - ¢o)exp [ - E] - = 0 8q8¢ kT 2 -00 (D.13) -1r +OOd 1~( )2 -00 q28 q - qo exp 2kT o [_~] _ J qo exp kT - ~ 8 2p ( _)2 q -1r d<p 282q q 2 [).q(q_qO)2] q f~:: dq exp [_ ).Q(i;;i )2] . (D.14) Letting x = V2kT Aq (q - qo) (D.15) the above equation becomes kT 8 2p(qo, ¢o) 2A q 82 q (D.16) Similarly, The average value of p then becomes (D.18) which was used in (5.17).