- Email: [email protected]
Molecular length dependence of optical properties of hydrocarbon oligomers
20 March 1998
Chemical Physics Letters 285 Ž1998. 160–163
Molecular length dependence of optical properties of hydrocarbon oligomers Yi Luo a , Patrick Norman b, Kenneth Ruud
b,1
˚ , Hans Agren
b
a
b
Department of Chemistry, UniÕersity of Copenhagen, DK-2100 Copenhagen Ø, Denmark Institute of Physics and Measurement Technology, Linkoping UniÕersity, S-58183 Linkoping, Sweden ¨ ¨ Received 5 December 1997; in final form 9 January 1998
Abstract It is demonstrated that common power law dependencies for the energy gap, the polarizability, and the hyperpolarizability can be obtained for shorter hydrocarbon oligomers if the molecular length is used as the basic structural parameter. q 1998 Elsevier Science B.V.
It has for a long time been recognized that conjugated molecules play an important role in the search for materials with large nonlinear optical ŽNLO. responses, and that modern theoretical techniques can be helpful in this search. Computations can pinpoint the microscopic origin of NLO properties and also reveal special structure to property relations. The Žhyper-.polarizabilities of conjugated polymers increase exponentially with increasing size of the oligomers until a linear dependence, or a saturation region, is reached. This has been confirmed by a number of experimental and theoretical works, see the review of Bredas ´ et al. w1x and references therein for a recent account. In the case of trans-polyenes most theoretical studies have been devoted to chain saturation employing both semi-empirical w1x and ab initio w2,3x methods. The chain length is often replaced by the number of repeat units or the number
1 Permanent address: Department of Chemistry, University of Oslo, P.O. Box 1033, Blinder, N-0315 Oslo, Norway.
of p electrons; for the investigation of a single polymeric species this is of minor concern since the two parameters are proportional to one another. However, those parameters cannot provide a direct comparison among different systems as shown by Heflin et al. w4x in a study on hyperpolarizabilities of trans- and cis-polyenes, and later in a study of the hyperpolarizabilities of trans- and diphenyl-polyenes w5x. Instead, the investigations of these species indicated that the molecular length is the overall determining factor for ground-state properties. This is in agreement with experimental observations for lowmolecular-weight compounds, which are considered to be useful as media for optical fibers when dissolved in organic solvents or matrix polymers w6x. In this paper we put the observations made for polyenes on firmer ground by comparing several types of hydrocarbon oligomers and explore the molecular length dependence of their optical gaps Ž Eg ., polarizabilities Ž a ., and hyperpolarizabilities Žg .. This is accomplished by using a well-tested level of electronic structure theory – the time-depen-
0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 0 2 6 - 8
Y. Luo et al.r Chemical Physics Letters 285 (1998) 160–163
dent self-consistent field method ŽTDSCF., also denoted the Random Phase Approximation ŽRPA. – and a common, and previously tested w5,7x, basis set for the wavefunctions. We employ the response theory formalism for obtaining molecular properties up to fourth order as presented in Refs. w8,9x. No specific assumption is made about the perturbing fields and the optical gaps, polarizabilities, and hyperpolarizabilities are obtained fully analytically. The method is matrix-direct and integral-direct, and implemented on parallel computers, enabling iterative solutions of the RPA equations for larger systems. It is well known that the longitudinal components of the polarizability and the hyperpolarizability dominate the optical responses w1x. The present paper is therefore confined to results for these longitudinal components. Several types of hydrocarbon oligomers are considered, namely trans-, cis-, and diphenyl-polyenes, polyacene, vinylene, and phenylene. Original calculations for trans-polyene, polyacene, and some of the short cis- and diphenyl-polyenes were presented in Refs. w5,9,10x. All molecular geometries are constructed based on building blocks of identical benzene and ˚ in the ethylene units. The C–C distances are 1.398 A ˚ for the single benzene units and 1.366 and 1.443 A and double bonds of the ethylene units. We consider ˚ i.e., well below the molecular lengths up to 20 A, saturation lengths for the quantities investigated w3,10x. The choice of a basis set for the hyperpolarizability calculations is crucial, and we take advantage of the extensive basis set investigations that have been performed for many organic molecules. Accurate RPA values for hyperpolarizabilities have been obtained using a 4-31G basis set w11,12x with additional polarizing and diffuse p- and d-functions on the carbons, both with an exponent of 0.05, see for instance Ref. w13x. The contractions for this basis set then become w8s5p1dr3s3p1dx and w4sr2sx for carbon and hydrogen, respectively. All calculations have been performed with the dalton quantum chemistry code w14x, and were carried out with 98% slave efficiency with up to 64 processors used in the calculations on different parallel platforms, including an IBMrSP2 cluster, a Cray-T3E and a Cray-Origin 2000 MPP machine. The number of p electrons and molecular length
161
dependences of the optical energy gap Ž Eg ., polarizability Ž a . and hyperpolarizability Žg . for all investigated species are illustrated in Fig. 1. The dependence on the number of p electrons dependence for all compounds roughly falls into three groups. The first group is the one containing only ethylene units, the trans- and cis-polyenes. The second contains the molecules constructed from benzene rings alone, the polyacenes and the phenylenes. The third is for molecules with a mixture of ethylene units and benzene rings, the diphenyl-polyenes and the vinylenes. As seen from the figure, it is difficult to use results from one class to predict those for another, and we conclude that the number of p electrons is not a good parameter for providing a unified description for all compounds. In contrast, when the molecular head-to-tail length L is adopted as an overall parameter, a unified picture for Eg , a and g appears. For all compounds considered here, a common power law length dependence is obtained: Eg s 10.6 Ly0 .38 " 0.02
a s 6.06 L1.65 " 0.07 g s 14.83 L4.19 " 0.10 Ž 1. By fitting the experimental data for conjugated oligothiophenes measured by Zhao et al. w16x, the power law exponents are found to be y0.3, 1.69, and 4.05, respectively. Our estimated exponents are thus in good agreement with these experimental results. The length dependence of the polarizability can be rationalized by some crude approximations: Let us first consider the sum-over-state expression for the long inplane component of the polarizability ²0 < x < n: ² n < x <0: ax xs2 Ý Ž 2. En n/0 where En is the excitation energy of the state < n:. By far the most intense transition in the one-photon electronic absorption spectra of polyenes is determined by the first optically allowed state 11 B u . The long in-plane component of the polarizability can therefore be well approximated by a two-state model, i.e. including only the ground and first excited state 11 B u . We can then rewrite Eq. Ž2. as 2 f 01 2 a x x f Ž ²0 < x <1: . s 43 2 Ž 3. Eg Eg
162
Y. Luo et al.r Chemical Physics Letters 285 (1998) 160–163
Fig. 1. Optical gap Ž Eg ., static polarizability Ž a . and hyperpolarizability Žg . versus number of p electrons Žleft. and molecular length Žright. for polyacenes Ždiamond., cis-polyenes Žsquare., diphenyl-polyenes Žcircle., phenylenes Žtriangle., trans-polyenes Žfilled square. and vinylenes Žstar.. The dotted lines show the obtained power law dependence, cf. Eq. Ž1..
where f 01 is the oscillator strength for the 11 B u state. We have shown in our previous study on polyenes w15x that there is an almost perfect linear relation between the oscillator strength for the 11 B u state and the length of the molecule. With the exponent for gap energy taken from the theoretical estimate above, we obtain a power law exponent for the
polarizability of 1.76, which is in good agreement with the theoretical estimate given above. The present results should be qualified with respect to some conditions. The first is that the calculations were carried out for molecular lengths that are well below the saturation length for the Žhyper.polarizabilities, and for which the optical gap has
Y. Luo et al.r Chemical Physics Letters 285 (1998) 160–163
not yet converged to its final Žand finite. value. Saturation corresponds to the bulk property after which an added length Žor volume. element only can make an additional linear contribution to the property, and after which the Žhyper-.polarizability thus scales with the length Žvolume. of the system. The second condition is that idealized geometries are used throughout and that the optical response can be enhanced by using the optimized geometries w10x. The oligomer convergence to the saturation length is indeed dependent on the use of optimized geometries as found for polyenes w3x and diphenylpolyenes w10x. However, the hyperpolarizability saturation lengths are still three to four times longer than the largest oligomers here investigated. A third condition is that calculations are performed in the static limit and if they were to be compared with experiment, dispersion should be taken into account. The dispersion can be assumed normal for reasonably small frequencies, as also predicted in a few cases w17x, and the present results should therefore be usable through simple extrapolation of the frequency. The direct vibrational contributions, which can be large in the static limit, can be neglected for the ESHG and THG processes at finite frequencies much larger than the vibrational frequencies w18x. Taking the abovementioned conditions into consideration, we believe that the predicted power law dependence of the optical energy gap versus molecular length can be of importance in applications. It means that a power law dependence for molecular properties versus the optical energy gap can be anticipated and that one can predict the polarizabilities by measuring the optical energy gap Žor the so-called absorption edge.. The power law exponent of y11.02 obtained from Eq. Ž1. for g versus Eg is in close agreement with experiment for some low molecular weight compounds w6x.
Acknowledgements Y. Luo appreciates the financial support provided by the Hellmuth Hertz Foundation. This work has
163
received support from The Research Council of Norway ŽProgramme for Supercomputing. and NSC at Linkoping University through grants of computer ¨ time.
References w1x J.L. Bredas, C. Adant, P. Tackx, A. Persoons, Chem. Rev. 94 ´ Ž1994. 243. w2x G.J.B. Hurst, M. Dupuis, E. Clementi, J. Chem. Phys. 89 Ž1988. 385. w3x D. Lu, B. Marten, M. Ringnalda, R.A. Friesner, W.A. Goddard III, Chem. Phys. Lett. 257 Ž1996. 224. w4x J.R. Heflin, K.Y. Wong, O. Zamani-Khamiri, A.F. Garito, Phys. Rev. B 38 Ž1988. 1537. ˚ w5x Y. Luo, P. Norman, D. Jonsson, H. Agren, Mol. Phys. 89 Ž1996. 1409. w6x H. Kanbara, H. Kobayashi, T. Kaino, N. Ooba, T. Kurihara, J. Phys. Chem. 98 Ž1994. 12270. w7x S.P. Karna, G.B. Talapatra, W.M.K.P. Wijekoon, P.N. Prasad, Phys. Rev. A 45 Ž1992. 2763. ˚ w8x P. Norman, D. Jonsson, H. Agren, P. Dahle, K. Ruud, T. Helgaker, H. Koch, Chem. Phys. Lett. 253 Ž1996. 1. ˚ w9x D. Jonsson, P. Norman, Y. Luo, H. Agren, J. Chem. Phys. 105 Ž1996. 581. ˚ w10x K. Ruud, P. Norman, D. Jonsson, H. Agren, T. Saue, H.J.Aa. Jensen, P. Dahle, T. Helgaker, to be published. w11x W.J. Hehre, W.A. Lathan, J. Chem. Phys. 56 Ž1972. 5255. w12x R. Ditchfield, W.J. Hehre, J.A. Pople, J. Chem. Phys. 54 Ž1971. 724. w13x J.K. Hurley, H. Lincshitz, A. Treinin, J. Phys. Chem. 92 Ž1988. 5151. w14x T. Helgaker, H.J.Aa. Jensen, P. Jørgensen, J. Olsen, K. ˚ Ruud, H. Agren, T. Andersen, K.L. Bak, V. Bakken, O. Christiansen, P. Dahle, E.K. Dalskov, T. Enevoldsen, H. Heiberg, H. Hettema D. Jonsson, S. Kirpekar, R. Kobayashi, H. Koch, K.V. Mikkelsen, P. Norman, M.J. Packer, T. Saue, P.R. Taylor, O. Vahtras, Dalton, an ab initio electronic s tru c tu re p ro g ra m , R e le a s e 1 0 , 1 9 9 7 ; s e e http:rrwww.kjemi.uio.norsoftwarerdaltonrdalton.html. ˚ w15x Y. Luo, H. Agren, K. Koch, P. Jørgensen, T. Helgaker, Phys. Rev. B 51 Ž1995. 14949. w16x M. Zhao, B. Singh, P.N. Prasad, J. Chem. Phys. 89 Ž1988. 5535. ˚ w17x P. Norman, Y. Luo, D. Jonsson, H. Agren, J. Chem. Phys. 106 Ž1997. 1827. w18x B. Kirtman, B. Champagne, J.-M. Andre, ´ J. Chem. Phys. 104 Ž1996. 4125.
Chemical Physics Letters 285 Ž1998. 160–163
Molecular length dependence of optical properties of hydrocarbon oligomers Yi Luo a , Patrick Norman b, Kenneth Ruud
b,1
˚ , Hans Agren
b
a
b
Department of Chemistry, UniÕersity of Copenhagen, DK-2100 Copenhagen Ø, Denmark Institute of Physics and Measurement Technology, Linkoping UniÕersity, S-58183 Linkoping, Sweden ¨ ¨ Received 5 December 1997; in final form 9 January 1998
Abstract It is demonstrated that common power law dependencies for the energy gap, the polarizability, and the hyperpolarizability can be obtained for shorter hydrocarbon oligomers if the molecular length is used as the basic structural parameter. q 1998 Elsevier Science B.V.
It has for a long time been recognized that conjugated molecules play an important role in the search for materials with large nonlinear optical ŽNLO. responses, and that modern theoretical techniques can be helpful in this search. Computations can pinpoint the microscopic origin of NLO properties and also reveal special structure to property relations. The Žhyper-.polarizabilities of conjugated polymers increase exponentially with increasing size of the oligomers until a linear dependence, or a saturation region, is reached. This has been confirmed by a number of experimental and theoretical works, see the review of Bredas ´ et al. w1x and references therein for a recent account. In the case of trans-polyenes most theoretical studies have been devoted to chain saturation employing both semi-empirical w1x and ab initio w2,3x methods. The chain length is often replaced by the number of repeat units or the number
1 Permanent address: Department of Chemistry, University of Oslo, P.O. Box 1033, Blinder, N-0315 Oslo, Norway.
of p electrons; for the investigation of a single polymeric species this is of minor concern since the two parameters are proportional to one another. However, those parameters cannot provide a direct comparison among different systems as shown by Heflin et al. w4x in a study on hyperpolarizabilities of trans- and cis-polyenes, and later in a study of the hyperpolarizabilities of trans- and diphenyl-polyenes w5x. Instead, the investigations of these species indicated that the molecular length is the overall determining factor for ground-state properties. This is in agreement with experimental observations for lowmolecular-weight compounds, which are considered to be useful as media for optical fibers when dissolved in organic solvents or matrix polymers w6x. In this paper we put the observations made for polyenes on firmer ground by comparing several types of hydrocarbon oligomers and explore the molecular length dependence of their optical gaps Ž Eg ., polarizabilities Ž a ., and hyperpolarizabilities Žg .. This is accomplished by using a well-tested level of electronic structure theory – the time-depen-
0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 0 2 6 - 8
Y. Luo et al.r Chemical Physics Letters 285 (1998) 160–163
dent self-consistent field method ŽTDSCF., also denoted the Random Phase Approximation ŽRPA. – and a common, and previously tested w5,7x, basis set for the wavefunctions. We employ the response theory formalism for obtaining molecular properties up to fourth order as presented in Refs. w8,9x. No specific assumption is made about the perturbing fields and the optical gaps, polarizabilities, and hyperpolarizabilities are obtained fully analytically. The method is matrix-direct and integral-direct, and implemented on parallel computers, enabling iterative solutions of the RPA equations for larger systems. It is well known that the longitudinal components of the polarizability and the hyperpolarizability dominate the optical responses w1x. The present paper is therefore confined to results for these longitudinal components. Several types of hydrocarbon oligomers are considered, namely trans-, cis-, and diphenyl-polyenes, polyacene, vinylene, and phenylene. Original calculations for trans-polyene, polyacene, and some of the short cis- and diphenyl-polyenes were presented in Refs. w5,9,10x. All molecular geometries are constructed based on building blocks of identical benzene and ˚ in the ethylene units. The C–C distances are 1.398 A ˚ for the single benzene units and 1.366 and 1.443 A and double bonds of the ethylene units. We consider ˚ i.e., well below the molecular lengths up to 20 A, saturation lengths for the quantities investigated w3,10x. The choice of a basis set for the hyperpolarizability calculations is crucial, and we take advantage of the extensive basis set investigations that have been performed for many organic molecules. Accurate RPA values for hyperpolarizabilities have been obtained using a 4-31G basis set w11,12x with additional polarizing and diffuse p- and d-functions on the carbons, both with an exponent of 0.05, see for instance Ref. w13x. The contractions for this basis set then become w8s5p1dr3s3p1dx and w4sr2sx for carbon and hydrogen, respectively. All calculations have been performed with the dalton quantum chemistry code w14x, and were carried out with 98% slave efficiency with up to 64 processors used in the calculations on different parallel platforms, including an IBMrSP2 cluster, a Cray-T3E and a Cray-Origin 2000 MPP machine. The number of p electrons and molecular length
161
dependences of the optical energy gap Ž Eg ., polarizability Ž a . and hyperpolarizability Žg . for all investigated species are illustrated in Fig. 1. The dependence on the number of p electrons dependence for all compounds roughly falls into three groups. The first group is the one containing only ethylene units, the trans- and cis-polyenes. The second contains the molecules constructed from benzene rings alone, the polyacenes and the phenylenes. The third is for molecules with a mixture of ethylene units and benzene rings, the diphenyl-polyenes and the vinylenes. As seen from the figure, it is difficult to use results from one class to predict those for another, and we conclude that the number of p electrons is not a good parameter for providing a unified description for all compounds. In contrast, when the molecular head-to-tail length L is adopted as an overall parameter, a unified picture for Eg , a and g appears. For all compounds considered here, a common power law length dependence is obtained: Eg s 10.6 Ly0 .38 " 0.02
a s 6.06 L1.65 " 0.07 g s 14.83 L4.19 " 0.10 Ž 1. By fitting the experimental data for conjugated oligothiophenes measured by Zhao et al. w16x, the power law exponents are found to be y0.3, 1.69, and 4.05, respectively. Our estimated exponents are thus in good agreement with these experimental results. The length dependence of the polarizability can be rationalized by some crude approximations: Let us first consider the sum-over-state expression for the long inplane component of the polarizability ²0 < x < n: ² n < x <0: ax xs2 Ý Ž 2. En n/0 where En is the excitation energy of the state < n:. By far the most intense transition in the one-photon electronic absorption spectra of polyenes is determined by the first optically allowed state 11 B u . The long in-plane component of the polarizability can therefore be well approximated by a two-state model, i.e. including only the ground and first excited state 11 B u . We can then rewrite Eq. Ž2. as 2 f 01 2 a x x f Ž ²0 < x <1: . s 43 2 Ž 3. Eg Eg
162
Y. Luo et al.r Chemical Physics Letters 285 (1998) 160–163
Fig. 1. Optical gap Ž Eg ., static polarizability Ž a . and hyperpolarizability Žg . versus number of p electrons Žleft. and molecular length Žright. for polyacenes Ždiamond., cis-polyenes Žsquare., diphenyl-polyenes Žcircle., phenylenes Žtriangle., trans-polyenes Žfilled square. and vinylenes Žstar.. The dotted lines show the obtained power law dependence, cf. Eq. Ž1..
where f 01 is the oscillator strength for the 11 B u state. We have shown in our previous study on polyenes w15x that there is an almost perfect linear relation between the oscillator strength for the 11 B u state and the length of the molecule. With the exponent for gap energy taken from the theoretical estimate above, we obtain a power law exponent for the
polarizability of 1.76, which is in good agreement with the theoretical estimate given above. The present results should be qualified with respect to some conditions. The first is that the calculations were carried out for molecular lengths that are well below the saturation length for the Žhyper.polarizabilities, and for which the optical gap has
Y. Luo et al.r Chemical Physics Letters 285 (1998) 160–163
not yet converged to its final Žand finite. value. Saturation corresponds to the bulk property after which an added length Žor volume. element only can make an additional linear contribution to the property, and after which the Žhyper-.polarizability thus scales with the length Žvolume. of the system. The second condition is that idealized geometries are used throughout and that the optical response can be enhanced by using the optimized geometries w10x. The oligomer convergence to the saturation length is indeed dependent on the use of optimized geometries as found for polyenes w3x and diphenylpolyenes w10x. However, the hyperpolarizability saturation lengths are still three to four times longer than the largest oligomers here investigated. A third condition is that calculations are performed in the static limit and if they were to be compared with experiment, dispersion should be taken into account. The dispersion can be assumed normal for reasonably small frequencies, as also predicted in a few cases w17x, and the present results should therefore be usable through simple extrapolation of the frequency. The direct vibrational contributions, which can be large in the static limit, can be neglected for the ESHG and THG processes at finite frequencies much larger than the vibrational frequencies w18x. Taking the abovementioned conditions into consideration, we believe that the predicted power law dependence of the optical energy gap versus molecular length can be of importance in applications. It means that a power law dependence for molecular properties versus the optical energy gap can be anticipated and that one can predict the polarizabilities by measuring the optical energy gap Žor the so-called absorption edge.. The power law exponent of y11.02 obtained from Eq. Ž1. for g versus Eg is in close agreement with experiment for some low molecular weight compounds w6x.
Acknowledgements Y. Luo appreciates the financial support provided by the Hellmuth Hertz Foundation. This work has
163
received support from The Research Council of Norway ŽProgramme for Supercomputing. and NSC at Linkoping University through grants of computer ¨ time.
References w1x J.L. Bredas, C. Adant, P. Tackx, A. Persoons, Chem. Rev. 94 ´ Ž1994. 243. w2x G.J.B. Hurst, M. Dupuis, E. Clementi, J. Chem. Phys. 89 Ž1988. 385. w3x D. Lu, B. Marten, M. Ringnalda, R.A. Friesner, W.A. Goddard III, Chem. Phys. Lett. 257 Ž1996. 224. w4x J.R. Heflin, K.Y. Wong, O. Zamani-Khamiri, A.F. Garito, Phys. Rev. B 38 Ž1988. 1537. ˚ w5x Y. Luo, P. Norman, D. Jonsson, H. Agren, Mol. Phys. 89 Ž1996. 1409. w6x H. Kanbara, H. Kobayashi, T. Kaino, N. Ooba, T. Kurihara, J. Phys. Chem. 98 Ž1994. 12270. w7x S.P. Karna, G.B. Talapatra, W.M.K.P. Wijekoon, P.N. Prasad, Phys. Rev. A 45 Ž1992. 2763. ˚ w8x P. Norman, D. Jonsson, H. Agren, P. Dahle, K. Ruud, T. Helgaker, H. Koch, Chem. Phys. Lett. 253 Ž1996. 1. ˚ w9x D. Jonsson, P. Norman, Y. Luo, H. Agren, J. Chem. Phys. 105 Ž1996. 581. ˚ w10x K. Ruud, P. Norman, D. Jonsson, H. Agren, T. Saue, H.J.Aa. Jensen, P. Dahle, T. Helgaker, to be published. w11x W.J. Hehre, W.A. Lathan, J. Chem. Phys. 56 Ž1972. 5255. w12x R. Ditchfield, W.J. Hehre, J.A. Pople, J. Chem. Phys. 54 Ž1971. 724. w13x J.K. Hurley, H. Lincshitz, A. Treinin, J. Phys. Chem. 92 Ž1988. 5151. w14x T. Helgaker, H.J.Aa. Jensen, P. Jørgensen, J. Olsen, K. ˚ Ruud, H. Agren, T. Andersen, K.L. Bak, V. Bakken, O. Christiansen, P. Dahle, E.K. Dalskov, T. Enevoldsen, H. Heiberg, H. Hettema D. Jonsson, S. Kirpekar, R. Kobayashi, H. Koch, K.V. Mikkelsen, P. Norman, M.J. Packer, T. Saue, P.R. Taylor, O. Vahtras, Dalton, an ab initio electronic s tru c tu re p ro g ra m , R e le a s e 1 0 , 1 9 9 7 ; s e e http:rrwww.kjemi.uio.norsoftwarerdaltonrdalton.html. ˚ w15x Y. Luo, H. Agren, K. Koch, P. Jørgensen, T. Helgaker, Phys. Rev. B 51 Ž1995. 14949. w16x M. Zhao, B. Singh, P.N. Prasad, J. Chem. Phys. 89 Ž1988. 5535. ˚ w17x P. Norman, Y. Luo, D. Jonsson, H. Agren, J. Chem. Phys. 106 Ž1997. 1827. w18x B. Kirtman, B. Champagne, J.-M. Andre, ´ J. Chem. Phys. 104 Ž1996. 4125.