J. Phys. Chem. 1992, 96,4325-4336
4325
Nonlinear Optical Phenomena in Conjugated Organic Chromophores. Theoretical Investigations via a ?r-Electron Formalism DeQuan Li,+ Tobin J. Marks,* and Mark A. Ratner* Department of Chemistry and the Materials Research Center, Northwestern University, Euanston. Illinois 60208 (Received: January 14, 1991; In Final Form: August 14, 1991)
This contribution explores the use of perturbation theory and the chemically oriented, computationally efficient PPP *-electron model Hamiltonian to describe the first three molecular hyperpolarizabilities and their interrelationships. Relationships between various nonlinear phenomena are treated as degeneracy factors, and a more general definition emerges from this analysis. Derivations of formulas describing second- to fourth-order susceptibilitiesare given for the monoexcited CI (MECI) and doubly excited CI (DECI) perturbation approaches, and numerical results are in good to excellent agreement with experimental values where available for a variety of second- and third-order nonlinear phenomena (second harmonic generation, frequency mixing, linear electrooptic effect, optical rectification, third harmonic generation, dc-induced second harmonic generation, degenerate four-wave mixing, Kerr effect). Additional analysis of chromophore architecturenonlinear optical relationships reveals that the observable second-order susceptibility, B,, is quite sensitive to intramolecular charge transfer and molecular distortions. This becomes even more pronounced in fourth-order phenomena since dinit is related to charge transfer in a cubic manner rather than linearly as in the case of Biip Third-order susceptibilities, ~ , , k l ,function exactly as uijin a centrosymmetric system and are dominated by the size and shape of the electron cloud. However, in a noncentrosymmetric environment, yijklhas a more complex behavior than either aijor Bijk, and charge-transferexcitations can make an important contribution.
I. Introduction Materials exhibiting highly nonlinear optical (NLO) response are currently of great scientific and technological interest for applications as diverse as optical telecommunications, signal processing, data storage, image reconstruction, logic technologies, sensor protection, and optical computing. Traditionally, such materials have been nonmolecular, extended inorganic solids such as LiNb03, KH2P04,and KTiOP04. However, recent results suggest that conjugated a-electron molecular and polymeric NLO materials hold great promise.* The attraction vis-&vis the inorganics lies in the inherent tailorability of molecular structures, greatly enhanced NLO response over a wide frequency range, ultrafast response times, low dielectricconstants, high laser damage thresholds, and superior film-forming/processing characteristk2 The design and synthesis of new molecular and polymeric NLO materials with optimized properties currently offer a major scientific challenge. Since the actual NLO properties of such materials depend heavily upon the characteristics of constituent chromophoric building blocks, it would be of great interest to develop straightforward,computationally efficient, and chemically oriented quantum-chemical descriptions of molecular NLO characteristics. Such an approach would be invaluable in better understanding the origin of molecular NLO characteristics and would be an asset in chromophore design. A number of theoretical approaches to calculating molecular optical nonlinearities have appeared in the literature, with two general types of model electronic structure Hamiltonians being used. The first class comprises more rigorous ab initio3 models, while the second is based on semiempirical modeh4v5oriented to understanding architectureperformance relationships and to guiding experimental efforts. The ab initio models require no semiempiricalparameters but do require extensive basis sets and, correspondingly, substantial computational efforts. For nonlinear optical properties of conjugated a systems, as for their linear optical properties? simple *-electron semiempirical models, especially the Pariser-Parr-Pople (PPP) model,’ have proven to be both efficient and highly ac~ u r a t e .In ~ this paper, we employ the PPP model to investigate systematically structureperformance characteristics for the first three nonlinear optical susceptibilities of some selected r organic chromophores. The actual calculation of response properties of molecules can be carried out (ignoring ground-state correlation effects beyond Hartree-Fock) using either coupled8 or uncoupledg HartreeFock Present address: Inorganic and Structural Chemistry (INC-4), Los Alamos National Laboratory, Los Alamos, NM 87545.
methods. In coupled Hartree-Fock, one includes the external applied field in the electronic structure calculation and then calculates the response properties by numerical or analytic differentiation of the energy or the polarizability with respect to the applied fields. This scheme is straightforward at zero frequency and has been applied quite extensively (indeed, &jk properties are calculated routinely in the Gaussian 88 and Gaussian 90 program packages). It has recently been extended to finite frequencySgd In uncoupled HartretFock, or sum-over-states approaches, one uses perturbation theory to express the nonlinear susceptibilities in terms of sums over excited states and then truncates these sums in accordance with the use of finite basis sets and a HartreeFock ground state. These methods offer easy computation of fre(1) (a) shen, Y. R. The Principles of Nonlinear Optics; Wiley: New York, 1984. (b) Zernike, F.; Midwinter, J. E. Applied Nonlinear Optics; Wiley: New York, 1973. (c) Abraham, E.; Seaton, C. F.; Smith, S.D. Sci. Am. 1984, 250, 85-93. (2) (a) Prasad, P. N.; Williams, D. J. Introduction to Nonlinear Optical Effects in Molecules and Polymers; Wiley: New York,1990. (b) Khananan, G., Ed.Nonlinear Optical Properties of Organic Materials. Proc. SPIE.Int. Soc. Opt. Eng. 1990,1147. (c) Messier, J.; Kajar, F.; Prasad, P.; Ulrich, D., Eds. Nonlinear Optical Effects in Organic Polymers; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1989. (d) Khanarian, G., Ed. Nonlinear Optical Properties of Organic Materials. Proc. SPIE-Int. Soc. Opt. Eng. 1989,971. (e) Heeger, A. J.; Orenstein, J.; Ulnch. D. R., Eds.Nonlinear Optical Propertiesof Polymers. Mater. Res. Soc. Symp. Proc. 1988.109. (f) Chemla, D. S.; Zyss, J., Eds. Nonlinear Optical Properties of Organic Molecules and Crystals, Academic Press: New York, 1987; Vols. 1.2. (g) Zyss, J. J. Mol. Electron. 1985,1,25-56. (h) Williams, D. J. Angew. Chem., In?. Ed. Engl. 1984, 23, 690-703. (3) (a) %kino, H.; Bartlett, R. J. J. Chem. Phys. 1986,85,976-989; 1986, 84,2726-2733; 1991,94,3665-3669. (b) Inoue, T.; Iwata, S. Chem. Phys. Lett. 1990,167 (e,566-570. (c) Hurst, G. J. B.;Dupuis. M.; Clementi, E. J . Chem. Phys. 1988,89 (l), 385-395. (d) Rice, J. E.; Am=, R. D.; Colwell, S. M.; Hardy, N. C.; Sanz, J. J . Chem. Phys. 1990, 93, 8828-8840. (4) (a) Li, D.; Ratner, M. A.; Marks, T. J. J . Am Chem. Soc. 1988,110, 1707-1715. (b) Li, D.; Marks, T. J.; Ratner, M. A. Chem. Phys. Letr. 1986, 131,370-375. (c) Garito, A. F.; Teng, C. C.; Wong, K. Y.; Enmmankhamiri, 0. Mol. Cryst. Liq. Cryst. 1984, 106, 219-258. (d) Dwherty, V.; Pugh, D.; Morley, J. 0. J . Chem. SOC.,Faraday Trans. 2 1985, 81, 1179-1 192 and referencestherein. (e) Morrell, J. A.; Albrecht, A. C. Chem. Phys. Lett. 1979, 61,4640. (f) Dirk, C. W.; Twieg, R. J.; Wagniere, G. J. J . Am. Chem. Soc. 1986, 108, 5387-5395. (5) (a) Oudar, J. L. J . Chem. Phys. 1977,66,2664-2668. (b) Oudar, J. L.; Chemla, D. S.J . Chem. Phys. 1977, 66, 2664-68. (6) Castellan, A.; Michl J. J. Am. Chem. SOC.1978, 100, 6824. (7) (a) Linderberg, J.; bhrn, Y. Propagators in Quantum Chemistry; Academic Press: London, 1974. (b) Murrell, J. N.; Harget, A. J. SemiEmpirical Self-Consistent-Field Molecular Orbital Theory of Molecules; Wiley: London, 1972. (8) (a) Zyss. J.; Berthier, G. J . Chem. Phys. 1982, 77, 3635-3653. (b) Subbaswamy, K. R.; Mahan, G. D. J . Chem. Phys. 1986, 84. 3317-3319. (9) Dykstra, C. E.; Jasien, P. J . Chem. Phys. Lett. 1984, 109, 388-393.
0022-3654/92/2096-4325$03.00/00 1992 American Chemical Society
4326 The Journal of Physical Chemistry, Vol. 96, No. 11, 1992
quency-dependent properties as well as straightforward physical interpretation and understanding; we therefore choose, for our calculations, to use sum-over-states methods with the PPP model Hamiltonian. The fundamental relationship describing the change in molecular dipole moment (polarization) upon interaction with light waves (oscillating external electric field E) can be expressed as Pi(wl)
C a i j ( - w I ~ ~ ) E j (+ ~2)
fl$(Bw)E$k
P,'"
COS 2U?
(4)
This can be contrasted with the more straightforward definition of eq 1. In the case of SHG, we then have
1
Ik
jkl
We present here a unique set of definitions and treat the relationships between different previous definitions as degeneracy factors. For SHG, most theoretical work follows that of Bloembergen (B)" and Ward (W),I2 from which polarizations can be expressed as jk
c6ijk(-wl;w2,03)Ej(02)Ek(w3)+
c jklm
Li et al.
Yijk/(-~1;~2,~3,~4)Ej(~Z)Ek(~3)E/(~4) +
pi(20)
jk
+
Oijk(-2w;w,w)EjEk cos2 wf
(5)
6 i , k / m ( - w I ; ~ Z , ~ 3 r ~ 4 , 0 S ) E j ( W 2 ) E k ( ~ 3 ) ~ / ( ~ 4 ) ~ m ( ~(1 ~)
Here P,(o,) is the polarization at frequency o1induced along the ith molecular axis, a the linear polarizability, 6 the quadratic hyperpolarizability, y the cubic hyperpolarizability, and 6 the quartic hyperpolarizability. The even-order tensors, 6 and 6, vanish in a centrosymmetricenvironmentwhile there is no environmental symmetry which requires all odd-order tensors, a and y, to vanish. The analogous macroscopic polarization arising from an array of molecules is given by eq 2 where x's are macroscopic suscep-
c
JKL
JK
6ijk(-2@;w1w)= 2fl;kw(-h;w,w)
Pi(3w)
+
I ;w2,w3,w4,ws)EJ(wz)EK(w3)Er(W4)E~w~)
+ ...
xl%
d%L
= NhifJfKk6ijk NhifJjKkf"Lffijk/
The electronicorigins of these nonlinear optical effects in highly conjugated media arise mainly from the delocalized r electrons. It is reasonable, therefore, to employ a model A Hamiltonian (PPP) to describe these nonlinear phenomena. Recently we reported a PPP studfib of chromophore architecture-frequency doubling efficiency relationships in a series of organic *-electron chromophores. The results were in favorable agreement with available theoretical and experimental data in the literature. In the present contribution, we redefine the nonlinear susceptibility tensor in a more general way and extend our quantum-chemical analysis to a wide variety of second-, third-, and fourth-order NLO phenomena. A number of these phenomena have not previously been subjected to such analysis. Experimental comparison with our results suggests that the approach (PPP model, sum over states) is useful quantitatively for B properties but is less accurate for 7 properties, especially in systems having substituents with large sigma polarizabilities.
II. Relatiolarhip between Various Nonlinear P b e n m ~ ,
jk/
$;(-30;U,W,w)~??$kE/
Sin 3wt
(6b)
and from eq 1 yijk/(3w;w,w,w)E/EkE/cos3 Of
pi(3W) =
(2)
tibilities. The macroscopic susceptibilities are related to the corresponding molecular terms a, 6, y, 6, etc. by local field corrections due to intermolecular interactions) and molecular number density (N)
(6a)
an additional, frequency-independent term also appears. For third harmonic generation, a degeneracy factor of 4 enters between the two definitions
xf~-wI;wZ,w3)ExwZ)EK(w3) +
Xl~~(-wl;o2,W3,W4)ExW2)EK(W3)EL(W4)
c xJ%d-w JKLM
This definition does not strictly correspond solely to SHG phenomena since a dc term is also involved. However, the advantages of using this definition become apparent when examining all nonlinear processes of the same order. When compared to the definition of B and W, a factor of two emerges for SHG, i.e.
(6~)
jk/
so that Y/jk/(-3w;w,w#) = 47;8(-3w;w,w,w)
(a)
(An additional term, proportional to yjjkr(-cd;o,o,o) also enters). Finally, one observes a factor of 8 difference for fourth harmonic generation, between the two sets of definitions where 6hijk/(-4W;W,O,W,W)
86&(-40;0,~,~,~)
In this section, we wish to derive the degeneracy factors for all second-order and third-order nonlinearities in a straightforward manner. The frequency dispersion should be included separately, as in eqs 1-6. The analysis follows that of Levine and Betheal" for the y i k/ properties. These degeneracy factors are largely a matter oidefinition (that is, whether one writes the perturbing field as real or complex) but must be properly defined for comparison with experiment. A. Secood-Order NonllnePrities. We will begin similarly to ref loa, with a schematic definition of first-order hyperpolarizabilities as
= @ET2(?) (7) where &(t) = cos w,t if we assume that there are n fields present. In this case, three electromagnetic fields suffice to describe the relationship among the second-order phenomena, namely frequency mixing (FMX), second harmonic generation, the linear electrooptic effect (LEO),and optical rectification (REC). We then have ET(t) Eo + El COS w i t + E2 COS w2t (8)
c:.,Es
Degeneracy Factors and the Definition of Nonlinear Sllsceptibility Tensors There has been some confusion in the literature concerning the definitional conventions for the nonlinear susceptibility x , used by theoreticians and by experimentalists. For second harmonic generation, depending upon the definition, a factor of 2 or a factor of enters when comparing nonlinear susceptibility tensors for second harmonic generation (SHG). Several authorslo have discussed these differing definitions and their interrelationship.
(10) (a) Levine, B. F.; Bethea, C. G. J. Chem. Phys. 1975,63,2666-2682. (b) Reference lb, p 34. (c) Singh, S.In CRC Handbook of Laser Science and Technology Vol. III, Optical Materials; Weber, M. J., Ed.;CRC Press: Boca Raton. FL, 1986; Part I, pp 1-228. (1 1) (a) Armstrong, J. A.; Bloembergen, N.; Ducuing, J.; Pershan, P. S. Phys. Rev. 1962.127, 1918-1938. (b) Bloembergen, N.; Shen, Y. R. Phys. Rev. 1964,133 (IA), A31-A49. (12) (a) Ward, J. F. Phys. Rev. 1965, 37, 1-18. (b) Orr, B. J.; Ward, J. F. Mol. Phys. 1971, 20, 513-526.
Nonlinear Optical Phenomena in Conjugated Chromophores
The Journal of Physical Chemistry, Vol. 96, No. 11, 1992 4321 6.00
Substituting eq 8 into eq 7
Y
6% = B S H d I E 2 m2wt = h@SH&IE2
COS
2wt
+ Pk2& (9a) 4.00
P&x
2 f l ~ x E i E 2COS
wit
= BFhfXElE2(coS (w1
COS w2t
+ w 2 ) t + cos (w1 - w2)tl
(9b)
(9c) = 2BLe&&l cos W l t It follows from eq 9a that the optical rectification coefficient is
3.50
3
,LX 0-
2.00
OREC
(9d)
v2BRBCE&-u
Comparing eq 9 with formal definitions of these susceptibilities (eq l), one can easily find that BFMX,
1 /2@SHG,
28LE0,
-
ek.
Figure 1. Frequency dependence of ox, and
oXYY
-
for various second-order 0 all phenomena nonlinear processes in aniline. Note that when hw converge to one unique dc value. 36
30
e6
8 eo 1s
(11)
'YET3
10
By setting
ET = Eo + E l
COS
wlt
+ E 2 COS 02t + E 3 COS w 3 t
(12)
and substituting eq 12 into eq 11, we obtain WAG
LEOE
(10)
'/ueRBC
or that the degeneracy factors for frequency mixing, second harmonic generation, the linear electrooptic effect, and optical d i c a t i o n are 1,2, 1/2, and 2, respectively. In other words, when one uses perturbation theory to calculate these phenomena, one should account for the corresponding degeneracy factors before comparing theoretical values with experimental data defined by eq 1 or 2. This can be verified by setting wI = w2 in the perturbation formula equation for frequency mixing. Instead of obtaining the second harmonic equation, one obtains the second harmonic formula multiplied by a factor of 2 (which is the degeneracy factor for SHG).When wi 0 (i = l, ...,n), all different value besecond-order nonlinearities should converge to one cause, at the dc frequency, mixing two zero frequencies should be no different than doubling one zero frequency (see Figure 1 where aniline is used as an example). B. Third-Order Nonllaerrities. The general definition of second-order hyperpolarizability is given by
#L
/
7
YHG
fi30
Pk2b
SHG
* Y T H d I E 2 E 3 m3 =
!&TH&&E~ COS 302 + (lower order terms) (1 3a)
Frequency (eV) Figure 2. Frequency dependence of yIxu and ywyyfor various third-order processes in aniline. At dc frequency (hw = 0), there is only one value for each individual third-order nonlinear tensor in all third-order nonlinear processes.
one unique Y~~~value (see Figure 2, where aniline is used as an example).
III. Construction of Monoexcited and Doubly Excited Singlet states
%YDFWMElE2E3{W ( h l
+ 0 2 ) + COS ( 2 0 1
- wz)l +
(lower order terms) (13d) and by setting wI = -u2 = w in eq 13, we have %U
74YIDRIEIE2E3
(13e)
cos Ut
Starting from a single-determinant ground state, there are two possible monoexcited states: a singlet excitation (In))and a triplet excitation (Int)). Similarly, there are six possible doubly excited states: namely two singlet (Iml)and Im2)),three triplet ( l m l t ) , Im?), and Imi)), and finally one quintet (lmq)). Only singlet excited states are required for optical properties. By choosing a normalized set of spin wave functions orthogonal to IS-1,Ms=O), one obtains (using straightforward operator algebra) the singlet excited state
Comparing eq 13 with the formal definitions of second-order hyperpolarizability (eq l), we obtain the following: j/4TTHG,
3YKms
3/2'YdcSHG,
3/4'YDFWMt
74'YlDRI E
Y' e:
(14) 1
In other words, the degeneracy factors for the third harmonic generation, dc Ken effect, deinduced second harmonic generation, degenerate four-wave mixing, and intensity dependent refractive 2/3, 4/3, and 4/3, respectively. index (ac Kerr effect) are 4, Again, this type of relationship can be verified by allowing wi to approach zero; all the third-order nonlinearities then converge to
-U@+i@lg)
fi
+ ja+iak)I
(16)
The square root of two enters from normalization, and lg) denotes the ground state; i, andj,+ are Fermion destruction and creation operators for spin a in space states i and j ; they obey the usual anticommutation relations [i,+,jd]+ = 6ad,6ip For double exci-
Li et al. TABLE I: Parameter Summery for PPP Calculations atom C(sp2) C(sp) C(sp3) H3=(C)
CH30 W(N) NH2 NMe, N=(O) N=(C)
CH3S N(sp)
19
eV 11.16 11.19 11.42 9.50 33.00 17.70 21.47 24.30 25.73 14.12 22.20 14.18
A7 eV 0.03
Yxx,
eV
bond C(Sp2)-C(sp2) 0.10 C(sp2)-C(sp) 0.58 C(sP2)-C(~p3) 0.0 H3-C 11.47 CH,O-C(sp’) 2.47 O(SP~)-N(SP*) 12.50 H2N-C(sp2) 7.50 Me2N-C(sp2) 8.97 02N-C(sp2) 1.78 N=C(sp)
13.05 CH3S-C(sp2) 1.66
C(sp)-N(sp)
Am
eV -2.318 -2.086 -1.182 -2.900 -1.808
-3.000 -1.753 -1.753 -2.10 -3.00 -1.00 -3.346
n,
1 1 1 1 2 1 2 2 2 1 2 1
1.397“ 1.46“ 1.52“ 1.10“ 1.37“ 1.21’ 1.40‘ 1.40“ 1.40’ 1.2W 1.7W 1.16“
“Ilavsky, J.; Kuthan. Collect. Czech. Chem. Commun. 1981, 46, 2687-2695. ‘Kiss, A. I.; Szoke, J. Acta Chim. Acad. Sci. Hung. 1976, 89, 337-346. Jsrgensen, P.;Linderberg, J. Int. J. Quantum Chem. 1970, 4 , 587-602. “Billingsley, F. P.,11; Bloor, J. E. Theor. Chim. Acta (Berlin) 1968, 1 1 , 325-343.
IV. *-Electron Calculations of First-, Second- and Third-Order Hyperpolarizabifities The PPP model *-electron Hamiltonian and attendant par~J~ ameterization have been discussed extensively e l s e ~ h e r e . ~ JIn this section, we discuss the application of this model Hamiltonian, using perturbation theory and the methods in Section I11 to calculate hyperpolarkabilitiesof various orders. The dipole matrix elements are calculated as outlined in the Appendix. Configuration interaction includes all monoexcited configurations and a number of doubly excited configurations adequate to assure apparent convergence in this manifold. The PPP parameters, taken from standard compilations, are summarized in Table I; no reparameterization has been done for these studies. A. Calculation of Polarizabilities and of Molecular Second Harmonic Generation and Frequency-Mixing Parameters. The polarizability, a,is important for recognizing trends in dynamic optical response and therefore for comparison with 8, y, and 6. Table I1 reports calculated polarizabilities for substituted benzenes and azulene. Delocalization of ?r electrons in one dimension can be significant in a: notice that a, H 2aYYfor azulene and that aYYN- 3/2a,, for p-nitroaniline; in both of these cases the polar axis polarizability is the larger component. The PPP model neglects all u contributions and takes an = 0 by symmetry; from PPP calculations to be less therefore we would expect aavg than aexptl. This is indeed in accord with the data in Table 11. An STO-3G calculation of a (w = 0) for nitrobenzene shows that !13) (a) Pople, J. A. Trans. Faraday SOC.1953,49, 1375-1385. (b) Pariser, R.; Parr, R. G. J. Chem. Phys. 1953,21, 466-471. (c) Pariser, R. J. Chem. Phys. 1953,21, 568-569. All computations reported in this study employ the Pariser-approximation ?,a, Jq,?,* d7 where 7, = x, y , or z . (d) Linderberg, J.; Ohm, Y. J . Chem. dhys. 1%7,d9,716-727. ( e ) Koutecky, J. J. Chem. Phys. 1%7,47, 1501-1511. (f) Jsrgensen, P.; Linderberg, J. Int. J. Quantum Chem. 1970,4,587-602. (14)(a) Ilavsky, J.; Kuthan Collecr. Czech. Chem. Commun. 1981,46, 2687-2695. (b) Kiss, A. I.; Szoke, J. Acta Chim. Acad. Sci. Hung. 1976,89, 337-346. (c) Billingsley, F.P., 11; Bloor, J. E. Theor. Chim. Acta (Berlin) 1968.11, 325-343. (d) Castellan, A.; Michl, J. J. Am. Chem. Soc. 1978,100, 6824-6827. (e) Downing, J. W.; Michl, J.; Jergensen, P.; Thulstrup, E. W. Theor. Chim. Acta (Berlin) 1974,32,203-216. (15)Li, D.; Marks, T. J.; Ratner, M. A. Mater. Res. SOC.Symp. Proc. 1989,134, 665-671 and references therein.
azzis only 12% of aYY or axxand that the latter two are roughly equal. The underestimate of a,then, arises mostly from our neglect of CI contributions. Overall, while there are variations in aij with substituent (p-nitroaniline is a factor of 2 greater than benzene), the mast striking observation is the nearly constant value of a that is responsive mostly to simple consideration of ?r and u charge cloud size (as is suggested by the units, volume, of ordinary polarizabilities). We have studied SHG phenomena extensively in the past few years, and therefore in this contribution, we only briefly touch upon systems previously examined in detail (Table 111). The present purpose is to compare experimentallysJ6J7and PPP-derived fly= values and to then calculate other types of NLO parameters for these same chromophores. fly, is defined as By,
( - ~ w ; w , w )= Bi
(22)
+ 1/3xjz,(@iii a,,
where fli = birr + 2(3-,) and i = the molecular dipole values shouli be exactly a factor of 2 larger direction. These than our previous results4 and those in all the publications using the Bloembergen and Ward definitions because of the degeneracy factor of 2. In normal experimental procedures for determining absolute second-order susceptibilities, the nonlinearity is measured versus that of a standard reference sample (e.g., quartz). For instance, if one uses a quartz crystal as a reference sample, d33 = 1.125 X lo9 esu corresponds to Bloembergen and Ward’s definitions, and xZzr= 2.25 X 1W esu corresponds to the definition proposed earlier in this paper. The confusion between the two definitions is difficult to sort out for several experimental data given in Table 111, because some authors do not report which definition was employed. It s&ms that in the case of pnitroaniline, the discrepancy between two experimental values PeXp= 34.5 X esu determined by two different and bcXp= 16.2 X research groups is due to this ambiguity. In Table IV, calculated sum frequency parameters for hal at 1.17 eV and hw2at 0.649 eV are given for a number of interesting charge-transfer-type conjugated organic chromophores. As expected, the PPP-derived values are close to those of corresponding SHG values. (16)(a) Nonlinear Optical Properties of Organic and Polymeric Materials; Williams, D. J., Ed.; ACS Symposium Series 233;American Chemical Society: Washington, DC, 1983. (b)See reference 4d. (c) Cheng, L.-T.; Tam, W.; Meredith, G. R.; Rikken, G. L.J. A.; Meijer, E. W. In ref 2b,pp 61-72. (17) (a) See references 3, 4b, 14,and 16. (b)Wu, J. W.; Helin, J. R.; Norwocd, R.A.; Wong, K. Y.; Zamani-Khamiri, 0.;Garito, A. F.; Kalyanaraman, P.; Sounik, J. J. Opt. SOC.Am. E Opt. Phys. 1989,6,707-720. (c) Wong, K. Y.; Garito, A. F. Phys. Rev. 1986,34, 5051-5058. (d) Pierce, B. M. Mater. Res. Symp. Proc. 1988,109,109. (e) Pemn, E.;Prasad, P. N.; Mougenot, P.; Dupuis, M. J. Chem. Phys. 1989, 91 (8), 4728-4732. (f) deMelo, C. P.; Silbey, R. J. Chem. Phys. 19%8,88(4),2567-2571,2558-2566, (g) Bishop, D. M. J. Chem. Phys. 1989,90,3192-3195.
Nonlinear Optical Phenomena in Conjugated Chromophores
The Journal of Physical Chemistry, Vol. 96, No. I I, 1992 4329
TABLE II: PPP-DerivedPdariubiiitiesfor Some Substituted Benzene Molecules in Units of molecule C6H6 CSHSN
polar axis none
0.649 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.649 0.00
X
Y Y Y Y Y Y Y Y
C6HSF
C6HsCl C&Br C~HSOH C6HSNH2
C6HsCN C6HSN02
P-NH~C~H~NO~ azulene
hw, eV 0.00
X
ffxx
7.980 8.232 8.113 8.123 8.127 7.863 8.027 8.266 12.687 12.123 31.686
cxp b axx
cm30 ,ppb
UYY
6.0
7.980 7.691 8.365 9.066 9.538 8.028 9.174 12.175 12.147 18.694 16.046
a-P
YY
aavg
6.0
5.320 5.308 5.493 5.730 5.888 5.297 5.734 6.814 8.278 10.273 15.911
W
c
10.02 9.21 9.86 11.93 12.93 11.16
12.48 12.05 12.33 17.1e 15.52f
Oalv is defined as alvg5 1 / 3 ( a x+xayy+ azz).Note that azz= 0 for the PPP model. bSchweig, A. Chem. Phys. Lett. 1967, 1 , 163-166 (value derivedfrom experiment at hw = 0.0). 'Meredith, G. R.; Buchalter, B.; Hanzlik, C. J . Chem. Phys. 1983, 78, 1543-1551 (experiments done at 0.643 eV = hw). dLide, D. R., Ed. CRC Handbook of Chemistry and Physics; 71st ed., CRC Press: Boca Raton, FL, 1990-1991, pp 193-209. cKarna, S. P.; Prasad, P. N.; Dupuis, M. J . Chem. Phys. 1991,94 (2), 1171-1181 (experiments done at h w = 2.10 eV). fBaumann, W. Chem. Phys. 1977, 20, 17-24 (experiments done at hw = 1.17 eV).
B. Calculation of Molecular Linear Electrooptic Effects and Optical Rectifcation Parameters. There are few experimental data for the hear electro-optic effect or optical rectification mainly because these experiments are more difficult to perform than SHG. Table V presents *-electron calculations of linear electro-optic effect and optical rectification data for the same organic chromophores as in Tables I11 and IV. Again, their values diminish slightly compared to those of SHG because of differences in the frequency dispersion. The symmetry properties
$fEC=
(e.g.9 @E;C
=
a,,,LEOE,
$EEC = fl)&OE,
and
TABLE III: Comparison of PPP-Derived8,- Values and Experimental B ,, Val- in Units of cm5 em-' molecule @"LPP) 0 &P) b.c hw, e~
= j3kEoE) (25)
all follow from the definitions. C. Calculation of Molecular Third Harmonic Generation Parameters. In this section, we present PPP-SCF-MECI-DECITHG calculations for the group of molecular organic liquids listed in Table VI. The average observable third-order susceptibility 4 is defined as H
The present agreement between experimentall* and theoretical values is in general good if full MECI and sufficient DECI are included. From these comparisons, one can appreciate the uncertainty in experimental values of microscopic response in determining the absolute molecular higher order tensors. In particular, the large relative uncertainty in the absolute molecular tensor comes mostly from the local field corrections. For a fourth-order tensor yijk/, the local field correction fir is required to the fourth power, and there are no good methods to determine fil experimentally. Thus, a 10% error infil can cause a 46%error in the absolute value of y , The most striking fact about the results of Table VI is that all of the chromophores show essentially the same THG response, in sharp contrast to the j3 properties just discussed. This suggests, as has been argued previously,19 that the y response arises, like a,largely from the extent of the charge cloud and is only weakly influenced by polarization effects. It also suggests that, in the absence of strong polarization, this contribution may be important in determining y, so that the PPP model may be inadequate. To explore this suggestion further, we now examine charge-transfer effects on yi,kb Five pairs of molecules with the same number of T electrons have been chosen as test cases (Table VII); note (18) (a) Meredith, G. R.; Buchalter, B.; Hanzlik, C. J . Chem. Phys. 1983, 78, 1533-1542. (b) J . Chem. Phys. 1983, 78 (3), 1543-1551. (19) (a) Li, D.; Marks, T. J.; Ratner, M. A. Muter. Res. Soc. Symp. Proc. 1988,109, 145-155. (b) Mater. Res. Soc. Symp. Proc. 1989,134,665-612.
2
N
W
N
0
1.02
1.06
1.17
1.84
0.79-2.46
1.17
4.55
1.97-4.6
1.17
34.4
16.2-47.7
1.17
11.7
14.3-17.5
0.656
13.26
13.4
0.656
29.3
21
1.17
36.3
16-42
1.17
298.1
225-295
1.17
45 1.9
450
1.17
213.1
180-260
1.17
466.8
470-790
1.17
2
B$zp)
Theoretical values are defined as discussed in this work (see text). For experimental j3, values, both definitions have been used by various authors (see text). 'Experimental data from refs 5b and 15-17.
that in each pair, one molecule is substantially more polar than the other. In pair I, the architecture changes in going to Dewar benzene more than double the predicted yxxxxvalue over that in the more polar [1.3.0] molecule. Similarly, in pair I1 the chargetransfer effects cannot overcome the unfavorable geometric influences on three-photon processes, and in addition, the thin bridge of cyclopropenylidene-cyclopentadiene provides little coupling between the x and y dimensions. As a result, the yxxyy value of cyclopropenylidene-cyclopentadieneis much smaller than that of its partner. However, when the structures become more similar, as in the case of azulene and naphthalene (111). charge-transfer effects emerge strongly, especially in yxxxx, and
Li et al.
4330 The Journal of Physical Chemistry, Vol. 96, No. 11, 1992 TABLE I V PPP-Derived Sum Frequency Generation (Frequency Mixing)' Parameters at hwl = 1.17 eV and hut = 0.649 eV in Units of cms esu-' molecule
Lx
Pxx*
8,""
P"",
P","
1.752 -0.093 -0.066 -0,053 5.533 -1.775 -1.683 -1.643 30.33
-3.511 -3.042 -2.843
137.3
-8.388 -5.920 -4.985
177.1
-7.038 -4.412 -3.428
O$Jl+NMe,
-25.16
1.588 -0.321 -9.078
TABLE VI: Comparison of PPP-Derived and Experimental TmG Values at hw = 0.649 eV in Units of esu exp" PPP exp' PPP molecule TTHG ~ ~ T H c molecule TTHC TTHG C6H6 3.85 6.26 C6HS1 8.19 8.52 16.85 C ~ H S N O ~ 5.37 5.49 C6HSN 3.36 6.14 C6HSCN 4.11 11.48 C6HSF 3.19 C6HsCl 4.34 7.23 C ~ H S N H ~ 5.71 6.90 C6HSBr 5.40 7.54 OExperimental data of ref 20.
TABLE MI: Charge-Transfer Effects on PPP-Derived T,,~/ Parameters at hw = 0.00 eV in Units of esu molecule
Lx
Yxxxx
Yyyyy
Yxiyy
Yavg
-17.89
-4.618
-2.689
-5.578
I -43.68 -81.35
"The sign of these j3 components is given following the definition of ref loa. The two-level model suggests that the sign of @ is the same as the sign of Aji, the difference in dipole moment between excited and ground states. Thus, only azulene, of the molecules in this table, is less polar in the excited than in the ground state.
TABLE V PPP-Derived Linear ElectroopHc Effect Parameters of Some Major Second-Order Components in UNts of lo-" cm5 esu-' molecule (ha= 1.17 e v )
x
ox,,
P,
PYYX
0.0795
0.0795 0.0621
-5.009
1.637
1.637
1.584
2.986
2.98
2.723
6.383
5.214
5.145
3.926
-103.9
14.17
OZN-NMe2
-129.5
5.145
O$N J-"z
a
22.21
-0.573
-0.573
ID
PYXY
-.1.649
-26.08
_11
IV
v
a 0-0
-1.831
-3.359
-10.45
-20.43
-22.94
-29.53
-103.5
-17.69
-60.88
-48.59
-249.5
-54.56
-31.49
-75.82
-36.35
-15.24
-32.42
-409.0
-19.81
-19.15
-93.41
-201.2
-17.28
-26.51
-54.30
-34.54
-33.38
-30.45
-95.26
e)=o=(3
-50.94
TABLE MIL Comparison of Variations in the Observable 9 with Architecture in Linear and Planar Conjugated Chromophores n linear ?ID planar 420 0
0
-12.63
-0.74
-29.16
-5.17
1.380
consequently the observable ti. is more than doubled in azulene. One would expect on the basis of charge-transfer effects that dithiafulvalenes should have larger yxxxxvalues than fulvalenes and tetrathiafulvalenes. In contrast, yxxxxof dithiafulvalene is approximately a factor of 2 smaller than that of fulvalene but much larger (4 times) than that of tetrathiafulvalene. This is because the first excitation of fulvalene is very low in energy (AI? = 1.45 in MECI calculations and 1.38 eV in DECI calculations) compared to 3.39 for dithiafulvalene and 3.29 eV for tetrathiafulvalene in MECI calculations and 2.75 for dithiafulvalene and 3.06 eV for tetrathiafulvalene in DECI calculations. In the case of dithiafulvalene and tetrathiafulvalene, one can conclude that charge-transfer effects will enhance the third-order susceptibilities provided that the geometries are identical and the first excitation energies are similar. Table VI1 shows clearly that, in general, the effect of charge transfer on the third-order susceptibilities (4)is significant, but far from dominant. In Table VIII, we compare the effects of dimensionality and conjugation length on observable 4 values. As shown in the table, ti. values in one-dimensional systems tend to be 1 order of magnitude larger than those in two-dimensional systems, arguing that low-dimensional systems are generally superior to two- or three-dimensional systems. This phenomenon of large y for low-dimensional systems was also noted by others in CNDO
-355.4
-51.06
-499.5
-68.38
-2672
-176.2
-2449
-386.8
calculations.17b Compared to charge-transfer effects, the lowdimensional character is clearly more important for the enhancement of ?. This result clearly justifies the great current interest in polyacetylenes as third-order nonlinear optical materials. D. Calculation of Molecular Dc-Induced Second Harmonic Gemration Parameters. Another straightforward experiment for
Nonlinear Optical Phenomena in Conjugated Chromophores TABLE I X Comparison of PPP-Derived and Experimental qMHc Values at hw = 1.17 eV in Units of lo-* esu C6H6
C6H5F
C6HsC1 C6HSBr C6H51 C6H5N
C6HSNH2 C6HSN02
2.34 2.2 2.44 -0.9 0.8 2.54 4.99 4.97 7.78 43.3
C6HSOH
3.4 4.9 4.6 5.1 5.8
6.59 7.42 8.12 9.21 5.90 7.81 18.68 5.83
4.1 5.8 3.6
TABLE X: Comparison of PPP-Derived and Experimental Four-Wave Mixing Values (Normalized to the Benzene foWM Value) ho, = 0.2 eV and hwq = 0.3 eV in Units of esu %Khi ??&Ma
1.oo 1.45 1.5 (0.65) 1.7 (0.74) 1.85 2.24 1.32 1.6 1.16
TABLE XI: Comparison of PPP-Derived and Experimental Optical Kerr Constants“ ( B , ) (Normalized to Benzene) at Frequency hw = 0.196 eV in Arbitrarv UNts
6.88
“Data of ref loa. bData of ref 23.
molecule
The Journal of Physical Chemistry, Vol. 96, No. 11, 1992 4331
normalized 1.oo 0.96 1.07 1.17 1.32 2.53 1.76 1.12 0.86
liquid benzene to1u ene nitrobenzene
PPP
expb 1.o 1.4 4.2
+
Br(?Ksrr)
1.oo 2.14 2.52
57.76 (5.78) 123.6 (12.37) 145.5 (14.56)
+
j yijji). bData of ref 26. “yKsrr ’ / 1 5 E i = , ( y i i jyijij
TABLE XII: Comparison of PPP-Derived and Experimental Intensity-Dependent Refractive Index (n2)“at hw = 0.605-0.638 eV (Normalized to Benzene) in Arbitrary Units ny* nPPP liquid lO%YP (normalized)b ?n2 (normalized) 1.oo 5.96 1.00 (3.63) C&& 28.6 f 1.6 1.05 5.09 1.29 (4.69) C6H5N 30.0 f 2.5 C6HsCI 50.7 f 4.6 1.77 6.39 0.86 (3.12)
“n2 = 2r/no ~ ( ~ ) ( - w ; w , - w , w ) ;qn2 = l/lsxij(yiijj+ yijij + yijji). bData of ref 27. 2.93 2.81 3.14 3.42 3.86 7.42 5.15 3.28 2.51
“Normalized to benzene. Data of ref 23.
measuring third-order nonlinearities is dc-field-induced second harmonic generation. Table IX compares experimenta1108,20 and theoretical Td&S0(-2w;w,w,o) values in units of esu at h w = 1.17 eV. The agreement between the experimental results and the *-electron calculation is rather good with the greatest discrepancy occurring for nitrobenzene, probably for the reason already discussed in Section C. The discrepancy between the two experimental values is large, at least partly due to the local field corrections (vide supra). By examining the results in Tables VI and IX,one can conclude that an experimental uncertainty factor of 2 is not unusual; the PPP-SCF-MECI-DECI calculations can predict the experimental results to within the same accuracy. E. CIllculatimof Four-Wave Mixing Parameters. Degenerate four-wave mixing occurs in all media and is a general, widely utilized technique in higher order NLO measurements and applications. Experimental four-wave mixing20 data are listed in Table X together with PPP calculated values. In the actual experiments, the outputs of two dye lasers having fundamental frequencies wIand w2 are adjusted such that w3 = 2wl- w2 is held constant while 6w = w1 - w2 is varied linearly in steps. By measuring the resonance frequency maximum and minimum, the nonresonant third-order tensors can be determined. The present *-electron calculations were carried out using low frequencies (hwl = 0.2 eV and ha2 = 0.3 eV) for each molecule. The resulting yFWM values were considered to be nonresonant and were chosen to compare with experimental values. It is difficult to obtain absolute values experimentally, and therefore all data are normalized to benzene. As with the THG and dc second harmonic parameters discussed above, the agreement between theory and experiment is good, with the largest discrepancy being less than a factor of 2; once again, only small differences are seen with substituted benzenes of different polarity. F. Calculation of Ken Effect Parameters. There are basically two types of Kerr effects: the dc Kerr effect, y(-w;w,O,O); and the ac Kerr effect, y(-w;w,-w,w), or the intensity-dependent refractive index (n2) phenomenon. In the dc Kerr effect, an optical (20) Oudar, J. L.; Chemla, D. S.;Batifol, E. J . Chem. Phys. 1977, 67, 1626-1635.
Kerr constant (B) defined in eq 2721is measured, where X is the
wavelength, nI, and nI are refractive indexes parallel and perpendicular to the dc electric field Edc. Using classical statistical B can be expressed as a power series in the number density
+
B = B(’) pB(2) To first order, B
+ ...
B(I),and B(’)can be described by
I
\
~
I
(28)
xtiy as
”
Relative experimental optical Kerr constants23(Br, normalized to benzene) are presented in Table XI, with theoretical B values obtained by summing over i and j in eq 29. It can be seen that the PPP model is adequate in describing dc optical Kerr effects, with the differences between theory and measured values being within a factor of 2. The ac Kerr effect involves the intensity-dependent refractive index, where the index of refraction is modulated by light rather than by a dc electric field. The total refractive index is given by n = no n21 (30)
+
where I is the intensity of the incident light and the intensitydependent refractive index n2 can be described by ~ ( ~ ) ( w ; w , - w , w ) as 2* n2 = -~(~)(-w;w,-u,w) n0
Here x ( ~can ) be evaluated in terms of T(-w;w,-w,w) by using the Lorentz model to correct for local field factors. In Table XII, we compare calculated and measured24normalized n2 values at a frequency hw = 0.605-0.638 eV. The agreement is good for pyridine, but the discrepancy increases to a factor of 2 due to the slightly lower number density of chlorobenzene and to local field ) value; such corrections modifying the experimental x ( ~(Kerr) (21) (a) Lewis, J. W.; OrHung, W. H. J . Phys. Chem. 1978,82,698-705. (b) Martin, F. B.; Lalanne, J. R.Phys. Reu. A 1971, 4, 1275-1278. (22) (a) Kielich, S.Acta Phys. Pol. 1966, 30, 683-707. (b) Kielich, S.; Piekara, A. Acra Phys. Pol. 1959, 18, 439-471. (c) Kielich, S.Acta Phys. POI. 1964, 26, 135-154. (23) Blaszczak, 2.;Dobek, A.; Patkowski, A. Acta Phys. Pol. 1973, A44, 15 1-1 54. (24) Song, J. J.; Levenson, M. D. J . Appl. Phys. 1977, 48, 3496-3501.
4332 The Journal of Physical Chemistry, Vol. 96, No. 1 1 , 1992
Li et al.
TABLE XIII: PPP-DerivedFourth Harmonic Generation Parameters for Chromophores Having Ck Symmetry (in Units of lo-‘* esu) hw, eV
molecule
8,XXXX
0.649
56.57
L x y y
axYYYY
ayyxxx
-0.2585
-4.497
-1.626
-2.801
-0.3276
-0.2918
-0.4389
-2.374
-3.181
0.649
-11.14
-0.2367
0.649
988.6
15.45
0.200
-546.86
-24.04
4.677
-28.56
0.200
330.2
35.66
4.449
34.84
-13.55
TABLE XIV PPP-DerivedFOUL Harmoh- Generation Parameters at ho = 0.2 eV (h Units of IO-’’
Lx
molecule
L x x x
L x x y
3305
-388
7996
-1542
8XXXYY
6XXYYY
6xyyyy
-15.53
-18.12
349.0
-58.86
-33.35
175.3
34.47
3.402
259.9
46.49
1.762
78.76
aYYYYx
0.8069
4.815
esc,
a,,,
dyxxxx
~YXXXY
6YXXYY
-414.7
64.32
-17.98
-15.09
~YXYYY
33.53
332.1
-52.59
-20.94
66.89
994.0
153.8
36.96
2.251
-2.350
728
231.2
47.88
0.9713
-5.517
M0’*N02
8746
15576 M
+
h
N
0
w
,
902.3
1605
-1723
-
corrections lie outside the scope of our isolated-molecule computa tions. G. Calculation of Molecular Fourth Harmonic Generation Parameters. In this section, we report the first theoretical calculations on fourth harmonic generation 6ijklm(40;w,w,w,w). As expected, 6Uk.m resembles Bi’k in such aspects as charge-transfer effects. The symmetry of hurth harmonic generation requires that any permutations of j k l m must not change the value of the fifth rank tensors 6,.klm.Aniline, nitrobenzene,pnitroaniline, and azulene all have a i-fold axis along y, and hence any tensors with an odd number of identicaly indexes will vanish (e.g., 6 bW,,, , , S = 0). Finally, Kleinman symmetryz5is also satisfizor as shown in Figure 3, at w 0. Major PPP-derived nonzero tensors are listed in Tables XI11 and XIV for some charge-transfer chromophores. Similarly to &,, aiijiiis the predominant tensor in the group of 35 = 243 tensors because of charge transfer along ith direction. The major contributions to this term, come from fg8Ap? or fgfsdApd and fJdAp8 terms where fBs andf,d defined in eqs A15 and A16 refer to singlet and doublet excitations, respectively, and A&) and AM$) are the change in dipole moment in the ith direction for single and double excitations, respectively. This explains why bjiiii will generally have a sign opposite to bdi,because &: will always have the same sign as Apsand the minus sign comes from the coefficient of perturbation expressions for 6 (compare eq A1 to eq A27). In general, 6h,,{/ scales similarly t o Pltk (compare the 6h,jt/ increases with the Btr increases for aniline, nitrobenzene, andpmtroaniline). Finally, t i e frequency dependence of 6hi/k, for p-nitroaniline is plotted in Figure 3. As expected, four-photon rmnances appear at frequencies as low as hw = 1.0 eV. H. Charge-Transfer Phenomena and Hyperpohrhbilities. From the perturbation picture and the results just given, it can be seen that two types of terms dominate in the description of hyperpolarizabilities. The first is the coupling between two different states, characterized as oscillator strength in the case of ground and monoexcited states. The second corresponds to the
-
( 2 5 ) (a) Kleinman, D. A. Phys. Rev. 1952,126, 1977-1979. (b) Wagniere, G . Appl. Phys. B. 1986, 41, 169-172.
80.0
40.0
ii 0.0
Lo
-40.0
-80.0’
I
o.do
I
I
I
I
I
I
*
1
0.dO
r
1
I
I
1
I
I
,
1
I
.do
,
.
I
1
1
1
I
I
I
1. 0
Frequency (eV) Figure 3. PPP-derived frequency dependence of ayyyy,,,ayxw &,, and 8,xyyy for p-nitroaniline.
coupling of any given state with the radiation field, which is described by the dipole moment of that particular state. When comparing the dipole moment change from a ground state to an excited state, one can characterize this second term as intramolecular charge transfer. In general, hyperpolarizabilities of any order can be understood qualitatively in terms of oscillator strengths and dipole moments of excited states. For polarizabilities a,the only important interaction is the oscillator strength between ground and monoexcited states, and thisdetermines the very simple behavior (e+, the variation of refractive index with size is monotonic). In Figures 4 and 5, we show the changes of a, and ayyas a function of the dihedral angle between the two planar aryl fragments in biphenyl and 4-(dimethylamino)-4’-nitrobiphenyl. In the cast of biphenyl, axxand aYyare invariant with twisting, and in the case of 4-(dimethylamino)-4’-nitrobiphenyl, a, (along the charge-transfer axis) decreases while cyyv increases slightly. These results are exactly as expected. On the other hand,
Nonlinear Optical Phenomena in Conjugated Chromophores
0,
The Journal of Physical Chemistry, Vol. 96, No. 11, 1992 4333
z
P a
20
10 124
d
....,...,,..,. 10 20 ab
, , I ,
I , , ,
4b
I , , ,
do
,,,, ,,,, ,,,,
do 50 do
Dihedral Angle (deg.)
do
F i p e 4. Plot of odd-order susceptibility tensors (aand y) as a function of the dihedral angle between the two planar fragments of biphenyl.
for second-order susceptibilities, two monoexcited states are involved (see Scheme I in the Appendix), and therefore fi is determined by the products of the transition oscillator strength between the ground state and monoexcited states and among monoexcited states themselves; this second term reduces to charge transfer when these two monoexcited states are identical. By examining the type of terms contributing to 6, one can easily generalize that these chargetransfer effects are more pronounced in fourth-order processes, because they appear not only in the single excitations, but also in the double excitations. This is clear from Figure 5 : both fix,, and,,,,,6 decrease rapidly when the communication between the two phenyl rings is turned off by a 90° dihedral angle. We also note that these charge-transfer effects dominate the tensor component along the corresponding charge-transfer direction, with the rest of the tensors being only slightly perturbed (CS,,,, flyy,, 6xxxyy,and 6yxxxychange very little upon twisting; see Figure 5 ) . Having the mixture of two types of oscillator strength terms (e.g., a-like) and one type of charge-transfer term (e.g., 8-like), y has the most complex behavior. If the molecular structure has a center of symmetry, and therefore no charge transfer, y behaves as a,as can be seen clearly in Figure 4. In noncentrosymmetric structures, the particular tensors along the chargetransfer direction can be enhanced. This chargetransfer enhancement has the same properties as in second- or fourth-order processes: as shown in Figure 5, yxxxxof 4-(dimethylamino)-4'-nitrobiphenyl decreases rapidly when the conjugation between rings is broken as the dihedral angle between the two fragments approaches 90°. The less significant influence of intramolecular charge transfer on the third-order observable susceptibility (T) has two explanations. First, although the particular tensor along the charge-transfer direction is enhanced, the other, less enhanced tensors are nonnegligible. Second, the observable is composed of tensors of all orientations ( x , y , and z) rather than just of those projected along the charge-transfer direction as in the case of the secondorder effect &.
0
Dihedral Angle (deg.) Figure 5. Plot of major susceptibility tensors up to the fourth order as a function of the dihedral angle between the two planar phenyl fragments of 4-(dimethylamino)-4'-nitrobiphenyl.
of the results in terms of the nature of the contributing excited states and the availabilityof the full frequency dependence. These computations all employed the PPP model Hamiltonian and the Pariser approximation for calculating dipole matrix elements. The PPP model has been used previously for calculating nonlinear optical properties, as well as simple absorption spectra, for aromatic molecules. PPP models have proven to be excellent for computing willator strengths and excitation energies, two critical components of the a,8, y, and 6 electronic properties. The agreement between calculated (PPP model sum-over-states) and experimental values is excellent for second-order and good for first-order and third-order susceptibilities; for fourth-order susceptibilities, there are no experimental or theoretical models for comparison. Polarity or charge-transfer effects are crucial for 6 and 6; however, the polarity effects are small for most tensor components of y. The importance of charge transfer in even-order hyperpolarizabilities (B and 6) yields substantial variation with such electronic structure variations as addition of donor or acceptor substituents. Conversely, y and higher even-order hyperpolarizabilities, like the polarizability a,are determined primarily by the size and shape of the electron network. Consequently, they generally show much smaller changes with substituents and vary only slightly (i.e., less than a factor of 3) acras the series of singly substituted benzenes (though, as Figure 5 shows, charge transfer can contribute significantly to y in some cases). Since the PPP model is for u electrons only, important uelectron responses will lead to errors in prediction of experimental quantities. For those properties dominated by charge transfer, @ and 6, such corrections are expected to be relatively minor, since most dynamic polarization arises from u electrons (this is why the PPP model describes optical transactions very well in u species). For a and y, especially in centrosymmetric systems, the magnitudes of the u contributions may be considerably larger, and the PPP model may become less accurate; this may well account for some of the y tensors computed for iodobenzene, where polarization of the iodine substituent, which should be important, V. Conclusions is neglected in the PPP treatment. The ground state in these calculations is taken as the singleWe have computed a, 8, y, and 6 properties using a determinant SCF state. Correlation effects in the ground state straightforward uncoupled Hartree-Fock scheme, in which these will of course cause errors in these calculations; such errors might response properties are calculated using sum-over-states expresbut the generally good experimental be important in some case~,3*~ sions. While other methods for computing these responses have ~ ~present ~ ~ ~ scheme - ~ ~ offers ~ ~ ~ several ~ ~ ~ - ~ ~comparisons here suggest that, at least for aromatics, such corbeen d i s ~ ~ s s e d , the rections are not of qualitative influence. advantages including facile, chemically oriented interpretation In summary, the choice of new nonlinear optical materials can be guided by fairly reliable, computationally efficient, and (26) Zerner, M. C.; Parkinson, W. A. J . Chem. Phys., in press. chemically oriented PPP u-electron descriptions of molecular (27) Lalama, S. L.; Garito, A. F. Phys. Rea A 1979, 20, 1179-1192. nonlinearities. (28) Rustagi, K. C.; Ducuing, J. Opr. Commun. 1974, 10 (3, 258. (29) Garito, A. F.; Heflin, J. R.; Wong, K. Y.; Zamani-Khamiri,0.Muter. Acknowledgment. This research was supported by the NSFRes. Soc. Symp. 1988, 109,91-102. (30) Karna, S. D.; Dupuis, M. J . Compur. Chem. 1991, Z2, 487-499. MRL program through the Materials Research Center of
4334
The Journal of Physical Chemistry, Vol. 96, No. 11, 1992
Li et al. SCHEME I1
SCHEME I
&Ap& Northwestern University (Grant DMR882157 1) and by the Air Force Office of Scientific Research (Contracts 8842-0122 and 90-0071). M.A.R. thanks the Dow Chemical Company for funding the Dow Research Professorship. We are grateful to Prof. A. Garito for pointing out a typographical error in ref 27 and to Mr. M. Todd for the STO-3G calculation of a.
SCHEMEI11
Appendix. Evaluation of Dipole Matrix Elements via Second Quantization Given the ground-state wave functions, in the perturbation approach, one needs next to evaluate the dipole matrix elements of the form (x,lPlx,,,) in order to calculate hyperpolarizabilities. In this appendix, we will discuss the evaluation of the matrix along with the eigen vectors Ix,,,) and eigenelements (xnli~x,,,) value ha,,, which correspond to the mth excitation. A. Second-Order Susceptibilities. The B Terms. Following Bloembergen” and Ward,I2the individual tensor components for second harmonic generation can be expressed in the following convenient form. Bijk(-2w;w,w) =
Pjk
ksHGl
Here we follow the notation of Lalama and gar it^,^' in which Ai-j is the coefficient of the j,+i,lg) term in the monoexcited CI state In). Similarly, we can describe the (nlWln’) terms as
dit = (nlWln’) =
y2c A~-jA~?[(dia+j, + (dis+js]Cs+tm$:)[l,+k&) + ij,k/
St
ls+k,lg)l
=
Ai+jAr‘( g(i+js+tl+klg)mi!) sf,ij,k/
x
= ij,k/ cA~4jA~‘[6ikm - ~6j/&)] ~’
(A51
We point out that similar results given by Lalama and Garito indicate, inappropriately, the absence of the off-diagonal term in eq A5 when i = k, j = I, and n # n’; this is simply a typographical error in that paper. We should also point out that in eq AS, the term 6~$j&s#?$) is defined as zero to avoid divergence in the perturbation expression of &jk when In) = lg). For second-order nonlinear susceptibilities, there is only one type of term
B Here ksHG = 2 is the degeneracy factor for SHG, Pjk is the permutation operator, +inand are matrix elements defined earlier, and hungcorresponds to the excitation energy from the ground state (Ig)) to the excited state (In)). This expression for B, k is obtained with the dipole approximation having the perturbing damiltonian
4%
H’= &F sin wt
(A21
This result is not valid near electronic resonances and contains no rotational or vibrational contributions. To evaluate the first-order hyperpolarizabilities, we need only calculate the dipole matrix elements (diln)and (nliln’) and the eigenvalue hw, because single excitations from the HartreeFock state are dominant for three-photon processes (Scheme I). We can describe matrix elements between the ground-state and monoexcited configurations as where Hh) = Cs+tm$) and m$) = Jqs(j)dh)q,(7) d?. qS(7) is the wave function of the sth molecular orbital. Using the expression obtained in Section I11 for monoexcited singlets In),we can rewrite eq A3 as
a
(drln)(nlrln’)(nlrlg)
(A6)
In order to have nonzero &jk tensors, it is required that all three matrix elements be finite. In the special case when n = n’, the above expression degenerates into
B
a
( A W 2 [ ( n l r l n )- ( A r k ) ]
(A71
In this situation, one needs to have a nonvanishing (gjdn) in addition to having a large dipole change ( ( g j d g ) = 0). In other words, if the transition from ground state 18) to the excited state In) has charge-transfer character (large dipole moment change) and this transition is not forbidden by symmetry, &jk will be enhanced by this particular transition. B. Third-Order Susceptibilities. The y Terms. Third-order processes are four-photon processes (Scheme 11). Therefore one needs to include doubly excited states as well as singly excited states in order to calculate second-order hyperpolarizabilities. There are two types of doubly excited singlets, Iml) and Im2),as indicated in eq 17-21, The doubly excited states can degenerate into pseudodouble excitations corresponding to one electron being excited twice by a singlet excitations (see Scheme 111). Therefore, there are three types of terms for y,jk/, third-order susceptibilities. type I: ySl= (dW (nlilml )(mlliln’) ( n l W (A81 type 11:
ys2 =
(diln)(nlilm2) (m2liln’) ( n l d g ) (A9)
type 111: yo = (diln)(n~iln’)(nIiln’’)(n’li(g)(A10) It is difficult to satisfy the condition of strong coupling among the ground state (Ig)),different monoexcited states ( I n ) ) , and doubly excited states (Im))simultaneously. The condition can be satisfied easily in the special case n = n‘= n”, In this situation,
4336 The Journal of Physical Chemistry, Vol. 96, No. 11, 1992
molecular charge transfer. This can easily be seen in the special case when conditions m, = m,’, m2 = m i , and n = n’ = n”are satisfied. and bu2 are linearly enhanced by the doubly excited charge-transfer effect (i.e. bu2 0: fgfdApgd) while 6,, and 6A
-
Li et al. are also linearly enhanced by the monoexcited charge redistribution effect (i.e., 6u3,6d a fgisdAp,). It is worth noting that bus scales in a cubic manner with charge-transfer effects in the monoexcitations (i.e., bus a fgsApg:), and this determines the stronger vector properties of 6 and the large influence of charge transfer.
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