Nonlinear Optical Materials - American Chemical Society

Chapter 7 Frequency-Dependent Polarizabilities and Hyperpolarizabilities of Polyenes Prakashan Korambath and Henry A. Kurtz 1

Downloaded by CORNELL UNIV on July 30, 2016 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0628.ch007

Department of Chemistry, University of Memphis, Memphis, T N 38152

Polarizabilities (α) and second hyperpolarizabilities (γ) as a function of frequency are calculated for a series of polyenes, H ( C H ) H with n rangingfrom2 to 20, by the TDHF method with AM1 parameterization. For the second hyperpolarizabilities, third harmonic generation, electricfield induced second harmonic generation, and intensity dependent refractive index quantities are calculated. The frequency dependencies are discussed and comparisons made amongst the different γvalues. The saturation behavior of these quantities is also examined and limiting values for αand γper subunit are computed. 2

2

n

Several computational studies have been done to explore the behavior o f the nonlinear optical properties o f polyenes as the length increases. M a n y early papers used a power-law expression (an ) to fit the polarizability and hyperpolarizability (1,2). Such a power-law behavior with a constant exponent is not adequate to describe the limiting behavior o f polyenes. B o t h experimental (3) and theoretical (4) evidence have shown that the polarizability and second hyperpolarizability approach linearity with large numbers o f subunits, i.e. the exponent approaches unity. The behavior is usually indicated by examining the value/subunit -- which approaches a constant at large n. Almost all previous theoretical work examined only the static values o f the polarizability and second hyperpolarizability. The goal o f this study is to examine the behavior o f the frequency dependent quantities. b

Current address: Department of Chemistry, Ohio State University, Columbus, OH 43210

1

0097-6156/96/0628-0133$15.00/0 © 1996 American Chemical Society

Karna and Yeates; Nonlinear Optical Materials ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

134

NONLINEAR OPTICAL MATERIALS

Method Each H ( C 2 H 2 ) H oliogmer, for n = 1 to 20, was first fully optimized using the A M I n

(5) parameter set and then the properties calculated. The frequency dependent optical properties were calculated using the T D H F (6,7) method which has been implemented by us in both M O P A C (8) and G A M E S S (9). polarizability (a),

The properties o f interest are the

first hyperpolarizability (P), and second hyperpolarizability (y).

These quantities can be defined from a series expansion o f the response o f a system to an external electric field (F) as

E(F) = E ( 0 ) - 2 : H F - ^ 2 > « i i - ^ I P V W i


i

F

i

Z

!

F

i.j

J !

i.j.k

-1

2r

H !

i,j,k,l

S H

F F F F i

j

k

I

- •

( 1 )

where the indices i , j , k, and 1 run over the Cartesian components (x, y, and z). One or more o f the external fields are usually frequency dependent (i.e. lasers) and each combination o f different fields leads to slightly different values for the coefficients in Equation 1. This leads to a classification o f the different optical properties o f interest which correspond to different experimental situations. The quantities calculated by our programs are listed in Table I. For a detailed definition o f this quantities, see a text book on nonlinear optics such as the one by B o y d (10). Table I. Quantities Calculated by M O P A C and G A M E S S Polarizability: Frequency Dependent Polarizability

a(-©;©) First Hyperpolarizabilities:

Second Harmonic Generation ( S H G ) Electroptic Pockels Effect ( E O P E ) Optical Rectification (OR)

P(-2CQ;(Q,(D)

p(-©;©,0) P(0;©,-©) Second Hyperpolarizabilities: y(-3©;©,©,©)

Third Harmonic Generation ( T H G )

y(-2©;©,©,0)

Electric Field Induced Second Harmonic ( E F I S H )

y(-©;©,©,-©)

Intensity Dependent Refractive Index (IDRI) or Degenerate Four-Wave M i x i n g ( D F W M ) Optical Kerr Effect ( O K E )

y(-©;©,0,0)

The programs automatically provide all components o f a , P, and most o f y in whatever molecular coordinate system the molecule was input. In order to provide a comparison with other work and to provide a unique set o f data, we present our results as "averaged" values according to the following definitions. CL

= —(

Y xzxz Y xxzz

Y yzyz Y yyzz

Y yxyx Y zzxx

Y zxzx Y zzxx

Y zyzy ) Y zzyy ) }

Values o f the first hyperpolarizability (p) will not be reported in this work as they are all near or equal to zero. Molecules must have an asymmetry i n their electronic distributions to exhibit P or a dipole moment (n). For large oligomers, the above averages are always dominated by a single component along the molecular axis, i.e. report these components.

or Yxxxx>

m

& other works often only

In the limit when all other components can be ignored,our

values can be related to the others by a = 1/2

and y = 1/5 Yxxxx-

Polarizability A plot o f the calculated a(-co;G)) versus co for each oligomer is given in Figure 1. These results show the following general trends — 1) for a given oligomer a increases (slowly at first and then more rapidly) until a pole (infinity) is reached which corresponds to an excitation energy; 2) at a given frequency, as the oligomer length increases, a increases; and 3) as the oligomer length increases, the excitation energy moves lower. The lowering excitation energies approach a limiting value corresponding to the band gap o f the ideal polyacetylene polymer. F r o m A M I optimized structures this value is approximately 2.0 e V and is in reasonable agreement with the experimental values (11). The main reason for this agreement is the very good geometries provided by the A M I parameterization for these systems. Bredas and co-workers (12) have shown that the optical properties o f these conjugated systems are very sensitive to the bond alternation and the A M I parameterization works very well for polyenes. In order to demonstrate the saturation effect for a , it is more convenient to look at the value per subunit, shown in Figure 2. In our work, this quantity is defined as ct/sub(n) = a(n) - a ( n - l ) . It is often defined by other workers as a(n)/n. These definitions are identical in the limit o f very large n but the incremental definition used here shows more rapid convergence to the limiting value and is a better approximation to the appropriate numerical derivative. A l l the curves for frequencies below the lowest limiting excitation energy are clearly approaching a constant at large n. A l l the curves for frequencies above the excitation energy diverge as n increases.



136


0

1

2

3

4

Energy (eV) Figure 1. Calculated polarizabilities as a function of photon energy (frequency) for polyene oligomers H ( C 2 H 2 ) H with n from 2 (lowest curve) to 21 (highest curve). n


5

7.

KORAMBATH & KURTZ Polarizabilities

& Hyperpolarizabilities

of Polyenes

137

F o r the below excitation energy curves, the limiting a/sub values have been estimated by others by fitting log(oc/sub), or ct/sub, to a polynomial in 1/n (13,14), a b e d ^v subunit - L—. ^- )J = a 7+n- "+ 7— n " 7 ' s

1

y(EFISH) > y(IDRI) as noted by others (6). A s with the polarizability, the saturation behavior is demonstrated by examining y/sub (defined as y(n) - y(n-l)). Figure 4 shows the behavior o f these values for the different types o f y. In this figure are all the curves that show convergence with increasing length plus one curve to illustrate divergent behavior. The curves that show convergence are those with energies less than the appropriate pole for the infinite (polymer) system, i.e. 1/3 the band gap for T H G and 1/2 the band gap for E F I S H and IDRJ. A t 20 subunits (40 carbons) the y/subunit values do not show complete saturation. Those values at 0.0 e V are essentially converged n=20 but as the energy gets closer to the pole it is clear that the y/subunit converges for much larger systems (n>20). This, like other theoretical works, is not in accord with the recent experimental estimate that the static (0.0 eV) g does not show saturation until nearn=120(3 l. >

The data shown in Figure 4 is used to find the limiting values o f y/subunit by fits to equation 7 in the same manner as the polarizability fits. The limiting values obtained from y/sub values are much less reliable than a/sub as the data are much further from saturation. Data in Table V I I gives the results o f our extrapolation predictions, again using the last 9 points in each curve.

Table V I : y/subunit limiting values IDRI EFISH Energy T H G 1.212 0.0 1.213 1.211 1.274 1.304 0.25 1.411 1.479 0.50 1.674 2.525 0.625 4.880 1.949 0.75 2.751 3.001 1.00 8.275

Conclusions These semiempirical calculations clearly show the proper saturation behavior of a and y for polyenes o f increasing length. These A M I results also agree quite well with the ab initio results (13) but further work needs to be done to understand the descrepancy with the experimental estimates. Acknowledgments Support for this work is gratefully acknowledged from the A i r Force Office Scientific Research (AFOSR-90-0010).


of

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References 1. De Melo, C. P.; Silbey, R. Chem. Phys. Lett. 1987, 140, 537 2.Samuel, I. D. W.; Ledoux, I.; Dhenaut, C.; Zyss, J.; Fox, H. H.; Schrock, R. R.; Silbey, R. J. Science 1994, 265, 1070. 3. Etemad, S.; Heeger, A. J.; MacDiarmid, A. G. Ann. Rev. Phys. Chem. 1982, 33, 443. 4. H. A. Kurtz, Int. J. Quantum Chemistry Symp. 1990, 24, 791. B. Champagne, J. G. Fripiat, and J.-M. Andre, J. Chem. Phys. 1992, 96, 8330. E. F. Archibong and A. J. Thakkar, J. Chem. Phys. 1993, 98, 8324. 5. Dewar, M. J. S.; Zoebisch, E. G.; Healy, E. F.; Stewart, J. J. P. J. Am. Chem. Soc. 1985, 107, 3902. 6. Sekino, H; Barlett, R. J. J. Chem. Phys. 1986, 85, 976. 7. Karna, S. P.; Dupuis, M. J. Comp. Chem. 1991, 12, 487. 8. The TDHF capabilities have been released in J. J. P. Stewart's MOPAC 93 from Fujitsu Ltd and MOPAC 7fromQCPE. The work in this paper is based on our own modified version of MOPAC 6. 9. Schmidt, M.W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S. J.; Windus, T.L.; Dupuis, M.; Montgomergy, J. A. J.Comput.Chem. 1993, 14, 1347 10. Boyd, R. W. Nonlinear Optics; Academic Press: San Diego, 1992. 11. Heflin, J. R.; Wong, K. Y.; Zamani-Khamiri, O.; Garito, A. F. Phys. Rev. B. 1988, 38, 1573. 12. Meyers, F.; Marder, S. R.; Pierce, B. M.; Brédas, J. L. J. Am. Chem. Soc. 1994, 116, 10703. 13. Hurst, Graham; Dupuis, M.; Clementi, E. J. Chem. Phys. 1988, 89, 385. 14. Kirtman, B. Chem. Phys. Lett. 1988, 143, 81; Kirtman. B. Int. J. Quantum Chem. 1992, 43, 147. RECEIVED December 29, 1995


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