Theoretical approach to the design of organic molecular and polymeric nonlinear optical materials

Chapter 2

THEORETICAL APPROACH TO THE DESIGN OF ORGANIC MOLECULAR AND POLYMERIC NONLINEAR OPTICAL MATERIALS Benoit Champagne Laboratoire de Chimie Theorique Appliquee, Facultes Universitaires Notre-Dame de la Paix, B-SOOONamur, Belgium

Bernard Kirtman Department of Chemistry and Biochemistry, University of California, Santa Barbara, California, USA

Contents 1. Introduction 2. Methodologies 2.1. Separation of Electronic and Vibrational Responses 2.2. Methods for Determining Electronic Polarizabilities and Hyperpolarizabilities 2.3. The Electron Correlation Problem: Wavefunction versus Density Functional Theory Approaches 2.4. Conventional Perturbation Approach for (Pure) Vibrational Hyperpolarizabilities 2.5. Nuclear Relaxation-Finite Field Approach to Vibrational Hyperpolarizabilities 2.6. Numerical Procedures and Basis Set Aspects 3. Applications 3.1. General Considerations 3.2. Second-Order NLO Organic Materials 3.3. Third-Order NLO Organic Materials 3.4. Role of the Medium Acknowledgments References

63 65 66 68 72 74 75 77 80 80 83 98 114 119 119

1. INTRODUCTION

experimental NLO observations. However, the modern explosion of activity in this field was initiated by the invention of the laser After several decades of intense research efforts the field of or- and the subsequent generation of second-harmonic light in quartz ganic nonlinear optical materials (NLO) remains a subject of much by Franken in 1961 [3]. Shortly afterward the first theoretical paper theoretical and experimental interest. In comparison with linear providing quantum chemical expressions for the molecular first hyrefraction, light scattering, and absorption, which are well-known perpolarizability j8 and second hyperpolarizability y appeared [4]. phenomena, the nonlinear response to an oscillating electric field Since then theory has evolved to the point where it now plays a is relatively unfamiliar and sometimes poorly understood. On the substantive role in understanding the various microscopic mechaother hand, the possibility of utilizing this response for optical in- nisms of NLO activity and their relative importance. In particular, formation processing, telecommunication, and integrated optics, the quantum chemical theory aims at developing models for eluas well as for chemical and physical analysis, has great potential, cidating new phenomena, explaining experimental observations, NLO processes include, for example, harmonic light generation, and developing structure-property relationships that can be used two-photon absorption, and field-induced modification of the re- to design materials with desired properties in a cost-efficient manfractive index. The discovery of the Kerr effect in CS2 [1] in 1875 ner. Recent advances that delineate the contribution of vibration and, a few years later, the Pockels [2] effect in quartz, mark the first and electron correlation will, no doubt, prove to be of value in that

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63

64

CHAMPAGNE AND KIRTMAN

regard as will new theoretical methods that are being developed to treat the effect of interactions with the surrounding medium. In this chapter our focus is on determination of the secondand third-order NLO properties, j8 and y, of conjugated organic molecules and polymers in the nonresonant regime. At the macroscopic level, these properties are known as the secondorder (x^-^^) and third-order (x^^'^) susceptibilities. For increasingly large oligomers tending toward the polymeric limit, the system is finite in the transverse and perpendicular directions but effectively infinite along the longitudinal direction. In the latter case, it is convenient to define the hyperpolarizability per unit cell, P/N or y/N, which, in the infinite chain length limit, is equivalent to the nonlinear susceptibility. We start with the sum-over-states (SOS) perturbation theory expressions that characterize the simultaneous electronic (j8^ and y^) and nuclear response of the system to a spatially uniform electric field. In writing this expression it is assumed that the effect of overall rotations can be ignored and that the states can be described by ordinary vibronic wavefunctions. Then a clamped nucleus (CN) approximation is applied and the total hyperpolarizability is separated into the usual electronic component, which includes zero-point vibrational averaging (ZPVA), plus a vibrational component (j8^ or y^) that arises from those SOS terms where at least one of the intermediate vibronic states corresponds to the electronic ground state. This separation is presented in detail in Section 2.1. In addition, the difference between the CN approximation and the "exact" SOS formulas is analyzed and shown to be quantitatively insignificant for typical second- and third-order conjugated organic NLO molecules. An important feature of these molecules is that the vibrational component can be of major significance. Furthermore, experiment can probe the importance of the two contributions by exploiting the time-scale partitioning of the NLO response [5]. Indeed, the vibrational response is generally one to three orders of magnitude slower than the electronic response. The electronic and vibrational responses, including their frequency dispersion, may be evaluated at many different levels of theory. Our emphasis will be on ab initio, as opposed to semiempirical, methods because of reliability issues as well as the possibility of systematic improvement by employing more extended basis sets. Furthermore, correlation plays a very important role in the properties and systems of interest in this review and the level of performance of ab initio many-body perturbation theory-coupled cluster (MBPT/CC) approaches, in particular, is now becoming well-established. In Section 2.2 we concentrate on MBPT/CC and other wavefunction methods for treating electronic (hyper)polarizabilities. The density functional theory (DFT) approach, which is currently unsatisfactory for extended systems, is examined in Section 2.3. The material in Sections 2.2 and 2.3 is also relevant for vibrational (hyper)polarizabilities because their evaluation is determined by how the electronic response properties vary with molecular geometry. In Section 2.4 the conventional perturbation theory formalism for the vibrational properties, which takes into account the (often very substantial) contribution of vibrational and electrical anharmonicity, is presented. This formalism remains the only way of treating frequency dispersion "exactly." However, there are now more efficient finite field (FF) methods for determining vibrational (hyper)polarizabilities in both the static and infinite optical frequency limits. These methods, which depend upon determining

field-dependent electronic properties at the field-dependent equilibrium geometry, are described in Section 2.5. The infinite optical frequency limit is useful because it generally provides a good approximation to the value at optical frequencies as will be demonstrated in Section 3.2.4. Finally, the replacement of vibrational normal coordinates by field-induced coordinates (FICs), which is also described in Section 2.5, allows one to combine the perturbation theory and finite field approaches in a manner that preserves the best features of each. In Section 3 the various aspects related to design of secondorder (p) and third-order (y) conjugated organic NLO materials are tackled. Some general introductory material, concerning (i) the chain-length dependence of y (in centrosymmetric oligomers) and of (3 [in donor-acceptor (D/A) molecules and in oligomers with asymmetric unit cells], (ii) the relationship between vibrational and electronic hyperpolarizabilities, (iii) the potential for enhancement of NLO properties in charged structures, and (iv) practical considerations, is given in Section 3.L These themes are then pursued in depth in Sections 3.2 and 3.3, which deal, respectively, with p and y materials. The chain-length dependence of the electronic hyperpolarizability is covered in Sections 3.2.2 and 3.3.1 along with the special role of bond length alternation (BLA) in this regard. For D/A molecules, in particular, the dependence of p^ on the nature of the linker is also discussed in Section 3.2.2 following a consideration of the effect of the D/A strength and its qualitative and quantitative characterization in Section 3.2.1. For oligomers with unit cell asymmetry, the chain length dependence of p^ is determined by an interplay between the degree of asymmetry and the degree of delocalization due to conjugation. This situation is analyzed in Section 3.2.3 using polymethineimine (PMI), which has a very large unit cell first hyperpolarizability value (in the infinite polymer limit), as a working example. We also discuss conformational as well as D/A substitution effects for PMI. Among the aspects related to the design of new NLO materials, we particularly concentrate on the vibrational versus electronic contributions. Many studies have already demonstrated that, for conjugated organic materials, the vibrational hyperpolarizability can be at least as important as its electronic counterpart for many NLO processes. This is in contradiction to an old affirmation that the vibrational hyperpolarizability at an optical frequency was negligible with respect to its electronic counterpart. This subject is covered in detail in Sections 3.2.4 (for p) and 3.3.2 (for y). Topics discussed include the type and frequency of the important normal modes, cases where the vibrational response is much larger than the corresponding electronic response, the role of anharmonicity, the importance of electron correlation, and an alternative description of the vibrational motion in terms of field-induced coordinates. A particular model—the two-state VB-CT model— that has been widely used to treat electronic and vibrational (hyper)polarizabilities is assessed in Section 3.2.5. Less conventional second-order and third-order NLO materials are described in Sections 3.2.6 and 3.3.3, including octupolar molecules, buckminsterfullerene, quadrupolar systems, and spirocompounds. Finally, another aspect that is beginning to come of age is charged systems. We will see in Sections 3.2.7 (for p) and 3.3.4 (for y) that charging can lead to very large enhancements of both the vibrational and the electronic hyperpolarizability. Individual chains bearing charged sohtons or polarons are considered both with and without the counterion, and cyanine cations are discussed as well.

65

THEORETICAL APPROACH TO DESIGN The role of the medium is reviewed in Section 3.4. It is critical to understand this role to make the Hnk between theoretical characterization of a single molecule and the macroscopic properties of NLO materials. Experimental measurements are often carried out in solution. The treatment of solvent effects based on placing the NLO solute within a cavity that is embedded, in turn, in a dielectric medium is detailed in Section 3.4.L Comparison with experiment relies on the introduction of field factors that we will see are often misinterpreted and not easily determined. For practical devices, solid state effects, discussed in Section 3.4.2, are of paramount importance. In the case of (3 materials, these effects have received a lot of experimental attention because of the asymmetry requirement but, theoretically they are usually evaluated by using the crude electron gas model. Initial calculations based on ab initio cluster models indicate the potential for large modification of the NLO properties due to electrostatic intermolecular interactions. Although these calculations can only be carried out for relatively small clusters, we will see that they provide the parameters necessary for a classical electrostatic interaction model based on distributed (hyper)polarizabilities that can be extrapolated to the macroscopic limit. Our aim is not to be exhaustive in detailing all existing techniques and mentioning all calculations performed on conjugated organic materials but rather to provide an overview of the state of our knowledge emphasizing areas that need further investigation and design directions that may prove fruitful in the future. Among the aspects we have not addressed are near-resonance NLO phenomena and conjugated chromophores containing transitionmetal atoms such as the phtalocyanines, porphyrins, and ferrocenyl derivatives. Throughout the chapter we have adopted the atomic unit (au) system. The corresponding conversion factors to SI and cgs-esu units are as follows:

system. The electric dipolar term is generally dominant. Consequently, except for a few studies where this term is negligible (see, e.g., [6]), the electric dipole approximation is adopted or, equivalently, the electric field is assumed to be spatially uniform. In that event, the total energy may be written as a function of the electric field and thefield-dependentdipole moment:

H = H=^-1

A(E')-dE' ^ A(E) E

After expanding fi(E) as a power series in E,

(2)

+ ^K^^^ ^ i 8 ^ ^ ^ ( - ^ c r ; (ou C02)E^(cc.i)E^(a>2) + iK^^) Yl y%y8(-^cr\ I3y8

a>i, <02, CU3)

X E^(wi)Ey(w2)E5(w3) + •

(3)

we may identify the first-order response as the linear polarizability and the second- and third-order response as the first and second electric dipole hyperpolarizability. These responses depend upon the (Cartesian) field directions, indicated by the Greek subscripts /3, y, and 6, as well as the (circular) optical frequencies w^, ^2, and W3 In Eq. (3) the subscript a denotes the direction of the molecular dipole moment—either permanent or induced—and co^ = 2/w/ gives its oscillatory frequency. K^^^ and K^^^ are such that the (3 and y associated with different NLO processes converge toward the same static value. The superscript 0 is a reminder that these properties are evaluated at zero electric field. The static, dc-Pockels 1 au of polarizability (a) equals L6488 x lO'^^ C^m^ J"! (dc_P), optical reactification (OR), and second harmonic generation (SHG) processes are associated with j8(0; 0, 0), P(-o); o), 0), = 0.14818 A^; 1 au of first hyperpolarizability (jS) equals 3.2063 x 10~^^ j8(0; -o),o)) and /3(-2ft>; a>, w), respectively, whereas the static, dc-Kerr, electric field-induced second harmonic generation (dcC^ m^ J-2 = 8.641 X 10-^^ esu; 1 au of second hyperpolarizability (y) equals 6.235377 x SHG or ESHG), third harmonic generation (THG), and intensitydependent refractive index (IDRI) responses are determined by 10-65 C^m^ J-^ = 5.0367 x 10-^0 esu. y(0; 0, 0, 0), 7(—w; w, 0, 0), 7(—2cu; o;, w, 0), 7(—3cu, ct>, a>, w), and y{—o)\o),-o),o)), respectively. Degenerate four-wave mixing (DFWM) and electrooptic Kerr effect (EOKE) are synonymous 2. METHODOLOGIES with IDRI and dc-Kerr, respectively. Sometimes the Taylor series There is more than one quantum mechanical approach to dealing in Eq. (3) is replaced by a power series (fractional coefficients rewith the interaction between electromagnetic radiation and mat- placed by unity) [7], which means that one shoud be careful to ter. The most common is the semiclassical approach used here specify the convention. wherein the particles are treated quantum mechanically while a For convenience one may suppress the vector or tensor nature classical treatment is applied to the radiation. In this approxima- of the various quantities in Eq. (3), as well as their frequency detion the effect of matter on the radiation field is ignored and the pendence, to obtain Hamiltonian is the sum of two terms, one representing the isolated molecule (or material) HQ, and the other being the radiationM(E) = / + a^E + i j S ^ E + ^ / E E E + • • • (4) molecule interaction term H^. If the wavelength is larger than the molecular dimensions, then H^ can be expressed as the multipole Then a combination of Eqs. (2) and (4) yields the energy expression expansion E{E) = EE=O-IJPE-ia^E-

Hi = -A • E - m X H - 2 .VE +

iiS^EEE-^yOEEEE+. • • (5)

(1) which leads to the following alternative definitions of the molecuwhere one recognizes successively the electric dipole term, the lar properties in terms of derivatives with respect to a field: magnetic dipole term, the electric quadrupole term associated with the field gradient, and so forth. In Eq. (1) the fields and field gradients are evaluated at the origin of the molecular coordinate

66

CHAMPAGNE AND KIRTMAN (0|MS^|fc)(A:|/LifO|/)(/|,.OA/,^)(^| W0|,

(d'EiE')\ ^ /
/3(E)

E'E'EE'E

L K M

(7)

k

I

(0|/ig^|A:)(A:|/.0^|/)(/|/l^^|m)(m|MfO|0)

+E'E'E'EE L

M

k

I

(a>^ - Wo-X^L/ - ^ 2 - ^ 3 ) ( ^ M m " ^ s )

^

,

^^ ^

+E'E'EEE'^

V ^E'4 / E ' . E V ^E^3 ; E ' = E

- l ^ g ^ JE'.E - V^E^ JE^.E

K

^'^

Equations (6)-(8) give rise to different computational procedures depending upon whether or not analytical derivatives are available. Finally, these expressions pertain to a molecule at a fixed geometry. When the equilibrium geometry changes induced by the applied field are taken into account the relationships become more complicated. Such considerations become important in electric field simulations of donor-acceptor molecules or solvent effects [8, 9] and are the basis for the finite field treatment of vibrational hyperpolarizabilities discussed in Section 2.5.

k

I rn

M

k

I

(0|/iQ^|/:)(A:|Af^|/)(/|/x^Q|m)(m|iLt;00 „

^Kk

- ^ 2 - W3)(^m - ^ 3 )

(0|;i^"|/:)(/:|^^"|/)(/|/x7|m)(m|/if"|0)

^ (0|iLtOO|^)(^|^OL,/^^/|^LO|^^^^,^(

oo„

k - (O(j){o)u - (02- (03)((x)m - W3) {0\fi^j^\k){k\fi^^\l){l\fi^\m){m\fif\0) K

L

II

k

k

m ^"^Kk

(O(j)i(Oi - 0)2 -

00 73 {(O/^ - (Oa)i(ou

I

- (02)i(Oii

(02)((0m

a>3)

+ (02)

,{0\fil^\k){k\fji^^\0){0\fi^\l){l\fif\0) k

{(OKk - (0(r){(Oi - W3)(w/ + (02)

I

{0\^JL^\k){k\^f\l){l\fjLf\m){m\fif\0)

+E'E'E' (o)]^ - (Oa)((Oi - a>2 - 6J3)(^m - ^3) /

m

,{0\fM^\k){k\fi^\0){0\fi^\l){l\,jif\0)

-E'E'

:i

(10)

(0)^ - (0(r){(Oi - 0)2)((Oi + 6>2)

Here {K\il\L) = fi^^ has been employed and the corresponding expression for j8 is given by BN-40. The electronic component corresponds to the terms which remain (for j8 see BN-39) e(SOS). X 'YaPyS (-^(T\ (01, 0)2,(02) e(+)SOS, . , e(-)SOS. . = yal3y8 ("^o"' ^ 1 ' ^ 2 ' ^3) + Ja^yS ("^o-^ ^ 1 , ^ 2 ' ^ s )

^^"^EP-a,2,3

E^E'EEE K L M k I fn

(+)SOS. X , (-)SOS. X = yapyS ( - ^ o - ; W i , 0^2, ft>3) + 7a|3yS (-ft^cr; tOi, &>2, ^ 3 )

- (^(T)i(ou

^

,

Much of the work presented here (and in several other subsections) has been carried out in collaboration with D. M. Bishop, who also has a chapter (co-authored with P. Norman) in this handbook [11]. The primary focus of their chapter is on small molecules whereas we are more concerned with large conjugated systems. For our treatment to be complete and self-contained there will be some dupUcation but, wherever possible, this will be avoided by referring to the chapter by Bishop and Norman (BN). Our starting point for separating the electronic and vibrational responses is the sum-over-states SOS perturbation theory expressions [10] for the first and second hyperpolarizabilities in the adiabatic approximation. The resulting expression for /3 is given by BN-34; for y we have

E'E'E'

L

^? ? ? ?

K

2.1. Separation of Electronic and Vibrational Responses

-h-'j:^-.X2,3

(ftifcjf - Wo-Xw/ - £02 - '03)('«Mm - ««3)

m

{0\^i^a^\k){k\fjif^\l){l\tl^^\m){m\fif^\0) ((OKk - (0(T)((OU

- (02-

(03)((0Mm " ^ 3 )

(0, 0\ixa\K, k){k, K\Up\L, l){l, L\fiy\M, m){m, M|Ag|0, 0) ((OKk - <^CT){(^LI

-Oil-

m)i(OMm

- 0)3)

.k,K l,L m,M

(0, Q\ka\K, k){k, K\iip\0, 0)(0, OI^^IL, /)(/, L\jx8\0, 0)

-E'E-

{(OKk - 0)(T)i(Ou

- (03)i(0u

+ (02)

(9)

-E'E'EE K L I

( 0 | # I A:) (A:| AFIO) (0| A?^ |/) (/l/xf^ |0)' ((OKk - (Oa)((Ou - (02){(ou + ^2)

(11)

If Ei(R^) is the electronic energy (including nuclear repulsion) in which the set P-o-, 1,2,3 consists of the 24 permutations of the at the equilibrium geometry of electronic state /, and ej is the pairs (-(x)(r,ct), (coi,P), (a>2, 7), and (CU3, 6). Here \K,k) = vibrational energy corresponding to<^|, then (t>f'i^K is a vibronic wavefunction wherein
v(SOS). ^aPyS

(-(^(T',

. 0)1, 0)2,

(^3)

but employing electronic rather than vibronic states. For the electronic component, the result depends explicitly upon the nuclear configuration R. After zero-point vibrational averaging over <^Q
67

THEORETICAL APPROACH TO DESIGN

also be shown that thefirst-orderZPVA can be used, in conjunction with finite field methods [18, 19], to recover part of the pure vibrational hyperpolarizability. The CN analogue of Eq. (14) for y^ can be found by substituting Eq. (13) into Eq. (10) and, then, doing a closure over vibrational states wherever they do not appear explicitly in the denominator. For instance, the first three terms on the right-hand side of Eq. (10)—which form the double harmonic approximation contribution—^become

[13] one obtains (see BN-41 for jS) e(CN), {-(OCT; (Oi, 6^2, W3) ''afiyS e(+)CN. . , e(-)CN. . = yaPyS ^ ~ ^ ^ ' "^1' ^ 2 ' ^3) + ya^yS ("^o"' ^ 1 ' ^ 2 ' ^3) h-3

EP-,1,2

E'E'Etol

,,OKr,KLr,LM,,MO fia (^^ H'y Mg (O)K(R)

- (Oa)i(OL(^) - ^2 - ^ 3 ) ( ^ M ( R ) " ^ 3 )

L/i: L M t^a ^p My Mg

-E'E'o (w;^(R) - Wo-)(w/^(R) - tU3)(a>^(R) + ft>2)

(14)

Although other choices are possible [14], the nuclei are usually clamped at the equilibrium ground state geometry (determined either theoretically or from experiment) and the ZPVA correction is, then, given by

Eo E( (OK - O^a) h-'E^-

•mm)f

(o)i - W2 - ^ 3 )

+J^{0\t,^\k)lk\ E' {(01

^E'"

r

/xf/ilfA^i"

- (x)2- (03)((0M - (03)

Mf^fM^" ((OK - 00(T)((OL

((Om - (03)

- ( 0 2 - (03)

(17) The complete expression is quite messy but at the double har= ip\y^(-a)(r; ^ i , W2, W3)(R)|0| - r^(-Wo-; 0)l, 0)2, ^3X^0) monic level the result is fairly simple (see below). (15) The accuracy of the CN approximation has been addressed by with an exactly analogous formula for j8. This correction behaves Bishop et al. [20]. Neglecting effects due to mechanical and eleclike the electronic component as far as its dependence upon the trical anharmonicity they have developed compact expressions for optical frequency and process is concerned. The corresponding the difference between this approximation and the exact SOS for(pure) vibrational properties behave quite differently as we will mula. Although frequency dependence was included we present see. In particular, the latter typically decrease upon going from only the static results because their assessment of the numerical repercussions for prototype conjugated molecules is based on the static to optical frequencies whereas the former increase. With the exception of diatomics, which are far from our present latter. It was assumed that vibrational energy differences are small concern, the ZPVA correction is determined in practice by means compared to those associated with electronic motions and, thereof perturbation theory (see also Section 2.3). Thus, the electronic fore, the energy denominator corresponding to Eq. (12) may be hyperpolarizability functions (as well as the vibrational potential) written through first order as are expanded as a power series in Q = R - RQ. The terms that are linear in Q are zeroth-order in electrical anharmonicity, the terms quadratic in Q arefirst-order,etc. Then, the expectation values are o ^ obtained by treating the anharmonic vibrational potential as a per-1 (18) turbation. The terms in the potential that are cubic in Q are firstA^ ^K order in mechanical anharmonicity, the quartic terms are secondorder, and so forth. Using this scheme the zeroth-order—or dou- A second relation, identical to (18) except that e^, {k'K' = ble harmonic—ZPVA correction vanishes by definition whereas kK, 00) is replaced by E/^/(R) - E/^/(R^,), is utilized as well. Afthe first-order (in electrical + mechanical anharmonicity) term is ter a derivation that is complicated, but contains no further asgiven by sumptions, the following first-order differences for y are obtained (Ay^ = Ay^(+)-A7^(-)): ^pZPVA ^ [P]0,l + [pjl,0 ^^ZPVA(_^^. ^^^ ^^^ ^^)

4

^

^ « = 4EP-M,2,3EE^E'E> A 0

/.2

A0 A0

K L M (16) for any electrical property P. In Eq. (16) the superscripts indicate the order of electrical and mechanical anharmonicity, respectively; (Oa is the circular vibrational frequency corresponding to the normal coordinate Qa, and F^bb is a cubic force constant. It is easy to demonstrate that the next nonvanishing contribution is third-order [15]. Even infirstorder the computations required are quite demanding so that applications have been limited to small molecules. However, the possibility of treating large conjugated molecules has been opened by the introduction of field-induced coordinates [16, 17] that reduce the number of mechanical anharmonicity constants to be evaluated from 0{f^) to 0{f) with f = 3N - 6 being the number of vibrational degrees of freedom. We will elaborate on this point further in Section 2.5, where it will

r(;.f)^(AfA^^Mr)' + (19)

A^ 1

2 E^-«^4,2,3EEE A0/A0X2 (QlAtr)^(Mf^)lQ)(QlMfMiQ|0) To

68

CHAMPAGhfE AND KIRTMAN ,OK..KO 2(0|M^^/.;^"|0)(0|(/t!;^)'(/it")'|0>

+

^1 K

L

uPK KO A*'

m

//Mi''

(20)

(A?)2

^T^0y =

^-EP-^-1,2,3E

I:E

0^2M0_

(M°°)' L

ho)a

M

2.2. Methods for Determining Electronic Polarizabilities and Hyperpolarizabilities

(A^2(A0,)

i^i^y-^f +I:'E1[ u^^'-^f)' ( A 0 . ) ( A 0 )2 (A0,)2(A0 ) J L

M

(21)

ho)a

where the derivative with respect to Qa, evaluated at R^, is indicated by a prime to the right of the parentheses. Similarly, for j8

^Pl

•M

4EP-^4,2EE'E'(O A | A ^ a

K

L

AO

(22)

EP- o-,l,2

A^a^r =

E L is:

+E

(A0)2(/ia.a)

orbital (LUMO) and the ^Ag state, which has the HOMO —> LUMO double excitation as its most important component, were utilized. The nature of the ^Ag state necessitates the use of a configuration interaction singles and doubles treatment and, hence, the excited state geometry optimization was carried out by a finite distortion procedure. One vibrational normal mode, a combination of bond length alternation and hydrogen wagging, makes the largest contribution by far (almost 50%) to y^^^N) j j ^ ^ resulting 3-state-l-mode model gives Ay^ = 14 au and Ay^ = - 7 au whereas y^^^N) ^ 2784 au and y^^^N) ^ ^QQ J^^^^ calculations suggest that corrections to the CN approximation will generally be negligible for nonresonant phenomena.

(23)

L The static longitudinal j8 and y of a small push-pull polyene, NH2—(CH=CH)3—NO2, and of butadiene were considered for a preliminary evaluation of the CN approximation. All calculations were done in a 6-31G basis. For the D/A polyene, the necessary excited state information—geometry, dipole moments, excitation energies, and transition dipole moments—^was calculated at the Configuration Interaction Simples (CIS) level using the 60 lowest-energy singlet states. In addition, an restricted HartreeFock (RHF) treatment was employed to determine the ground state vibrational normal mode frequencies and coordinates. Only two excited electronic states were retained owing to their dominant contribution; they give j8^^^^^ = 4882 au. Similarly, 99% of the total CN j8^ value of 9824 au is recovered by limiting the sum-over-mode expression to 9 modes. Within this 3-state-9-mode model, Aj8^ and A/3^ attain values of 0.9 au and -148 au, respectively. For rran5-butadiene, two excited electronic states again proved sufficient for qualitative purposes even though they give somewhat less than 50% of y^<^^N) ^^ ^^m case, the lowest-lying ^Bu state, described mainly by the transition from highest occupied molecular orbital (HOMO) to lowest unoccupied molecular

There are many procedures for evaluating electronic polarizabilities and hyperpolarizabilities and various ways of classifying these procedures. One convenient division is between SOS methods and techniques based on differentiation of the energy or dipole moment with respect to external fields. Propagator or response function approaches, where the usual SOS expressions are recast into the form of a superoperator resolvent, Ue in the first class although the formalism and strategies are different. The second class contains classical "wavefunction" approaches as well as density functional theory methods based on determining the total electron density. Due to the plethora of papers dealing with the development of suitable exchange-correlation (XC) functionals for computing linear and nonlinear responses by DFT, we treat this subject separately in Section 2.3. SOS methods consist in evaluating the quantities that appear in Eq. (14) (or in the analogous equations for a, j8) and, then, carrying out the indicated summations for the desired set of electric field frequencies. The crudest of the SOS schemes, referred to as the uncoupled Hartree-Fock (UCHF) method, approximates the exact wavefunctions by Slater determinants composed of (fieldfree) Hartree-Fock orbitals and the exact energies as the sum of the corresponding orbital energies. Thus, the excitation energies are given by orbital energy differences between unoccupied and occupied one-electron states [21]. In this scheme the fieldinduced modifications of the electron-electron interactions are not accounted for and the SOS expressions reduce to summations over molecular orbitals. Although it has been shown in ah initio calculations that this approach strongly underestimates the linear and nonlinear responses, it remains in use for large molecules (see, e.g., [22]) where improved treatments are out of reach and it is also the basis of Hiickel-type semiempirical methods (cf. [23]). At best, only qualitative aspects can be rehably investigated at the UCHF level. The next step forward, generally, is to improve the state descriptions using configuration interaction (CI) with either single excitations (CIS) or single and double excitations (CISD), etc., or a multireference version of these schemes. With the exception of investigations on small systems [24] or studies on several individual TT-conjugated compounds [25] SOS-CI procedures have been applied primarily at semiempirical levels (see, e.g., [26a] for j8 and [26b] for y) with the goal of defining the essential excited states. However, these approaches suffer from size inconsistency due to truncation of the CI expansion not only with respect to the excitation level but also with respect to the number of excited states (within the given basis set) at each level. The UCHF and

69

THEORETICAL APPROACH TO DESIGN CIS (all excited states) wavefunctions are size-consistent but all other levels of excitation, short of full CI, are not. This limitation is particularly problematic when considering the dependence of the NLO response as a function of the system size. Size-consistency problems that arise from truncating the number of excited states in the SOS expression have been demonstrated in semiempirical Pariser-Parr-Pople UCHF calculations on increasingly large polyene chains [27]. Further investigation of these truncations is needed for models that correlate the electron-hole (e-h) pairs formed upon excitation. The SOS-CIS approach has had its greatest success in computing the trends in behavior of j8 for a large range of organic molecules (see, particularly, the review by Morley [28]; see also [29]). When considering y it is essential to include double excitations [30]. To avoid truncation with respect to the number of states at the double excitation (and higher) level, several elegant approaches, including the correction vector (CV) [31] and the density matrix renormalization group (DMRG) [32], have been proposed. Application of these schemes has been limited primarily to semiempirical treatments [CNDO or ZINDO for the CV method and the Hubbard model for the DMRG procedure], which raises questions concerning reliability and internal consistency. As pointed out by Kanis et al. [29] the semiempirical model Hamiltonians used are often parametrized on the basis of optical properties calculated at the CIS level. On the other hand, these approaches offer computational advantages for studying both the resonant and nonresonant frequency-dependent NLO response. The main advantage of the SOS approaches is that the interpretation of the second- and third-order responses is simplified in cases where there are just a few dominant excitation channels. For j8, the simplest scheme limits the SOS expression to a unique excited state [33]. In the resulting two-state approximation (TSA) the static j8^ is given in terms of spectroscopic quantities as

to describe the effect of a solvent on the cleavage of ionic bonds. In the VB-CT model the electronic wavefunctions of the ground and excited states are each written as a combination of the limiting covalent and charge-transfer (zwitterionic) wavefunctions, ! and (t>2 respectively. Thus, ^ g = cos 001 + sin d(f>2

(25)

^ e = sin 001 - c o s 002

(26)

where the mixing parameter 6 is determined by the stationary condition on the energy, which gives sin 20 = 2r/AEge

_^

(A/x,ge)/ge

a^(0)

2^l^ ge

AE,'ge

2t^^al Hi ange

P^(0) =

^ ^ (sin 20)2 : 2AE,'ge

6jLt|e[Mee

3A/xL r. — ^ i ^ (COS 20) (sin 20)2 2A|e

— ^ (cos 20) (sin 20)4

Here figc is the transition dipole moment between the ground and excited state, AEge is the corresponding excitation energy, A/i,ge is the difference in dipole moment between the ground and the excited state (^tee - Mgg)? and /ge = (/Xge)^AEge is the oscillator strength. The TSA has received considerable attention during the last two decades because the first hyperpolarizability of most TT-conjugated push-pull systems is predicted with reasonable accuracy by considering just the lowest-lying longitudinal chargetransfer excited state. From semiempirical CIS calculations, it has been shown that such a treatment generally overestimates jS^ by a factor of 2 whereas the inclusion of a third state corrects the estimates to give 80% or more of the full CIS value [29]. However, the third state has not been characterized as yet so that one must explore the set of low-lying states to find it. In the TSA the ground state is often taken to be the HartreeFock wavefunction whereas the excited state corresponds to the HOMO —> LUMO transition. In general, of course, correlation can be included in either or both of these states by the CI methods described previously. Another simple way of doing this is through the semiempirical two-state valence-bond charge-transfer (VBCT) model. This model was first proposed by MuUiken [34] for investigating structural and spectral properties of molecular donoracceptor complexes. It has also been used in a form called the empirical valence bond (EVB) method by Warshel and coworkers [35]

=

=6

(AEge)3

(28)

Aie A^ ^ge

(24)

(AEge)2

^ i ^ (sin 20)^ At

•f^ggi

6t^Alil2V

/(O) =

i8"(0) = 6

(27)

with F = H22 - H i i and t = H^ (Hi] = {0/|H|0y)). Given the wavefunctions of Eqs. (25) and (26), electronic (hyper)polarizability expressions can be obtained by substituting into Eq. (24) and its analogues [36, 37] or, equivalently, by following the FF procedure [38, 39],

2

(A^ge) (figer

cos 20 = F/AEge

(29)

.A

24[/i|e(/^ee - Mgg) - Mgel A^ ^ge ^

= —1II2 |5(^Q5 20)2 - 11 (sin 20)2 Age

2Age

i^r5(cos20)2-ll(sin2e)^ 16?3

(30)

[In deriving Eqs. (28)-(30) it is assumed that 1^12 = 0 so that /i-ge = /(A)ai2)/Age and A/i-ge = V(^^ll2)/^ge with A/;ti2 = M22 - Mil-] It is possible to estimate values for the parameters that enter into Eqs. (28)-(30) from various experimental measurements [40] and thereby estimate the (hyper)polarizabilities. However, the most useful aspect of the treatment probably Hes in the universal relations among a^(0), p^{0), 7^(0), and the mixing parameter 0, which are represented in Figure 1 for compounds ranging between the covalent 0 = 0 and zwitterionic (0 = IT/2) forms. The division into three regions in the figure provides a convenient classification scheme that is also relevant for setting strategies to maximize the electronic NLO response of push-pull conjugated molecules. We note that the VB-CT model has been extended to quadrupolar and octupolar molecules [41, 42] and to vibrational hyperpolarizabilities as well [37, 39, 43, 44]. For the latter case, however, the assumptions that are made have been called into question [45]. In fact, it has been shown that one cannot choose a consistent set of parameters for both the electronic and vibrational hyperpolarizabilities [37] (see Section 3.2.5).

70

CHAMPAGNE AND KIRTMAN Region A

Region C

Region B

Hi = -A • E

a"

0.0

\ \

/

** —/ ^ ^* / V

\

/

0=71/4

9=0.0

erties. One simply adds the perturbation potential

•

T* *^

0=71/2

Fig. 1. Schematic behavior of a^, j8^, and y^ versus Q obtained from the two-state VB-CT model and division into three regions according to their sign and ampHtude. Reprinted from D. M. Bishop et al., /. Chem. Phys. 109, 9987 (1998). Copyright 1998, American Institute of Physics, Woodbury, NY.

To remove the two-state limitation Soos and Ramasesha [46] have developed a diagramatic valence bond CI method (DVB) that utilizes the CV technique. Their procedure takes into account all ionic and covalent states that can be constructed within a given basis. These states cannot be simply categorized with respect to the usual molecular orbital CI scheme; they contain, for example, all excitation levels. Because of the large number of states, application of DVB has been limited to modestly sized conjugated systems within the semiempirical Pariser-Parr-Pople (PPP) framework. Although a two-state approximation has also been used for the second hyperpolarizability within the VB-CT model, this is undoubtedly insufficient. A three-state treatment has been proposed, originally for push-pull systems [47]: 7^(0) = -24

+ 24

(Atge)^(A/ige)2

(AEge)3

+ 24

(Mge)^(/^eeO^

(AEge)3

(31)

(AEge)2(AEge/)

The first term on the rhs is the negative (N) term, the second is known as the dipolar (D) term, and the third is often referred to as the two-photon term (T). For centrosymmetric systems the D term vanishes and, thus, a positive r^^^) ^^^^ ^^ ^^^ ^^ ^^^ ^ ^^^^- ^^ the case of polyacetylene considerable effort has been devoted to identifying the "essential" e^ = m Ag excited state state [48, 49]. Because of the difficulty in specifiying e and e' a priori the threestate approximation is commonly improved by summing over e^ in the T term [50] or over a limited number of the excited states e and e^ [49]. The expression on the rhs of Eq. (31) can also be rewritten in terms of approximate polarizabilities of the ground and excited electronic states [51]. On this basis Nakano and Yamaguchi, using the three-type treatment, have come up with a convenient classification scheme to rationalize the cicumstances under which one can expect to find a large y^. The second major class of computational procedures relies on the evaluation of field derivatives of the dipole moment or the energy to determine polarizabilities and hyperpolarizabilities. These derivatives may be obtained either analytically or by numerical, that is, finite field (FF), techniques. Static FF methods are certainly the most straightforward of all to apply. The RHF version was first proposed by Cohen and Roothaan [52] for atomic prop-

(32)

to the Hamiltonian, where E is a static electric field vector of a given magnitude, and solves the self-consistent coupled HartreeFock problem in the same manner as for the field-free case. From the computed value of the dipole moment (or the energy) for fields of different magnitude and direction one can obtain the expansion given in Eq. (4) [or Eq. (5)] by numerical differentiation. This procedure is general in the sense that it can be applied with any quantum mechanical method for which the fielddependent energies (or dipole moments) can be determined. Although one cannot differentiate numerically with respect to a timedependent field it is still possible to use the FF method to obtain y(-w; 0), 0, 0), for example, as the first derivative of P(—o); co, 0) or the second derivative oi a{-oj; co) [cf. Eq. (8)] provided, of course, that one or both of these frequency-dependent properties is available. Even when there are no static fields involved in the property of interest—/3(-2a>; a>, w) is an example—it is possible to do the computation by taking the numerical derivative of an appropriately defined lower-order (in this case a) frequencydependent property [53]. It is important to note that the property values calculated from Eqs. (4) and (5) are guaranteed to be the same only if the Hellmann-Feynman theorem is satisfied, which, in turn, requires that the wavefunction be fully optimized (within the given basis) according to the variation condition. When the latter requirement is not met (e.g., MP2, CCSD) numerical differentiation of the energy is preferred. As will be detailed in Section 2.6, the FF approach can suffer from numerical accuracy problems, especially when the order of the derivative is high. Other potential difficulties to bear in mind are (1) the perturbation potential V is unbounded and (2) the wavefunctions of interest correspond to metastable states that are generated by using an incomplete basis set [54]. Analytical differentiation is required for time-dependent fields and, in the static case, is an alternative to the FF method. At the ab initio level the time-dependent Hartree-Fock (TDHF) approach [55, 56] and its static equivalent, the coupled-perturbed Hartree-Fock (CPHF) procedure [57] (also called CHF procedure), are the most commonly used. In this approach one determines the order-by-order response of the density matrix to the perturbation potential given in Eq. (32), but modified so that the electric field may have dynamic as well as static components. The Hartree-Fock perturbation equations are solved in each order via an iterative self-consistent-field (SCF) procedure and, then, the frequency-dependent properties are evaluated from the product of the dipole moment matrix m and the density matrix D. Thus, for example, 7ai3y5(-^o-; ^ 1 , ^ 2 . ^ s ) = -Tr[ma;D^^^(o>i, (02, c«>3)]

(?^)

where the sub(super)scripts a, /3, y, 5 indicate the particular Cartesian component and the number of these sub(super)scripts corresponds to the order of perturbation theory. It is also possible to reduce the order of perturbation theory required, and thereby improve computational efficiency, even though the expressions are more complicated [56]. Correlation may be added to the TDHF treatment by various means. One way is by many-body perturbation theory. Wormer and coworkers [58] use a double perturbation expansion of the

71

THEORETICAL APPROACH TO DESIGN dipole moment expectation value to obtain a{-a); (o) through second order in the correlation potential. They call their method SDT-MBPT(2) because triple, as well as single and double, excitations must be taken into account. However, "apparent" correlation effects due to the self-consistency requirement on the TDHF wavefunction are included only through second order. By introducing full self-consistency Hattig and Hess [59] generate the TDMP2 method. The approach of Rice and Handy [60, 61] is similar except that, starting with a pseudo-energy, they develop frequency-dependent relations that can be numerically differentiated to yield the next-order frequency-dependent hyperpolarizability. However, their approach omits the second-order poles that are present in the more accurate treatment of Hattig and Hess. Detailed formulas are given by Rice and Handy at the MP2 level and calculations are reported for jS^^^C —0); 0), 0) and ^MP2^_2^. ^^ ^ ) of several small molecules. Itoh and coworkers [62, 63] adopt the same starting point as Rice and Handy but remove an assumption made in the pseudo-energy treatment. This leads to their quasi-energy derivative (OED) formalism, which reduces to the TDMP2 method described previously for a^^^{-(o; co). The generalization to hyperpolarizabilities, as well as to the coupled cluster (CC) and CI methods, has also been given.

9.00x105 0.4

0.6

^0) in eV Fig. 2. TDHF/6-31G Dispersion of y£(-a)o-; wj, ^2, ^3) (in atomic units) for the most common NLO processes. Reprinted from B. Champagne et al., Int. J. Quantum Chem. 70, 751 (1998) by permission of John Wiley & Sons, Inc.

where co^ = CD^ -\- J^i ^j- The important point here is that A (or A^) is a constant that depends upon the molecule [76-78] but In polarization propagator, or response theory, approaches is independent of wi, (02 (or ^1,(^2, ^3). For the major NLO [64], the poles and residues associated with the hyperpolariz- processes cj^ = k x w^; in the case of the electronic second hyability superoperator resolvent determines the entire frequency perpolarizabilities, for example, /:(THG) = 12, A:(ESHG) = 6, spectrum. Several different streams that lead beyond TDHF or, A:(DFWM) = 4 and A:(dc-Kerr) = 2. Thus, in the low-frequency equivalently. Random Phase Approximation (RPA) can be dis- regime, the y^ values for these processes will occur in the ratinguished. One of these is the development by J0rgensen and tio determined by the value of k, as illustrated in Figure 2 for coworkers—followed by Agren and his collaborators—of multi- Sii2H26. We will see later that the vibrational hyperpolarizabiliconfigurational self-consistent-field (MCSCF) approximations for ties do not follow this rule and their contribution can invert the the Hnear, quadratic, and cubic response functions [65]. Very ordering. Divergences from the simple ratio rule given previously recently, full CI calculations have been reported [66]. Another are expected as co increases (for polysilane chains, see [79]), parstream consists of a hierarchy of CC response methods elabo- ticularly in the vicinity of resonances. Recently, Hattig [78] has rated by J0rgensen and coworkers [67]. Both the MCSCF and CC extended the dispersion relations through eighth-order in (o and approaches have been extended to the determination of excited has shown that a single fourth-order coefficient suffices to destate polarizabilities [68] as well as multiphoton absorption inten- scribe the first hyperpolarizabilities [i.e., electrooptical Pockel efsities [69]. In connection with CC approaches it is important to fect (EOPE) and SHG] as well as the second hyperpolarizabilities include the equation-of-motion method [EOM-CCSD] of Bartlett with one or more static fields. The latter accounts for EOKE and ESHG but not THG or DFWM for which one additional term, in and collaborators as well as its Cl-like approximation [70, 71]. either case, is required. Thus, the entire low-frequency spectrum Correlated methods that include frequency dispersion are difcan be determined from a few frequency-dependent calculations. ficult to apply to NLO systems of practical interest due to their Alternatively, one can evaluate the coefficients a priori. General substantial computational requirements. With the exception of a schemes for this purpose have been elaborated by Hattig and J0rfew studies on prototype nitroanilines [72], these techniques have gensen [80] employing the CCS, CC2, and CCSD techniques and mostly been employed on small (two- to five-atom) molecules. For more recently by Larsen et al. using full CI theory [81]. the simple diatomic molecule HF there is still a significant discrepThere is one further type of approximation that is commonly ancy between experiment and theory even when the vibrational made to eliminate frequency-dependent correlation calculations contribution is taken into account [71, 73, 74a]. In the case of butadiene, which is the simplest linear polyene, the importance of altogether. This is done by assuming that the TDHF results capture the essential frequency dependence. Thus, one possibility is electron correlation has been a subject of much debate [74b, 75]. In the low-frequency region of the spectrum the computational to use the TDHF A and/or A^ or, more generally [74], to multieffort involved can be reduced by taking advantage of the expan- plicatively scale the static values correlated (CORR) according to sion formulas for the diagonal components [as well as the "aver- the rule age" quantities, jSy, J8_L, y\\, and y^ (see BN 5-26 for their defini^ C O R R ( ~ ^ O - ; wi,a>2) tions)] that is, = ^CORR(0;0'0)

l3^(-(0a-\ cui, cc>2) = i8^(0; 0, 0 ) [ l + Aa)l + 0(a>^)]

(34)

/ ( - c u o r ; 0)1, (02, co^) = / ( O ; 0, 0, 0)[H-^'ft>^ + O(cu^)] (35)

^TDHF(~^O"^^1>^2)

(36)

i8^PHF(0;0.0) A similar multiplicative or percentage correction can be written for the second hyperpolarizability. Alternatively, one can use an

72

CHAMPAGNE AND KIRTMAN

additive correction [60],

the calculation is done with an adequate basis set, the DFT result may or may not be better than Hartree-Fock. Nonetheless, because functionals are still rapidly evolving [88, 92] it is likely that = ^CORR(0;0,0) significant improvements will be made. When it comes to spatially extended systems, which is the typ(37) CPHF (0;0,0)] ical circumstance for NLO materials, the DFT overestimate can From the expansion of Eqs. (34) and (35) through second order, become quite dramatic. In Figure 3, for example, we present a the additive correction is preferred if ^cORRiScORR(0' 0» 0) = plot of the longitudinal j8 (actually, f3/N) for the series of push^TDHFJ^TDHFCO; 0> 0)- The vahdity of these correction proce- pull polyenes N H 2 - ( C H = C H ) A ^ - N 0 2 . These calculations were dures has been tested for a few small systems and, with the ex- done in a split valence 6-31G basis (and at the MP2/6-31G geomeception of formaldehyde and acetonitrile [60, 82], the percentage try) [93]. Although inadequate for short chains, there is good empirical evidence [94-96] as well as logical argument [97] indicating (multiplicative) estimate has been well supported [83]. that such a basis gives semiquantitative accuracy, at the very least, for extended chains. Likewise, higher level ab initio calculations 2.3. The Electron Correlation Problem: Wavefunction [96, 97] show that MP2 gives a reasonable estimate of the correlaversus Density Functional Theory Approaches tion effect even if it somewhat overshoots the mark. From the figElectron correlation often plays a primary role in determining hy- ure, then, it is clear that the two DFT methods are seriously in erperpolarizabilities, particularly of conjugated molecules with large ror at the longer chain lengths. In the infinite chain limit p should NLO properties. Because these molecules tend to be large the con- saturate [98, 99] and, therefore, p/N should approach zero. This ventional wavefunction approaches described in Section 2.2 are follows from the fact that the influence of the substituents is localdifficult to apply. Although linear scaling techniques may lead to ized more or less to the chain ends. The Hartree-Fock and MP2 a more favorable situation, it is natural to consider density func- curves exhibit a maximum, as required, but the DFT curves appear tional theory (DFT) methods as an alternative. The results of to depart catastrophically from the correct behavior. (In fact, the DFT studies on atoms and small molecules in the early 1990s [84] DFT curves do go through a maximum at about A^ = 30 in STOproved to be sufficiently encouraging that further investigations of 3G calculations.) A similar situation occurs for y as may be seen in Figure 4, medium-sized conjugated organic molecules were undertaken [85] and, now, systems as large as Cgo have been treated [86]. Along- where we present results for unsubstituted polyene chains (at the side the applications there have been major improvements in the Hartree-Fock geometry). The quantity plotted is the ratio with reexchange-correlation functionals based on their ability to repro- spect to the Hartree-Fock value. As the chain length increases, duce molecular energetics. For polarizabilities and hyperpolariz- the MP2:HF ratio rapidly becomes independent of the number abilities special attention has been paid [87, 88] to ensuring the of (CH=CH) units and saturates at a value of 1.6 whereas the correct asymptotic behavior of these functionals. With the devel- DFT:HF ratios are larger and are not yet saturated at A'^ = 16. opment of DFT response theory [89] a rigorous treatment of the These MP2 calculations extend previous results [100] determined over the range 12 > A'^ > 6. The DFT value obtained with the frequency dependence of NLO properties has become available. For small, or compact, molecules there are several functionals SVWN functional, which is somewhat too large at A'^ = 10, overthat allow one to calculate a with an accuracy that is comparable shoots the MP2 value for C30H32 by a factor of about 20. Furtherto conventional post-Hartree-Fock methods [88, 90]. The situa- more, decreasing the bond length alternation in C20H22 leads to a tion is not as satisfactory, however, for j8 and y [91]. In the case of qualitatively incorrect decrease in y [101]. In fact, a plot of y verhyperpolarizabilities the value is commonly overestimated and, if

o o

^MP2/RHF -D-B3LYP/RHF -^SVWN/RHF

1; 1^ PL,

O

H

J3

I-^ o

•a c

o


Number of unit cells

Fig. 3. Evolution of the longitudinal first hyperpolarizability per unit cell, Pl^(N)/N, of N H 2 - ( C H = C H ) A ^ - N 0 2 as a function of the number A^ of CH=CH units for different methods (MP2/6-31G geometry and 6-31G basis set for the properties).

8

10

12

Number of unit cells Fig. 4. Longitudinal second hyperpolarizability of increasingly large polyene chains given as the ratio y(X)/y(RHF), where X = MP2, SVWN, and B3LYP. All property calculations were carried out in the 6-3IG basis at the B3LYP/6-311G* geometry.

73

THEORETICAL APPROACH TO DESIGN sus bond length alternation incorrectly exhibits a maximum using conventional DFT methods [102]. If the B3LYP functional were used instead our experience indicates that there would be an improvement but only a relatively small one. Building in the correct asymptotic behavior with the LB94 potential does not help at all and the story remains the same with all commonly used functionals. The one feature that does make a significant difference [101] is the fraction of Hartree-Fock exchange included in, say, the B3LYP functional. It is possible to tune that fraction (upward from the conventional 20%) so as to produce agreement with the MP2 calculation of y—at least in C2oH22- Unfortunately, the same prescription does not work for a, in which case even 100% HartreeFock exchange is insufficient. Nonetheless, this observation does highlight the pivotal role of the exchange functional (X) in describing polarization properties. Another insight into the failure of DFT is afforded by the charge distribution along the polyene chain induced by an applied field. Here we use charges obtained

.2

o o

by a simple MuUiken population analysis that is adequate for a qualitative comparison between DFT and Hartree-Fock or MP2. The top panel in Figure 5 pertains to C40H42 and the bottom panel to NH2-(CH=CH)6-N02. In either case the field is 2.0 x 10"^ au; for the push-pull polyene it is directed along the longitudinal axis so that electron density is transferred from the donor to the acceptor. It can be seen that the DFT curves correspond to uniform transfer of charge across the chain whereas the nearly identical Hartree-Fock and MP2 curves exhibit an inflection at the chain center. Thus, there is excessive charge transfer in the DFT calculation or, equivalently, inadequate screening. This is consistent with the overestimate of the calculated hyperpolarizabilities and is due to the shortsightedness of conventional X functional as discussed further later in this chapter. An important point to note is that the failure of DFT observed for TT-conjugated systems is due to the extended (in space) nature of these systems rather than the 7r-conjugation itself. We know this because the same errant behavior occurs for linear molecular hydrogen chains [103,104]. Correlation calculations of a and y have been carried out [105, 106] on such chains (i^intra = 2.0 au; Winter = ^-0 ^^) ^sing larger basis sets and higher level methods than for the substituted and unsubstituted linear polyenes. This removes any doubts concerning the level of correlation treatment and the basis set. The resuhs for H^g in Table I show large DFT overestimates similar to those seen earher in this section. It is worth noting that the error in the DFT value for a [107] is significantly reduced when the X functional is improved (SVWN -^ B3LYP, PBE -^ PBEO) or, for a reasonably good X functional, when the C functional is omitted (VSXC -> VSX).

o

Table I.

Comparison of Results for a and y of (H2)9 Obtained Using Various Methods and Basis Sets

-d I-H

4

8

12

Unit cell position

a (au)

0.10]

o P-H I-H


'2 -d o

Method

3-21G

6-31G

6-311G**

6-311 ++G**

CHF MP2 MP4 CCSD(T)

217 195 174 162

— — — —

222 217 205 200

222 218 206

SVWN B3LYP PBE PBEO VSXC VSX

—

— — — — — —

—

365 293 338 286 367 291

280

— — — —

292

— — — —

KLI

—

261 ±1

.2 7(10^ au)

13

Unit cell position Fig. 5. Field-induced unit cell charges as a function of the unit cell position along the H-(CH=CH)2o-H (top) and NH2-(CH=CH)6-N02 (bottom) backbones computed using a 6-3IG basis set. Top reprinted from B. Champagne et al., /. Phys. Chem. 109, 10489 (1998). Copyright 1998 American Institute of Physics, Woodbury, NY.

CHF MP2 MP4

279 332 285

288 346 302

296 391 382

i\r\\n
) V WIN

4300

KLI

700 ± 100

74

CHAMPAGNE AND KIRTMAN X and C potentials as well as a linkage between improvements in the X potential and improvements in the C potential [112]. Thus, it could take some time before the precise nature and the origin of the field-dependent component of the DFT potential is known. Meanwhile the KLI results suggest the addition of a term having the same form as the potential due to interaction with an external field E, but with E replaced by a screening field Escr, that points in the opposite direction. One mijht expect that the magnitude of Escr, which is proportional to E, would depend upon the polarizability and that implies a self-consistent treatment. Whether current efforts along these lines will prove fruitful remains to be seen.

z(a.u.)

Fig. 6. Constituent parts of the KLI potential of (H2)9 in an electric field with a6-311 ++G** basis. The H atoms are at alternating distances of 2 and 3 au (central pair at ±1.0 au). Reprinted from S. J. A. van Gisber2.4. Conventional Perturbation Approach for (Pure) gen et al, Phys. Rev. Lett 83, 694 (1999). Copyright 1999, by the American Vibrational Hyperpolarizabilities Physical Society. Because of the key role of the X functional, DFT calculations using the accurate X-only potential of Krieger, Li, and lafrate (KLI) [108] were undertaken [104] for Hig in the 6-311 ++G** basis. This potential exhibits many of the same features as the "exact" X-only potential [109] and, in Heu of the latter, provides a basis for further analysis of the DFT error. Using the KLI potential the overshoot in y is markedly reduced from over 1000% to 83%, whereas the corresponding reduction in a is from 77% to 27%. For interpretive purposes the DFT exchange energy may be written in the general form [110] Ex = -

dridr2p(ri)

[g(ri,r2)-l]

p(r2)

The SOS formula for the CN vibrational second hyperpolarizability, y^ = yV(CN)^ ^^^j^ ^^^ derived as indicated in Section 2.1 by the procedure leading to expression (17). To proceed further Bishop and Kirtman (BK) [12] make two assumptions. One is that in the nonresonant regime optical frequencies may be neglected in comparison with electronic transition frequencies (see also Section 2.5). When this is done expression (17) becomes (0\aap\l){l\ayS\0)

h-^Z^.o-,l,2,3 .

/

+IE'

(38)

|ri - r2l

+S'

which leads to a partitioning of the X-only potential into a hole term and a response term, that is. i;x(r): :4«l^(r) + 4^^P(r)

(39)

with hole/ X

f J

r

J^(r,ri)-1]

(40)

|r-ril and resp

(r)

-\lh-

dv2

p(ri)p(r2)

Bg{ri,T2) Sp(r)

0)1 — 0)2 — 0)2

mtia\k}(k\ppysm 0)]^ — 0)a

{0\PaPy\m){m\,jLs\0) 0)m - W3

(42)

= [a^] + [fjLp]

where IJL, a, ^ are electrical properties of the ground electronic state that depend upon the normal coordinates Q, and the square bracket notation on the rhs should be obvious. The second assumption is that the power series expansions of these properties about the equilibrium position, for example.

(41)

Plots of ux, ^x^^, and v^^^ associated with the KLI potential for Hi8 in the presence of a uniform applied field in the longitudinal direction (z) are given in Figure 6. The amplitude of the oscillations in i;^^^^ is invariant along the length of the chain. On the other hand, i;^^^ shows a linear variation that partially counteracts the potential due to the applied field UE and, thereby, reduces the induced polarization. The fact that this ultra-nonlocal component of v-^"^ is missing from conventional X potentials is the reason for the excessive polarization found in ordinary DFT calculations. Although the KLI treatment yields a first characterization of the field-dependent term in the DFT potential it would be worthwhile to compare with the "exact" X potential and, in addition, the nature of the field-dependent correlation (C) term remains to be clarified. Initial studies [111] of field-free energetics show that, if the "exact" X potential is used, conventional C potentials worsen the agreement with experiment. This is consistent with the reduced error in the DFT value for a obtained in Table I when the C component of the VSXC potential is removed. It is also consistent with a fortuitous cancellation of errors between conventional

««. = «», + E(?)e« + ^E(^)e«& + •

(43)

and also the corresponding expansion of the vibrational potential, ^ = ^^ + I E

^iQl

+ \ E"^abcQaQhQc+ • • •

(44)

are convergent. In Eq. (43) we have used the notation {afi/a) = {^(^a^/^Qa), {afi/ah) = {daocfi/dQa). As far as the properties are concerned, the harmonic approximation is obtained by truncating the expansion after the linear term in Q. From a perturbation theory perspective the quadratic term is considered to be first-order in electrical anharmonicity, the cubic term is second-order, and so forth. In Eq. (44) there is no linear term due to the equilibrium condition and, thus, the harmonic approximation is obtained by truncating the series after the quadratic term. The cubic term is considered to be first-order in mechanical anharmonicity, the quartic term is second-order, and so forth. Treating the electrical and mechanical anharmonicity, by ordinary double perturbation theory one can write the square-bracket

THEORETICAL APPROACH TO DESIGN quantities in Eq. (42) as a sum of harmonic and anharmonic terms. The complete expressions for jS^ and y" are given by Plsy

= [fictf + [M«]" + • + [|a^]I + [|a3]"I + .

(45)

and lO II 'aPyS = [fipr + [fjipr + •••

+ [a^f + [a^f + ••• + [fiW

+ llJ-^af^ + ••• (46)

in which we have used the abbreviations []^ = []^'^, []^ = []0,l _|_ []l,0^ []II ^ []2,0 _^ []l,l ^ []0,2 ,^here the superscripts indicate the order in electrical and mechanical anharmonicity. In fact, the notation [X]"''" means nth order in electrical anharmonicity [(n-j) is the number of times the (2+;)th derivatives of an electric property appear, with ; ranging between 0 and n — 1] and mth order in mechanical anharmonicity [(m — ;*) is the number of times a cubic (; = 0) (F^bcX a quartic (; = 1) (Fatcd)^- • • force constant appears, with ; ranging between 0 and m - 1]. As can be seen from Eqs. (45) and (46), half of the [X]^'^ terms vanish due to symmetry. The first two lines of Eq. (46) come directly from Eq. (42), whereas the last two lines are from the remaining terms in Eq. (10) that were not explicitly considered previously. An exactly analogous derivation starting from BN-44 leads to our Eq. (45). In the harmonic approximation (n = m = 0) the first hyperpolarizability consists of just a single [^ta]^'^ term and the second hyperpolarizability is the sum of two square bracket terms, [Q,2J0,0 ^jjjj [/ij8]^'^. Compact formulas for these terms, as well as the [X]^ and [X]" contributions are available in BN (see BN, Table X) and in the literature [113]. The sums in Table X of BN run over the 3N — 6 (3N — 5 for linear molecules) normal coordinates. To evaluate the vibrational hyperpolarizabilities in the double harmonic approximation we require these coordinates, and the corresponding vibrational frequencies, as well as the normal coordinate first derivatives of the electric properties (evaluated at the equilibrium geometry). For the first-order anharmonic contribution the cubic force constants and second derivatives of the properties are needed in addition. As we will see in the next section, it is unnecessary (with one possible exception) to go beyond first order to obtain an accurate approximation to the vibrational hyperpolarizabilities at ordinary optical frequencies. (In the static limit, however, second-order anharmonicity terms are also required.) For the systems of interest here the calculations have most often been done at the ab initio Hartree-Fock level. This is a compromise between semiempirical methods that can be unrehable [114] and computationally demanding correlated ab initio treatments. Nevertheless, a few studies, mainly devoted to small systems, have shown that electron correlation effects can be quite important [115-117]. Although trends among different chemical families, for example, HX and CH^X4_„ with X = halogen, remain mostly unchanged upon including electron correlation at the MP2 and CAS levels, for NLO systems this remains an open question. The vibrational force constants and electrical property derivatives are determined either analytically, by the CPHF method or the analogous MP2 procedure, or numerically using finite differences. Usually, at the Hartree-Fock level, all harmonic quantities are evaluated analytically except {dfi/dQ)^, which is determined

75

by calculating j8 with the addition of different fractions of the normal coordinate to the equilibrium geometry (finite displacement method). At the MP2 level of approximation, (da/^Q)o also requires a finite displacement procedure due to the lack of available codes for analytical evaluation. For mechanical and electrical anharmonicity contributions, the number of finite displacement calculations that must be carried out to evaluate all the cubic force constants and second-order derivatives of the electric properties is so large that no results for NLO materials have yet appeared in the literature. However, that situation is about to change with the introduction of field-induced coordinates (FICs) [16,17]. For it has been shown that to a very good approximation (see infinite optical frequency approximation in the next section), a small number of FICs suffice to reproduce exactly the results of a complete first-order normal coordinate treatment. The first-order derivatives of the electrical properties with respect to normal coordinates are directly related to the infrared, Raman, and hyper-Raman intensities evaluated within the double harmonic oscillator approximation. Thus, in principle, one can use experimental data to estimate the zeroth-order vibrational hyperpolarizabilities. This so-cal\Qd semiempirical approach has been employed in the past to deduce the vibrational polarizability and hyperpolarizabilities of numerous small molecules (for further discussion, see [118]). More recently, it has been apphed by Zerbi and collaborators for conjugated organic NLO materials [119]. The same general strategies that are employed in the calculation of p^ and y^ can be applied to determine the first-order ZPVA correction to the corresponding electronic properties. However, the ZPVA computations are more difficult. From the second term on the rhs of Eq. (16) we see that the calculation of A^^^^"^, for example, requires the second derivatives of the electronic property whereas the first derivatives are sufficient to obtain y^. If the Hessian (actually, only the diagonal elements) can be determined in the presence of an applied field, then the (numerical) third derivative with respect to the field gives the desired property derivatives with respect to normal coordinates in the static limit. On the other hand, no new quantities are needed to evaluate the first term on the rhs of Eq. (16), which means, in particular, that FICs can be utilized [17, 120] to reduce the number of computations. Finally, the same scaling techniques used to obtain the frequency dispersion of the electronic properties [60, 74, 82, 83] can be employed for the ZPVA as well. Once the first-order ZPVA has been determined one can obtain the second-order corrections to j8^ and y^ [18, 19] by the finite field relaxation method to be described in Section 2.5.

2.5. Nuclear Relaxation-Finite Field Approach to Vibrational Hyperpolarizabilities If a static electric (pump) field (E) is imposed on a molecule at the field-free equilibrium geometry R = RQ, and the positions of the nuclei are allowed to relax, then there will be a shift of the equilibrium geometry to R = Rg. The nuclear relaxation R E - RQ contains information about both the mechanical and electrical anharmonicity [121,122] that can be extracted to give vibrational hyperpolarizabilities. This is done by determining the effect of the nuclear relaxation on the equilibrium value of the electrical properties given by (AP)RE=P(E,RE)-P(0,RO)

(47)

76

CHAMPAGNE AND KIRTMAN

where the first argument of P denotes the field acting on the system and the second argument specifies the geometry [although we only use P{^", RE/) with E'^ = E^ in principle any combination can be realized]. It is useful to expand (AP)Rg as a power series in E but before doing so we need to specify P more precisely. P may be static or frequency-dependent; it may be the pure electronic property or it can include the ZPVA correction and/or other vibrational terms. The simplest choice is the static electronic CN value P^, in which case the expansions of jx, a, and j8 may be written

(in fact, only the [fi^^ term is nonvanishing). From this pattern it is not surprising that /3^(^)(-2ftj; o), (o) and y^^'^)(-3w; w, w, a>) vanish in the w -> oo limit. The one major process not included here is the intensity-dependent refractive index (IDRI) [or degenerate four-wave mixing (DFWM)]. In this instance only the harmonic [a^]^ contributes to y ^ ^ L ( - w ; cu, - w , for example, contains not only 126]: dc-Pockels, the static electronic second hyperpolarizabihty but also a vibralr..^.iO 2v(r) (57) i8^^^^(-c.; a>, 0)a,^oo = ^ [ A t < ^ 0 tional term due to the relaxation of the molecular geometry. We observe that the electronic properties are determined by the in- OR, stantaneous response to a probe field in the sense that the nuclear (58) ie^^^)(0; (o, -ct>),,_^oo = ^ [ i ^ « ] L o configuration is fixed. Thus, one might guess that the vibrational SHG, term in g2 is related to y^o-yg(—^; ^> 0? 0)a;_>cx), where o)/^ is (59) i8^(^H-2w; 0), w)a,-^oo = 0 the probe field (infinite frequency = no nuclear realaxation) and O/y, 0/5 are pump fields. In fact, as noted by Bishop, Hasan, and dc-Kerr, Kirtman (BHK) [123], one obtains the sum of the zeroth- and firsty^(^)(-cc>; 0), 0, 0)a,^oo = ^ [ « ^ ] L o + it/^/^lLo order square bracket terms in Eq. (46) evaluated at the infinite op(60) tical frequency limit. The result of taking the o) -^ oo limit is to retain only what Elliott and Ward [124] have described as enhanced ESHG, terms, which dominate the full response by virtue of the condition y^(^)(-26>; (o, CO, 0)a>^oo = l[f^Pt=0 (^1) wa/ft> < 1 for all vibrational frequencies cu^. Following along the IDRI, same lines the entire set of coefficients is given by y^(r)(-o>; 0), 0), -a>)a,^oo = 5 [ « ^ ] L o (^^) v(r), (51) a i = a ^ ^ ( 0 ; 0 ) + a;;^^(0;0) THG, yV(r)^_3^. ^^ ^^ a))a>->oo = 0 (63) v(r)^ (52) 6l = i8^^^(0;0,0) + i8;^^;(0;0,0) Another choice for P is the zero-point vibrational average of the v(r) (53) static electronic property or, alternatively, just the ZPVA correc^i = r^^r5(0;0'0^o) + C ^ ; 5 ( 0 ; 0 ' 0 ' 0 ) tion itself. Following exactly the same procedure as before [18], but ,v(r) , (54) now with the ZPVA correction, one simply makes the substitutions ^2 = i8^^y(0; 0, 0) + /3;^^;(-cu; a>, 0)a,-^oo ^v(r)

82 = yl^b^^'^ 0' 0,0) + r;';;^^-^' ^' o, o)c.^oo (55) with ^3 = 7a^y6(0; 0, 0, 0) + y^'^y^{-2o>: O), o>, 0)a,^oo

(56)

In Eqs. (51)-(56) all quantities are evaluated at RQ and we have used the superscript v(r) to indicate that just the nuclear relaxation contribution is included (nr and NR are alternative notations). It turns out that in taking the limit cu ^ oo the [^^] square bracket terms in Eq. (46) are eliminated from the expression for y^^ ^ ( - w ; w, 0, 0). As a result we see that y^^^ g ( - w ;ft>,0, 0)co-^oo contains the lowest-order perturbation terms of each square bracket type that is present in the complete vibrational property expression. The same is true for all the NR properties in Eqs. (51)-(56). For the static /3^^^\ the terms through first order are present whereas only the harmonic [ixar term contributes to /3^^^)(-w; w, 0). In the case of the second hyperpolarizabihty, the static y^^^) contains terms through order II; for y^^^)(-ft>; o), 0, 0), the highest order is reduced to I; and for y(^){-2a); 0), 0), 0), there is a further reduction to zeroth-order

/JLI(E, R E ) ^

[Z^a](E, R E )

etcetera

(64)

[Z^Jia] = {^ll)-^l% = ^^^t

ZPVA

(65)

ai,bi,...,g2-^

a[,b[,...,g'^

(66)

and in Eqs. (48)-(50). Then the coefficients flp Z?p . . . , ^3 are given by expressions analogous to those in Eqs. (51)-(56) except that the static electronic properties are replaced by the static ZPVA corrections and the NR polarizabilities and hyperpolarizabilities are replaced by the remainder of the vibrational contribution [for this purpose the ZPVA correction is considered to be an electronic term (see the following formulas)]. Thus, the relations corresponding to Eqs. (54)-(56) are ^2 = imapyiO;

0, 0) + ^l^^{-(o',

0), 0)

g2 = [Zy]al3yS(0; 0, 0, 0) + T^^^sC-w; « , 0, 0) gj = [Zy]„jey8(0; 0,0, 0) + Ta^gyS^^^'^' '^' '^' 0)w^oo

(67) (68) (69)

THEORETICAL APPROACH TO DESIGN

77

generates p'^^^H-co; co, 0) and y^(^^(-a>; ft),0,0); and the corresponding relation for j8(-2w; co, co) with 0^^^\-2o); o), o)) = [fiaf on the Ihs gives y'^^^H-lco; co, w, 0). Equation (77) with «curv(_^. ^ ) - [2a]I -h [/i^]" yields ^'"'''^(-0); w, 0) = [Z^f -h [)Lta]" and yC^^(-a>; co, 0, 0) ^ [Zy]^ + [a^]" -h [^jS]" + [ti^a]™ «; = [Za]«^(0; 0) + a^'^(0; 0) (70) [cf. Eq. (74)]. Finally, from these examples it should be obvious what happens if j8^^^(-2co; cu, w) is employed on the Ihs of b[ = [Z/3]«^^(0; 0, 0) + i8^'^^(0; 0, 0) (71) Eq. (77). Values for the coefficients ai, bi, gi, hi, g2y ^^^ S3 ^^ g[ = [ZyU^ysiO; 0,0,0) + y^pysi^', 0,0,0) (72) Eqs. (48)-(50) are obtained by numerical differentiation. Because Although it is possible to go to higher order, normally the first- the usual fitting errors are exacerbated in this case by inaccuraorder approximation [cf. Eq. (16)] to the ZPVA correction will be cies in the field-dependent geometry optimizations, a low default threshold must be used for both the residual atomic forces and employed. In that event, the SCF calculation, especially if one is interested in the terms (73) involving higher-order field derivatives. Furthermore, in carrying ^ ^ J S T ^ " ^ ' ^ ' ^)(o->oo = [ M ' « ] " - > O O out the field-dependent geometry optimizations it is required that the field-free Eckart conditions be strictly enforced [16]. In our ^irs^-^; (o,o,o)co^oo = [«¥i^oo + [i^i8]"^oo work we have also employed a Romberg quadrature scheme to re+ [M^«]L"-.oo (74) move higher-order contaminations in the fitting procedure. The FF relaxation coefficients can be decomposed into the individual (75) static square bracket quantities provided [a^]^ is available from yLpy8^~^^'^ ^' ^' 0)w^oo [MiSlL^^oo perturbation theory and, of course, the electronic components are (In [18] the superscript v' in Eqs. (73)-(75) is replaced by "curv." needed as well. Indeed, Z?2 gives the [fna]^ term directly and, in This is strictly correct in the OJ ^- oo limit but, in general, P^ = combination with bi, yields [JJ?]^. Similarly, g^ provides [/AJ8]^, pcurv _ [2/>] or P = i'^ -h P^^'^ + P^^^.) Note that the square which, together with g2 and [a^]^, gives [fi^a]^ and, then, [/JL f^ brackets on the rhs of Eqs. (73)-(75) are exactly the same type as in may be obtained from gi. A separation into electrical and mechanthe corresponding NR contribution [cf. Eqs. (57), (60), and (61)] ical anharmonicity contributions, however, is not possible within but they are the next higher nonvanishing order of perturbation this scheme. theory. A computational simplification of the NR finite field method, For the small molecules that have been tested [116,126] it has been found that the a> -^ oo value is reasonably accurate at op- particularly useful for large molecules, can be achieved by introtical frequencies. There is a pattern to the results obtained: with ducing field-induced vibrational coordinates known as FICs. In just one or two exceptions the magnitude of the average [a^]^ general terms, the FICs are determined [16] by the difference beand [fJbP] contributions to Eqs. (60)-(62) decreases upon going tween the field-free and field-dependent optimized geometries. from an infinite to a finite optical frequency. This is what would A rigorous analytical formulation has now been presented [17] that happen if, as suggested by BK perturbation theory, the frequency shows that a small number of these coordinates suffice to repro; factor (1/w^) in each (co -^ oo) vibrational term were replaced duce the results of an NR finite field calculation in which all vibraby — (l/w^) to yield the finite frequency correction. In considering tional degrees of freedom are taken into account. The number of the w -> 00 approximation one should bear in mind the additional FICs required does not depend upon the size of the molecule. Utiassumption [see immediately prior to Eq. (42)] that o) is negligible lizing this feature, together with the BHK perturbation method, compared to electronic transition frequencies. It follows that the leads to an effective combination of both (perturbation theory and static electronic polarizability and hyperpolarizabihty are used to NR finite field) approaches. Thus, in addition to the computacalculate [a^]^ and [fip]^. Because these electronic properties are tional advantages, one can readily separate harmonic from anhargenerally larger in magnitude at optical frequencies, this second monic terms as well as effective mechanical from electrical anharmonicity contributions. Very recently [120] the FIC analysis has approximation will tend to offset the first one. been extended to calculation of the first-order ZPVA correction If one wants to eliminate the infinite optical frequency approx[cf. Eq. (16)] and, from Eqs. (67)-(72), the corresponding curvaimation, this can be done within the nuclear relaxation-finite field ture terms. This has made it possible for the first time to determine approach by employing dynamic properties in Eq. (47). Using the these vibrational contributions for a prototypical molecule of intotal frequency-dependent hnear polarizability for illustrative purterest as an NLO material. It has also led, at least in this case, to poses Bishop and Kirtman [19] have derived the following simple a separation of the total property into two rapidly convergent perrelations: turbation sequences. One sequence begins with the pure electron property and is followed by successively higher-order ZPVA cor[a^(E,RE) + a^^^^(E,RE)] rections; the other begins with the NR contribution and is followed = p ( 0 , R o ) + a^(^^(0,Ro)] by all the remaining perturbation terms. in which j8^ = /3^^^^ + jS^' and similarly for y^. In the infinite frequency limit the ZPVA correction term vanishes but that is not so for the static a^', $^\ y^' given by the (now obvious) analogue of Eqs. (51)-(53):

+ [p^(0, Ro) + P'^^'HO, RO)]E + ...

(76)

p ^ ^ ( E , RE)] = p ^ ^ ( 0 , Ro)] + [i8^^^(0, RO)]E + • • • (77) Exactly analogous relations hold if a is replaced by j8. Starting with a^(-(o; 0)) and a^^^\-(o; o)) = [^a^jO ^^ ^^^ jj^^ ^f ^^ ^^^^ ^j^^

2.6. Numerical Procedures and Basis Set Aspects In many cases, the electronic or electronic+NR (hyper)polarizabilities are evaluated via a finite difference scheme using lower-order properties determined at different field amplitudes. For instance.

78

CHAMPAGNE AND KIRTMAN

7 can be obtained from either of the following two expressions: ^ [a(2^E) + a ( - 2 ^ E ) - 2a(0)]

yO,k

(2^E)2 ;8(2*E) -

^(-2*E)

2^+lE

^''^

with /: > 0. Accurate first- and second-order derivatives require that the field ampHtude be small enough to avoid contamination from the higher-order hyperpolarizabihties. However, the smaller the field ampHtude, the lower the accuracy of y because the number of significant digits in the field-dependent (hyper)polarizability difference decreases. A good compromise has to be found by choosing the amplitude of E and the range of k values before a Romberg-type procedure (see, e.g., [127]; for an application, see [128]) can be successfully applied to remove the higher-order contaminations. Then, successive improvement of the j8 (y) estimates starting from the j8^'^ (y^'^) value can be calculated using the general iterative expression ,

Table XL Romberg Table for Evaluation of Static Electronic Second Hyperpolarizabihty of All-Trans Hexatriene at the CPHF Level with Triple-Zeta (TZ) Basis Set". Comparison between the First Derivative [of JSLCG)] and Second Derivative [of a^CO)] Approaches as a Function of the Convergence Threshold on the Density Matrix Elements

AP yP-'^,k _ AP -1

p-l,k+l (79)

where p > 1 is the order of the Romberg iteration. This procedure results in an accuracy of 1.0 au for the second vibrational and electronic hyperpolarizabihty of/>-nitroaniHne or hexatriene and is typically better than 0.1% for the hyperpolarizabihties of 20- to 50-atom systems. Table II exemplifies the improvement in accuracy with the order of the Romberg iteration. It also shows that (i) the higher the order of the derivative, the higher the required accuracy of the property to be differentiated, (ii) the higher the accuracy, the better the Romberg convergence, and (iii) the lack of convergence of the Romberg procedure provides a clear indication that the field amphtudes are too small. Because the field effects increase with the size of the system—especially for 7r-conjugated systems—the larger the molecule, the smaller the required field amplitudes and the larger the higher-order contaminations. The error associated with the level at which electron correlation is treated, and the basis set truncation error, are present in all ab initio electronic structure calculations. Accurate predictions of the electrical properties of molecules require the choice of a suitable basis set. As evidenced by recent discussions concerning extrapolation to the complete basis set limit (see, e.g., [129]) this is not an easy aspect to resolve. For small systems, which can be studied with a wide range of extended basis sets and at different levels of electron correlation, it has been shown that high-accuracy polarizability and hyperpolarizabihty results necessitate the use of diffuse and polarized basis functions because the external part of the electronic cloud is primarily responsible for the linear and, especially, nonlinear responses. The scrupulously careful work of Marouhs and coworkers on a set of small molecules (H2O, CO2, SF6, O3, CO, CH4, HCN, HCP, HCl, CF4, H2O2, ^C4H6,...) is certainly the best proof of this requirement (see, e.g., [130]; see also [75e]). Dykstra and coworkers have also detailed the importance of incorporating diffuse and polarization functions rather than doubling or tripling the valence functions [131, 132]. Sadlej and coworkers have designed extended basis sets for computing polarizabilities (Pol) [133, 134] and hyperpolarizabihties (HyPol) [135] but they are often difficult to apply to small conjugated systems like transbutadiene due to convergence difficulties. In the case of conjugated organic NLO materials, the use of extended basis sets is prohibitive because of the computation time

P_ k

0

1

2

3

First derivative of p (density matrix convergence threshold of 10-9 and IQ--12) ' 0 1 2 3 4 5 6 7

69,953.2 69,968.3 70,028.8 70,270.2 71,229.8 74,946.3 86,086.0 33,880.1

69,948.1 69,948.2 69,948.3 69,950.3 69,991.0 71,233.1 103,488.0

69,948.1 69,948.2 69,948.2 69,947.6 69,908.2 69,082.8

69,948.1 69,948.2 69,948.2 69,950.2 69,963.2

First derivative of (3 (density matrix convergence threshold of 10-6) 0 1 2 3 4 5 6 7

70,069.2 70,005.3 70,066.7 70,277.8 71,238.4 74,966.8 86,071.3 33,898.8

70,090.5 69,984.8 69,996.3 69,957.6 69,995.6 71,265.3 103,462.0

70,097.5 69,984.0 69,998.9 69,955.1 69,911.0 69,118.8

70,099.3 69,983.8 69,999.6 69,955.8 69,923.6

Second derivative of a (density matrix convergence threshold of 10-12) 0 1 2 3 4 5 6 7

69,950.0 69,956.3 69,988.3 70,109.2 70,590.4 72,477.3 78,983.1 71,901.1

69,947.9 69,945.6 69,948.0 69,948.8 69,961.4 70,308.7 81,343.8

69,948.1 69,945.4 69,947.9 69,948.0 69,938.2 69,573.0

69,948.1 69,945.4 69,947.9 69,948.0 69,944.0

Second derivative of a (density matrix convergence threshold of 10-9) 0 1 2 3 4 5 6 7

69,937.8 69,953.1 69,987.5 70,108.8 70,590.3 72,477.3 78,983.1 71,901.1

69,932.7 69,941.6 69,947.1 68,948.3 69,961.3 70,308.7 81,343.8

69,932.1 69,941.2 69,947.0 68,947.4 69,938.1 69,573.0

69,932.0 69,941.1 69,947.0 68,947.5 69,943.9

Second derivative of a (density matrix convergence threshold of 10-6) 0 1 2 3 4 5 6 7

99,612.5 77,450.0 71,933.6 70,600.6 70,734.0 72,504.6 78,960.1 71,908.4

107,000 79,288.8 72,377.9 70,556.1 70,143.8 70,352.8 81,310.7

"The reference field amplitude is 0.0004 au.

108,847 79,749.5 72,499.4 70,583.6 70,129.9 60,622.3

109,309 79,864.6 72,529.8 70,590.8 70,138.0

79

THEORETICAL APPROACH TO DESIGN Table IV. CPHF Static Electronic Longitudinal First Hyperpolarizability (in Atomic Units) of Short a,a>-Polyene Chains Calculated with Different Atomic Basis Sets"

Table III. CPHF Static Longitudinal Electronic Second Hyperpolarizability of Short Polyene Chains Calculated with Different Atomic Basis Sets^ C4H6 TL

(^")

N02-C2H2-NH2

C10H12 y£(0) (100 au)

NO2-C6H6-NH2

6-31G

379.2 (-1.0)

Sm%.9

6,833 (-75.2)

9,445 (-18.8)

6-31G*

308.5 (-19.5)

4303.5 (-5.5)

6-31G*

6,225 (-77.4)

8,669 (-25.5)

6-311G*

310.2 (-19.0)

4212.5

(7.5)

6-311G*

8,025 (-70.9)

9,055 (-22.2)

6-31G+pd

466.0

(21.7)

5590.9

(22.8)

(9.8)

6-31G*+pd

379.1 (-1.0)

4706.0

(3.4)

(2.3)

6-311G*+pd

392.0

(2.4)

4689.8

(3.0)

Sadlej

391.5

(2.2)

6-31G

6-31G+pd

30,250

(9.6)

12,774 11,899

(10.0)

6-31G*+pd

29,532

(7.0)

Sadlej

27,691

(0.3)

TZ

10,868 (-60.6)

11,124 (-4.4)

TZ

467.0

(21.9)

5518.5

(21.2)

TZ+ELP(s)

13,530 (-51.0)

11,272 (-3.0)

TZ+ELP(s)

474.6

(23.9)

5581.0

(22.6)

TZ+ELP(sp)

460.2

(20.2)

5503.6

(20.9)

TZ+ELP(spd)

383.0

—

TZ+ELP(sp)

27,486 (-0.4)

12,799

TZ+ELP(spd)

27,597

11,632

(10.0)

—

4552.4

cc-pvdz

7,391 (-73.2)

8,834 (-24.1)

cc-pvdz

277.9 (-27.4)

3953.7 (-13.2)

cc-pvtz

9,180 (-66.7)

9,173 (-21.1)

cc-pvtz

301.3 (-21.3)

4063.1 (-7.5)

aug-cc-pvdz

23,431 (-15.1)

11,324 (-2.6)

aug-cc-pvdz

380.4 (-0.7)

4505.0 (-1.0)

aug-cc-pvtz

25,589 (-7.3)

aug-cc-pvtz

362.9 (-5.2)

d-aug-cc-pvdz

27,590 (-0.03)

— —

d-aug-cc-pvdz

359.6 (-6.1)

d-aug-cc-pvtz

27,436 (-0.6)

— —

a c ^ ^ ToW^ TTT f^f a -l
^The 6-31G and (aug)-cc-pvXz basis sets are standard basis sets. 6-31G-f pd is the basis set designed by Hurst et al. [94] for polyene chains where one set of p and d diffuse functions (exponent = 0.05) has been added to the standard 6-31G. In 6-31G*+pd the same p and d diffuse functions as in 6-31G4-pd are used. Sadlej refers to the Pol basis set designed for correlated static polarizability calculations of small molecules [133, 134]. The triple zeta (TZ) Dunning-Huzinaga basis set [144] has been augmented with diffuse functions from the electrical property (ELP) basis set of Liu and Dykstra [131]. In this scheme, (s) refers to one s function (exponent = 0.05/0.06 for C/H) on each atom, (sp) refers to (s) -Itwo sets of p functions (exponent = 0.03 and 0.005/0.9 and 0.1 for C/H) on each atom, (spd) refers to (sp) + three sets of d functions (exponent = 0.9, 0.13, and 0.02) on each carbon atom. MP2/6-311G* infinite chain [145] geometries have been chosen to remove effects due to the chain length dependence of the unit cell geometry. The numbers in parentheses are, in percent, the difference with respect to the TZ-!-ELP(spd) basis set (in bold) that has been chosen for the similarity of results with d-augcc-pvtz whereas the computational cost is much smaller and enables the study of longer chains.

and also because it poses intrinsic convergence difficulties related to the unboundedness of the dipole moment operator as well as a basis set linear dependency problem [54]. On the other hand, rather small atomic basis sets perform well when it comes to evaluating the longitudinal component that dominates the (hyper)polarizability tensor of spatially extended systems. This results from the fact that deficiencies due to the limited number of functions on any one atom are counterbalanced by functions located on neighboring atoms. It has been shown on a number of occasions that the 6-31G basis set yields fairly accurate values (<20% error) for the electronic properties of model conjugated systems [94-96, 100, 136-138]. Table III shows the improvement of 6-31G electronic polarizability and second hyperpolarizability estimates as a polyene chain is extended. Table IV demonstrates a smaller basis

function exponents of the 6-31G-Hpd, 6-31G*-l-pd, and 6-311G*+pd basis sets are 0.10 (0.07). For the electrical property (ELP) basis set, the exponents of the 5 function on the N and O atoms are 0.06; for the two p functions, the exponents are equal to 0.04 and 0.006 (0.05 and 0.007) for N (O), whereas for the three sets of d functions they are 0.9, 0.13, and 0.02 for N and O atoms. MP2/6-31G geometries have been adopted.

set dependence of j8^ of push-pull polyenes than y^ of polyenes and the importance of adding a set of ^-polarization functions. The hyperpolarizability density analysis of Nakano and collaborators [139-142] provides another justification for the suitability of small basis sets to describe polarization in the direction along which a molecule is linearly extended. For small molecules, contour plots of the third-order electron density derivative (associated with y^) display elaborately structured shapes that are strongly dependent upon the basis set (as well as electron correlation) whereas model hydrogen chains—a prototype extended system—exhibit simple and regular contours. A comparison in Table V of basis set effects upon the electronic versus vibrational second hyperpolarizability of small and medium size molecules shows that the vibrational component is much less dependent upon the addition of diffuse functions than its electronic counterpart. On the basis of these results on hexatriene, the 6-3IG basis set performs better than the Dunning triple-zeta (TZ) basis set. In the case of small molecules (acetylene, ethylene, and ethane), although larger basis sets than 6-31G are needed, the effect of the additional functions is much larger for the electronic than the vibrational properties [143]. With the exception of J-polarization functions, the impact of adding functions to the 631G basis is also small for \mf^^^^ and the Vmf^^^^^lP\{^) ratio (Table VI). For N O 2 - C 4 H 4 - N H 2 , going from'6-31G to 631G* increases [M^IL'^^Q by 17%, decreases j8f^(0) by 29%, and increases their ratio by 65% (Table VI). Nevertheless, this large

80

CHAMPAGNE AND KIRTMAN

Table V CPHF Static Longitudinal Vibrational (Double Harmonic) Second Hyperpolarizability of All-Trans Hexatriene Calculated with Different Atomic Basis Sets in Comparison with Their Electronic Counterpart^

Table VI. CPHF Static Longitudinal Vibrational (Double Harmonic) First Hyperpolarizability of NO2—C4H4—NH2 Calculated with Different Atomic Basis Sets in Comparison with the Electronic Counterpart^ r

t^/3]L;l=0M

[«']L;1=0M

nO,0

7£(0) (au)

/

X

1^1(0) (au)

[^«]L;1==0//5L(0)

NO2-C4H4-NH2

^C6H8 6-31G 6-31G* 6-311G* 6-311G**

-1769 -2869 -2278 -2495

89,490 87,414 97,410 98,172

6-31G 6-31G*

3319 3876

1508 1071

2.20 3.62

6-311G*

3751

1015

3.70

6-31G+pd

4122

1897

2.17

6-31G*+pd

4528

1205

3.76

6-311G*+pd

4262

1143

3.73

TZ

4032

1896

2.13

TZ+ELP(s)

4078

1925

2.12

69,948

TZ+ELP(sp)

4158

1873

2.22

54,435 48,488 53,712 59,024

6-31G+pd

-1008

117,788

108,658

6-311G*+pd

-2455 -3739

116,070

93,345

117,708

93,865

Sadlej TZ

2060

107,547

TZ+s

1588

109,284

73,940

TZ+ELP(pd)

4343

1127

3.85

TZ+sp

-692

119,669

102,370

TZ+ELP(spd)

4307

1128

3.82

114,595

92,638

96,575 103,182 120,389

52,762

TZ+spd cc-pvdz

-3485

cc-pvtz

-3182

aug-cc-pvdz

2512

55,968

cc-pvdz

3422

945

3.62

cc-pvtz aug-cc-pvdz

3616 4382

976 1132

3.70 3.87

90,701

^The properties have been evaluated on the ground state geometrical structure that is optimized for each basis set. See Table III for description of the basis sets.

effect is strongly attenuated for the CHO/OH pair as well as for longer chain polyene linkers. Work by Luis et al. [120] has demonstrated that basis set effects due to inclusion of diffuse functions are rather small, not only for the electronic and double harmonic vibrational (3 and y of a typical push-pull 7r-conjugated system, but also for the ZPVA and the lowest-order anharmonicity corrections. This confirms the adequacy of using the 6-31G basis or, in some cases, 6-31G* to address semiquantitatively the relationship between electronic and vibrational hyperpolarizabilities of extended TT-conjugated systems. However, it is important to draw attention to the fact that larger basis sets may be needed when dealing with nonplanar systems.

CHO-C4H4-OH 6-31G

709

1428

2.01

6-31G*

582

1335

2.29

6-311G*

567

1384

2.44

6-31G-Hpd

832

1639

1.97

6-31G*-Hpd

684

1539

2.25

TZ

814

1651

2.03

TZ-hs

814

1657

2.04

TZ+sp

806

1641

2.04

TZ+spd

633

1494

2.36

cc-pvdz cc-pvtz aug-cc-pvdz

542 544 632

1315 1384 1500

2.43 2.54 2.37

^The longitudinal axis is defined parallel to the main inertial axis. The properties have been evaluated on the ground state geometrical structure that is optimized for each basis set. See Tables III and IV for description of the basis sets.

3. APPLICATIONS 3.1. General Considerations Consider, as an example, the set of linear polyenes C4H6, C6H8, CgHio,... ,C2N^2N-h2 in the all-trans configuration. Because these oligomers are centrosymmetric the first hyperpolarizability will vanish. Calculations of the static longitudinal electronic second hyperpolarizability at the CPHF/6-31G level yield [146] y£(0; 0, 0, 0)/N = 0.3,1.8, 5.4,12.0,... x 10^ au for N = 2, 3, 4, 5 , . . . . In comparison, the values for the corresponding saturated polyethylene {C2M^4N-{-2) oligomers are 7^(0; 0, 0, 0)/ N = 1.3,1.9, 2.3, 2.6,... x 10^ au. Clearly, ^£(0; 0, 0,0)/A^ per carbon atom grows much more rapidly with chain length for the linear polyenes than the saturated polyethylenes, and similar behavior is found for the linear response. This difference is due to

the electron delocalization in the linear polyenes associated with TT-conjugation. As the linear polyene chain is extended, the monotonically increasing plot of y£(0; 0, 0,0)/A^ versus A^ eventually saturates to the infinite polymer, that is, polyacetylene (PA), limit, which is three orders of magnitude greater than the corresponding polyethylene value. If one is interested just in the limiting value, then a plot of y£(0; 0, 0, 0)[N] - y£(0; 0, 0, 0)[A^ - 1] = Ay£(0; 0, 0, 0)[A'^] versus N will converge much more rapidly (see Fig. 7) to the same result because chain end effects tend to cancel in taking the difference. To define Ay£(0; 0, 0, 0)/[A^] it is necessary to specify the two oligomers involved, which we indicate in square brackets. Ordinarily this parameter will be suppressed because the choice is obvious.) For short polyene oligomers other

81

THEORETICAL APPROACH TO DESIGN

0.0 10"

12

16

20

24

Number of (CH=CH) unit cells

Fig. 7. Chain-length dependence of the static longitudinal electronic second hyperpolarizability per unit cell calculated as either Ayi^[N] or yi^m/N. These are CPHF/6-31G results obtained at the B3LYP/6-311G* ground state geometry of the infinite chain.

components of the second hyperpolarizability tensor may be comparable to 7^(05 0, 0, 0)/N, although they are all smaller even for butadiene. However, the longitudinal component quickly becomes dominant with increasing A^. Details may vary according to the basis set, treatment of electron correlation, and the properties of the particular system—the degree of bond length alternation along the conjugated backbone is a critical ingredient—^but the behavior just described is general for the second electronic hyperpolarizability of all quasi-linear conjugated centrosymmetric oligomers (see also polypyrrole and polydiacetylene chains [147,148]. For obvious reasons these are known as gamma, or third-order, NLO materials. To generate a first hyperpolarizability one must remove the center of symmetry. Starting with a centrosymmetric oligomer this can be done most simply by capping one end with a donor (D) group and the other with an acceptor (A) group. Two D/A pairs that are often used are NH2/NO2 and N(CH3)2/CHO. Another possibility is to replace the symmetric repeat unit of the oligomer by an asymmetric unit. An example of the latter is — (CH=N)—, which is isoelectronic with - ( C H = C H ) - . There are other types of asymmetric structure but the two just mentioned are the ones that will be emphasized in this review. Both contain oligomeric segments, so it is meaningful to talk about the dependence of the static longitudinal electronic first hyperpolarizability, pf^iO; 0, 0), on the chain length. For beta, or second-order, NLO materials the soUd state packing is of crucial importance. If the three-dimensional architecture is centrosymmetric it will destroy the second-order NLO response. Solid state packing can be important for y materials as well, but the effect arises from quantitative rather than qualitative (i.e., symmetry) considerations. The evolution of pf^(0; 0,0) with increasing chain length is determined by the asymmetry as well as the electron derealization. In contrast with ^£(0; 0, 0, 0)/N the magnitude of i8f^(0; 0,0)/N generally goes through a maximum as the number of monomer units is increased (Figs. 8 and 9). Particularly when an asymmetric repeat unit is involved, the sign of j8£(0; 0,0)/N may be different for long chains than it is for short chains and the pure longitudinal component may not become dominant until the chain is quite long [149].

Number of ( C H = C H ) unit cells Fig. 8. Evolution of the static longitudinal electronic first hyperpolarizability (in atomic units) of NH2—(CH=CH)7vr—NO2, as a function of A'^, calculated as either APl[N] or Pl[N]/N. These are CPHF/6-31G results obtained at the MP2/6-31G ground state geometry from [212]. 0


'^Z 0 0 •a^ p-^

200.0

0.0

-200.0

b0 s0

^u

-400.0

0
^3

-800.0

•^ «^

W)N 0 c3

-1000.0

-1200.0

•s& ^s

^ -1400.0

r^

CO >,

^

4

8

12

16

20

Number of ( C H = N ) unit cells

Fig. 9. CPHF/6-31G static longitudinal electronicfirsthyperpolarizability (in atomic units) per unit cell of a polymethineimine chain, with a bond length alternation of 0.126 A, versus the number of (CH=N) unit cells, calculated as either Apf^[N] or (5l[N]/N Reprinted with permission from B. Champagne et al., /. Chem. Phys. A 101, 3158 (1997). Copyright 1997 American Chemical Society.

Although there are exceptions [37, 150, 151], theoretical and experimental evidence has begun to accumulate indicating that static vibrational hyperpolarizabilities of quasi-linear conjugated (and other) species are often of the same order of magnitude as the corresponding static electronic hyperpolarizabilities [37, 39, 95,125,148,150,152-170]. For molecules ranging from push-pull substituted benzenes and stilbenes to polyenovanillins and other polyenes, oligorylenes, or oligoacenes, Zerbi and coworkers have carried out indirect measurements that seem to confirm this observation. In the experimental determination, j8^(0) is obtained by combining infrared and Raman intensities (along with vibrational frequencies) whereas j8^(0) is estimated from electric fieldinduced second harmonic generation at an optical frequency. For

82

CHAMPAGNE AND KIRTMAN

the third-order process, 7^(0) is obtained from Raman intensities, by neglecting the [/AJ8] term, whereas r^(0) is also estimated from ESHG at an optical frequency. The typically close agreement (measured and calculated) between i8^(0) and p^(0), as well as 7^(0) and 7^(0), has led Zerbi and coworkers to speculate that these two quantities are manifestations of the same underlying physical phenomenon connected with the degree of bond length alternation along the conjugated chain of these systems. Indeed, on the one hand the degree of BLA in conjugated systems is the critical determinant of electronic delocalization and resulting electronic hyperpolarizabilities [149, 171] whereas, on the other hand, vibrational motions associated with the BLA, make the leading contributions to the vibrational hyperpolarizability [16, 125, 148, 152, 153, 155, 158-160, 164]. The smaller the BLA, the greater the electronic derealization and the larger the effect of changes in the BLA due to vibrational motions. Thus, the static electronic and vibrational contributions to the hyperpolarizability will tend to vary in tandem. For /3^(0) and /3^(0) an attempt has been made to prove [43] the approximate equality of j8^(0) and p^{0) but, as detailed in Sections 3.2.4 and 3.2.5, the relationship between these two properties is more complex than originally thought. In fact, calculations on (3 materials reveal that j8^(0)/j8^(0) depends significantly upon the nature and length of the oligomer linking the D/A pair and that, contrary to )3^(0), both inductive and mesomeric effects due to the D/A pair must be considered in accounting for the variation in i8^(0). When the strength of the D/A pair is sufficient to create a cyanin-like structure, we will see that the vibrational j8 and y can strongly differ from their electronic counterparts. For unsubstituted oligomers, we will detail in Section 3.3.2 how the 7^(0)/7^(0) ratio changes with BLA and with the nature of the conjugated chain (TT versus a). It will also be shown for specific compounds (stilbazolium merocyanines, charged solitons, substituted squaraine dyes, etc.), that one can expect vibrational first hyperpolarizabilities that are larger by one order of magnitude or more than their electronic counterparts. Vibrational response times are slower than electronic response times because of the difference in transition frequencies. If we assume that the limiting electronic excitation energy is on the order of a few electron-volts, then the ratio of the response times can vary from a factor of roughly 8 (e.g., in the case where AE = 2 eV and 0) = 2000 cm~^) to about 8000 (low-frequency modes of about 10 wavenumbers, AE = 10 eV). As detailed mostly in Sections 3.2.4 and 3.3.2, in most instances the key vibrations he in the high-frequency range (1500-2500 cm~^) and, therefore, they will contribute to the hyperpolarizability on a time scale that is useful for practical devices. Furthermore, as proposed by Kamada et al. [5], experiment can take advantage of the time-scale partitioning to probe the importance of the two contributions. So far our discussion pertains specifically to neutral (and isolated) species. It is well known that the y materials discussed in this chapter can be charged in the solid state by doping with an electron donor or acceptor (and by other means as well). The electrical conductivity due to the localized geometric structure that is formed about the charge on the chain has been investigated extensively, but the NLO response has just been tackled. Initial calculations, as well as speculation based on related studies, indicate that the resulting electronic and vibrational 7L(0) may be much larger than for the corresponding neutral chain segment. The relationship between the vibrational and electronic components may also be very different. Because the charge structure is localized, the enhancement factor must go through a maximum as a function

of chain length. This leads to the possibility of tuning the NLO properties by varying dopant concentration and the dopant nature and size that will influence the amount of charge transferred and its localization. In principle, neutral j8 materials can be charged in the same manner as y materials. The NLO properties of beta systems containing charged oligomers have scarcely been examined but preliminary results suggest that further investigation is warranted. A discussion of charged species will be presented in Sections 3.2.7 and 3.3.4. As far as practical applications are concerned, a wide variety of devices based on NLO properties have been envisioned. The requirements for the materials that will be employed are diverse as well. One fundamental common feature is a sufficiently large NLO susceptibility. In this respect, j8 materials are in a more advanced state than y materials. As noted earlier, an important issue for the former is preservation of the microscopic asymmetry on a macroscopic scale. This has been done successfully using organic crystals where hydrogen bonding, or cocrystallization (with active or inactive compounds), or addition of substituents (chiral or achiral) controls the molecular orientation and guarantees a large x^'^^ • Another possibility is the use of centrosymmetric molecules, such as the quadrupolar squaraine dyes that, because of the absence of dipolar interactions, are easier to assemble asymmetrically [172,173], using Langmuir-Blodgett (LB)films,for example, so as to yield a large x^^^- Nevertheless, in most cases, part of the ;^^^^-potential is lost because the packing is not ideal. Still NLO susceptibilities that are one or two orders of magnitude larger than inorganic dielectrics and semiconductors, such as LiNb03 and KTiOP04, have been achieved. The organic crystals are not ordinarily suitable for SHG devices, however, because of strong absorption of the generated double harmonic light. On the other hand, organic chromophores dispersed in a polymeric host, and poled polymers, are already practical for use in electrooptical switches and modulators. In this case thermal stability of the poling and other practical problems, such as the density of molecules, still remain. Langmuir-Blodgett films and self-assembled monolayers (SAM) represent another approach to controlling the macroscopic order and asymmetry that has not yet come fully to fruition. Yet a further consideration is phase matching conditions on the refractive indices for the incoming and outgoing waves that make a different packing required to maximize different NLO processes. For j8 materials, the primary task of theory is to develop an understanding that will lead to diverse materials with a large microscopic optical nonlinearity, that can be sufficiently retained under macroscopic conditions and will satisfy such auxiliary requirements as weak absorption of the SHG beam and thermal stability. For 7 materials there are also important requirements apart from large NLO susceptibility. These requirements include high laser damage thresholds as well as mechanical, thermal, and environmental stability. Anisotropic heating due to high laser power input can create large changes in the refractive index thereby masking the optical Kerr effect. In addition, depending upon the NLO process, excessive two- and three-photon absorption must also be avoided. In contrast with j8 materials, sufficiently large microscopic susceptibilities have not yet been produced. Although TT-conjugated organic polymers have the largest NLO responses observed, they are still an order of magnitude or two short of the goal. Thus, the theoretical search for guiding principles that will allow this goal to be realized continues to be the highest priority. In this subsection we have provided a brief overview of second- and third-order NLO property calculations for quasi-

THEORETICAL APPROACH TO DESIGN linear 7r-conjugated oligomers-polymers and organic molecules. Some of the particular topics treated were (i) the evolution of the property with chain length, (ii) the relative importance of vibrational versus electronic components and the connection of both to BLA, (iii) the potential for enhancement of the NLO response due to formation of charged structures, and (iv) the current status with respect to practical requirements. Within this overall picture there are many ramifications that account for the complexity of the subject as well as the opportunities for further important advances. We now turn to a detailed exploration of j8 materials in Section 3.2, to 7 materials in Section 3.3, and,finally,medium (solid state or solution) effects in Section 3.4. 3.2. Second-Order NLO Organic Materials Only non-centrosymmetric systems can exhibit a nonzero first hyperpolarizability. As mentioned earlier, we will concentrate on those (j8) materials where the asymmetry is introduced by endcapping an oligomer with a D/A pair and/or by constructing an oligomer from an asymmetric building block. So far, most of the j8 materials can be cast in the first category. Even if the constituent molecules or polymers have a nonvanishing j8 the first nonlinear response of the bulk, x^'^\ can be zero as a result of a centrosymmetric crystal packing. In addition to crystal symmetry, the unfavorable relative orientation of chromophoric entities within the unit cell will, typically, lead to a partial or complete cancellation of the nonlinear property. Some of the strategies for overcoming detrimental crystal packing effects were noted in Section 3.L These will be elaborated, along with others, in Section 3.4, where we deal with the role of the medium. Our focus in this subsection will be on second-order NLO properties of individual chromophores. Numerous experimental and theoretical studies dealing with push-pull conjugated systems, that is, conjugated systems endcapped with D/A pairs, have investigated the effect of the D/A pair as well as the nature and length of the conjugated linker. This subject was thoroughly reviewed a few years ago by Kanis et al. [29], Verbiest et al. [174], and ourselves [175]. Here we bring these reviews up to date.

83

provide a useful guideline, words of caution are in order. As we have noted [175] (i) the quality of the fit varies depending upon the linker, (ii) there is ambiguity in the choice of a, and (iii) in the theoretical investigations there is an open question about the role of geometry. These aspects will be discussed here. The first set of a parameters that were used are the para substituent constants op originally related by Hammett to the variation of the dissociation constant of ;7-substituted benzoic acids in water at 25 °C [181]. These op parameters involve a combination of inductive (oi) and mesomeric or resonant (OR) effects (i.e., oTp = cri + (7R) that are not separated in most studies of the correlation with p. However, this single-parameter model is of limited utility for strongly coupled disubstituted systems where p^ is dominated by the large charge-transfer contribution. To characterize the enhanced D/A character of substituents that interact resonantly, dual a^/a~ parameters have been determined for A/D neglecting the (often) small inductive contribution. The relevance of these dual parameters is evident for instance from the work of Li et al. [178], where the slope of the calculated Pariser-Parr-Pople curve for i8vec(0) versus ap changes with the nature and strength of the acceptor group. Although it would be preferable to use a parameters that vary continuously between the nonresonant and resonance-dominated limits, the unbiased a^ parameter for noninteracting pairs and the dual (o-p/o-p) parameters pertain only to the two extreme cases. As a result different choices are made by different authors. Cheng et al. [180] assume that resonance is dominant for all D/A pairs. On the other hand, a binary choice is made between o-p /o-p for strong pairs and o-p for weak pairs by Jacquemin et al. [179]. The deviations from a Unear relation obtained by Ulman [177] and the difference in slope for para substituted nitro- and methoxy-benzenes [180] show that the dual parameters depend upon the resonating group from which they have been extracted. These parameters also depend upon external conditions (e.g., the solvent) and on the manipulation of the experimental thermodynamic quantities from which they are derived. Nevertheless, it is noteworthy that constants derived from thermodynamic or spectroscopic data can account qualitatively, and often quantitatively, for p^ variations due to chemical substitution.

It turns out that the binary choice of Jacquemin et al. [179] leads to a very good linear fit for a (CH=CH)4 linker where, in contrast to experiment and other calculations, the geometry has been artificially frozen for all D/A pairs. This raises the issue of To enhance the development of new efficient second-order NLO separating pure electronic contributions to the susceptibility from materials, it would be useful if one could estimate, prior to synthe- those that arise as a result of geometry relaxation. With a strong sis or even theoretical calculation, the value of p associated with D/A pair the relaxation will cause a large change in the BLA of the a given D/A pair substitution. This requires (i) a basis for under- linker and, thereby, have a substantial effect on p^ [182]. For weak standing the relationship between D/A strengths and the p value D/A pairs the effect is small but, since they have little influence on and (ii) a quantitative treatment of this relationship. the plot of jS^ versus o-, a good linear fit will be obtained with or For conjugated linkers calculations have revealed [163, 176- without geometry relaxation. Similarly, a goodfitfor a set of strong 179] an almost linear relation between jS^ (or P\QC = A • i^^/lAL D/A pairs will be obtained if the BLA is approximately the same in which p^ = J2j=x,y,z Plj ^^^ lAl is the norm of the dipole in all cases [182]. With the exception of the (CH3-S)-substituted moment) and the difference of the experimentally derived Ham- compound, the correspondence between the AMI P^^^^) and the mett a parameters for the para substituent effect of a given D/A a'^l(T~ parameters of disubstituted pyrazines has been found satpair. Similar relations have been deduced for the magnitude of isfactory using the AMI optimized geometry [183]. For monosubthe experimentally derived SHG [180], which is dominated by its stituted benzenes. Champagne [163] has considered the geometry electronic component [see Eq. (59)]. The difference in a param- relaxation due to the D or A and has proposed the use of cri and eters quantifies the asymmetry or strength of the D/A pair; the ^R (^p = ^ R + ^ ) to improve the correlation for substituents with slope of pf^ versus a measures the polarizability of the interven- large oi parameters. As could have been expected from the imporing TT-electron charge distribution. Although this relationship does tance of the resonant charge-transfer contribution to p^ inductive 3.2.1. Characterization of the Relation Between p^ and the D/A Strength—Assessment of Different Predictive Parameters

84

CHAMPAGNE AND KIRTMAN

effects were found to be a factor of 4 smaller than resonance effects. A different conclusion holds for the vibrational counterpart to j8^, as will be discussed later. The corresponding fit found for the dipole moment (/IL = -2.88ai - 1.56o-^) explains why Zyss had to remove the inductive (o--electron) contribution to /^L to obtain a good correlation between the latter and J8L [184]. Subsequent to the work of Ulman [177] and Morley et al. [176] the nearly linear relation between o-p and (3^ was considered a primary tool for NLO design. However, in 1993, it was discovered by Marder et al. that stronger acceptors can diminish nonlinear optical response in simple D/A polyenes [185]. These authors found experimentally that, for a given number of double bonds, the /3(0) values of the dimethylamino polyene aldehydes are greater than those of the corresponding dimethylamino dicyanovinyl molecules even though dicyanovinyl (o-p" = 1.20) is known to be a better acceptor than the aldehyde group (a^ = 1.04). In fact, a simple four-orbital TSA suggests that, for a given conjugated bridge, j8^(0) will go through a maximum as the strength of the D/A pair is increased [186]. Another property that, like i8^(0), is related to the strength of the D/A pair is the BLA along the polyene backbone. Starting with the unsubstituted polyene, the BLA decreases as the D/A strength is increased until it passes through zero at the regular cyanine geometry and, after changing sign, increases in magnitude while approaching the zwitterion limit. Thus, the BLA can be used instead of the D/A strength as the key parameter relating /3^(0) [as well as a^{0) and 7^(0)] to the structure of the D/A polyene [187]. Alternatively, one may use the mixing parameter 6 of the VB-CT model for this purpose (cf. Section 2.2). In Figure 1, ^ = 0 corresponds to the unsubsituted polyene (BLA = 8 = maximum), 0 = 77/4 corresponds to the cyanine form (BLA = 0), and 6 = 77/2 to the zwitterion structure (BLA = -8 = minimum). It follows that the BLA will depend upon the mixing parameter roughly as 8 cos(20) [36]. In comparison, j8^(0) in Figure 1 is proportional to (cos 20) (sin 26)^, which means that it vanishes for the covalent, cyanine, and zwitterion forms while passing through a maximum between the first two (covalent -> cyanine) and a minimum of equal magnitude between the last two (cyanine -^ zwitterion) [37-39]. For a particular push-pull polyene (or other conjugated linker) the question is, where is the molecule located with respect to the maximum (or minimum) along the curve of j8^(0) versus 6 [and, likewise, for a^(0), 7^(0)]? Contrary to the experimental measurements, gas-phase AMI FF calculations [185] indicate that the dimethylamino/aldehyde and dimethylamino/dicyanovinyl polyenes both lie to the left of the maximum in Figure 1; that is, the stronger acceptor corresponds to a larger /3^(0). It now appears that the apparent contradiction is explained by the effect of the solvent that stabilizes the zwitterionic with respect to the polyenic form. This solvent-induced enhancement of the D/A strength, which has been demonstrated experimentally [188] and theoretically [189] indicates that the solvent can play a significant role in optimizing NLO properties. However, a CISD/CV/INDO study based on the Onsager reaction field (see also Section 3.4) by Albert et al. [190] has shown that, even for the most polar solvents, these effects are rather limited and therefore that the major changes in electronic and NLO properties result from the strength of the D/A pair. Moreover, the importance of the solvent effect depends primarily upon the degree of stabilization of the chargeseparated form in solution. Analogous studies in soUd state media are still isolated and limited to electronic effects [191-193] so that one cannot yet draw general conclusions.

Even for an individual molecule very recent calculations on substituted hexatrienes [9,37] show that doubling the number of D and A groups can shift the structure from the polyenic region into the region where P^(0) is either small or large and negative (see the last two entries of Table VII). These RHF/6-31G calculations also verify that the D/A-induced passage between these structures is accompanied by a continuous change of BLA along the chain leading to a reversal of bond order. An elegant theoretical approach to varying the strength of D/A pairs has been proposed by Marder and coworkers [171, 188]. It consists of simulating these pairs by an external uniform static electricfield.As the amplitude of the field is increased the electronic structure and geometry changes from the polyenic to the zwitterionic form. Like Hammett's a parameters, as well as the BLA and the VB-CT 6, one can use the amplitude of the externalfield,Es, to describe the variation in /3(0) as a function of the D/A strength. Since /x-(O), a(0), and 7(0) are determined at the same time this led to a convenient unified description of all the dipolar properties in terms of successive higher-order derivatives with respect to Es. Unfortunately, a careful ab initio investigation by Kirtman et al. [9] reveals that, for linear polyenes, the Es that reproduces the jjL^iO) and j8^(0) of a real molecule is not appropriate for either a^(0) or 7^(0) of the same molecule. In fact, there is often no field that will reproduce the latter two properties. Furthermore, even for /x^(0) and j8^(0) the required Es depends significantly upon the nature and length of the conjugated linker. This limitation emphasizes the importance of including field nonuniformity and/or bonding interactions to reproduce the complete D/A effect as foreshadowed by earlier studies of Chen and Mukamel [194]. Finally, Kirtman et al. [9] note that the calculation of higher-order properties from lower-order ones by differentiation with respect to Es automatically introduces vibrational contributions, thereby making the analysis of the field simulation scheme more compHcated than initially assumed. It does turn out, however, that for any given property (/x, a, j8, 7) the electronic and vibrational curves (vs. Es) are qualitatively similar in shape to one another and to the corresponding BLA. Other approaches to characterize the strength and asymmetry of a given D/A pair and predict the j8 values have also been proposed based upon theoretically determined electronic properties of the D and A or of the D/A conjugated system. The difference between the Coulomb on-site energies, aj) — a/^, has been used by Marder et al. [186]. On the other hand, Suzuki et al. [195] and Sheng et al. [196] employ quantities derived Table VII. CPHF/6-31G Static Longitudinal Electronic First Hyperpolarizability for D/A-Substituted Hexatrienes in Comparison with the Average BLA along the Conjugated Backbone Molecules H-CH=CH-CH=CH-CH=CH-NH2 N02-CH=CH-CH=CH-CH=CH-H CHO-CH=CH-CH=CH-CH=CH-OH N02-CH=CH-CH=CH-CH=CH-NH2 P02-CH=CH-CH=CH-CH=CH-N(CH3)2 (N02)2-C=CH-CH=CH-CH=C-(NH2)2 (P02)2-C=CH-CH=CH-CH=C-(NH2)2

BLA (A)

j8[(0) (au)

0.121 0.120 0.118 0.097 0.078 0.044 -0.026

972 1502 1774 4084 6380 219 -2889

85

THEORETICAL APPROACH TO DESIGN from the HOMO and LUMO orbital energies of the D/A substituted system. For instance, Sheng et al. [196] define the effectiveness of the donor (acceptor) by 8^ (^D), which is the difference between the LUMO (HOMO) orbital energy of the acceptor(donor-)substituted polyene and that of the unsubstituted polyene. Then, they take the difference 5DA = ^D - ^A and,finally,to have a quantity that vanishes for the cyanin structure (BLA = 0) utilize ^DA ~ ^DA-^DACcyanin). The comparison ofj8^ versus BLA and ]8^ versus 5j^^ curves made by Sheng et al. based on AMI calculations shows to what extent these phenomenological relationships can be used and how they compare with the VB-CT method. Although a qualitative understanding of D/A effects upon the first hyperpolarizability has been reached using several different parameters (a, BLA, 6, Eg, 5^^). semiquantitative (position of the maxima and minima) and quantitative predictions remain difficult. Indeed, Es, cr, 6j^^ are not sufficient to determine j8^ from knowledge of the D/A pair because other aspects such as chain length dependence and interactions between the D/A pairs are not accounted for. As far as the parameter 6 is concerned, there has not yet been adequate testing in this regard (see, however, [40]). All things considered, the BLA appears to be the best parameter for determining whether one is near an extremum or zero in P^ (or P^). Nevertheless, as noticed by Albert et al. [190] the asymmetry in the terminal substituents may limit the accuracy of the predictions based upon the BLA. Indeed, due to the difference in the strength of the substituents, the zero-j8 and cyanine (BLA = 0) limits do not coincide. 3.2.2. Dependence ofP^ on the Nature and Length of the Linker in D/A Systems Although the D/A pair is required to get a nonzero j8|^ value, the conjugated linker plays a critical role in determining the magnitude of the first hyperpolarizability. The susceptibility of an oligomeric linker depends upon both its nature and its length. Although breaking the conjugation along the chain may have a substantial effect (see Table VIII) length is the most important variable from the viewpoint of maximizing jS^ per unit length (for which we use Pj^/N) or, almost equivalently, jS^ per unit volume. From the first investigation of the chain-length dependence of J8L it was anticipated that there would be an enhancement due to increasing chain length. The existence of a power law relationship between jS^ and the length of the linker (or the number of conju-

gated multiple bonds) has been verified many times for systems containing up to about 10 multiple bonds (see Ref. [197] for comparison between experimental and theoretical "b" parameters). Nevertheless, since the CNDOVSB work of Morley et al. in 1987 [198] on D/A polyphenyls, it has been known that jS^ per unit volume will go through a maximum as the length of the chain is further increased. Furthermore, in the long chain limit jS^ must saturate because the D and A moieties cease to communicate with one another. Over the years there have been both experimental [199-205] and theoretical investigations of the chain-length dependence of jS^. The majority of the calculations are of the semiempirical variety [98, 99,198, 206-209] although there are also a few ab initio studies [128, 179, 210-212]. Semiempirical methods are advantageous because of the computational efficiency and the fact that electron correlation effects are implicitly included. On the other hand, their reliability is questionable, particularly with respect to dependence upon chain length. There are several reasons this is so. First, in most cases the CI space is restricted to monoexcited configurations even though it has been shown that inclusion of doubles can lead to substantial changes in jS^ depending upon the size and nature of the conjugated linker [208]. In addition, the excited states are usually restricted to afixednumber (^100-300) of those that are the lowest in energy. This disadvantages the longer chains with respect to shorter ones and may be the cause of the nonmonotonic behavior found by Morley and coworkers [98, 208] for j8^ or jS^/AT. The role of higher-order substitution configurations as well as the effect of truncating the excited state manifold is exemplified in Table IX by comparing the early CNDOVSB/CIS results of Morley [207] on a,co-A/',A/^'-dimethylamino,nitro polyynes of increasing length with those obtainedfiveyears later using the correction vector method, using all singly and doubly excited configurations derived from an active space composed of 10 7r-orbitals [209]. Not only the magnitude of P^Q^{-2O}\ CO, a))/N but also the location of the maximum is quite different. It is of interest to compare semiempirical with ab initio results. For N H 2 - ( C H = C H ) A ^ - N 0 2 the static Pl = PliO) has Table IX. Comparison between Two Semiempirical Schemes for Computing j85ec(~2aj; (o, a))/N of Q;,aj-A/^,A/^^-Dimethylamino,nitro Polyynes^ A^

Table VIII. CPHF/6-311G** Calculations on Model Push-PuU Systems Containing Linkers of Different Length and Conjugation along the Chain^ Molecules N02-(CH=CH)3-NH2 N02-(CH=CH)5-NH2 N02-(CH=CH)-CH2-(CH=CH)-NH2 N02-(CH=CH)2-CH2-(CH=CH)2-NH2 N02-(CH2-CH2)3-NH2

/35ec(0) (102 au) 21.9 46.0 1.7 11.3 -0.3

(80)

/3|. = «L''

4

10 12 14 16 18 20

/35ec(-2w; (o, (o)/N—CIS 70.8 78.7 76.6 71.5 66.2 61.1 47.9 44.7 36.3

j85ec(-26^; 0), (x))/N—ClSD 67.1 82.9 97.4 109.7 121.0 130.1 138.2 145.1 151.6

^The value per triple bond is given in 10^ au for hco = 0.656 eV. The CNDOVSB/CIS calculations [207] include --100-300 excited states, whereas '^The results correspond to RHF/6-311G** geometries. jSvec gives the pro- the CNDOVSB/CV/CISD treatment [209] includes all single and double jection of the vector (Px, Py, Pz) on (/i^, Mj, A^z), that is, JLI • i8/|/I|. excitations from 10 active 7r-orbitals.

86

CHAMPAGNE AND KIRTMAN

Table X. Static Longitudinal Electronic First Hyperpolarizability of NH2/NO2 Push-PuU Conjugated Molecules with Polyenic or Polyynic Linkers^ CPHF/6-31G

MP2/6-31G

MP4/6-31G

N02-(CH=CH)-NH2 N02-(CH=CH)2-NH2

379 1798

1096 4640

1058 4627

N02-(C=C)-NH2 N02-(C=C)2-NH2

654 1999

1113 3312

988 2931

^The values are given in atomic units and correspond to MP2/6-31G optimized geometries. Data from [128].

been determined using ab initio methods by Jacquemin et al. [212] whereas Morley [98] has calculated the norm of the second-order 1^0 response, j8 = [jS^ + jS^ + jS^]^/^ using the semiempirical CNDOVSB/CIS technique (AMI geometry optimization). The norm is dominated by the diagonal longitudinal component. In the ab initio MP2(RHF)/6-31G calculations (MP2/6-31G geometry) pf^(0)/N is a maximum 2it N = 13 (11) and the value at the maximum is 6.5 (5.3) times that at N = 2; the corresponding semiempirical parameters are N = 12 and 4.3, respectively. Including frequency dispersion modifies the comparison between CNDOVSB/CIS and MP2(RHF)/6-31G because the excitation energies of the most important excited states decrease with chain length. As a consequence, there is a displacement of [jS/A^lmax toward longer chains and a further enhancement with respect to the A^ = 2 values. For example, the value of A^Max for /3^(-2w; o),(o) at A = 1907 nm is 14 (15) at the MP2(RHF) level using the MP2/631G ground state geometry and about midway between 12 and 18 for C]^JDOVSB, judging from the reported data. The corresponding [i8/Ar]max values are MP2 = 221 x 10^ au (RHF = 128 x 10^ au) and CNDOVSB = 150 x 10^ au (after multiplying by 2 to account for the difference of convention), which, in order, are 8.4 (12.6) and 8.0 times that at N = 2. Thus, the semiempirical AMl/CNDOVSB/CIS results compare rather closely to MP2. This is due, at least in part, to the similarity of the MP2/6-31G and AMI values for the BLA, which, in the middle of the polyene backbone, are 0.072 A for the former and 0.097 A for the latter. From a quantitative point of view, including electron correlation in the ab initio treatment can substantially affect 13^(0) by altering the BLA along the linker backbone—a key parameter as discussed earlier—and by modifying the electronic structure at the optimized geometry. As an example we may consider the case of NH2-(CH=CH)io-N02. The CPHF/6-31G value of i8[(0) increases by 93% due to shifting from the RHF/6-31G geometry to the MP2/6-31G geometry with a smaller BLA [212]. (A similar geometry-induced, 92% increase is obtained when pf^iO) is computed at the MP2, rather than CPHF level.) Moreover, at either geometry, Pf^iO) is nearly doubled upon going from the CPHF to the MP2 treatment. As a result of both correlation effects, treated at the MP2 level, there is an overall increase by a factor of 3.71 in the value of i8[(0). The effect of electron correlation depends upon the chemical nature of the system and may alter conclusions drawn at the Hartree-Fock level of investigation. This happens when comparing the electronic first hyperpolarizability of NH2/NO2 push-pull molecules with polyene versus polyyne linkers as shown in Table X.

All calculations were done at the MP2/6-31G optimized geometry. The CPHF results suggest that, for a fixed number of carbon atoms, the polyyne-based compounds will have a larger Pf^(0) but the MP2 (as well as MP4) calculations lead to the opposite conclusion [128]. In Table X the differences between MP2 and MP4 are modest and, certainly, much less than the differences between CPHF and MP2. Although there are not many studies where more sophisticated electron correlation treatments and more extended basis sets are apphed to chainlike molecules, those that have been done indicate that correlation corrections beyond MP2/6-31G will generally be less than about 25%. A very recent investigation substantiating this point has been reported by Jacquemin et al. [96]. In the latter paper it was also found that the frequency dependence of the correlated jS^ is given satisfactorily by the multiplicative scaling procedure of Eq. (36), which requires only the static correlated jS^ along with dynamic TDHF values. For j>-nitroaniline and other substituted benzenes the electron correlation effect is in the same direction, but somewhat smaller than for the D/A polyenes [213-217]. In these cases, a 15-50% increase in jS^ has been found within the MP2 or MCSCF scheme. For;?-nitroaniline, Sim et al. [215] have found that electron correlation has little effect on the geometry. While keeping in mind the limitations of the CNDOVSB/CIS approach, as we have seen this method affords a useful starting point for comparing various conjugated oligomeric linkers with regard to their potential for producing a large P^Q^.(0)/N. A compendium of results for jS^ecC^)/^ versus N obtained by Morley and coworkers, using the strong D/A pair N(CH3)2/N02, is given in Table XI. For small chain lengths (N = 4, 6) p%c(^)/N increases in the order furan > phenyl > pyrrole > thiophene > acetylene (polyyne) > cumulene > ethylene (polyene). When the chains are lengthened beyond N = 6, the polycumulenic linker gives rise to a much larger P^QC(^)/N than the polyenic linker. The ordering of the nonaromatic linkers for N = S, PyQc{0)/N (polycumulene) > I3^QC(^)/N (polyene) > jSyecCO)/^ (polyyne), is consistent with the fact that better electron conjugation is associated with a smaller BLA along the chain. For aromatic linkers, as compared to polyene linkers, there is a reduction in Pyec(^)/N that is connected with the aromatic stabilization energy (ASE). On the other hand, the polyaromatic compounds are more stable to external conditions, which has obvious practical advantages. These data for the polyaromatics were obtained assuming planarity; inter-ring torsion in the equilibrium configuration will lower the hyperpolarizability by interrupting the conjugation. In some cases, breaking the conjugation may increase the NLO response. When a conjugated system is rotated about one of the TT-bonds the excitation energies undergo strong bathochromic shifts. Using the example of ethylene it has been shown that this leads to an enhanced y^ [220]. Moreover, in unsymmetrically substituted conjugated systems the dipole moment of some excited states strongly increases as the rotational angle approaches 7T/2 as a result of charge separation (see, e.g., [221]). This so-called sudden polarization [222] accounts for the enhancement of the first hyperpolarizability in 3,5-dialkyl-2',6'-dialkyl-4-quinopyrans [223] where the ground state is twisted by an amount that depends upon the size of the alkyl substituents. Because of the small excitation energies involved (<1 eV) applications based on these materials will focus on infrared absorbers rather than optical frequency doublers. One may also expect that the vibrational contribution to /3 will be nonnegligible.

87

THEORETICAL APPROACH TO DESIGN Table XL

Chain Length Dependence of PyQc(^)/N (in Atomic Units) for Various Conjugated Linkers and the Same N(CH3)2/N02 D/A Pair as Obtained by CNDOVSB/CIS Calculations^ N

ASH

2

4

6

8

10

BLA

[218, 219]

N(CH3)2-[CH=CH]2iv-N02^

93.3

284.5

467.3

546.5

571.2

0.09

N(CH3)2-[C=C]2iv-N02^

56.0

57.9

48.8

42.1

25.2

0.18

H,N(CH3)2=[C=C]2iv=N02,H^

19.0

73.6

283.8

723.5

2375.6

0.0

— — —

N(CH3)2-[Ph]^-N02^

36.3

27.5

20.4

15.7

0.04

21.4, NA

N(CH3)2-[Pyk-N02^

43.7

38.0

28.5

21.8

0.05

20.7, 25.5

N(CH3)2-[FU]A,-N02^

27.8

17.4

12.3

9.3

0.13

11.8,19.8

N(CH3)2-[Th]jv-N02^

53.9

56.0

42.8

34.7

— — — —

0.09

18.6, 22.4

^Standard geometrical parameters from crystallographic data and planar conformation have been used. The BLA reported is the average value along the backbone (in angstroms) and the aromaticity stabilization energies (ASE) in kilocalories per mole are from [218, 219]. ^Ref. [198]. ^Ref. [207]. ^Ref. [98]. ^Ref. [206].

Because the largest l>JLO responses [measured by (jSvec/AOmax] are expected for polyenic (and polycumulenic) linkers, whereas chemical stability is ensured by aromatic moieties, it is not surprising to find that many attractive compounds for ISH^^O devices are built by combining these two aspects, that is, a short or medium polyenic linker capped by one or two substituted aromatic rings. A/^,Ar'-dimethylamino,nitro-stilbene (DANS) is a prototype of these compounds as are D/A diphenylacetylenes. Let us consider what happens when a short polyene or polyyne segment is inserted between two aromatic rings. This directly extends the conjugation length and may also indirectly increase the effective conjugation by reducing the inter-ring torsion angle. It has been demonstrated experimentally [180, 200] that such an insertion leads to an increased j8vec(-2w; w, w). FF/MNDO calculations (AMI geometries) by Barzoukas et al. [224, 225] show an increase in jSyecCO) of 20% in going from 4-A/^,A'^'-dimethylamino4'-nitrodiarylacetylene to 4-A/^,Ar^-dimethyiamino-4'-nitrostilbene if the compounds are planar. If the torsion angle between the ring planes is fixed instead at 7r/4 radians, then j8vec(0) of the stilbene compound decreases by 48%, whereas the value for the diarylacetylene species is reduced by only 22%. This has been interpreted as being due to the presence of two perpendicular TTbonds in the acetylenic moiety, which prevents a large decrease in electron conjugation. From RHF/6-311G** calculations on these compounds [168] it is found that, in the gas phase, the acetylenic species is planar whereas the torsional angle is 17° for the stilbene molecule. As a result the jSygcCO) values are similar for both compounds. In solution, one should also consider the ease of rotation around the C - C bonds for the tolan derivative. It has also been calculated that the (B^^iO) enhancement due to extending the conjugated C = C / C - C or C = C / C - C segments between aryl groups is larger for the former [226] than it is for the latter [227]. Other conjugated segments such as N = N [228-231] and (o--conjugated) Si(]VIe2)-Si(]VIe)2 [232] have also been studied both experimentally and theoretically. The azo compounds are stable under polymerization conditions and are calculated to possess p^ values that become larger than their stilbene analogs when o) increases

Table XII. Relative jS^ec Values for Different D/A Heteroaromatic trans-StilbenQS with Respect to the Parent D/A trans-StilbQnQ as Obtained by Various Quantum Chemical Methods CPHF/4-31G NH2/N(CH3)2 NO2

Static^

INDO/1 CPHF/6-311G** Correction vector, /lo) = 0.1 eV^ Static^

Phenyl

Phenyl

1.00

1.00

1.00

Pyrrole

Phenyl

1.70

1.75

1.35

Phenyl

Pyrrole

0.85

1.05

0.67

Pyrrole

Pyrrole

1.36

—

0.91

^AMl full geometry optimization, N(CH3)2, from [235]. ^AMl full geometry optimization, NH2, from [236]. ^RHF/6-311G** full geometry optimization, NH2, from [168]. due to smaller excitation energies (in the static limit, their values are similar; the predicted ordering depends upon the computational method). The ^-conjugated Si(Me)2-Si(Me)2 bridge improves transparency while reducing the charge transfer between the D/A moieties. Following the study of Kanis et al. [233], optimizing the NLO response of mixed (7--/7r-conjugated compounds will require modulating both the a- and TT-D/A character of the substituents. Adequate choice and positioning of the heteroaromatic rings in push-pull heteroaromatic stilbenes can also help in maximizing the jS^ec value. Indeed, due to their intrinsic donor or acceptor character, these aromatic rings can either enhance or dampen the D/A character of the substituents. This was first evidenced in experiments by Wong et al. [234] and, then, theoretically rationalized in terms of electron excess or deficiency on the carbon atoms of these rings [235, 236]. For instance, because the pyrrole moiety is a better donor than the phenyl ring, the 13%^ value will be maximized by placing the 1^2 donor group on the pyrrole while the nitro group is attached to the phenyl. Table XII shows that these

CHAMPAGNE AND KIRTMAN auxiliary D/A effects are more or less accounted for in all treatments but that both the quality of the geometry and the method used to calculate /3 are quantitatively important. Instead of aromatic rings, nonaromatic cyclic olefins can act as strong auxiliary D/A and thereby increase the first hyperpolarizability. Indeed, using a cyclopentadiene ring with an attached acceptor and a cycloheptatriene ring with an attached donor, it was found by Zhu et al. [237] in semiempirical calculations that p^ is enhanced by a factor ranging between 2 and 5 (depending upon the D/A) with respect to the analogous compound containing benzene rings. Other D/A dipolar systems of interest that have been theoretically characterized are the retinal derivatives [238, 239], sulfurcontaining compounds [202, 240, 241], the merocyanines [190], the heterocyclic pyridinium betaines [242, 243], and spiro compounds [244]. 3.2J. Interplay between Asymmetry and Conjugation: Pofymethineimine Chains as Working Example It has been suggested that oligomers and polymers built from asymmetric units could be of interest for NLO applications because they can present first hyperpolarizabilities per unit length of magnitude similar to the best D/A systems [245]. In particular, various empirical, semiempirical, and ab initio investigations of polymethineimine (PMI) oligomers (see Fig. 10) have shown that (i) for sufficiently large oligomers, (Sf^/N is similar to [p/N]max of NH2/NO2-substituted polyene chains, (ii) the specific chain-length dependence of jS^ and pyN can be rationalized in terms of unit cell and chain end asymmetries, (iii) there is a substantial impact of chain conformation, and (iv) the first-order nonlinear response may be tunable by end-capping the PMI chains with D/A pairs. About 10 years ago, Albert et al. [246] carried out calculations on small PMI chains (N = 2-5) and found that p^i-lo); o),a)) has a magnitude similar to that of push-pull polyenes substituted with moderately strong D/A pairs. It also has the same sign as jjLx and evolves more or less as the third power of the chain length. A short time later, Tsunekawa and Yamaguchi [247, 248] pointed out that replacing carbon atoms by nitrogen in small NH2/NO2-substituted polyene chains results in a decrease of the electronicfirsthyperpolarizability that was attributed to a blueshift of the absorption maximum. Furthermore, they found that making the replacement at even-numbered positions (counted from the NO2 acceptor group) leads to molecules with a smaller fx but larger p than the case where the nitrogen atom is substituted at odd-numbered positions. However, the more complicated dependence of JSL/AT upon the length of the resulting PMI chains— which shows a maximum for medium-size oligomers—^wasfirstevidenced in a model Hiickel study [249]. This was later confirmed by semiempirical PPP [250] and ab initio Hartree-Fock [251] calculations. A plot of ^Pl(0)[N] or pl{0)[NyN versus A^ of unsubsituted all-trans PMI chains (see Fig. 11) reveals a characteristic "dromedary back" shape with a change of sign (except for very large BLA) before convergence toward the asymptotic limit Aj8|^(0)[oo] (= [pl[N]/N]N^oo)' The sign of p^ is associated

^

^

^

Nv

^

Fig. 10. Model polymethineimine (all-trans) chain.

with a negative longitudinal dipole moment for all chain lengths (the positive longitudinal axis points toward the right in Fig. 11). For PMI oligomers both the chain end asymmetry and the unit cell asymmetry contribute to j8^. As the chain is lengthened, the chain end contribution is expected to saturate eventually, whereas the unit cell contribution will evolve linearly at large N. Inspection of Figure 11 shows that, at large N, the asymmetry of the linker backbone induces a P\{!S) of sign opposite that caused by the chain ends at small A'^, thereby resulting in a hump at medium chain lengths. The magnitude of Pl^{0)/N at the polymer limit is much larger than at the maximum that occurs at intermediate N. Both asymmetry contributions are influenced by electron derealization as can be seen from the dependence upon BLA (Fig. 11). A smaller BLA increases the electron derealization and is associated with a larger Aj8|^ (or pyN) at the maximum (defined by A^max), a larger A^max, and a much larger magnitude A/3L[OO]. This behavior has been characterized by approximating pf^(0)[N]/N as a simple product of two functions; one arising (primarily) from electron delocalization and the other (primarily) from asymmetry: N Here (l-^-m^/N) is the asymmetry function, in which m^ measures the relative importance of the chain ends with respect to the backbone. The delocalization function [mi + m2 tanh((Ar - m3)/m4)] accounts for the remaining chain-length dependence including the inflection point followed by the gradual increase toward the asymptotic limit. In this function (mi + m2) is the infinite polymer limit of Pi^/N [comparison between mi + m2 and the extrapolated A)3L(0)[OO] values in Table XIII shows that the preceding function gives a global fit that is useful for quantifying trends but does not give an accurate asymptotic limit]; m^ largely determines the position of the inflection point; and mi/mi and m4 are associated with the amplitude and rate, respectively, of the main evolution with chain length. Thefittedparameters for the trans-transoid (TT) structure listed in Table XIII show that the smaller the BLA

Number of unit cells (N) Fig. 11. Effect of BLA (Ar) on the chain-length dependence of ^p\[N} for all-trans PMI chains. All values were obtained by the CPHF/6-31G method and are given in atomic units; Ar is the difference between the single and double bond lengths along the backbone (see Fig. 10). Reprinted with permission from B. Champagne et al, /. Chem. Phys. A 101, 3158 (1997). Copyright 1997 American Chemical Society.

89

THEORETICAL APPROACH TO DESIGN the larger the value of A^ at the inflection point (larger m^) and the larger the relative asymmetry of the chain ends with respect to the backbone asymmetry (larger |m5l). For BLA in the range considered here (0.100-0.220 A), the asymptotic Aj8^(oo) v a l u e obtained by extrapolation—increases in magnitude with electron derealization (i.e., with decreasing BLA) and, at the same time, the backbone asymmetry becomes relatively less important. In addition to BLA, the pair of backbone atoms can also be varied to optimize the second-order NLO response. Greater atomic alternation (i.e., difference between the atoms) within a given row of the Periodic Table is associated with a decrease of electron delocalization. This is detrimental to pf^ [179] but, at the same time, such alternation creates the backbone asymmetry that is responsible for nonzero jS^ in the first place. Therefore, it is worth investigating other asymmetric unit cell oligomers and polymers such as those depicted in Figure 12 in order to span new regions of delocalization-asymmetry space. Another important variable is the chain conformation. A change in conformation can increase or decrease the magnitude of Aj8£(oo) (see Table XIII) and possibly change the sign as well. With respect to all-trans chains, the relative chain end asymmetry is reduced for the trans-cisoid, cis-transoid, helical, and ghde plane structures. The trans-cisoid conformation possesses the largest CHF/6-31G Aj8f^(oo) value. It is 45% larger than Aj8^(oo) for the all-trans conformer (BLA = 0.126 A) whereas the cistransoid and helical trans-cisoid conformers have values that are.

respectively, 53% and 33% smaller in magnitude. The small energy difference between the different conformations [252] means that substitution and crystal packing effects could modulate their relative stability and thereby change the NLO response. Indeed, the conformation of the PMI oligomers synthesized by Wohrle [253] remains a mystery. From gas phase geometry optimizations [252, 254,255] the helical trans-cisoid conformation and the trans-cisoid conformation modified by a glide plane operation were found to be energetically similar whereas the planar all-trans, cis-transoid, and trans-cisoid structures are less stable. Several theoretical works have tried to identify the conformation by simulating the IR [255, 256] or X-ray photoelectron spectra [257]. With the exception of the work of Hirata and Iwata [255], who eliminated the all-trans conformation, none were conclusive. These studies [254-256] also showed the importance of electron correlation on both the ground state geometry and electronic properties [136]. For the all-trans conformers electron correlation leads to an increase in electronic derealization, which results in an increase of Apf^[oo] by a factor of 7.7 [245], and, at the same time, the hump at small chain length disappears due to a smaller relative chain end asymmetry. The large MP2/6-31G value of ^|3l[oo] = 13 x 10^ au stresses once more the wide potential of these systems for NLO applications. Indeed, for comparison, the value of [P/N]max for a,o)amino-nitropolyenes obtained at the same level of approximation (MP2/6-31G//MP2/6-31G) attains 15 x 10^ au [212].

H 1

H 1

B-^^^B 1 H

1 H

Fig. 12. Model asymmetric-unit-cell polymers. Table XIII.

Parameters of the Fitting Function [mi + m2 tanh((Ar - W3)/W4)] x (1 -H m5/A0 for Pl^[N]/N, and Extrapolated A^J^(oo), of Different PMI-Based Systems^

TT/Ar = 0.100A TT/Ar = 0.126A Tr/Ar TT/Ar TT/Ar TT/A7'

= = = =

0.150A 0.190A 0.220 A 0.100A<^

CT/Ar = 0.127A TC/Ar = 0.11lA

Ai8f^[oo]

mi

m2

m3

m4

ms

mi -\-m2

-871 -470 -274

-959 -552

11.5 10.8

6.3 7.0

-9.8 -7.9

-1830 -1022

-355 -222

10.0 7.4

-6.3

-629

-156 -3791

6.2 9.5

7.8 10.0 10.1 7.5

0.907 0.853 0.772

-3.3 -1.6 -1.4

-326 -220 -6798

0.468 0.436 0.793

- 1 3 , 040 ±327

54

3.8 8.9

5.1 11.2

746 -1473

12.81 0.999

688 ± 3 5 -2,112 ±75

3.7

12.4

-1.1 -2.6 -1.4

-764

0.033

-988 ± 2 8

7.7

7.9

-0.8

-258

-0.291

-0.2

-1532

0.250

-1,461 ±51

-532

0.733

-1,461 ±51

-104 -64 -3007 692 -736

HEL/AA' = 0 . 1 1 2 A

-25

-737 -739

GPL/Ar = 0.094 A

106

-364

NH2-(N=CH)Ar-N02

-306

-1226

1.9

4.0

N02-(CH=N)A^-NH2

-225

-307

4.2

3.1

-41

mi/mi

- 2 , 7 5 3 ±129 -1,461 ±51 -869 ± 2 8 -434 ± 7 -294 ± 2

—

^The parameters m^, mi, mi +m2, and A^£[oo] have dimensions offirsthyperpolarizabilities and are given in atomic units, whereas the other quantities are pure numbers. See [211,251] for a description of thefittingand extrapolation procedures. XT, CT, TC, HEL, and GPL stand for the following structures: all-trans or trans-transoid, cis-transoid, trans-cisoid, helical, and trans-cisoid modified by a glide plane, respectively. The NI^/N02-substitutedchains possess the all-trans conformation. All calculations were performed at the CPHF/6-31G level of approximation except for those marked with a C to indicate correlated MP2/6-31G.

90

CHAMPAGNE AND KIRTMAN

By capping PMI chains with the NH2/A^02 pair, the hyperpolarizabiHty can be enhanced and the maximum in P'\^/N made more pronounced [211]. Indeed, although jS^ is always less than that of the corresponding capped linear polyene linker, the fairly small difference in some instances may be worth the improved transparency for practical purposes. At small chain lengths, the asymmetry of the NH2/NO2 pair acts in concert with the linker when the NO2 group is placed at the nitrogen end of the oligomer. For longer chains one would expect the reverse placement to be optimal and this turns out to be the case. In the infinite chain limit, of course, the placement does not matter because one approaches the value for the unsubstituted PMI polymer in either event. As expected (see Table XIII), when the acceptor is placed at the nitrogen end the m^, asymmetry parameter sharply decreases whereas it increases when the acceptor is placed at the carbon end [211]. In the latter case, the magnitude of i8^(A^rnax)/A^max is multipHed by 34 compared to the unsubstituted PMI. In another study it was shown that substituted PMI oligomers have larger jS^ values than their polyenic analogs if the D/A is weak and the D is attached to the nitrogen end [179]. This has been explained by the fact that asymmetry then becomes the limiting factor rather than delocalization. Although in push-pull systems one or two excited states dominate the full second-order response, the situation is quite different for PMI oligomers where there is a band of charge-transfer states generated by the charge-transfer excitation within each —(CH=N)— unit. When the chain is small, a simple few-state UCHF/PPP scheme [250] can account for the maximum in P\^/N but this treatment rapidly fails when the size of the chain is increased. However, within the INDO/CIS procedure the two-state approximation reproduces the chain-length dependence of jS^ quahtatively and it is found that the change of sign in jS^ is accompanied by a change in sign of the difference in dipole moment between the ground andfirstexcited state [258]. Small chains behave like push-pull systems where the important excitationinduced charge transfer occurs between the two extremities. In large oligomers, on the other hand, the charge is mainly redistributed between adjacent carbon and nitrogen atoms at the center of the chain with the redistribution decreasing toward the chain extremities. 3.2.4. Vibrational versus Electronic First Hyperpolarizability We turn now to the vibrational hyperpolarizabilities of the j8 materials discussed in this review. As we have already seen in Section 2.4 the vibrational contribution is expected to be negligible at optical frequencies for SHG but not for the dc-P or OR effects. For the latter cases, in the NR approximation [cf. Eqs.(57), (58)], i8^(w) will be about 1/3 the static double harmonic value. Figures 13 and 14 compare the RHF/6-31G vibrational and electronic longitudinal first hyperpolarizabilities of a typical NH2/N02-substituted polyene at different optical frequencies for the dc_P and SHG phenomena. The dispersion curve for Pf^ follows the 6t>L Taylor series expansion of Eq. (34) for low frequency and becomes infinite at about 6.1 eV, which is the location of the first dipole-allowed excitation. On the other hand, for frequencies above the vibrational resonances i8L(^) saturates rapidly to the infinite optical frequency limit of zero and of ^[fioi]^, ^^Q for the SHG and dc_P processes, respectively. Recently, experimental and theoretical evidence has begun to accumulate indicating that the static i8^[i8^(0; 0, 0) = iSV(O)] and

"•—electronic

dc_P

•electronic SHG

• • -vibrational

dc_P

• • - v i b r a t i o n a l SHG

0.0

0.4

0.8

1.2

1.6

2.0

(3100 nm)

(1550 nm)

(1033 nm)

(775 nm)

(620 nm)

hv (eV)

Fig. 13. Frequency dispersion of the electronic and vibrational longitudinal first hyperpolarizabilities of NH2—(CH=CH)2—NO2 for dc_P and SHG processes obtained using the TDHF and double harmonic CPHF techniques with the 6-31G basis set. With the exception of the static limit, the vibrational first hyperpolarizability below hv = 0.4 eV (~3300 cm~^) has not been plotted because this region corresponds to vibrational resonances.

a

I

•

electronic clc_P

—O—electronic SHG - • -vibrational dc_P

i!

- D - v i b r a t i o n a l SHG

1000.0

B 5b 14.0

(hv^)^(eV^)

Fig. 14. Evolution of the electronic and vibrational longitudinal first hyperpolarizabilities of NH2-(CH=CH)2-N02 for dc_P and SHG processes as a function of the square of coj^ = ITTV]^ to display the relation between the electronic quantities. See Figure 13 for further details.

static j8^[j8^(0; 0, 0) = j8^(0)] are often of the same order of magnitude in push-pull TT-conjugated compounds [37, 150, 156-158, 163, 164, 259], although there are important exceptions to be discussed later. For molecules ranging from push-pull substituted benzenes and stilbenes to polyenovanillins and other polyenes, Zerbi and coworkers [156,157,259] have found that the calculated /3^(0)/]8^(0) ratio is close to unity. They have also carried out indirect measurements that tend to confirm this fact. In the experimental determination, j8^(0) is obtained by combining infrared and Raman intensities (along with the vibrational frequencies) whereas P^(0) is estimated from electric-field-induced second harmonic generation (ESHG) at an optical frequency. The close agreement between /3^(0) and (3^(0) has led Zerbi and coworkers to speculate that these two quantities are manifestations of the same underlying physical phenomenon. Indeed, they have at-

THEORETICAL APPROACH TO DESIGN 4U0.U

91

tempted to prove [43, 260] the approximate equality of j8^(0) and P^* (a.u.)= -51.496 -572.52 a^° N(CHJ^ j8^ (0) based on the two-state VB-CT model with a single dominant » 3 2 300.Q Rs 0.93428 vibrational mode. In addition to questions regarding the model 200.(^ ([37]; see also Section 3.2.5) there are further assumptions in their CI CH/^ ^ treatment that have been criticized [45]. Subsequently, some of OMe* % -j ^ 100.0 F these assumptions were removed by Kim et al. [39], who showed SI 0.0 that the model leads to a value for j8^(0)/j8^(0) that depends on is the electronic characteristics of the push-pull compound. Using CHCH . ' ^ typical parameters they estimate the maximum ratio to be slightly 2 ^^ ^ -100.0 S,Gk larger than unity. On the basis of an essentially identical model PaineUi [44] was able to support the conclusions of Kim et al. and 2 -200.0 also to demonstrate that j8^(0) is proportional to the polaron bind• NO ^ «5 -300.^ .6 ing energy, which is a measure of the strength of the electron-0.2 0 0.2 0 -0.4 phonon coupling. Recently, Torii et al. [165] have employed a similar two-state model to conclude that j8^(0)/j8^(0) < 0.5 but several questionable new approximations were utilized. •N(CHp^ PJ (a.u.)= -134.27 -524.6 o^" As we will see in Section 3.2.5 the VB-CT model leads to inconR= 0.83775 sistent results if the same parameters are used to calculate the entire set of polarization properties for a given molecule [37]. Thus, this model (as currently applied) cannot provide a satisfactory unI't derstanding of the relationship between j8^(0) [which depends on ^ - , •CHOH, "CN a^(0) and /i^(0)] and j8^(0). Furthermore, several studies have noted examples where j8^(0) is more than an order of magnitude a© CF. larger than j8^(0). Included among these systems are the merocyaCl» Of) U nines [150], compounds with two donors and two acceptors [37], 3 OOMe-^ 0 Qt charged polyenes [168], and polymethineimine oligomers [252]. NO • • CO^Hl The fact that j8^(0) can be so much larger than /3^(0) has •CHO aroused interest in structure-property relationships that will lead 0.2 0.4 to the simultaneous optimization of both of these properties. Thus, studies of /3^(0) and the j8^(0)/i8^(0) ratio have been carried out for several famihes of analogous or homologous push- Fig. 15. CPHF/6-31G /3^(0) (top) and /3^(0) (bottom) values as a funcpull TT-conjugated systems. One of these deals with substituent tion of o-^. Reprinted from B. Champagne, Int. J. Quantum Chem. 65, 689 effects in monosubstituted and disubstituted benzenes [163]. In (1997) by permission of John Wiley & Sons, Inc. this case it turns out that, in the double harmonic approximation, the i8^(0)/j8^(0) ratio computed at the CPHF/6-31G level is slightly larger than unity and almost constant with respect to the Q/J8L(0), where [^ca]^. ^^Q is the double harmonic apsubstituent(s). Nevertheless, the calculations for monosubstituted [^Jia]l benzenes plotted in Figure 15 indicate that the variations in J8L(0) proximation for $1^(0), ranges between 1 and 3. However, in the are better described by the unbiased mesomeric constant (a^) cyanin region where pf^iO) and $1^(0) are both small the ratio of than the corresponding variations in p\^{0). To quantify the induc- the vibrational to the electronic value can become very large betive contribution i8L(0) and i8L(0) were fitted by least squares to cause the two curves are not exactly coincident. This is the case for PL = Picn + PRC7-R, yielding )8^(0) = -376.1c7i - 424.1o-^ and the (NH2)2/(N02)2-substituted hexatriene. J8L(0) = —127.5(71 - 530.7(7^. From thesefitsone can readily see The relative importance of the lowest-order anharmonicity that induction plays an important role in p\^(0) whereas Pf^iO) is contribution, as computed by the FF/NR scheme in the static limit dominated by resonance effects. (i.e., [^L^][^. ^^o/t^^^L; (0=0^ ^^ ^^^^ Siven in Table XIV. For the As in the case of the electronic first hyperpolarizability, it is of polyenic structures this ratio varies between 0.15 and 0.45, but it interest to characterize the behavior of 0^ when the strength of the increases by an order of magnitude in the cyanin regime. Using D/A pair(s) become(s) stronger and the structure evolves from the FICs, it has been found that the major anharmonic component is polyenic to the cyanin and zwitterionic forms. Table XIV summa- electrical in nature, that is, [M^IL'^^^Q ^^^her than [ix^]^^^^^ [17]. rizes the RHF/6-31G double harmonic and first-order anharmonic At typical optical frequencies both the harmonic and anharmonic contributions to 1^1^(0) for a set of D/A hexatrienes [9, 37]. As contributions to jS^ are reduced. In the infinite optical frequency charge transfer increases from zero the BLA decreases, reverses limit for the dc_P (or OR) process, the harmonic term becomes sign, and then increases in magnitude as the zwitterion limit is 1/3 its static value whereas the anharmonic term vanishes. The approached. The general behavior of i8L(0) with respect to these adequacy of the infinite optical frequency approximation can be variations in the BLA is the same as i8^(0); that is, starting at the gauged by using the unique or dominant oscillator model, in which polyenic limit both increase at first, go through a maximum, and case then decrease to zero at the cyanin structure. For negative BLA 0,0 the first hyperpolarizability reverses sign but, otherwise, the shape [ixa] L; (o 1 1_2/M2 of the curve is the same with i8^(0) approaching zero in the zwitte(82) 1+ 0,0 J 3 3\o) ) [l-((o/a)a)^ [fia] rion limit (cf. Fig. 1). For most compounds in Table XIV the ratio L; w=0

92

CHAMPAGNE AND KIRTMAN

Table XIV

RHF/6-31G Static Longitudinal Vibrational First Hyperpolarizability for D/A-Substituted Hexatrienes (in Atomic Units) in Comparison with the BLA: Harmonic [jjia]^. ^ ^ Q and First-Order Anharmonic [fi^]]^. ^ ^ Q Contributions^

Molecules

BLA

^f^<.=o

[M«]L;

=O/^L(0)

CM']L;.=0/[^<;a>=0

H-CH=CH-CH=CH-CH=CH-NH2 N02-CH=CH-CH=CH-CH=CH-H CHO-CH=CH-CH=CH-CH=CH-OH

0.121 0.120 0.118

1,527

1.57

0.16

2,508

1.67

0.18

3,357

1.89

0.19

N02-CH=CH-CH=CH-CH=CH-NH2

0.097

8,928

2.19

0.45

P02-CH=CH-CH=CH-CH=CH-N(CH3)2

0.078

16,017

(N02)2-C=CH-CH=CH-CH=C-(NH2)2

0.044

9,768

(P02)2-C=CH-CH=CH-CH=C-(NH2)2

-0.026

-8,085

2.51 44.6 2.80

-0.92 2.46

^The compounds are given in increasing order of charge transfer between the D and A.

[/i r

3il,0

3il.O

3 [1 - (ft>/cOfl)2]

3 \ (o

r

(83)

3 «J

6000.0

3iO,l

[;^-

3i0,l

[M- U(o=Q

(84) [1 -

iw/u>a)^2

where coa is the harmonic frequency of the dominant vibrational normal mode. For typical A and o)a values of 1064 nm and 1600 cm~^ the o)a/o) ratio is about 0.2. This reduces the factor of 1/3 in the infinite optical frequency approximation for the har- M monic term to 0.30, whereas the electrical and mechanical anharmonicity components in Eqs. (83) and (84) are reduced by factors of almost 40 and 600, respectively, with respect to their static values. According to this analysis, the [fJt?]]^. oj/^t^^^h- w ^^^^^ ^^^ ^^^ Number of (CH=CH) units dc-P effect in (P02)2-C=CH-CH=CH-CH=C-(NH2)2 will decrease from 2.5 to 0.2 whereas for the other compounds it be- Fig. 16. Evolution of the RHF/6-31G values of PliO)/N and PliO)/N comes completely negligible. A similar treatment can be carried for push-pull NH2—(CH=CH)jv—NO2 compounds with the number of out for SHG but the bottom line is that, according to the infinite (CH=CH) units A^. frequency approximation, jS^is expected to be insignificant. Like its electronic counterpart the vibrational first hyperpolarizability of increasingly large linkers end-capped by a D/A pair sat- tions a r o u n d the sp^ spacer (when present), and (iii) inter-ring urates toward an asymptotic value characteristic of the linker due torsions, all of which are expected to be strongly influenced by to the decreased interaction between the D/A pair. Similarly, the medium effects. Thus, the high-frequency modes are more perresponse per unit cell, or length or volume presents a maximum as tinent to the real system (the calculations were done on individual shown in Figure 16 for the prototypical N02/NH2-a:,ft>-substituted chains) and they are also of greater interest from a practical point polyenes. The maximum in li^(0)/N occurs at A^ = 8 compared to of view. Once the contribution from the low-frequency modes has A^ = 10 for its electronic analog whereas the ratio $1^(0)/p^{0) is been subtracted out there is little variation in jSyec /P\QC(^) ^^^ 1.6 at the point where the sum is a maximum. From Figure 16 one the linkers considered in Table XV, particularly if one focuses on can also see that $1^(0)/N decays more rapidly toward its asymp- the longer chain results. Nonetheless, it is worthwhile to examine totic value than j8^(0)/A^. the role of the various structural factors. The nature of the linker plays a critical role in determining The first point concerns the effect of varying the BLA. Changthe magnitude of the vibrational first hyperpolarizabiUty as well ing from the polyenic (BLA ~0.1 A) to the polyynic (BLA '^0.2 A) as p^ and the j8^//3^ ratio. Some CPHF/6-311G** results for linker reduces both 0^(0) and /3^(0), but has a slightly larger efD/A-substituted conjugated linkers, taken from [168], are summa- fect on the former thereby lowering the j8^(0)/j8^(0) ratio a small rized in Table XV. In analyzing this data it is useful to separate amount. The same conclusion has been drawn using basis sets the total sum over modes (SOM) value of jSyecCO) into a con- other than 6-311G** [261]. Kogej et al. [262] have found from fitribution from normal vibrations with o) < 100 cm~^ and with nite field simulations that inserting an sp^ defect at the center of 3V;>100. 0) > 100 cm-1 - ov;<100. j8v^t'''''(0) and jSyec "'"''(0) are the corresponding the chain to break the conjugation can markedly increase the elecnotation. For the former the vibrational response time is at least tronic first hyperpolarizability provided the D/A strength is large two orders of magnitude slower than the electronic response time, enough. For the NH2/NO2 pair, inserting a CH2 or SiH2 moiety assuming the electronic process is characterized by transitions in in a polyene segment decreases /3^(0). In fact, the effect on j8^(0) the range 1-3 eV. Furthermore, these low-frequency modes in- is much more than on j8^(0) thereby leading to a substantial involve mostly (i) out-of-plane torsions of the NH2 group, (ii) mo- crease in the ratio (see Table XV). However, the reason /3^(0) is

THEORETICAL APPROACH TO DESIGN Table XV

93

CPHF/6-311G** values (in 10^ au) of I3%^i0) and j8^ec(0) for a,w-Nitro,amino-Substituted Linkers [168]^ iS^ec(O)

^;;ec'''(0)

/35ec(0)

/35ec(0)

NH2-linker-N02

i8^ec(0)

(CH=CH)2 (CH=CH)3 (CH=CH)4

35.1

3.99

4.04

76.9 131.5

3.51 2.86

3.69 3.04

(CH=CH)5

191.7

2.58

2.77

(C=C)2

24.7

3.29

2.52

(C=C)3

43.4

2.77

2.19

(C=C)4

67.5

2.58

2.01

C H = C H - C H 2 - C H = ' CH

18.4

( C H = C H ) 2 - C H 2 - ( C H = =CH)2

80.5

10.7 7.16

6.58 2.79

C H = C H - S i H 2 - C H = CH

23.9

(CH=CH)2-SiH2-(CH= =CH)2

50.4

4.79

Ph (p-NA) Th

14.0 19.0 18.3

2.52

2.52

3.17 2.62

3.15 2.62

52.6

6.36

4.60

(Ph)2 (Th)2

33.3

2.17

1.35

72.3

3.62

2.01

(Py)2

46.1

2.59

2.18

Ph-CH=CH-Ph

106.0

2.91

1.50

Th-CH=CH-Th Py-CH=CH-Py

181.3

5.40

1.96

134.3

4.05

2.58

Ph-N=N-Ph

45.8 69.2

1.30 1.98

1.30 1.95

Th-CH=CH-Ph

-4.7

-0.12

1.87

Ph-CH=CH-Th

82.2

2.16

1.78

Py-CH=CH-Ph

86.1

1.75

1.61

Ph-CH=CH-Py

58.9

2.42

2.26

Py cis-CH=CH-CH=CH

Ph-C=C-Ph

17.1

6.02 3.28

ov;>100, «The corresponding /3^ec(0)//35ec(0) and p;hc'''''iO)/Pt^ .(0) ratios are given in the last two columns.

affected less (especially in the longer chains) is primarily because of the large contribution from vibrations with co < 100 cm~^ It is interesting that a substantial increase in /3^(0) can be induced in these sp^ defect systems by switching on an external electric field of a specific magnitude [262]. Certainly further investigation is warranted. For aromatic linkers both the electronic and vibrational j8(0) decrease with respect to the analogous cis- or /ran^-butadiene linker, but j8V(0)/j8^(0) remains in the range 2-4. This ratio has also been calculated for A^-(4-nitrophenyl)-(l)-prolinol (NPP), in which case it is somewhat larger than 2, and for 3-methyl 4-nitropyridine 1-oxide (POM), where the value is slightly larger than unity [164]. In the case of dimeric linkers with NH2/NO2 as the D/A pair aromatic stabilization may explain the variations of j8^(0) in the order pyrrole > thiophene > phenyl. On the other hand, for j8^(0) [and jS^(0)7)8^(0)] the order is thio-

Fig. 17. Orientation of the dipole moment, as well as the electronic and vibrational static first hyperpolarizability vectors with respect to the molecular axis for the quinonoid (top) and aromatic-protonated (bottom) forms of the 4-[2-(l-methyl-4-pyridinio)ethenyl]phenolate merocyanine dye [150]. The length of the arrow is proportional to the magnitude of the vector. The origin of the Cartesian frame has been placed at the center of mass.

phene > pyrrole > phenyl. For biphenyl the relatively large torsion angle diminishes its electronic and vibrational first hyperpolarizabilities. Bithiophene and bipyrrole both exhibit similar j8^ and j8^/j8^ provided the low-frequency modes are ignored. The insertion of a short polyene or polyyne segment between aromatic rings, to increase conjugation directly and indirectly (by reducing the torsion angle), does not lead to substantial variation in the ^v;>l00(Q)/^e(Q) ratio although both the numerator and denominator increase substantially. It seems that j8^ is almost independent of the segment (CH=CH, C=C, or N=N) that is inserted in the biphenyl compound. Although j8^(0) varies by more than a factor of 2, most of the variation is associated with the lowfrequency modes in the planar Ph—N=N—Ph linker. The auxilary donor-acceptor effect of the aromatic rings themselves (see also Section 3.2.2) is similar for both j8^(0) and j8^(0). However, the inter-ring torsion angle does influence the ratio with or without the o) < 100 cm~^ vibrations excluded. As a result, although the largest j8^'>l^^(0) in all linkers examined is obtained for N H 2 - P y - C H = C H - P y - N 0 2 the largest total value including /3^(0) is given by N H 2 - P y - C H = C H - P h - N 0 2 . The interconversion of stilbazolium merocyanine dyes from the quinonoid to the aromatic form generates a substantial variation of the i8^(0)/j8^(0) ratio (Fig. 17). Stabilization of the aromatic form has been realized in polar solvents and is accompanied by a negative solvatochromism [263, 264]. For modeling purposes the aromatic form has been induced by adding a proton on the oxygen atom. At the RHF/6-311G* level of approximation the PyQc(^)/P%c(^) ratio is very large for the quinonoid form due to a small electronic contribution whereas, upon protonation (taking the center of mass as origin), the electronic component increases (in absolute value) by nearly one order of magnitude and the i8^ec(0)/iSvec(0) ^^^tio is reduced to 1.9 (see Table XVI). Although of smaller magnitude, similar effects have been obtained upon replacing the phenyl by a thiophene ring [150].

94

CHAMPAGNE AND KIRTMAN

Table XVI.

CPHF/6-311G* Values (in 10^ au) of jS^ecCO) and jS^ecCO) for the Different Forms of the 4-[2-(l-Methyl-4-pyridinio)ethenyl]phenolate Merocyanine Dye^

Merocyanine Protonated merocyanine

i8?ec(0)

i8^ec(0)

10.4 -91.2

229.0

22.1

-172.7

1.9

(a)

i8^ec(0)//8?ec(0)

(b)

^Data from [150].

An important point to note is that PyQc(^) and i8^ec(0) reverse sign between the quinonoid and aromatic forms. By taking advantage of hydrogen-bonding interactions, this sign change has been used to build cocrystals where the first hyperpolarizability vec(c) tors are parallel and constructively reinforced, whereas the dipole moment vectors annihilate each other [265]. Based on a simple additive model an upper bound of 1.32 has been estimated for the jSyec(~^; ^j ^)/PwQc(~^''> ^ ' 0) ratio of the antiparallel dimer from CHF/6-311G* calculations [150]. This supports the experimental fact that the electrooptic coefficient calculated from the measured SHG coefficient is consistently smaller than the measured electrooptic coefficient [266]. Additional compounds deserve to be investigated to better understand and optimize the sub- Fig. 18. Atomic displacements for individual vibrational coordinates used stantial jS variations associated with the aromatic-to-quinonoid in- to characterize the vibrational longitudinal first hyperpolarizability of NH2—(CH=CH)3—NO2; the length of the arrow is proportional to the terconversion. Similar aromatic to quinonoid interconversion can displacement: (a) a priori effective conjugate coordinate; (b) most conalso be induced upon doping as we will discuss in Section 3.3.4. tributing mode to [iJia]^^; (c)first-orderFIG ix\)' PMI chains constitute another class of compounds with very large j8^(0)/j8^(0). The most notable case occurs for trans-cisoid chains distorted by a glide plane operation where it has been found for the polyene and, for the aromatic species, over 75%. It is enthat at N = 14 (the largest oligomer treated) the double harlightening to compare this with the contributions due to the indimonic term [iJLa]^^, W ^ Q / ^ ^^ ^-^^ times larger than /3L(0) and vidual normal modes. In the case of the polyene the normal mode this term is still increasing linearly with chain length [252]. Fur- shown in Figure 18b is the largest single contributor to the double thermore, the first-order anharmonicity correction (for N = 6, harmonic $^(0). However, this motion gives only 30% of the total f^^^L a;=o/[^^^L- (0=0 ^ ^'^^^ ^^^^^ ^^^ ^^ ^^ ^^ ^^^^^ ^^ ^^^ ^^^ D/A polyenes. Because the number of vibrational degrees of freedom grows linearly with the number of atoms, there are many modes to take into account in the SOM expression for j8^(0) in molecules of interest as NLO chromophores. Of course, some of these modes may not contribute significantly. The ultimate simplification would occur if just a single vibrational motion proved to be dominant. For conjugated molecules, especially push-pull polyenes, it has been suggested by Zerbi and collaborators [155,157] that the collective motion associated with the BLA would satisfy this criterion. They refer to the coordinate describing this motion as the effective conjugation coordinate (ECC). A precise definition of the ECC has not been given. Although it is often assumed to be a normal coordinate, that turns out to be too restrictive. Recently, a simple implementation of the FF/NR method (cf. Section 2.5) has been formulated [16] that allows one to determine the contribution of any such internal coordinate, or set of internal coordinates, to j8^. This method was applied to two different a priori choices for the ECC in /7-nitroaniline and NH2 - ( C H = C H ) 3 - N O 2 . The best results were obtained for the modes drawn in Figure 18a, where each hydrogen atom is given the same longitudinal displacement as the atom to which it is attached (as opposed to moving the hydrogen atoms in a way that keeps all angles fixed). Using this definition of the ECC over 95% of the complete SOIVI value was recovered

value and it takes the next four most important normal modes before even the 75% level is reached. In the case of/>-nitroaniline, the largest single contributor represented in Figure 19b provides 57% of the total value whereas addition of a second normal mode gives a result (74% of total) that is comparable to th& a priori ECC [158]. It remains to be seen how successfully the notion of an a priori ECC can be extended to other 0^ molecules (the percentage error in y^ for the same two examples is considerably larger than the error in j8^). To a large extent this issue is muted by the fact that one can now readily compute a single first-order field-induced coordinate [17] that will exactly reproduce the total longitudinal j8^. This FIC is shown in Figure 18c for NH2 - ( C H = C H ) 3 - N O 2 . Clearly, it is quite different from the corresponding a priori ECC and it remains to be seen whether an intuition regarding the form of the FIC can be developed. Although it is highly relevant for quantitative predictions, the inclusion of electron correlation has only been addressed very recently thanks to the lowered computational cost of the FIC-based FF/NR procedure [120, 212]. For the N H 2 - ( C H = C H ) A r - N 0 2 oligomers with N = 2,3, electron correlation corrections evaluated at the MP2/6-31G level increase the electronic component by a factor of 3 whereas the vibrational counterpart decreases by about 10%. This has a big impact on the P'^/P^ ratio, which is reduced to less than 0.20 for the dc-P effect. The magnitude of higher-order electron correlation corrections and the degree to

95

THEORETICAL APPROACH TO DESIGN (a)

of such treatments relies heavily on the choice of molecular parameters. To provide an assessment of the VB-CT modoi perse Bishop et al. [37] have derived a simple set of parameter-independent relations between the vibrational and electronic contributions to the static diagonal polarizabilities and hyperpolarizabilities. Their relations are a consequence of the fact that, within the two-state VBCT model, the normal coordinate derivative of an electrical property is proportional to the next higher-order property:

(b)

(85) (86) (87) where (88)

Ai = V12 (c)

In deriving these equations it is assumed, as usual, that A)Lti2 ^^^ t are independent of the nuclear geometry. Using Eqs. (85)-(87) the following expressions for the diagonal components of the static vibrational (hyper)polarizability tensor can be obtained within the double harmonic oscillator approximation: 3N-6 {dix^ldQif

a"(0)= Y.

^

= Ba^(0)a^(0)

(89)

i=l

3A^-6

Fig. 19. Atomic displacements for individual vibrational coordinates used to characterize the vibrational longitudinal first hyperpolarizability ofpnitroaniline; the length of the arrow is proportional to the displacement: (a) a priori effective conjugate coordinate; (b) most contributing mode to [fia]^; (c)first-orderFIG {x\).

i8^(0) = 3

^ /=1

{dii^ldQi){da^ldQi)

-]

/(O) = 4

= 3Ba^(0)j8^(0)

(90)

3Ar-6 {dp.^ldQi){dp>^ldQi) ^ /=1

^^^{da^ldQi){da^ldQi) which they will modify the structure-property relations deduced thus far has yet to be established. To our knowledge, due to the substantial computational needs, there exists only one study where the importance of the ZPVA correction to j8 of push-pull 7r-conjugated compounds has been evaluated. This has been achieved for NH2 - ( C H = C H ) 3 - N O 2 by taking advantage of efficiencies afforded by the use of FICs [120]. In the static limit it was found that the p^^^^/p^ ratio ranges between -0.05 and -0.08 at the RHF level with different atomic basis sets. Although jg^^^A appears to be relatively unimportant in this particular instance, that conclusion may not be general and, again, electron correlation as well as frequency dispersion will be required for quantitative estimates. 3.2.5. Assessment of the VB-CT Model for Relating Electronic and Vibrational Hyperpolarizabilities From ab initio calculations on several famihes of push-pull TTconjugated systems various trends have been identified that relate electronic and vibrational first hyperpolarizabilities in terms of geometrical and/or electronic structure parameters. For further understanding and generalization of these relations a simple analytical scheme, such as the two-state VB-CT model presented in Section 2.2 for the electronic hyperpolarizability, could be very helpful. Indeed, an extension of this model to vibrational hyperpolarizabilities has been carried out by Kim et al. [39] and a very similar approach has been taken by Painelli [44]. However, the success

i=l = 4Ba^(0)/(0) +

^? 3Bp^(0)P^(0)

(91)

with 3N-6 ^2

^=E^

(92)

Note that all vibrational modes are included in this treatment. The behavior of each vibrational contribution with respect to its electronic counterpart is displayed in Figure 20 as a function of the mixing parameter 6. By combining expressions (89)-(91) in pairs to eliminate B one then obtains the parameter-free relations

(i87a^)/(;8Va^)=3

(93)

([Mi3fV)/(rV«')=4

(94) (95)

Ab initio Hartree-Fock tests of Eqs. (93)-(95) were conducted on a set of D/A 7r-conjugated molecules spanning the polyenic, cyanin, and zwitterionic regions (cf. Fig. 1) [37]. The calculated results, displayed in Table XVII, are in most instances far from the values predicted by the VB-CT model. This indicates that the model as currently formulated cannot be valid for both vibrational and electronic properties. Of course, it may be possible to remove

96

CHAMPAGNE AND KIRTMAN Electronic Vibrational

§^ O N

II

00 O

0^ o.oi 9=0.0

0=71/2

0=TC

The qualitative behavior of the vibrational properties as a function of the mixing parameter 6 can be determined from the VB-CT model by combining Eqs. (89)-(91) with Eqs. (28)-(31). Upon doing so it turns out that the curves for a^(0), 0^(0), and the two components of 7^(0) (i.e., [a^]^'^ and [/x)S]^'^) have the same general shape as their electronic counterparts in Figure 1. This has been numerically exemplified by Kim et al. [39] and is consistent with most of the results presented in Section 3.2.4. It seems that the variation of the vibrational properties with the amount of ionic-covalent mixing is not as sensitive to the details of the model as the ratio of vibrational and electronic values for a given 0. Finally, the derivative da^ jdQi in Eq. (86) can, alternatively, be written as a sum of two terms obtained by differentiating either the numerator or the denominator of Eq. (28). It follows from the VB-CT model that the former should be exactly twice the latter [37], and CIS/6-3 IG ah initio calculations for the first molecule in Table XVII are in reasonable agreement with that prediction for the nine most important vibrational normal modes. On the other hand, these results are inconsistent with the model of Castiglioni et al. [43, 260], which neglects the larger numerator term. 3.2.6, NonconventionalDIA NLO Materials

Chromophores as Second-Order

As classified by Wong et al. [267] there exist several other types of interesting D/A chromophores for NLO applications. One of these is the case of multiple D/A chromophores. We have seen in J fc -10000.00=0.0 0=7C 8=71/2 Section 3.2.1 that double substitution can shift the structure from the polyene to the cyanin region. Theoretical and experimental work performed by Spreiter et al. [268] on multiply-substituted 1.0 10' two-dimensionally conjugated tetraethynylethenes indicates that i8vec(-26;; a),o)) can be as large as in DANS. Moreover, the occurrence of a dominant nondiagonal tensor element is a factor that A,,*-^"*-!^,,^^ ^^, ^ '^ • • -* ^ ^ liTfc a 0.0 10" is expected to improve the phase matching. ^v / t / t Since 1990 it has been recognized that octupolar molecules 1 \ .S.N such as l,3,5-triamino-2,4,6-trinitrobenzene (TATB) [269-273] can « ^ ^ — ^ 11 1 ^ § -1.0 10' 1 exhibit second-order NLO responses as large as their dipolar 1 1 1 I Electronic 1 analogs. Furthermore, the absence of a dipole moment has opened 1 1 new strategies for building two-dimensional crystalline structures \\ /' Vibrational [nK°'° -2.0 10* that display large macroscopic second-order NLO responses [274]. \/ This relatively late discovery is related to the absence of an elec• • • Vibrational [a^]^'^ | tric field-induced SHG signal for octupolar molecules because the o -3.0 10* O measured quantity is the projection oi ^ on fx and ^c = 0 whereas 0=7C 0=0.0 0=7C/2 C/5 the hyper-Rayleigh scattering (HRS) technique measures an oriFig. 20. Comparison between the ^-dependence of the electronic and vientational averaged j8 [271, 275]. To our knowledge, most of the brational contributions to the static longitudinal polarizability and hyperinvestigations on octupolar systems have been restricted to TATBpolarizabilities as obtained from the two-state VB-CT model with typical parameters; r = - 4 x 10"^ au (-1.09 eV), A^t^ = 4 au (10.2 D), type compounds, to the simple tricyanomethanide anion and to {dVjdQ) = 1.5x10-^ au (1.25x10-^ eV A"! amu-^/^), A = 3x10-^ au, analogs of crystal violet [270, 271, 276]. As in the case of D/A w = 6 X 10~^ au (1317 cm~l). It has been assumed that B = 3 (A/o))^ to dipolar compounds, the j8 response has been shown to depend account for the contribution of several modes. upon the solvent polarity [277]. Some of our own yet unreported RHF/6-31G calculations listed in Table XVIII demonstrate that, again, like D/A dipolar compounds, the vibrational contribution to the j8 of octupolar molecules can be significant. some of the assumptions while still retaining much of the simplicity To avoid absorption of the second harmonic signal, Bahl et al. of the VB-CT scheme. Some directions worthy of investigation are [278] have proposed substituting non-covalent through space D/A (i) include a second excited state that, in addition to being critical for y, is required to obtain quantitative accuracy for (3 (see Sec- interactions for through bond 7r-conjugation. For similar /3 values, tion 2.2), (ii) allow for the coupling between the VB and CT states, it turns out that the through space D/A chromophores exhibit a t, and A/Li-12 as well, to depend on the normal coordinates, and smaller Acutoff• Furthermore, their small dipole moment favors the (iii) remove the restriction fjii2 = 0, which, among other things, creation of asymmetric macroscopic structures. We will come back to this strategy later when discussing third-order NLO materials. causes all the properties to vanish at ^ = 0 and ir/l.

97

THEORETICAL APPROACH TO DESIGN Table XVII.

RHF/6-31G Vibrational-to-Electronic Polarizability and Hyperpolarizability Ratios [see Eqs. (53)-(85)] for a Set of Push-Pull TT-Conjugated Systems from [37]

Molecule Ideal VB-CT 1 -Amino-6-nitrohexa-1,3,5 -triene l-Formyl-6-hydroxyhexa-l,3,5-triene l-A/^,N-Dimethylamino-6-phosphonohexa-l,3,5-triene l-Ammoniohexa-l,3,5-triene-6-carboxylate l-Phosphinohexa-l,3,5-triene-6-sulfinate l,l-Diamino-6,6-dinitrohexa-l,3,5-triene 4,4-(/7-Methylpyridyl)-l,l-dicyano-l,3-butadiene 4-Methylpyridone (or p-methylpyridone) l-Amino-4-nitrobenzene (or p-nitroaniline) l,l-Diamino-6,6-diphosphinohexa-l,3,5-triene

Table XVIII.

([;L.iS]0'0/a^)

([a2]0,0/^v)

(yV«')

(i8V«')2

3

4

3 7.67 10.54 7.28 4.68 0.79 41.7 -25.0 -4.34 6.07 3.74

26.82

4.05 2.87

103.3

4.76

14.09

-2.47

28.20 11.07

-2.59

1671.0

-14.9 24.4

370.88

-55.4 4.89

56.77 18.89

24.67

15.00

RHF/6-31G Values (in Atomic Units) of Tensor Components of the Static Electronic and Vibrational First Hyperpolarizability of Octupolar Molecules^ Pyyy

l,355Triaminobenzene 1,3,5-Trinitrobenzene l,3,5Triamino-2,4,6-trinitrobenzene

-'xxy

291 319 854

-P.xyy 66 0 500

Pvvv

-'xxy

82 221 783

t^xxx

t^xxyy

19 0 458

^Pxxx = 0 in 1,3,5-trinitrobenzene because the structure is planar whereas the amino groups in 1,3,5-triaminobenzene distort from their planar conformation. The y-3xis corresponds to one of the C2 axes.

The substituted calix[4]arenes, where the four chromophores are oriented in the same direction, are among the most popular multi-dipolar chromophoric assemblies because this should lead to large dipole moments with high polar order and stability. Semiempirical calculations have verified that the dipole moment is enhanced whereas the p^ value per chromophore decreases due to TT-TT Stacking [279, 280]. Helical structures have also been found to display interesting second-order NLO responses. On the one hand, the nonzero p^ values of unsubstituted helices that arise from their chirality can be substantially enhanced upon adding D/A pairs [281]. In addition to the D/A strength the magnitude of the twist angle also influences j8^. On the other hand, the chiral contribution to /3^ may become very large on the macroscopic scale [282]. These chiral and achiral components have a mixed electrical and magnetic origin [283]. 3.2,7. Doping Effects An interesting possibility that exists for the j8 materials we have been discussing is the introduction of charges through chemical doping. Charged systems have not received much attention because their hyperpolarizability was not experimentally accessible until development of the HRS technique. The circumstance where a charge is localized on an end group has been considered [174, 284] because of the potency of the resulting end group as a donor or acceptor. In cases such as the protonated Schiff base of

the retinal chromophore [239] or stilbazolium dyes [285] charging is associated with extremely large j8 values. For organic acids it has been found, using the ZINDO correction vector approach including solvent effects through a self-consistent reaction field [286], that the negatively charged base form possesses a larger P^(-2(o; a),(o) than the neutral acidic form and the increase was correlated with a redshift of the absorption maximum. In contrast with the case of end group charges, there are no experimental studies on push-pull molecules where the charge is primarily localized on the linker. Unsubstituted (oligomeric) molecules that are charged (by doping) along the backbone have been studied extensively—not for their NLO properties, but primarily in connection with their electrical conductivity. As we will see in Section 3.3, ab initio calculations reveal the possibility of large-y NLO enhancement in these oligomers and polymers due to the charge and associated geometric rearrangements. Thus, it is logical to see what happens for D/A substituted oligomeric chains under similar circumstances. Recent calculations on singly charged (positive) polyene linkers substituted with D/A = NH2/NO2 found increases in p^ compared to the neutral by up to a factor of 2.8 at AT = 8. This is consistent with the fact that the excess charge is delocalized over the entire molecule and the BLA is reduced to zero over almost the entire carbon backbone for N ranging between 2 and 8. Because of the increase in p^ the ratio j8^(0)/j8^(0) is reduced (see Fig. 21); it varies from 0.57 at AT = 2 to 0.63 at A/^ = 8, whereas the corresponding values for the neutral are 2.20 and 1.74 [168, 175]. Al-

98

CHAMPAGNE AND KIRTMAN 3.3. Third-Order NLO Organic Materials

Third-order NLO compounds may or may not have a center of inversion. In the former, the first hyperpolarizability vanishes due B g to symmetry whereas for the latter both second- and third-order effects occur. Many conjugated oligomers and polymers, such as -neutral chains u o polyacetylene and polyparaphenylene, belong to the first category. o S Exceptions include oligomers that are asymmetric regardless of -radical cation chains the chain length (e.g., PMI) and those that are asymmetric only 2a for chains containing an odd number of oligomers such as PTh ^1 and polysilane (PSi) in the planar all-anti configuration. In the latter case p/N will approach zero as the chain is lengthened. We will focus here on oligomers and polymers that do have a center of inversion. Of particular interest will be the chain-length dependence, the effect of bond length alternation, the relative magnitude of Number of CH=CH units Fig. 21. Chain-length dependence of the j8^/j8^ ratio as a function of the vibrational contribution, and the effect of doping. In addition the charge of the system. Reprinted from B. Champagne and B. Kirtman, to large second-order NLO response, push-pull 7r-conjugated sysChem. Phys. 245, 213 (1999). Copyright 1999, with permission from Else- tems may also exhibit substantial third-order NLO response [owing to the dipolar terms (D) in Eq. (31)]. We will discuss their povier Science. tential for NLO applications considering both electronic and vibrational contributions. Although most of the ab initio calculations to be reviewed here 1.6 10*1 were done by the CPHF method there are also some correlated —•—neutral treatments and studies of the frequency dispersion that will be dis1.4 10* -Q—charged cussed. Because the investigated systems are spatially extended, 1.2 10* good accuracy can be achieved using a split valence basis despite CJ N the fact that such a basis is inadequate for small oligomers (see '3*3 1.0 10* Section 2.6). •a o

^I

d -^

3,3.1. Chain-Length Dependence and Bond Length Alternation Effects upon y^ in One-Dimensional Oligomers

4.0 10^ 2.0 101

4

5

6

7

8

Number of C H = C H units Fig. 22. Chain-length dependence of the longitudinal static (electronic + vibrational) first hyperpolarizability as a function of the charge of the system.

though /3^ itself decreases significantly due to charging at smaller N there is no change SLXN = 1, 8. Thus, the total hyperpolarizability [/3^(0) + i8^(0)] shows a marked increase for the longer chains (Fig. 22). Thesefindingsare indicative of the effects that may arise due to doping. However, they are not definitive, most importantly because the counterion has not been taken into account, but also because electron correlation was ignored. To our knowledge no comparable studies have been undertaken for other linkers [such as polythiophene (PTh)] or for oligomers (such as PMI) with an asymmetric repeat unit. Because the nature of the charged structure in unsubstituted PTh (and other heteroaromatic or aromatic) oligomers is different from the linear polyenes (see Section 3.2) the behavior of jS^ and jS^ may be different from that previously described when this oligomer is used as the linker in a D/A molecule. The effect of charging PMI oligomers (with or without D/A capping) remains to be determined and the same is true of cumulene linkers that, as neutrals, give exceptionally large pf^ in semiempirical [209] and ab initio [287] calculations.

The first hyperpolarizability of oligomers or polymers with a center of inversion vanishes due to symmetry. Thus, the second hyperpolarizability becomes the leading nonlinear term in the expansion of the induced dipole moment as a function of applied field. Because y/N (as well as a/N) increases with oligomer chain length until an asymptotic limit is reached (Fig. 7), we shall be interested either in extended chains or in the polymer limit. Two main approaches have been adopted to evaluate the third-order NLO response of polymers. Thefirst,which will be employed here mainly, is called the oligomeric approach. It consists of evaluating the properties for increasingly large oligomers and, if necessary, extrapolating to the infinite chain length limit. As discussed later, extrapolation raises issues regarding the choice offittingfunctions, the amount of oligomer data and the accuracy of this data. The second approach is the band structure, or crystal orbital, method, which takes advantage of the one-dimensional periodicity of these systems [288]. Special provision must be made for dealing with the nonperiodic potential describing the interaction between the polymer and the external field. At the UCHF level, an appropriate scheme for computing the static and dynamic i!!ia\[oQ], ii^p\[oQ\, and A7L[OO] was presented quite some time ago by Genkin and Mednis [289]. This scheme has been implemented using Huckel theory [290] and it has been verified that the band structure values agree with infinite oligomer results [288, 291, 292]. However, higher-level treatments have been slow to appear. An ab initio finitefieldCPHF procedure has been proposed [293] but the resuks reported are inconsistent with oligomer calculations [294]. Earlier attempts were limited to the electronic linear response [295]. Most

THEORETICAL APPROACH TO DESIGN recently, the Genkin-Mednis (GM) procedure has been formally extended to yield a fully analytical GM/CPHF treatment [296]. Implementation of this treatment, along with further theoretical steps to include electron correlation and vibrational hyperpolarizabilities, is being pursued. Thus far, ab initio hyperpolarizabilities for polymers are available only through the oligomeric procedure, which leads to a consideration of the chain-length and/or delocalization-length dependence. Some qualitative features can be obtained for y^ from the simple free electron and Hiickel approaches [288]. The free electron model predicts that y^ will increase as L^ whereas the Hiickel approach for a nonalternant polyene chain predicts an L^^ dependence. Introduction of alternancy in the Hiickel treatment is required for y^/L to converge in the asymptotic (L -^ oo) limit. With regard to the evolution of electronic structure and properties as a function of chain length, Hiickel (and PPP) calculations [297] show that one should think of a new unit cell as being inserted at the center of the chain rather than at either end. This is not only true for the electronic—and vibrational—response but also for geometrical parameters [146]. Thus, it is preferable to estimate the hyperpolarizability of the infinite polymer per unit cell by taking the difference yi^[N] - yj^lN - 1] = AyL[^] instead of dividing the value for the longest chain by the number of unit cells (see Fig. 7). The former quantity converges much more rapidly with chain length to the polymer limit because the end chain effects are mostly removed. However, the hyperpolarizability per unit cell converges much more slowly than the energy per unit cell or the geometry and this means that an extrapolation is usually required (see, e.g., [298] as well as [146, 148]). One mitigating factor is that the degree of convergence is determined by the number and arrangement of the conjugated bonds rather than the number of monomer units. A simple argument to that effect may be made using PA as an example. If the unit cell is taken to be [-(CH=CH)-]2, rather than -(CH=CH)-, then saturation will be reached with a chain containing only half as many unit cells. It is important to realize, however, that the larger unit cell will yield fewer data points for extrapolation, thereby reducing the accuracy. A reasonably satisfactory extrapolation procedure is now available [146, 149]. The first step is to choose a variety of fitting functions. Those that have been used include a power series in 1/N of variable order (often <4), the exponential form a - btxp(-cN), the solution of the logistic equation a/[l-\-b Qxp(-cN)], and Pade approximants. In addition, the property per unit cell is sometimes replaced by its logarithm in carrying out a power series fit. It is our experience that no one form is best in all cases but the exponential form and the logistic equation generally work well and are consistent with one another. In the second step, the data set is obtained by taking all points between a maximum N (A^max) and a minimum N (A^min)- For a fixed A^max and fitting function, N^i^ is varied to determine the range over which the extrapolated hyperpolarizability (per unit cell) is stable, that is, nearly constant. This is repeated varying A^max and the function. The entire set of stable results obtained in this manner yield a mean value for the hyperpolarizability as well as its uncertainty. Because of the stability requirement the number of points always substantially exceeds the number offittingparameters. This contrasts with the approach taken by Dalskov et al. [299], who use the same number of points asfittingparameters. The lack of stability they observe in polynomial fits for a^ is easily understood by recognizing that for a good fit an excess number of points is required.

99

One pitfall in extrapolating is the difficulty of obtaining a smooth dependence of ^y[N] upon chain length, particularly for long chains. This is because of errors in the various numerical procedures that are involved, such as convergence of the SCF and CPHF iterations as well as numerical differentiation. In addition to the presence of numerical errors, the lack of a rigorously justifiedfittingfunction has led to the consideration of sequence transformations as a means of improving extrapolations by detecting and utilizing regularities in the input data [299, 300]. Some of the transformation algorithms show promise [300] of being sufficiently robust to produce accurate results despite the limitations that are typical of oligomer calculations. In the meantime, the extrapolated value of CPHF/6-31G y^ per unit cell given later in this chapter for polyacetylene (PA) was determined [146] by applying stability requirements to polynomial fits of y\[N]IN, even though Ay^C^l converges much more rapidly with chain length. PA is the prototype Tr-conjugated polymer. It has attracted numerous theoretical studies of its second hyperpolarizability not only for this reason but also because it possesses the largest y^i^) per unit length that has been computed or measured for an existing chemical system. In addition, as we will see, the calculations have many important aspects that have called for investigation and, in fact, continue to do so. The first ab initio study of the chain-length evolution was due to Hurst et al. [94], who carried out CHF/6-31G calculations of the all-trans linear polyenes from C2H4 to C22H24. This work was later extended [146] to chains containing up to 44 carbon atoms and, using the procedure described previously, extrapolated to obtain a polymeric value of ^yi{^)[oQ\ = 6.91 lb 0.39 x 10^ au. Frequency dispersion for the most common NLO processes has also been determined by applying the analogous TDHF approach [301], for chains through C30H32. In this case the ratio of the frequency-dependent to the static value [i.e., yf^ico)/yf^{0)] was extrapolated to the infinite chain limit. The lowest optically allowed electronic transition frequency was estimated from the frequency-dependent linear polarizability calculations to be about 2.8 eV. At a frequency well below that, that is, hoj = 0.6 eV (A = 2071 nm), the EOKE, IDRI, dc-SHG, and THG values for [y£(w)/y£(0) - 1] are 0.105, 0.221,0.365, and 1.021, respectively. These values, nonetheless, deviate significantly from the exact low-frequency dispersion formula given by Eq. (35), which gives (based on EOKE as the standard) 0.105, 0.210, 0.315, and 0.630, indicating that A = 2071 nm is outside the low-frequency regime for dc-SHG and THG of infinite polyacetylene chains. A study of the frequency dispersion was also performed by Luo et al. [302] using of a double direct RPA scheme and it is consistent with the preceding analysis. Figure 23 describes how strongly a reduction of BLA enhances the longitudinal second hyperpolarizability per unit cell. Whereas the RHF/6-31G BLA is about 0.112 A, electron correlation at the MP2/6-311G* and B3LYP/6-31G* levels reduces the BLA to 0.06 ± 0.01 A [145], which is close to the experimental value [303]. For a BLA of 0.061 A, extrapolation of the CPHF/6-31G hyperpolarizabilities leads to a polymeric AyL(0)[oo] value of 3.58 ± 0.03 X 10^ au [304], which is more than 5 times larger than that obtained for the RHF/6-31G geometry. Besides these indirect geometrical effects, electron correlation also directly modifies the density matrix and, hence, the response to an external electricfield.Although the correlated Ay£[A^] converges slowly with chain length, determination of the static correlated hyperpolarizability of the polymer (or of long chains) turns out to be feasible because the ratio of the correlated yL(0) to

100

CHAMPAGNE AND KIRTMAN Table XIX.

A7£(0)[oo] Values Determined by Various ab Initio Methods at Several BLA

BLA (A)

Method

Ay£(0)[oo] (au)

Ref.

0.112 0.112 0.112 0.112 0.112 0.117^ 0.061 0.112 0.112 0.112 0.085 0.085 0.061

CPHF/6-31G CPHF/6-31G* CPHF/6-31G+PD CPHF/6-31G RPA/6-31G CPHF/6-31G CPHF/6-31G PS-GVB/6-31G UCHF/6-31G MP2/6-31G MP2/6-31G MP4/6-31G Current best estimate, see text

8.0 X 10^ 5.8 X 10^

Hurst et al. [94] Hurst et al. [94] Hurst et al. [94] Kirtman et al. [146] Luo et al. [302] Hamada [324] Champagne [304] Lu et al. [322] Fanti and Zerbetto [325] Toto et al. [100] Toto et al. [100] Toto et al. [100] Champagne [304]

7.4 X 10^

6.91 ±0.39 xlO^ 1.1 X 10^^ 1.0 X 107

3.58 ± 0.03 xlO^ 2.3 X 10^^ 4.2 X 105^ 1.3 X 107 2.0 X 107 1.8 X 107 6.3 ± 1.5 X 10^

^Obtained by averaging the two extrapolated values. ^A'^ = 11 geometry. '^Obtained by multiplying by 6 to account for the difference of convention in defining y. ^TL^O)[A^]/N value estimated from the almost converged result atN = 80.

about 8%. Although enlarging the basis set can make a big difference for small oligomers, studies at both the CHF and MP2 —•—0.065A 1 levels show that the 7^ (larger basis set)/7^(6-31G) ratio always —D—0.085A gets closer to unity when the chain length increases [lOOj. From 1.2 10* [lOO] we can predict an upper limit of 20% for the uncertainty ^<—0.105A / in Ay£(0)[oo] due to the 6-31G basis set limitations. Accounting —*-0.125A 1 '§ & 8.0 10^ for all effects [BLA = 0.061 A, 7£(0)[MP2]/y£(0)[RHF] = 1.80, and 7£(0)[]V[P4]/yf^(0)[]VIP2] = 0.92], we obtain a current best estimate of A7£(0)[oo] = 6.3 ib 1.5 x lO'^. Table XIX summarizes •-a ^ l^y\j!^)\o6\ values calculated at different ah initio levels of approx•gj - | 4.0 10' imation. It is interesting to observe that, although correlation increases TL^O) substantially in polyacetylene and other quasi-linear TT-conjugated polymers (discussed later), it decreases the linear polarizability a\{^) (although by a relatively small amount). FiFig. 23. Longitudinal static electronic second hyperpolarizability per unit nally, our conclusions about the magnitude of the correlation efcell, A7L(0)[A^], of polyacetylene chains as a function of the number A^ of fect should still be regarded as tentative, because the preferred unit cells and of the bond length alternation. The values were computed at CCSD(T) treatment has not yet been done with sufficiently exthe CPHF/6-31G level of approximation. tended basis sets. The correlation calculations that we have discussed so far pertain to the static limit. Including frequency dispersion turns out to the Hartree-Fock y\{^^ converges much more rapidly with chain be a difficult proposition because, on the one hand, calculations of length than either the numerator or denominator by itself [100]. In frequency-dependent hyperpolarizabilities at correlated levels are PA, for example, the ratio [100] converges (for JVIP2/6-31G versus still limited to small molecules and, on the other hand, the scaling RHF/6-31G) to within less than 2% of the infinite chain value at approximations described in Section 2.2 do not perform well when A/' = 5 (i.e., decapentaene) whereas the RHF/6-31G A7£(0)[A^] tested on the smallest oligomer, that is, rran^-butadiene. At a typiis less than 85% of the infinite chain value even at A^^ = 22. cal optical frequency (A = 694.3 nm, ho) = 0.0656 au = 1.78 eV) At the frozen RHF geometry the direct correlation effect, as de- multiplicative scaling, based on the EOIVI-CCSD/POLU-+ static termined at the MP2 level, leads to an increase of l^y\i!S)\oQ\ correlated value, overshoots the frequency dispersion of 7^ by 20, by 90%. It is noteworthy that—at least for PA chains—the di- 24, 51, and 281% for the EOKE, IDRI, ESHG, and THG prorect increase of Ay£(0) depends little upon the BLA. For in- cesses, respectively [75d]. In the same order the errors introduced stance, in the case of C20H22, varying the BLA from 0.065 A to by additive scaling are - 1 , -2,5, and 101%. The tendency to over0.125 A leads to an MP2/RHF ratio ranging between 1.79 and estimate 7^(a>)/y^(0) is due to the fact that the Hartree-Fock dis1.82 [102]. Higher-order electron correlation effects, which are in- persion is too large, which, in turn, may be connected to an undercluded at the MP4 level, reduce the preceding MP2 estimates by estimate of the excitation energy for the first dipole-allowed ex1,6 10*

it

^I

/•

,,J^^

101

THEORETICAL APPROACH TO DESIGN Table XX.

yf^iO) Values for C16H18 Determined by Various Semiempirical and ab Initio Methods

BLA (A)

Method

yliO)(au)

Ref.

Kirtman et al. [146] Toto et al. [100] Jacquemin et al. [27] Nakano et al. [315] de Melo and Silbey [307] Nakano et al. [315] Nakano et al. [315] Kurtz [311] Shuai and Bredas [313] Nakano et al. [315] Shuai et al. [314] Shuai and Bredas [316] Shuai and Bredas [316]

0.112

CPHF/6-31G

4.1 X 10^

0.112

MP2/6-31G

7.7 X 106

0.08

UCHF/PPP

5.0 X 10^«

0.10

UCHF/INDO

1.0 X 10^«

0.11

CPHF/PPP

6.3 X 10^^

0.10

CPHF/PPP

6.1 X 10^^

0.10

CPHF/INDO

2.9 X 10^^

0.11

CPHF/AMl

6.7 X 10^^

0.11

SOSA^EH/CISD

1.3 X 10^«'^

0.10

MP2/PPP

7.2 X 10^^

0.10

SOS/MRDCISD/PPP

5.8 X 10^^

0.10

SOS/MP2-SD/PPP

1.0 X 10'^

0.10

SOS/EOM-CCSD/PPP

5.9 X 10^

^Obtained by multiplying by 6 to account for the difference of convention in defining 7. ^Obtained graphically (10% uncertainty). ^Obtained by multiplying the average 7 by 5.

cited state. This confirms the initial conclusion drawn from EOKE calculations by Sekino and Bartlett [83a], who also mentioned that the triplet instability in ^ran^-butadiene may cause the TDHF/RPA excitation energy to be too low [305]. Because the Hartree-Fock description of the optically allowed excited states becomes worse for longer chains, the overestimate could become worse as Bartlett and coworkers argue. Thus, a study of the effect of correlation on dispersion in longer polyenes is sorely needed. We have concentrated so far on the ab initio Hartree-Fock and post-Hartree-Fock treatments. In principle, semiempirical procedures can be applied to determine static properties, and through the TDHF and SOS approaches, frequency dispersion as well [26b, 306-316]. The first semiempirical study to provide an extrapolated polymeric result was performed at the FF/RHF/INDO level [308] and gave a value of 2.3 x 10^ au for Ay£(0)[oo] corresponding to a BLA of 0.10 A. At that time it was noted that the analogous CPHF/PPP results of de IVlelo and Silbey [307] were 4-6 times larger. By comparing with the values in Table XIX, we see that the FF/RHF/INDO Ar£^(0)[oo] is also quite a bit smaller than the ab initio CPHF and MP2 results. Applying other semiempirical schemes Kurtz [311] obtained extrapolated (using the FF method with A^ = 9-20) Ay^(0)[oo] similar to FF/RHF/INDO: 0.94 x 10^ au for MNDO (BLA = 0.106 A from [317]) and 1.41 x 10^ for AMI (BLA = 0.11 A). Multiplying his values by 5 (to obtain the longitudinal component) gives Ay£^(0)[oo] in better agreement with CPHF/6-31G than with MP2/6-31G calculations for similar BLA. The MlSfDO and AMI calculations were later extended to account for frequency dispersion by Korambath and Kurtz [318]. For butadiene, their frequency-dependent TDHF/AMl results are at least a factor of 2 smaller than the EOM-CC values of [75d]; on the other hand, for A^ = 10 and ^w = 0.5 eV their IDRI, dcSHG, and THG values of [y^((o)/y^(0) - 1] are larger than the corresponding TDHF/6-31G results of Hasan et al. [301], which, it is claimed [83a], overestimate the frequency dispersion. Nakano et al. [315] used the PPP Hamiltonian to evaluate the effect of electron correlation on the static third-order longitudinal NLO re-

sponse of the polyacetylene polymer. For a BLA of 0.10 A they obtained values of 4.6 x 10^ au and 6.4 x 10^ au at the CPHF and MP2 levels of approximation, respectively. Although this method correctly gives an increase in A 7 L ( 0 ) [ < ^ ] due to correlation, both the magnitude of the increase (40%) and the absolute y values are smaller than those obtained with ab initio techniques for the same BLA. Additional tests of using semiempirical Hamiltonians, in conjunction with CPHF, TDHF, and MP2 treatments, are needed to tell us which are preferable and to what extent they can be expected to give reasonable results for chemical systems that are out of the reach of ab initio calculation. A number of semiempirical calculations have been carried out with the SOS approach. In that event, the accuracy of the oligomeric and polymeric values depends upon (i) the extent to which the configuration space is truncated leading to a less accurate description for longer chains, (ii) whether sufficient electron correlation is included to reasonably describe the essential two-photon excited states [26b, 310, 312], (iii) the Hamiltonian parametrization, and (iv) whether all non-size-consistent terms are cancelled or not [316, 319]. Some of these aspects are illustrated numerically in Table XX, where various semiempirical and ab initio results are compared for C16H18. It is important to compare the theoretical A7[oo] with available experimental data. If the effect of the solvent (reaction + cavity field factors) is, as a rough approximation, assumed to be negligible for a very long polyacetylene chain (see Section 3.4.1), then the measured value corresponds to an isotropic average y per unit cell (i.e.. Ay) obtained for isolated chains. From THG measurements A'y(-3w; co, w, w) = 20 x 10^ au at ho) = 0.65 eV [320] whereas, from another study, a lower bound of 12 x 10^ au was derived for A7(-2w; w, co, 0) at ho) = 0.93 eV [321]. Dividing our best estimate for the static longitudinal value by 5 to obtain the average, multiplying by a factor of 2.1 to account for frequency dispersion in either case [301], and neglecting the small vibrational contribution for both dc-SHG and THG, one obtains a theoretical average value per unit cell of 26 x 10^ au, which is of

102 Table XXI.

Polymer PA

PY

CHAMPAGNE AND KIRTMAN Ab initio Values of the Asymptotic Longitudinal Static Electronic Second Hyperpolarizability per unit Length for Several 7r-conjugated Polymers Containing Only Carbon and Hydrogen

BLA(A)

Method

Ay£(oo)/<2 (au)

0.112

CPHF/6-31G

148 X 10^

0.085

MP2/6-31G

418 X 104

0.117

CPHF/6-31G

214 X 10^

0.220

CPHF/3-21G

216 X 10^

0.220 0.166

CPHF/SVp CPHF/3-21G MP2/6-31G

178 X 10^ 532 X 10^

MP2/6-31G

418 X 10^

0.166 0.104

PDA

PBT

103 X 10"^

Ref. Kirtman et al [146] Toto et al. [100] Hamada [324] Archibong and Thakkar [326] Archibong and Thakkar [326] Toto et al. [332] Toto et al. [330] Toto et al. [330] Hamada [324] Perpete et al. [148]

0.167

CPHF/6-31G

879 X 10^

0.167

CPHF/6-31G

521 X 10^

0.231/0.104

CPHF/4-31G

345 X 10^

0.225/0.091

CPHF/6-31G

943 X 10^

0.225/0.088

CPHF/6-31G

475 X 10^

Kirtman and Hasan [327] Hamada [324] Perpete et al. [148]

0.128/0.066

CPHF/6-31G

228 X 10^

Perpete et al. [148]

the same order of magnitude as the experimental data. However, the current theoretical estimate must be regarded as preliminary because there are a number of aspects that need further attention. These include (1) higher-level treatment of electron correlation and frequency dispersion, (2) more complete examination of the solvent effect, particularly the chain-length dependence and local field factors (including those used to extract the reported experimental value), (3) effect of conformational disorder caused by steric hindrance of side groups and by temperature (a study based on the VB-CT model has found that such disorder causes y^ of PA to be enhanced by a factor of about 5 [322] whereas the coupled quantum oscillator model predicts an effect in the opposite direction [323]), (4) determination of the ZPVA correction, and (5) consideration of geometric and other defects. Apart from PA, ab initio calculations have been carried out on a number of polymers having simple unit cells. The extrapolated static 7L P^'' ^^^it length, ^yf^[oo]/a with a the unit cell length, is given in Table XXI (along with the PA value) for a subset of these consisting of the nonaromatic, 7r-conjugated polymers polyyne (PY), polydiacetylene (PDA), and polybutatriene (PBT), which, like PA, contain only C and H atoms. There are some correlated values in the table but, for the most part, the results have been obtained at the CHF level. This is sufficient to address the order of magnitude and some general trends. The singletriple (S/T) bond pattern of PY chains is associated with a larger BLA than the single-double (S/D) bond pattern in PA whereas PDA combines single-double (S/D) and single-triple (S/T) bond length alternations. For PDA chains there is a PBT resonance form with the BLA pattern: single-double (S/D) and double-double (D/D). Although the polydiacetylenic structure is preferred by 11 kcal mole~^ per C4H2 unit [148], side-chain substitution by polar groups and crystal packing interactions can favor the PBT structure. The ^yl[oo]/a values for PA, PY, PDA, and PBT, calculated at the same level (CPHF/6-31G at RHF/6-31G geometry) and with the same extrapolation procedure, are in the ratio 1.0:0.35:0.32:15.4, which closely parallels the reverse order of their

BLA [148]. (If one defines the BLA of PDA as the average of the two S/D and S/T values there is a switch of order with PY but both have essentially the same BLA.) An identical ordering was also obtained for Aa^[cx)]/fl with ratios of 1.0:0.86:0.69:3.28. Although Hamada's estimates are systematically larger by almost a factor of 2, his conclusions for PA, PY, and PDA are similar [324]. Calculations performed by Archibong and Thakkar [326] on much stronger alternating PY chains and by Kirtman and Hasan on more alternating PDA chains [327] further demonstrate the relation between Ayf^(oo) and BLA. For polycumulene oligomers the BLA tends toward zero. Thus, it is not surprising that calculations on a chain of 12 carbon atoms (the longest considered) yield values of TL^^) ^^^^ ^^^ ^^^^ ^^ ^^^S^ ^^ magnitude as the corresponding C12H14 polyene, although the signs are opposite [328]. Longer cumulene chains and inclusion of electron correlation effects are needed for more definitive conclusions. For polyene chains further investigation of conformers other than the planar all-trans one should be carried out. Studies of the conformational effect have been limited, thus far, either to the linear response [329] or to small oligomers [26b, 310]. Toto et al. [330] have shown that electron correlation effects are especially important for PY. In this case the 1VIP2/RHF ratio converges slowly to an infinite polymer limit of ~8. Satisfactory convergence has been achieved by carrying out the CHF calculations at the correlated geometry, in which event the ratio is much smaller (-^2); indicating that the major part of the correlation effect is due simply to the change in geometry. [Earlier MCSCF calculations on short PY chains [331] actually showed decreases in yC ^0)—by about 15% in C8H2 for example—arising from the direct (i.e., electronic as opposed to geometric) correlation contribution (at an RHF geometry).] After including both the direct and indirect effects due to correlation at the MP2 level, the PA/PY ratio for ^.yi^[oo]/a becomes equal to unity whereas the BLA of PY is still 0.022 A larger than PA. This may seem inconsistent with the analysis presented previously, but the PY value should be twice as large, for a given BLA, because it has two perpendicular systems

103

THEORETICAL APPROACH TO DESIGN Table XXII.

Polymer All-trans PSi

PPy

PTR

Asymptotic Longitudinal Static Electronic Second Hyperpolarizability per Unit Length Obtained by Various Methods for a Selection of Conjugated Polymers Containing Heteroatoms BLA(A)

PAc

^yl{oo)/a

Ref.

(au)

CPHF/6-31G

691 X 10^ 617 X 10^ 984 X 10^

0.022/0.052

CPHF/6-31G

201 X 10^

Toto et al. [147]

0.022/0.052

MP2/6-31G

279 X 10^

Toto et al. [147]

0.014/0.027"

SOS/MRD-CISD/INDO

217 X 10^^'^

Beljonne and Bredas [338]

0.030/0.033"

SOS/MRD-CISD/INDO

676 X 10^^'^

Beljonne et al. [339]

CPHF/AMl

257 X 10^^

Choi [340]

SOS/MRD-CISD/INDO

476 X 10^^'^

Adant et al. [335]

CPHF/MNDO

935 X 10^/

Lu and Lee [336]

CPHF/6-31G CPHF/6-31G

0.0 0.0 0.0

— Th42q^

Method

BLA = 0.064

—

Hasan and Kirtman [95] Perpete et al. [167] Hamada [334]

"AMI BLA value for the central unit of the heptamer. ^~^Estimated from yi^[N]/Na by accounting for the difference of convention in the definition of y and using a standard unit cell length. The values of A^' and the standard unit cell length a (in atomic units), are as follows: b)N = l,a = 6.50; c)N = S,a = 7.25; d)N = 13, a = 7.25; c)N = S,a = 7.25; f) N = S, a = 4.58. In d) the estimate was done graphically. In f) the average value was multiplied by 5 to obtain the longitudinal component. ^Obtained by multiplying by 6 to account for the difference of convention in defining y. '^Dimer of 2,5-bis(2-thienylenemethylidene)thiophene that contains four (two) aromatic (quinonoid) rings.

of delocalized 7r-electrons. (The same consideration should be invoked in comparing PY with PDA.) Obviously, the role of electron correlation in determining TLC^)' particularly in connection with geometrical structure, remains to be further elucidated. Because the formation of solitons, polarons, and bipolarons in doped -TT-conjugated systems may be accompanied by a substantial reduction of the BLA along the conjugated backbone, the preceding resuits foreshadow the large enhancement of y^^^) obtained in such cases (Section 3.3.4) although this may be only part of the cause. In addition to 7r-conjugated systems, the o--conjugated polysilanes [333] are also of interest for NLO applications. Even though poiysilane chains are fully saturated compounds, the dependence of their linear and nonlinear response, as well as excitation energies, on chain length is more similar to a Tr-conjugated chain than it is to the carbon analog, polyethylene. Nevertheless, as shown in Table XXII, the asymptotic yf^iO) per unit length is at least one order of magnitude below that of Tr-conjugated compounds. On the other hand, polysilanes possess improved transparency over TT-conjugated systems at typical second harmonic frequencies. By analyzing the frequency dispersion of the TDHF linear response. Champagne et al. [79] have located the lowest dipole-allowed transition of the infinite polymer at 6.50 ± 0.07 eV whereas, for the infinite PA chain, it is at 2.8 eV. The conformation of polysilanes can be varied by changing substituents and this flexibility may be tuned to optimize the NLO responses. A recent CPHF/6-31G study by Hamada [334] has shown that the all-trans structure possesses the largest ^yl[oo]/a; this value decreases by 39, 88, and 89%, for the 7/3 helix, trans-cis, and alternating trans-cis conformers, respectively. In the next section we will see that the polysilanes also exhibit vibrational optical nonlinearities that are comparable to their electronic counterparts, which is typical of conjugated systems. Polyaromatic systems are particularly attractive to experimentalists owing to their stability and therefore, have drawn the attention of theoreticians. However, the comparisons necessary to

deduce structure-property relationships are difficult to make because of the diversity of computational methods employed. Among the conclusions that can be drawn from Table XXII are the following: (i) polypyrrole possesses a smaller Ayf[oo]/a value than polythiophene; (ii) adding quinonoid moieties into polythiophene oligomers (Th4a2q) does not enhance the third-order NLO response [335]; (iii) polyacenes (PAc) are interesting because convergence has not yet been reached for A^ = 8 and the large value of ^.yf[oo]/a reported [336] is just a lower bound (for the given method of calculation). From RPA/6-3H-+G calculations on small oligomers Luo et al. [337] conclude that yi[N]/N increases in the order thiophene > furan > pyrrole. With respect to the first hyperpolarizability of D/A systems, this inverts the order of pyrrole and furan (see Table X in Section 3.2.3). Geisler et al. [341] have studied oligomers of thienyleneethynylenes and thienylenevinylenes. Their SOS/MRD-CISD/INDO results reveal that the former have smaller y^(0) but, due to a smaller gap, y^(w) increases faster with dispersion. Comparison with the results on oligothiophenes obtained at a similar level of approximation [339] demonstrates the limited interest of adding double or triple carbon-carbon bonds between the thiophene units. Finally, we note that further enhancement of y for aromatic oligomers should be expected. This is evident, for example, from the experimental and theoretical investigations of Ohta and coworkers [5, 342,343] into the heavy atom effect on furane and tetrahydrofuran derivatives. Other studies have shown that the presence of sulfur atoms or S-S-S bridges can lead to increased second hyperpolarizability [142]. Alternatively, it is possible to improve transparency usually to the detriment of y^. This has been accomplished, for example, using exotic structures like polyspiroquinoids where 1,4cyclohexadiene rings are linked through tetrahedral carbon atoms. At the TDHF/AMl level of approximation, Abe et al. [344] predict a much larger bandgap than for the polyenes, with the trade-off being a decrease in yf^/a of at least two orders of magnitude.

104

CHAMPAGNE AND KIRTMAN

Another aspect of modeling new third-order NLO materials consists in determining the second hyperpolarizability of their constitutive fragments. Thus, Keshari et al. [345] have evaluated the static and dynamic y^ of basic heterocyclic structures at the TDHF level using a 4-31G basis set augmented with diffuse/> and d functions. It was found that the calculated y^ occurred in the order thiophene > pyrrole > furane > pyridine > pyrazole > 5-tetrazine; benzothiazole > oxazole, and they were able to show a qualitative relationship with the HOMO-LUMO gap: that is, the smaller the gap, the larger y^. 3.3.2. Vibrational versus Electronic Second Hyperpolarizability The first investigations of the vibrational second hyperpolarizability of oligomers adopted the double harmonic oscillator approximation where 7L(0) is the sum of [a^]^^ and [fifi]^^. We have seen in Eqs. (60)-(62) that, when the infinite optical frequency approximation is also apphed, one or both of these square brackets also determines the value of the double harmonic yj^ for each NLO process. In fact, DFWM depends upon [a^]^^,

•

7.0 10*]

dc-Kerr (v)

- • -dc-Kerr (e) 1

—O— DFWM (v)

- n - DFWM (e)

—*—ESHG(v)

- * - ESHG (e)

—»«~THG (V)

-M-THG(e)

6.0 10*

1

^

O 5.0 10* 4.0 10*

:fl?E = Es:::S::: •

i

offi^^ o r-l

^

3.0 10* 2.0 10*1

a o 1.0 10^ 0.0 lO'f-

too

o

0.4

0.6

0.8

1.0 .

1.2

1.4

1.6

1.8

2.0

h-1

hv in eV Fig. 24. Frequency dispersion of the RHF/6-31G electronic (e) and vibrational (v) longitudinal second hyperpolarizability of Si4Hio [348]. Both the double-harmonic [a^]^^ and [^1^8]^' and first-order anharmonicity [/Lt^aJ^ contributions are considered.

[MJSIL provides ESHG, dc-Kerr depends upon a combination of the two, and the THG vibrational response is zero. Later, firstand second-order anharmonicity terms were evaluated using the FF/NR scheme. In the infinite optical frequency approximation there are first-order terms of the form [/X<^Q:]L that contribute to dc-Kerr but to no other NLO phenomena [cf. Eq. (60)]. For the static process there are, in addition, second-order corrections of the form [fi^]^^. The first indication that y^ may be important in oligomers came from calculations on o--conjugated polysilane rather than TT-conjugated polyacetylene chains [95]. This RHF/6-31G study of all-trans trisilane and pentasilane found that (i) both the [a^JjJ.^^Q and [/XJSJJ^'.^^Q contributions are larger in magnitude than 7L(0) ^^^ (^0 they are of opposite sign. Consequently, the ratio of y^to TL^O) varies substantially according to the NLO process. In the case of trisilane (pentasilane), ratios of 0.50 (0.26), -0.53 (-0.35), and 3.12 (1.91) were obtained for dc-Kerr, ESHG, and DFWM, respectively. Because \y^{-(o; o), w, -(o)\, \y^(-2co; co, o), 0)| > \y^{—(i)\ (o, 0, 0)| this study indicates that the vibrational component does not necessarily increase with the number of static fields. Furthermore, the ordering of the vibra- Fig. 25. End-of-chain (top) and collective (bottom) in-phase H-wagging tional component according to NLO process is clearly different mode in Sii4H3o (552 cm~^ and 624 cm~^). The length of the arrows is from that of the electronic component (particularly if one also directly proportional to the atomic displacement. The transverse displaceconsiders THG). In a subsequent study these calculations were ex- ments (not seen on this scale) are either one (top) or two (bottom) orders tended to longer chains [167] to determine the properties for the of magnitude smaller than the longitudinal displacements. Reprinted from infinite chain limit. At the RHF/6-31G level, the A[a^]^^^^^[oo] E. A. Perpete et al.,/ Chem. Phys. 109,4624 (1998). Copyright 1998, Amer-

ican Institute of Physics, Woodbury, NY.

and ^[^lp]^^^^Q[oo] values are 300 x 10^ and -159 x 10^ au, respectively. In the static limit, the Raman term amounts to 65% of y^whereas the infrared-hyper-Raman contribution is -34%. Consequently, adopting the infinite frequency approximation, the rj[/7£(0) ratio is -0.09 and 0.43 for ESHG and DFWM, respectively. For dc-Kerr the ratio is 0.05 in the double harmonic approximation, but upon including thefirst-orderanharmonicity contribution this increases to about 25%, showing that anharmonicity can be quite important. In the case of Si4Hio the frequency dispersion of yljs illustrated in Figure 24 for the common NLO processes. The largest normal modes contributions to [/I/3]L'. ^^Q in polysilane oligomers originate from the end-of-chain and collec-

tive H-wagging vibrations that involve in-phase motion of the hydrogen atoms along the backbone (see Fig. 25) [95,167]. The endof-chain H-wagging mode has an RHF/6-31G frequency that tends toward 552 cm~^ in the long chain limit. At the same time its contribution per unit cell approaches zero. On the other hand, the contribution of the collective H-wagging mode becomes more and more important as the chain length is lengthened and is estimated to contribute more than 90% of A[)Ltj8]j^'. ^^^^[oo] in the polymeric limit where its RHF/6-31G vibrational frequency is 607 cm~^ The large values found for (o'/iL/^GH-wagging)o ^^^

105

THEORETICAL APPROACH TO DESIGN

Table XXIII. RHF/6-31G [^^IL'^^^Q/'^L^^^ ^^^^° ^^ ^ Function of the Length of the Polyacetylene (PA), Polyyne (PY), Polydiacetylene (PDA), or Polybutatriene (PBT) Chain

Fig. 26. Longitudinal atomic displacements corresponding to the LAM-1 for Si2oH42 (co = 41 cm~^). The length of the arrows is directly proportional to the atomic displacement. The hydrogen atoms follow the Si atoms to which they are attached. The transverse displacements (not seen on this scale) are one to two orders of magnitude smaller than the longitudinal displacements. Reprinted from E. A. Perpete et al., /. Chem. Phys. 109, 4624 (1998). Copyright 1998, American Institute of Physics, Woodbury, NY.

Number of C atoms

PA

PY

2

-7.59

20.44

4

2.65

1.47

6

1.64

1.07

8

1.45

0.99

10

1.38

0.96

12

1.35

0.95

14

1.33

0.94

16

1.32

0.93

18

1.31

20

1.30

22

1.30

24

1.30

26

1.29

28

1.29

oo

1.28

0.92

PDA

PBT

2.76

-2.08

1.11

6.82

0.98

4.02

0.91

3.59

0.89

3.45

0.88

3.40

0.87

3.37

0.86

3.34

Table XXIV. RHF/6-31G [t^^fi^^^^Q/iot^fi^^^^Q Ratio as a Function of the Length of the Polyacetylene (PA), Polyyne (PY), Polydiacetylene (PDA), or Polybutatriene (PBT) Chain Number of C atoms

Fig. 27. Atomic displacements for x\ (top) and ^^2 har (t^ottom) longitudinal FICs that respectively determine the [/Lti8]j_^. ^^Q ^^^ t^^^L- w=0 ^^^" tributions to y^ of Si6Hi4. The length of the arrow is directly proportional to the atomic displacement.

(
PA

PY

2

0.028

-0.056

4

-0.022

-0.009

6

-0.020

0.003

8

-0.012

0.006

10

-0.004

0.007

12

0.004

0.007

14

0.011

0.006

16

0.017

18

0.023

20

0.029

22

0.032

PDA

PBT

-0.012

-0.029

0.006

-0.050

0.020

-0.044

0.030

-0.036

shown in Table XXIII, RHF/6-31G calculations for four homologous series of typical 7r-conjugated oligomers (PA, PY, PDA, and PBT) show that the Raman term {[oP']{^. ^ ^ Q ) is of the same order of magnitude as its electronic counterpart whereas the The normal mode that contributes the most to [//.jS]^'.^ _Q infrared-hyper-Raman term represented in Table XXIV by the ([a^]^'. ^ ^ Q ) may be compared with the single field-induced coratio [/i/3]L. ^^^Q/[OL^]{^. ^ ^ Q ' ^^ ^^ ^ ^ ^ ^ ^^^^ importance. Thereordinate, A'I'(A'2 har^ ^^^^ reproduces the square bracket quantity fore, within the double harmonic oscillator approximation, the exactly. Both FICs are plotted in Figure 27 for Si6FIi4 and one value of 7 L for the various NLO processes is given accurately by can immediately recognize their dominant H-wagging and LAM-1 27j^(-w; 0), 0, 0) = T L ^ " ^ ' ^ ' ^ ' ~ ^ ) = it^^^L w=0 whereas character. the magnitude of yj^(-2w; w, w, 0)/7£(0) is less than or equal to There are several studies where the double harmonic approximation for the vibrational second hyperpolarizabihties in TT-conjugated oligomers has been evaluated by the sum-overmodes perturbation theory approach [125, 148, 160, 347]. As

0.03. As in the case of TCORR/'^'CPHF ^^^ ^^^^^ ["^^L;^a>=o/^L(0)

converges with chain length much faster than either the numerator or denominator by itself. Thus, relatively short oligomers are

106

CHAMPAGNE AND KIRTMAN — • — dc-Kerr (v) - • - dc-Kerr (e) 1 5.0 lO'

#

f

N

S

f 1

4.0 10''

/

/

/ -r/

-HH—DFWM(v)

-D-DFWMCe)

—*— ESHG (V)

- * - ESHG (e)

—»<—THG (V)

- X - T H G (e)

"* 43'sT^

|

1

.- "•^ Fig. 29. Schematic drawing of the longitudinal components of the most contributing vibrational normal modes to [a^]^. ^^Q in PA chains: ECC-. like (top) and H-wagging (bottom) contributions. The length of the arrows is proportional to the displacement.

2.0 10''>

q ^

i [ 1.0 lO'

a ^3 p

0.0 10°

"3) fl O

-l.O 10^ 0.4

1.0

1.2

1.4

hv in eV

Fig. 28. Frequency dispersion of the RHF/6-31G electronic (e) and vibrational (v) longitudinal second hyperpolarizability of C24H26 for different NLO processes [348]. Only the dominant [OJ^JL term is considered in 7^.

sufficient to estimate the importance of the vibrational contribution to y. For PA and PY, the vibrational ratio calculated for the 12-carbon oligomer (A^^ = 12) is less than 5% larger than the asymptotic value whereas, for PDA and PBT, the overestimation is less than 20%. PBT possesses the largest [«^]L'. ^^QI^^^ P^^ ^^^^ length just as it possesses the largest y\{!S)\od\, In fact, the vibrational (Raman) values for PA, PY, PDA, and PBT closely parallel the corresponding y\i!S) in their dependence on the BLA. In the asymptotic Hmit the static -f jy^ ratios (which are dominated by [OI^IL'^^^Q) are 1.28, 0.92, 0.86, and 3.34 for PA, PY,

PDA, and PBT, respectively. This indicates that y^ is not simply Fig. 30. Atomic displacements for x\ (top) and A'2'har (bottom) longitua manifestation of the same underlying physical phenomenon as dinal FICs that respectively determine the [JU-JSIL'. (O=0 ^^^ ^^^^L- w=0 ^^^" y^, which was originally suggested by Zerbi and coworkers [152, tributions to 7^ of C6H8. The length of the arrows is directly proportional 153, 155], for in that event the ratio would be invariant. The preceding values of the static 7^/7^ provide a double harmonic esti- to the atomic displacement. mation for the same ratio in the case of the dc-Kerr and DFWM processes (infinite optical frequency approximation). For the former one simply multiplies by 1/3 and for the latter by 2/3. From from the ECC motion only by end effects (see Fig. 29). In PA Figure 28, which shows the frequency dispersion of both the elec- chains, there are two of these mixed quasi-ECC/H-wagging nortronic and vibrational second hyperpolarizabiUties of C24H26 for mal modes. At the RHF/6-31G level of approximation, in the long the dc-Kerr and DFWM processes, we can see the effect of the chains the lower-frequency mode (o) = 1320 cm~^) contributes infinite optical frequency approximation. For a typical wavelength about 20-25% of the total [«^]L-^^^O whereas the contribution of of 1240 nm (fico = 1.0 eV), T L ^ " ^ ' ^ ' 0' 0)/rL(~^' ^ ' ^' ^^ ^^^ the higher-frequency mode (co = 1860 cm~^) amounts to about 7 L ( - ^ ; ^ J ^? - ^ ) / T L ( - W ; ^» ^j —^) are 0.33 and 0.58, respec- 60-65% [160]. In the former the carbon and hydrogen motions tively. If the infinite optical frequency values are used for 7^, then are parallel; in the latter they are antiparallel. It is of interest to these ratios are changed to 0.36 and 0.59. On the other hand, if the compare the ideal ECC with the FIC ^^2 bar ^^^^ completely dedynamic 7^ is then replaced by the static value the ratios become termines [«^]L-^^^O- ^^^ ^^^ ^^ shown in Figure 30 for a small 0.43 for dc-Kerr and 0.86 for DFWIVI. PA chain containing three double bonds along with the FIC Xi In these vr-conjugated oligomers, the most important vibrawhich completely determines [fip]j^. ^^Q [17]. In the case of hextional normal modes are in-phase-collective vibrations that involve BLA variations coupled with H-wagging (H-C-C bend) dis- atriene the ideal ECC contributes 80-90% of the total vibrational placements. Apart from the large H-wagging motions, end effects response [16]. The dominant modes in PY, PDA, and PBT are also associated and small transverse displacements are the major differences between these collective A: = 0 modes and the ideal ECC introduced with BLA motions along the conjugated backbone (because there by Zerbi and coworkers [152-154] wherein all CH groups move are fewer hydrogens per carbon than in PA, H-wagging is less imin phase and by an identical magnitude along the longitudinal axis. portant). Due to the different numbers of atoms per unit cell there Although the H-wagging component often dominates the most im- are three such modes in PDA and PBT [148] but only one in PY portant normal modes, the magnitude of (^ai^/dQa)o is primar- [347]. In the PY polymer the RHF/6-31G frequency of this one ily determined by the quasi-ECC component [125], which differs mode is about 2580 cm~^ and its contribution amounts to 95%

107

THEORETICAL APPROACH TO DESIGN o u

^^//-^

^

15.0

4 ^

o

?()()

— • - f^

-^^— PDA

—•— PC

— O - PA

-^<— PBT

- e - PSi divided by 5 —^^ PPy |

— H - PMI divided by 10

10.0

5.00 --Q===—===&——^-t_^--^^

om

=

=

7

^

£

-5.00

n 3 3

-10.0

^

N. Fig. 33. Evolution of the [jLt^aJ^. ^ /or 2i0,0 ratio for various oligomeric series versus the number of heavy atoms in the backbone Nj^. This ratio characterizes the importance of the anharmonicity correction to Fig. 31. Evolution of the y^^\-(o; (o, -co, (o)a)^oo/yi^iO) ratio for var- dc-Kerr y^ in the infinite optical frequency approximation. All values were ious oligomeric series versus the number of heavy atoms in the backbone obtained at the RHF/6-31G level. Nj^. All values were obtained at the RHF/6-31G level. y^^\-2a); co, o), 0)a>^oo/rL(^)' ^^^^ ^^^ completely determined at the double harmonic level of approximation. For the oligomers not previously considered one can readily see that the vibrational contribution to DFWM is either larger than 7^(0) (PMI and PPy) o or, when smaller than YLC^)' certainly not negligible (PC). On n the other hand, the ESHG contribution is negligible [smaller than o 0.06 X 7^(0)] for all three oligomers. It is important to point out that, contrary to PA, PDA, PBT, PC, and PSi, the ground state geometry of PMI and PPy chains is nonplanar. As mentioned in Section 3.2.3, the trans-cisoid-like conformation modiI S 3 fied by a ghde plane operation is the most stable for PMI whereas PPy chains are helical. However, in the solid state, compounds such as oligopyrroles adopt a planar conformation due to crystal packing forces. Assuming that these forces are strong, the vibrational response was reevaluated with the chains restricted to executing planar motions. When a^ symmetry is enforced, the Fig. 32. Evolution of the y^^''{-2=0 are about one order of magnitude smaller than typical electronic transition frequencies for these compounds and, hence, the vibrational response time is expected to be about one order of magni1 + 2 r«2i0,0 tude slower. Champagne et al. [328], were able to use the NR/FF scheme (see Section 2.5) to evaluate the lowest-order anharmonicity con(96) tributions to yj^ in eight homologous series of oligomers consist^« JL; a>=0 ing of those aheady discussed (PSi, PA, PY, PDA, and PBT) as well as polycumulenes (PC), PMI, and polypyrroles (PPy). Their As detailed in the previous paragraph, the second term on the investigation was limited to chains containing a maximum num- right-hand-side square bracket is always small (<20%) except in ber of ( N A = ) 12 heavy atoms in the conjugated backbone, which the case of PSi chains. The third term is, therefore, a measure of was shown previously to yield a good estimate of the asymptotic the importance of the first-order anharmonicity contribution with y^/y^ ratio. Figures 31 and 32 display the chain-length depen- respect to the Raman term. Figure 33 shows that, for most sysdence of the two ratios, y^^\-(o; co, -o), o))(o^oo/yi^(0) and tems, the relative magnitude of the anharmonicity correction is o

108

CHAMPAGNE AND KIRTMAN

less than 5%. For nonplanar PPy chains it is about (-)10% for the trimer but that value increases in magnitude with chain length. In the case of PSi, the first-order anharmonic contribution is similar in magnitude to the double harmonic Raman contribution. For small oligomers containing four to six Si atoms electrical anharmonicity is much more important than mechanical anharmonicity [17, 348] and the anharmonicity is crucial for determining the vibrational dc-Kerr response because the two hamonic terms tend to cancel one another. Finally, in PMI the [fJL^a]]^ contribution dominates the vibrational dc-Kerr response. Again, the importance of the torsonial motions has been characterized by forcing the system to remain planar. In that event, the [/^^a]L. co=o/'^^^^^L^(o=0 ratio decreases to 0.05 and 0.41 for the longest oligomer of PPy and PMI, respectively. Thus, although the anharmonicity correction is negligible for PPy it remains significant for PMI. The ratio -a>; 0), 0, 0)a>-^oo to 7 L ( ^ ) ^^ plotted in Figure 34. Due 2 00

7 T

to [oi ]j^. ^^Q or to [iLt^alL or to both contributions, in most instances this ratio is somewhat larger than 0.30 (PY, PA, PDA, PSi, PPy, planar PMI); in some cases it is very large (PBT, PMI). The second-order anharmonicity term [fi^]^, which, in the infinite optical frequency approximation, contributes only to the static y^, has also been estimated using the NR/FF method. As shown in Figure 35 this contribution is small for PY, PA, PBT, PC, and PPy. For PDA, it is still increasing at Np^ = 12 and, therefore, one can-

not assume it will be inconsequential for longer chains. For PSi, it amounts to roughly 50% of y£(0) whereas for PMI it is one order of magnitude larger. Again, forcing the system to remain planar reduces this contribution considerably. Indeed, for the longest PPy and PMI chains, [/A^]}^. ^^^/yliO) decreases from 0.075 to 0.053 and from 11 to 0.07, respectively. At the present time including electron correlation effects is a tedious computational task, although the advent of field-induced coordinates appears to promise rehef in the future [17]. For short PY and PA chains the conventional SOM approach has been used at the MP2/6-31G level to determine the Raman contribution to the double harmonic longitudinal y^ [347, 349]. For the polyyne oligomers C4H2, C6H2, and C8H2, correlation at the MP2(6-31G) level scales the RHF(6-31G) value by factors of 0.736, 0.847, and 1.054, respectively [347]. At the same time correlation increases 7 L ( 0 ) SO that there is an overall reduction in ["^IL-^^^Q/'^L^^^ ^^ 32% (C4H2), 38% (C6H2), and 44% (C8H2) of the Hartree-Fock ratio. Longer chains need to be investigated to determine the magnitude of this effect in the polymeric limit. In the case of PA chains, the results given in Table XXV show that [oi^]j^. ^ ^ Q decreases by an almost converged (wrt chain length) factor of 3 when electron correlation is included [349]. Again, most of the contribution can be attributed to vibrational normal modes containing quasi-ECC motion as an important com-

o d q d V

O

C

Q

^.£ i3

-^00

Fig. 34. Evolution of the y]^^^ ( -co; (0,0, O)(o-^oo/yi^i^) ratio for various Fig. 35. Evolution of the [/W-'^IL. (^^o/yf^i^) ratio for various oligomeric oligomeric series versus the number of heavy atoms in the backbone A^A- series versus the number of heavy atoms in the backbone A^^- ^ 1 values All values were obtained at the RHF/6-31G level. were obtained at the RHF/6-31G level.

Table XXV.

2.0,0

RHF/6-31G versus MP2/6-31G [«^]L^^^O ^^^ '^L^^) ^^ ^^^^^ ^^ ^^^^^^ ,2n0,0

r£(0) Number of C atoms 4 6 8 10

RHF

[a^lL; (0=0 MP2

RHF

MP2

6,306

2381 X 10

16,741

6601

54,435

1539 X 10^

89,489

3068 X 10

21,745 X 10

5859 X 10^

31,525 X 10

1088 X 102

60,035 X 10

1636 X 10^

83,007 X 10

2831 X 10^

109

THEORETICAL APPROACH TO DESIGN ponent. Although electron correlation, as evaluated at the MP2 level, leads to a decrease of the vibrational frequencies of these modes by 6-12%, there is a factor of 3 decrease in [^^11'. ^ ^ Q ^^^^ comes from a reduction in (^a/dQ)Q. The effect of electron correlation on {da/dQ){) is riot simply a reflection of the effect on a itself, which decreases by a modest 10-15%. By varying artificially the BLA of PA oligomers and computing the polarizability at different levels of approximation it has been confirmed that the slope of the curve for a versus BLA is much smaller at the MP2 than RHF level [102]. At the same time there is a substantial increase in the electronic second hyperpolarizability. Consequently, the value of [^^IL'^^^Q/'^L^^) ^^ QHio(CioHi2), for example, is reduced by correlation (at the MP2 level) from 1.45 (1.38) to 0.19 (0.17). 3.3.3. Third-Order (Electronic + Vibrational) NLO Properties of Substituted Conjugated Systems and Buckminsterfulkrene Adding substituents to conjugated systems will modify their thirdorder NLO response. Although substitution along the conjugated backbone is of interest we will focus again on the case where D/A pairs are added to the chain ends. The strength of the D/A pair, as well as other factors such as the length and nature of the linker, influence the BLA along the conjugated backbone and the latter property correlates with changes in the NLO response. For D/A pairs of increasing strength we have already pointed out that the system evolves from the polyene limit through the cyanine structure to the zwitterionic limit. These BLA variations correspond to substantial changes in the magnitude and sign of both the electronic and vibrational second hyperpolarizabilities according to the general pattern shown in Figure 20 for the two-state VB-CT model. Similar geometry changes underlie the effect of donoracceptor doping upon y (see Section 3.3.4) and play a critical role in symmetrically substituted compounds such as squaraine dyes and symmetric cyanine cations. The role of D/A strength is nicely illustrated by considering the same set of D/A hexatriene molecules that were discussed earlier in connection with the second-order NLO response. CHF/631G calculations of y^^O) for these systems are reported in Ta-

Table XXVI.

ble XXVI. As the BLA decreases from the polyene value, corresponding to an increase in the strength of the D/A pair, yl^i^) initially increases. Using two NO2 donors and two NH2acceptors brings the molecule into the cyanin regime, resulting in a negative TLC^)- Finally, when the two NO2 groups are replaced by the stronger acceptor PO2, T^CO) remains negative but decreases in magnitude. This indicates that the latter molecule is on the threshold of the zwitterion region. As noted previously these variations follow the behavior of 'y£(0) obtained from the two-state VB-CT model. Similar changes are calculated for the vibrational contribution to DFWM using the perturbational SOM scheme. In the polyene region, the y^ (-^5 o), ~o), o))(o-^oo/yi^(^) ratio is rather insensitive to D/A substitution and is close to unity (Table XXVI). However, in the cyanine regime, where y^^^) becomes negative, the ratio is also negative because the vibrational contribution remains positive for the molecules considered here. In such a case the total (electronic + vibrational) response can be very small. The vibrational contribution to ESHG follows the same sign changes as its electronic counterpart in these seven molecules. In this case the electronic response is dominant in the polyene region but the vibrational response is larger by a factor of 4-5 in the two compounds with the smallest BLA. For the dc-Kerr process, the vibrational response calculated with using the FF/NR technique is roughly a factor of 3 larger than ESHG for the cyanine-like compounds whereas, for the polyene structures, both electronic and vibrational contributions are of similar magnitude. These calculations also show that the first-order anharmonicity contributions, which arise mostly from electrical (as opposed to mechanical) anharmonicity, make an important contribution to the dc-Kerr vibrational effect [17]. In considering the comparison made here between y^ and y^(0) one should not forget that, in the usual range of optical frequencies, y^ will typically be 2-3 times larger than its static value. The effect of electron correlation on yL(0) has scarcely been investigated. For ;?-nitroaniline, Sim et al. [215] showed that the MP2 value is about twice as large as the CPHF result and that including electron correlation changes yL(0) directly but not indirectly via modification of the ground state geometry. On the other hand, for short NH2/NO2 substituted polyenes and polyynes [128] going from the RHF/6-31G to the MP2/6-31G geometry leads to an increase of about 40% in y£(0), which is little influ-

,/(^) RHF/6-31G 7^(0) and yl^'^^-coa; ^ i , (02, W3)iu->oo/yL(0) Ratios as a Function of the NLO Process for D/A-Substituted Hexatrienes in Comparison with the BLA^ ,v(r) yL \-a)a',

Molecule H-CH=CH-CH=CH-CH=CH-NH2 N02-CH=CH-CH=CH-CH=CH-H CHO-CH=CH-CH=CH-CH=CH-OH N02-CH=CH-CH=CH-CH=CH-NH2 P02-CH=CH-CH=CH-CH=CH-N(CH3)2 (N02)2-C=CH-CH=CH-CH=C-(NH2)2 (P02)2-C=CH-CH=CH-CH=C-(NH2)2

BLA

(10^ au)

0.121

92

0.120

171

0.118

193

0.097

322

0.078

469

0.044

-235

•0.026

-111

0)1, C02, W3)a>^oo/yL(0)

DFWM

ESHG

dc-Kerr

1.02 0.95 0.95 1.15 1.15 -0.77 -1.74

0.08 0.13 0.13 0.26 0.06 3.99 4.62

0.81 (17) 0.91 (19) 0.97 (24) 1.76 (38) 1.05 (52) 13.8 (45) 10.4 (20)

^The compounds are given in increasing order of charge transfer between the D and A. The percentage of anharmonicity contribution to the dc-Kerr vibrational response is given in parentheses in the last column. Data from [9,17, 37].

no Table XXVII.

CHAMPAGNE AND KIRTMAN CPHF versus Correlated y^ (in 10^ au) of NH2/NO2-Substituted Short Polyenes and Polyynes as a Function of the Geometry [128]^

Linker CH=CH CH=CH-CH=CH CH=CH-CH=CH-CH=CH C=C C^C-C^C

Geometry

CPHF

MP2

MP3

MP4

RHF MP2 RHF MP2 RHF MP2 RHF MP2 RHF MP2

74 103 666 865 3432 4354 198 255 1123 1416

141 (90) 204 (98) 1,737 (161) 2,412 (179) 9,124 (166) 12,519 (178) 386 (94) 541 (112) 2,212 (97) 3,122 (120)

142 (89) 206 (100) 1,809 (172) 2,469 (185) 9,419 (174) 12,909 (196) 371 (87) 516 (102) 2,039 (82) 2,881 (103)

212 (186) 321 (212) 2,163 (225) 3,052 (253) 10,661 (211) 14,760 (239) 415 (110) 579 (127) 2,276 (103) 3,158 (123)

"The 6-31G basis set has been used for determining all geometries and second hyperpolarizabilities. The variations (in percent) with respect to the CPHF values are given in parentheses.

enced by chain length or level of calculation (Table XXVII). The direct effect of electron correlation, however, is much larger. Indeed, the MP2/CPHF ratio (at fixed geometry) is about 2.6 for the for polyene linkers with two or three C=C double bonds. Going from MP2 to MPS has a negligible effect but electron correlation effects brought in at the MP4 level increase yf^iO) still further. The overall increase for N H 2 - ( C H = C H ) 3 - N 0 2 [MP2 geometry; MP4 yL(0)] with respect to the Hartree-Fock result is 330%. For both of the polyynes treated (C2 and C4) the corresponding overall increase is a little less than 200% regardless of the level of the correlation treatment. Finally, by using the correction vector approach and an INDO/1 Hamiltonian, Albert et al. [350] were able to compare Full Configuration Interaction (FCI) calculations with CIS, CISD, CISDT, and CISDTQ for ^ran^-octatetraene and p-nitroaniline. Although one should remain careful about the carryover from semiempirical to ab initio Hamiltonians, their results indicate that double excitations are important for 7 L ( ^ ) ^^^ ^^^ triple or quadruple excitations. This study confirms the limitation of the CIS treatment to generate the excited state properties found in an earUer CNDOVSB work on D/A-substituted polyenes and polyphenyls [208]. In the latter investigation, the CIS scheme was shown to underestimate yf^(—3(o;o),o),o)) with respect to the CISD treatment. To our knowledge, there has been only a single study that has adressed electron correlation effects on the y^ of push-pull molecules [120]. In this investigation of N H 2 - ( C H = C H ) 3 - N 0 2 it was found that the correlated MP2/6-31G yl^i-lw; co, o), 0) is 136% larger than the RHF/6-31G value, whereas y\{-o)\ o), 0,0) is 51% larger and TLC-O; 0,0,0) is smaller by 6%. As in the case of the second-order NLO response, the effect of the nature of the conjugated segment upon y has also been investigated, although at a more restricted scale. Starting from ;7-nitroaniline [137, 215, 351] Keshari et al. [352] have shown that (at the TDHF level with an augmented double-^ basis set) inserting a CH=CH group between the benzene ring and NH2 leads to a substantial increase of y^ (dominated by its longitudinal component). For these compounds, the changes in y^ upon changing the donor substituent parallel those in the j8^ values. This is consistent with molecules belonging to region A in Figure 1. Although solvent effects and vibrational contributions are neglected, their results are in good agreement with the experimental data of Cheng

Table XXVIII. y^{-3co; co, (o, a>)[Azoarene]/y^(-3w; w, w, co) 4-Dimethylamino,4^-nitro-stilbene] as a Function of the Nature of the Aromatic Rings as Calculated by Albert et al. for hcD - 0.65 eV [229] Using the CNDOVSB/CV/CISD Scheme Molecule

Ratio

N(Me)2 - P h - N = N - P h - N 0 2 N(Me)2-Th-N=N-Ph-N02 N(Me)2-Ph-N=N-Th-N02 N(Me)2-Fu-N=N-Ph-N02 N(Me)2-Th-N=N-Th-N02 N(Me)2-Fu-N=N-Fu-N02

1.20 2.05 2.18 2.90 4.98 1.22

et al. [180, 200] on para-disubstituted benzenes as well as 4- and j8-substituted styrenes. Morley and coworkers [208, 209] have determined 7 L ( ~ 3 ^ 5 O), (X), (o) of NMe2/N02-ct,w-substituted polyenes, polyphenyls, and polyynes as a function of chain length. Due to substantial resonance effects, interpretation is difficult. However, it turns out that D/A-substituted polyynes possess larger yf^(—3o)',o),o),(o) values than their polyphenyl analogs but smaller than the polyenes. By adopting the CNDOVSB/CV/CISD approach, Albert et al. [229] calculated the y^(-3ft>; w, OJ, w) for several azoarenes. At hco = 0.65 eV (A = 1907 nm) they found that 7^(-3a>; co, o), co) [calculated as (yxxxx + 2xxyy + 2yyxx + yyyyy)/^] of 4-dimethylamino,4'nitro-phenylazobenzene is larger by 20% than its ethylenic bridge analog (4-dimethylamino,4'-nitro-stilbene) (Table XXVIII). This reversal of order with respect to static values obtained at the CPHF/6-31G level (Table XXIX) was related to the lower first excited state. For the azo compounds, they also found that replacing the two phenyl rings by two thiophene rings increases y^(-3co', CO, CO, co) by about 400% whereas the same operation with two furane rings has a much smaller consequence (Table XXVIII). This result and others reported in Table XXVIII are difficuh to rationalize either in terms of aromaticity or auxiUary D/A because frequency dispersion effects are very large for THG at this optical frequency.

111

THEORETICAL APPROACH TO DESIGN Table XXIX.

RHF/6-31G y^(0) and y£(-cuo-; 0)1,(02, (O3)aj^oc/yl{0) Ratios as a Function of the NLO Process for NH2/NO2-Substituted Linkers [37,158, 354] 7L

Linker CH=CH-CH=CH-CH=CH Ph (p-NA)

DFWM

ESHG

dc-Kerr

322

1.15 0.42 1.00 0.53

0.26 0.22 0.28 0.17

1.10 0.64 1.06 0.60

873

CH

+'^

(03)0)^00/y^iO)

(10^ au)

71 1023

/ra«5-Ph-CH=CH-Ph (fra«5-stilbene) /ran5-Ph-N=N-Ph

(-Wo-; w i , 0)2,

CH

^ CH

NH/^

^CH

CH ^CH

CH ^CH

PH

^NH-

.^c"._ CH

^CH

Fig. 36. Resonance forms of cyanine cations in equilibrium with the soliton-defect NH2-substituted polyene chain.

Nakano et al. [353] have studied the effect on 7^ of replacing neutral polyene chains. Using the CISD/INDO approach for sysH atoms with NH2 or NO2 groups along the backbone of PDA tems containing 10 7r-electrons, Pierce [355] calculated that the chains containing up to 42 C atoms. On the basis of CHF/INDO cyanine cation y^Jfi) is negative and about one order of magnicalculations (using the geometry of the unsubstituted PDA back- tude larger than it is for the corresponding polyene. As the chain bone), they determined that the asymptotic Ay£[oo] value can be is lengthened it is to be expected that the influence of the terenhanced substantially by an adequate positioning of these D/A minal NH2 groups will decrease and that the structure and (hysubstituents. Indeed, an increase by almost a factor of 4 is obtained per)polarizabilities of the cyanine cations will begin to resemble when every other CH=CH group is substituted by two donors and those of a polyene chain bearing a positive soliton defect (see the ones in between by two acceptors, thereby creating a PDA Fig. 36). Although the ClO^ counterion was not taken into acchain with high and low electron density backbone regions. count the calculated negative sign of y^Jfi) is in agreement with The nature of the linker also influences the importance of the four-wave mixing (FWM) study of bis(dimethylamino)penta-, the vibrational contribution to the second hyperpolarizability. For hepta-, and nona-methine dyes by Johr et al. [356]. In the static frethe most common NLO processes, Table XXIX compares the quency limit y^ is positive for small chains but negative for longer yl^i-coo-; a>i, ^2, (O3)(o-^oo/yi^(0) ratios of substituted polyenes chains (>5 carbons or >8 7r-electrons). Johr et al. attributed with substituted aromatic compounds. Going from all-trans hexa- the sign change to the combination of a slowly increasing positriene to a benzene linker reduces the Raman contribution and, tive a--electron contribution and a more strongly decreasing (altherefore, the DFWM vibrational TL by almost a factor of 3. The gebraically) negative 7r-electron contribution. Because for longer importance of the infrared-hyper-Raman term decreases less. In chains—the extent of which remains to be assessed—the structure the case of the fran^-stilbene linker, the y^/y^ ratios are similar to will resemble that of the soliton, one may expect y^ to decrease the polyene but it is important to reemphasize the role of non- in magnitude and subsequently become large and positive (see planarity. The torsion angles between the phenyl rings and the also Section 3.3.4). Along these lines, it is interesting to point out CH=CH spacer are 12° and 18°; moreover, the low-frequency vi- that further experimental studies on thiacyanine dyes and mesobrational modes (co < 100 cm~^) contribute 59% of [a^]?'^ n azathiacyanine dyes (BF^ is the counterion) of different lengths and 45% of [iw,j8]j^'. ^^Q [354]. Neglecting these low-frequency con- have enabled the observation of a saturation of y for systems contributions, the 7L(~^O-; 0)1, 0)2, W3)a)^oo/rL(^) ^^^^^ becomes taining six double bonds [357]. A preliminary RHF/6-31G double 0.41, 0.15, and 0.51, for DFWM, ESHG, and dc-Kerr, respec- harmonic oscillator estimation of 7^ of small cyanine cations shows tively. The analogous azo compound is planar and exhibits vibra- that [AtjS]^. ^^Q is also negative and is almost one order of magtional/electronic ratios similar to;7-nitroaniline. Thus, the general nitude larger than ^£(0) (Table XXX). Consequently, the ESHG effect of replacing the polyenic linker by aromatic moieties is to and dc-Kerr vibrational second hyperpolarizabilities are large with decrease the importance of the vibrational contribution to y with respect to yl^^). Thus, the sign and magnitude of [^JSIL-^^^Q ^^^ respect to its electronic counterpart. the small cyanin cations resembles that of the charged soliton (see Because the magnitude of the BLA along the conjugated back- also Section 3.3.4). However, in contrast with the charged soliton, bone is a key parameter in determining NLO properties, it is not [«^]L'. ^^0 ^^ smaller than the static electronic hyperpolarizabihty. surprising that cyanine cations, NHj=CH-(CH=CH)Ar-NH2 Symmetric substitution on neutral (-CH=CH-) chains (as (see Fig. 36), exhibit different second hyperpolarizabilities than opposed to the charged cyanines, which have an odd num-

112

CHAMPAGNE AND KIRTMAN

Table XXX.

RHF/6-31G TL^O) and yf^i-coo-', wi, (02, W3)a>^oo/7L(0) Ratios as a Function of the NLO Process for Cyanine Cations versus Polyenes and NH2/NO2-Substituted Polyenes of Similar Size .v(r) TL ( - ^ O - ; 0)1, (02, (O3)o)-^oo/yl(0)

r£(0) Molecule NH+=CH-[CH=CH]2-NH2 NH+ =CH-[CH=CH]3 -NH2 H-[CH=CH]3-H H-[CH=CH]4-H NH2-[CH=CH]3-N02 NH2-[CH=CH]4-N02

Table XXXI.

(103 au)

DFWM

ESHG

dc-Kerr

-93 -443

-0.41 -0.41

1.82 2.31

3.44 4.41

54 218

1.09 0.97

-0.008 0.004

0.56 0.47

322 1004

1.15 1.05

0.26 0.28

1.09 1.08

RHF/6-31G yf^iO) and yf^i-coo-', w^, C02, W3)a>^oo/7L^O) Ratios as a Function of the NLO Process: (Top Part) Comparison between Dipolar and Quadrupolar Polyenes; (Bottom Part) D-TT-A-TT-D Quadrupolar Systems [170] ,v(r) TL ( - ^ O - ; wi, W2, W3)W-^OO/7L(0)

r£(0) (10^ au)

DFWM

CH2=CH-CH=CH-CH=CH-CH=CH2 CH2 =CH-CH=CH-CH=CH-NH2 CH2=CH-CH=CH-CH=CH-N02 N02-CH=CH-CH=CH-CH=CH-NH2 NH2-CH=CH-CH=CH-CH=CH-NH2 N02-CH=CH-CH=CH-CH=CH-N02

218 92 171 321 121 262

0.97 1.03 0.96 1.15 0.95 0.82

CH30-(CH=CH)2-CO-(CH=CH)2-0=CH3^ CH30-C6H4-C402-C6H4-0=CH3^

362 213

0.71 4.61

Molecule

ESHG

dc-Kerr

0.004 0.08 0.13 0.26 0.09 0.11

0.47 0.67 0.74 1.09 0.65 0.64

0.15 -4.89

0.65 -7.48

^1,9-Dimethoxy 1,3,6,8-tetranonene 5-one. ^Cyclobutane-1,3 dione 2,4-bis(4-methoxyphenyl). her of carbons) has also been receiving some attention of late ([358]; [359] and references therein). For typical D and A moieties (NMe2, CHO, NO2), when the number of CH=CH groups in the linker (which may also contain phenyl rings) is smaller than 5, bis-acceptor substitution leads to larger measured THG and dc-SHG y than bis-donor substitution whereas the opposite holds for longer polyenic linkers. It has also been found that a steeper increase of y with chain length is obtained for bis-donor compounds than for push-pull systems and that bisacceptor molecules vary the slowest of all. The effect of the nature of the TT-conjugated chain upon y^ has been studied at the CPHF/ST0-3G level by Nalwa et al. [360] for bis-amino compounds; they determined that the third-order NLO response varies in the order PA > FY > polyazine ( N - C H = C H - N ) > PMI > ( C H - N = N - C H ) . Table XXXI shows that, for short polyenic linkers, the push-pull NO2/NH2 compound exhibits the largest y£(0) (compared to NO2/NO2 or NH2/NH2) as well as the largest yl^{-a)a', ^ i , (02, o)3)(o^cx)/yl(0) ratio [170]. For the symmetrically substituted all-trans polyenes, the Raman term is of the same order of magnitude as yL(0) whereas the infrared-hyper-Raman term is much smaller. Some symmetrically substituted squaraines are calculated to have substantial vibrational second hyperpolarizability. Table XXXI reports the y^(-Wo-; coi, ^ 2 , o)2)(o->oo/yl(^) ratios

for two D-TT-A-TT-D quadrupolar systems. For 1,9-dimethoxy 1,3,6,8-tetranonene 5-one these ratios are similar to the NH2/NO2 symmetrically substituted compounds whereas for cyclobutane-1,3 dione 2,4-bis(4-methoxyphenyl) the magnitude of the [M-JS]^. ^ ^ Q and [^^IL'^^^O terms is much larger than y^CO). The infraredhyper-Raman term is negative but the Raman term is positive. At the double harmonic oscillator level, this leads to a y ^ ( - w ; CO, 0, 0)a;-^oo/ yf^CO) ratio of -7.48. These ab initio data contrast with the conclusion of the three-state VB-CT model study of Hahn et al. [42], which concludes that y^ will be almost neghgible wrt y^ in linear quadrupolar molecules. The large and negative [IJLP]J^^^Q contribution of the squaraine dye is also associated with a very small BLA along the conjugated path [170]. Again, the most significant vibrational normal modes contain an important BLA character. Further ab initio investigations of both the electronic and vibrational second hyperpolarizability contributions are required for a complete interpretation of the large negative ESHG, DFWM, and THG second hyperpolarizabilities that have been experimentally determined for squaraine dyes [361] and their oligomers [362]. Spiro compounds constitute another type of interesting material for NLO applications where calculations reveal that the vibra-

THEORETICAL APPROACH TO DESIGN

113

causing the charge to be trapped in a region opposite the dopant. Likewise the unpaired spin will be trapped by geometry relaxation downstream. The region occupied by the localized charge is known as a charged soliton and the localized radical is a neutral soliton. Across the length of chain where the (charged or neutral) soliton is located the BLA changes sign. Thus, the BLA within the soliton is reduced in magnitude from that of the remainder of the polyene chain. If a second acceptor atom or molecule is added, ^^0,0 00 then the radical electron of the neutral soliton is transferred, givforming a spiroconjugated dimer, y®(0), [a^]^^Q, and [MJS] J^Q ing rise to a chain containing two charged solitons. The presence of are multiplied by factors of 2.13, 9.73, and 2.07, respectively these charged solitons is the origin of the very large enhancement [365]; (iii) the combination of (i) and (ii) leads to an increase in of conductivity that has been found for doped PA [379]. Because y^(0) -\- y"^{-co; 0), -0), ft>)w^oo in going from the monomer to of the reduced BLA one might also anticipate an enhancement of the dimer of almost 338%; (iv) the substantial increase in [a^l^^^o NLO properties depending upon the dopant concentration. This is due to low-frequency normal modes involving distortion of the possibility is now beginning to receive some attention. bridge carbon atom tetrahedral configuration; (v) including solTo study the NLO properties of an individual charged soliton vent effects within the semiclassical solvation model (for simpli- one may consider a polyene chain containing an odd number of fied compounds where the 2-amino-7-nitrofluorene moities have carbons so that the BLA on either end is of opposite sign. Such been substituted by l-amino-4-nitrocyclopentadiene groups) does chains have been examined by de Melo and Fonseca [380] along not change these predictions [364]. with chains containing two charged solitons (referred to as a bipoBuckminsterfuUerene (C6o), which is of interest for many of its laron) and an even number of carbons. Their CPHF/6-31G calproperties, has also drawn attention in connection with its third- culations reveal that 7L(0) for singly charged C^H^_^2 ( ^ o^^) order NLO response. As reviewed by Nalwa, there is considerable chains is negative for small N, goes through a minimum at N = 17 scatter in the experimental as well as theoretical data reported thus and then begins to rise steeply as N continues to increase. For far [366]. The most recent, and most accurate, calculations support the longest chain they considered (N = 23) the value of 7L(0) is the relatively small experimental values found by Geng and Wright less than that of the neutral polyenes C22H24 and C24H26. How[367], who determined an upper bound of y^ = 410 x 10^ au in ever, it is clear from their shape that the two curves [yf^iO) vs. N] an FWM experiment. From their TDDFT investigation, carried will cross. On general grounds it can be argued that, because the out with the asymptotically correct LB94 potential, van Gisber- charged soliton is a localized entity, the curve for the solitongen et al. [86] obtained 7^(0) = 65 x 10^ au. On the other hand, bearing chain will go through a maximum (per carbon) and then studies by Agren and coworkers, using the RPA scheme and split- decrease toward the curve for the neutral polyene. That maximum valence basis sets augmented with diffuse functions, give values of is of interest because it determines the optimum dopant concen109 and 114 x 10^ au [368, 369]. In comparison with PA chains tration, assuming the concentration is low enough that the solithis is a small third-order NLO response; it is comparable to 7^(0) tons on a long chain do not affect one another. CPHF/6-31G calof all-trans decapentaene [146]. The relatively small value is un- culations through C43H4"^ [381] show that 7£(0)/A^ is still rising doubtedly a consequence of the compact spherical, as opposed to sharply aXN = 43 and, at that point, the value is already 2-3 times quasilinear, shape of C6o- Considering the vibrational contribu- larger than PA. For a pair of solitons the longest chain treated was tion to y brings experiment and theory in closer agreement. In the C^QH^2~^ i^ which case 7£(0)/A^ is comparable to the value found double harmonic oscillator and infinite optical frequency approxi- for a chain of similar length bearing a single soliton. Again, the rate mations the 7^/7^(0) ratio, estimated from a combined HF/DFT of growth indicates that the maximum will be well beyond A^ = 30. approach [370], varies from 0.01 for dc-SHG to 0.64 for dc-Kerr to Clearly, it is desirable to consider longer chains and to evalu1.26 for DFWM. For FWM, in the co-co^ domain investigated by ate the effect of electron correlation, which could play a very imGeng and Wright, the calculated 7^(-2cu + co^; co, -w\ w)/7^(0) portant role (see later). Another critical aspect is the influence of ratio ranges between 0.1 and 0.4. the dopant counterion. Some initial CPHF/6-31G results for dopThe list of y materials mentioned here is not exhaustive owing ing with atomic lithium to form a negatively charged chain have to time limitations. Although we hope that most of the key fac- been obtained by Champagne et al. [151]. It should be noted at tors that influence y have been discussed some have undoubtedly this point that negative and positive soHtons are expected to have been omitted. One of the aspects we have not addressed is near- similar properties (at least, at the Hartree-Fock level) and this has resonant phenomena that depend upon the lifetime of the excited been confirmed in semiempirical calculations [309]. For a dopant state(s) involved [371]. Other aspects not studied include the anal- atom at the center of a C23H25 chain (the longest considered) the ysis of hyperpolarizabilities in terms of hyperpolarizability density, value of 7L(0) was found to be about 150 times larger than that moments, and charges [139, 372, 373], as well as conjugated chro- obtained for C23H^5. However, it is premature to draw definitive mophores containing transition-metal atoms such as the phtalo- conclusions until longer chains are examined. cyanines, porphyrins, and ferrocenyl derivatives. Apart from 7^ one might expect that doping of PA would also have a significant effect on 7^. In fact, for the isolated-charged3.3.4. Doped Polymers soliton-bearing chain two major changes occur in comparison with the neutral polyene [382]. One of these is that [a:^]^ /N becomes When polyacetylene is doped with an electron acceptor (acceptor), much larger; for C2iH2"3 and C2iH^3 the enhancement factor veran electron can be transferred from (to) the polymer to (from) the dopant. Electronic structure calculations [374-378] indicate sus C22H24 is 6.1. Even more striking is the increase in the magthat the geometry of the resulting radical-cation chain will relax. nitude of [^ti8]L (this quantity is negative), which goes from betional contribution may play a dominant role. DFWM measurements (in THF) have shown that the second hyperpolarizability of 2,2'-diamino-7,7'-dinitro-9,9^-spirobifluorene is about one order of magnitude larger than its monomer, 2-amino-7-nitrofluorene [363]. So far, theoretical investigations have not been able to reproduce this substantial enhancement. On the basis of RHF/321G and RHF/6-31G calculations Luo et al. [364] have found the following: (i) y^(0)[dimer] ^ 2y^(0)[monomer]; (ii) upon

114

CHAMPAGNE AND KIRTMAN

ing negligible compared to [«^]L ^ in the neutral polyene to being almost 5 (6) times larger in C2iH^3 (C2iH^3). This major effect on [/I/^IL could, perhaps, have been foreseen from the linear polarizability results of Champagne, et al. [383], who found that [ljr]j^ IN increases by well over two orders of magnitude upon going from C28H30 to C29H3"^. However, this is all without considering the dopant counterion, which tends to pin the charge. Indeed, the pinning effect is quite substantial [151]. For the longest chain considered, C23H25Li, [0:^]^' /N is reduced by more than a factor of 5 and is now almost 5 times larger in magnitude than [/>IJ8]L . It has also been found that the first-order anharmonicity term [^^a\^ , which contributes to the dc-Kerr effect in the infinite frequency approximation, is about twice the magnitude of [At/3]^'^nCi5Hi7Li. Besides considering longer chains it is of interest to examine other dopants that will exhibit greater charge transfer than atomic lithium. For the vibrational TL? ^^ particular, it would be worthwhile studying more spatially extended dopants to see if the pinning effect could be reduced. Finally, although some models have been proposed, to our knowledge there have been no calculations of the three-dimensional solid state structure of PA in the low doping regime and, of course, no calculations of the NLO properties of this structure. The all-trans polythiophene polymer, in contrast with PA, is potentially of practical significance. Doped samples are easily prepared and remain stable for years. As in PA the removal or addition of an electron leads to a geometrical relaxation that stabilizes the system and traps the excess charge. However, there is also an important difference from PA. Neutral PTh has a benzenoid structure [384] with two double bonds within each heterocyclic ring and a single bond connecting the rings. The structural rearrangement that accompanies the electron transfer leads to a local region of quinonoid character with a double bond connecting the ringsflankedby a benzenoid structure on either side. Thus, the BLA on either side of the local region is the same but there is a change in sign as one goes across each of the boundaries. This type of local structure is known as a polaron. A substantial effort has been directed toward theoretical studies of the structure and energetics of charged oligothiophenes. Early semiempirical calculations [385] suggest that the polaron is stable with respect to an undeformed structure by roughly 0.1-0.2 eV. The magnitude of the energy difference that is determined depends on whether the dopant is a donor or an acceptor and whether the dopant counterion is simulated or not. In the doubly charged system a bipolaron is formed, that is, a doubly charged region, rather than two separate polarons, according to the semiempirical treatments. Estimates of the stability of the former with respect to the latter vary from 0.25 to 0.45 eV. More recent ah initio and semiempirical calculations [386-388] have led to a number of important refinements: (1) The bipolaron structure is more stable than two separate polarons if, and only if, the dopant counterions are taken into account. (2) The bipolaron structure is more stable than two polarons if, and only if, the dopant pair is located above adjacent rings or they are located one ring apart. (3) Including the dopant counterions strongly localizes the region over which the geometric structure is distorted. (4) From a methodological point of view it is essential to include correlation in describing these systems. Neither RHF nor unrestricted Hartree-Fock (UHF) is adequate but spin-unrestricted DFT is satisfactory.

THG measurements have, very recently, been carried out on doped PTh [389]. For similar reasons as in the case of PA one would expect an enhancement of y£ and y^ ^^^ ^^ doping. To our knowledge, however, no NLO calculations have been carried out. There have been treatments of the vibrational spectra of positive bipolarons by Ehrendorfer and Karpfen [386] that are relevant for 7L- III their extensive semiempirical and ah initio calculations, which were done without including dopant counterions, they find a large enhancement of Raman intensities (depending upon the chain length) together with an even more striking increase in infrared intensities. This Raman enhancement is primarily a geometrical effect associated with the transition from a benzenoid to a quinonoid structure whereas the infrared enhancement is due primarily to the charge. Thus, the vibrational properties of bipolarons in PTh are consistent with those determined for charged solitons in PA and, in qualitative terms, one expects that y^ will behave similarly in the two cases. The results just described assume that the Th rings are in a planar anti conformation. Although the calculated equilibrium geometry actually deviates from planarity by torsional displacements about the C-C inter-ring bonds, measurements on neutral PTh in the crystalline soUd [390] indicate that the polymer is all-anti, presumably due to crystal packing forces. Thus, solid state medium effects could play an important role in determining the NLO properties. Clearly, the treatments of doping that have been undertaken so far have just begun to scratch the surface as far as NLO properties are concerned. In addition to a more complete investigation of PA and PTh numerous other polymers, such as polypyrrole, poly-/?phenylene, poly-/7-phenylenevinylene, among others, are worthy of consideration. Although poly-p-phenylene may not have been studied quite as thoroughly as PTh, similar structural calculations are available [387, 391, 392]. In addition, the existence and nature of bipolarons in 7r-stacks of poly-;7-phenylene, as suggested by some experiments [393], has been studied theoretically by Xie and Dirk [394]. They carried out an AMI treatment of a doubly charged vertical stack of up to 14 benzene rings with a 3.0 A separation and do, indeed, find a bipolaronic structure where the rings at either end are benzenoid in character whereas all other rings tend to be quinonoid. Consistent with this picture is the fact that the positive charges are localized on the penultimate benzene molecules in the stack. Furthermore, the calculations by Xie and Dirk of the orientationally averaged static y^ show an enhancement (with respect to a single ring) by 5-6 orders of magnitude for the largest stack, without leveling off as the number of benzenes is increased. It remains to be seen what effect the dopant counterions will have on these results. In a related vein the singly charged ;7-nitroaniline antiparallel stacked dimer has been examined in an ah initio treatment [395]. In this case a UHF calculation leads to a structure where the charge is localized on one ring. The dominant component of j8^ (origin at the center of mass) remains along the longitudinal intra-ring direction with a small increase in the value per molecule versus the neutral. 3.4. Role of the Medium Thus far our focus has been on individual organic molecules and polymer chains. Of course, in any system of practical interest these species will interact with the surrounding medium. Quite a bit of theoretical attention has been devoted to molecules in solution, particularly those of the push-pull variety. This is relevant for experimental measurements if not for practical devices. On the other

115

THEORETICAL APPROACH TO DESIGN hand, interactions in the soHd state, which are important for device applications, have received much less attention. We will begin with solutions and then move on to the solid state.

Eiocal = E R + E c

(98)

ER = (fRo + fRi)/isolute = fRMsolute

(^9)

Ec = fcE

(100)

with

3.4.1. Solvent Effects The effect of solvation on molecular properties has become a very active area of investigation in the past several years. Most treatments are based on the Onsager reaction field cavity model [396] or its generalization. In this model the solvent is treated as an isotropic polarizable continuum, described by a macroscopic dielectric constant, and the solute molecule is embedded in a cavity within that continuum. The permanent plus induced solute dipole moment Asolute (^^<^> ^^ modern theory, higher-order multipoles) polarizes the solvent and, thereby, creates a reaction field, ERQ = feo ^solute that back reacts with the solute dipole according to the potential ^Ro = -/^solute • ERQ

The local field at the solute molecule may be represented as a superposition of the reaction field and cavity field:

(97)

In quantum mechanical treatments Asolute is the dipole moment operator and FRQ is added to the Hamiltonian of the isolated molecule to form an effective solute Hamiltonian. Even in the absence of any external fields the static solute dipole will set up a static reaction field. Because the magnitude of ERQ depends upon the value of the static solute dipole the resulting effective Schrodinger equation must be solved self-consistently. This eventually yields frequency-dependent solute (hyper)polarizabilities associated with the solute molecule in the presence of its own static reaction field. These have been called solute hyperpolarizabilities (a^o\, jSsoi, and 7sol) ^Y Wortmann and Bishop [397]. The reaction field per unit dipole may be characterized by a reaction field tensor fR^ [398] that depends upon the size and shape of the cavity as well as the dielectric constant of the solvent. For an ellipsoidal cavity the reaction field tensor is diagonal with simple expressions for the diagonal elements [399]. If more realistic shapes are used the field can be simulated by a charge distribution on the surface of the cavity giving rise to the apparent surface charge (ASC) [400, 401] and conductor screening models (COSMO) [402]. The ASC treatment is employed with the socalled polarizable continuum model (PCM) that has recently been put into a convenient integral equation formalism (lEF) [401]. When a macroscopic electric field E is imposed on a solvated system it will be screened in part by the solvent, thereby producing a field at the solute (called the cavity field) E(^. In addition, there will be new components of ER, ERJ = fRiAsolutc ^^^ to induced solute dipoles (and, in modern theory, higher-order multipoles) oscillating at the frequency of E or a multiple of that frequency [403]. In quantum mechanical methods based on the selfconsistent reaction field (SCRF) and PCM-ASC methods (as well as their various approximations), the solute (hyper)polarizabilities are normally determined in the presence of the complete (static + induced) solute reaction field [403-407]. These have been called cavity hyperpolarizabilities (^cavity i^cavityj and 7cavity) and, like the so-called solute hyperpolarizabilities, the cavity hyperpolarizabilities constitute another type of (reaction field) dressed hyperpolarizabilities. Work by Agren and coworkers [404] has shown that the cavity and solute hyperpolarizabilities are related by an expression involving reaction field factors and solute or cavity polarizabilities [404].

and where fc is the cavity field tensor. For ellipsoidal cavities this tensor is diagonal and the diagonal elements are known as cavity field factors. The cavity field factors are a function only of the dielectric constant and the shape of the cavity [399]. Through first order the total field acting locally on the solute molecule is given by the product of a local field tensor L and the external field. Elocal = LE

(101)

Assuming there is sufficient symmetry, L will be diagonal with each element (i.e., the local field factor) being the product of a cavity field factor and a reaction field factor [L(w) = fc(co)FR(w)] [397]. The latter, in turn, is a function of the solute polarizability (as well as other solute properties) and, of course, all quantities depend upon the frequency of the field(s). Finally, the effective (hyper)polarizability, which corresponds to the molecular polarization induced by the external fields, is obtained by multiplying the solute (hyper)polarizability with appropriate local field factors. There is one such factor for each external field plus a factor for the induced oscillating dipole of the solute, for example, for SHG, /3eff(—2ft); co, co) = L(2ct))L{o))L{(jo)I3^Q\{—2O); CO, (O) = FR(2ft;)[FR(ft))fc(c.))] X [F^{a))fc((o)]Pso\(-2oj; co, co) (102) An alternative expression for the effective SHG first hyperpolarizabihty is obtained by multiplying the cavity hyperpolarizability with the appropriate cavity field factors: j8eff(-2ft>; co, (o) = fc(w)fc(w)i8cavity(-26>; (o, (o)

(103)

These effective (hyper)polarizabilities are the directly measured quantities. In relating theory to experiment one has to be careful about properly accounting for the field factors. If the calculated quantities are cavity hyperpolarizabilities, then only cavity field factors should be used whereas, for solute hyperpolarizabilities, both cavity and reaction field factors have to be included. Moreover, it has been shown by Cammi et al. [405] that simple— spherical or ellipsoidal—cavity field factors are often inadequate to properly account for the shape of the cavity. Second, one also has to be careful in interpreting experimental data to ascertain whether the effective hyperpolarizability ygff is reported or localcavity field factors have already been introduced. Experimental data are often corrected for local field effects (to derive solute hyperpolarizabilities) using the usual Lorentz and/or Onsager formula. However, it has been observed [397] that the Lorentz formula implicitly includes a cavity factor in L(2ft>) that should not be present because, to first order, there is no cavity factor at harmonic frequencies [e.g., in L(2(o) of Eq. (102)]. Finally, because there are several local-cavity field factors involved in converting from dressed to measured hyperpolarizabilities any systematic error in the individual factors will become magnified in the process. Nevertheless, as observed by Macak et al. [404], these errors often

116

CHAMPAGNE AND KIRTMAN

fall within the uncertainty of the theoretical treatment and do not affect major conclusions. There are many issues that arise in calculating cavity (hyper)polarizabiHties. These include the size and shape of the cavity [406-409], the treatment of solute electron density lying outside the cavity [410-414], the importance of taking into account solute multipoles [415] if the ASC treatment (which is an all-order multipole approach) is not used, the role of Pauli repulsion and dispersion [416], and/or the necessity of including nearby solvent shells within the cavity [417, 418]. In particular, for the latter point, the use of polarizable centers has been proposed by van Duijnen and coworkers [417]. Despite some of the difficulties in carrying out reUable calculations noted here it seems that including the full reaction field for donor-acceptor solutes can lead to increases by a factor of 2-3 or more in both the electronic and vibrational first hyperpolarizability [189, 217, 419-422]. Thus, it is important to take into account solvent effects in evaluating the potency of individual NLO chromophores based on solution measurements. When dealing with D/A compounds, a primary effect of the solvent is to stabilize the zwitterionic form and, from the VB-CT model, we see that this can change /3 and y in either direction. For nonpolar solutes there can be a significant effect that has a different origin. A TDHF-ASC study of small polyacetylene oligomers has revealed that low-dielectric-constant solvents induce substantial increases in the static and dynamic y^ [423]. In this case BLA effects are negligible and the increase is associated with a bathochromic shift of the lowest dipole-allowed electronic excitation. It was also found that electrostatic effects are much more important than dispersion and PauH repulsion. To get effective hyperpolarizabilities, the Ec/E ratio was computed assuming an ellipsoidal cavity that, for ^ran^-butadiene, gives 7£(0) [effective]/y£(0) [cavity] between 1.37 and 1.72 for s = 2 and 8, respectively. For comparison, at s = 8, the ratio obtained by the method of Cammi et al. [405] is 2.14. For all-/ran5-decapentaene, this y£(0)[effective]/y£(0)[cavity] ratio decreases to 1.16 (e = 2) and 1.28 (s = 8). Finally, in the long chain limit the ratio will tend toward unity. Later, using a semiclassical solvation model Luo et al. [424] have shown that the enhancement of the isolated molecule y^ due to dipole-dipole electrostatic interactions decreases with increasing chain length. For instance, the increase in y£(0)[cavity]/y£(0)[isolated] diminishes from about 2.3 to 1.9 and to 1.5 for chains containing 10, 20, and 40 carbon atoms, respectively. Because these results pertain to perfectly regular and linear chains, the next step in addressing the experimental data will be to take into account the conformational disorder. 3.4,2. Solid State Effects The effect of the medium on NLO properties in the solid state has been studied much more sparingly than solvent effects. This is a crucial issue for second-order materials because electrostatic interactions between dipolar molecules in a crystal favor a centrosymmetric, or nearly centrosymmetric, orientation that tends to cancel the first hyperpolarizability. It has been shown [425] that this tendency, together with phase matching requirements, will typically reduce the magnitude of the second-order response by a factor of 3-4. To alleviate this situation one can attempt either to reduce or to overcome the dipolar interaction. One strategy for reducing the interaction is to add substituents. A typical example is 3-methyl-4-nitroaniline (MNA), where a methyl group

is substituted for a ring hydrogen in/>-nitroaniline (PNA) [426]. A second strategy is cocrystallization as in the stilbazolium merocyanine case [265, 266, 427], or co-crystallization accompanied by proton transfer [428]. A third strategy is to create structures with low packing density, for example, self-assembled monolayers and Langmuir-Blodgett films [429]. Yet another strategy is to use weak donor-acceptor pairs—POM is an example—^because then the dipole moment and electrostatic interactions will be small [430]. Another approach is to consider chiral entities or substituants [282]. In general these approaches will weaken, or dilute, the hyperpolarizability of the individual molecules at the same time that they partially alleviate the bulk orientation effect. Thus, there is a gain on the one hand but a loss on the other. The most popular method for overcoming dipole-dipole interactions is the poled-polymer technique. Although large macroscopic susceptibilities are produced in this manner [431, 432] the resulting materials are thermodynamically unstable and often have poor crystallinity as well as poor long-term orientational stability. An alternative is cocrystallization with one- or two-dimensional inorganic structures [433]. An example [434] is given by the reaction of Pbl2 and ^ran5-4-[4-dimethylamino]-styryl-l-methylpyridinium iodide, which produces anionic [Pbl3]~ chains coupled to Nmethyl-stilbazolium cations that are completely aligned along one crystallographic direction. Still another alternative is afforded by taking advantage of hydrogen bonds to obtain a parallel, rather than antiparallel, alignment. This occurs, for instance, in crystals of A^-(4-nitrophenyl)-(l)-prolinol [435] where the NLO chromophores are in vertical stacks with the immediate neighbors (within the stack) horizontally translated. A handful of theoretical calculations have been done to elucidate and guide the development of second-order sohd state NLO materials. Using the ZINDO method, Di Bella et al. [191] found that the longitudinal (3^ is maximized for a stacked pair of PNA molecules when they are in a slipped cofacial arrangement such that the donor of one molecule is almost directly above the acceptor of the other. On the other hand, in crystalline MNA the nearest neighbor molecular pairs are related to each other by a twofold glide plane operation, but the calculated first hyperpolarizability [436] is reduced by 28% compared to the simple additive value. The latter result, obtained by a split valence Hartree-Fock supermolecule calculation, is reproduced fairly well by the simple electrical treatment of Dykstra et al. [437] in which the molecules are modeled by point dipoles (permanent + induced) located at the center of mass. In contrast with MNA and stacked PNA, urea crystals exhibit H-bonding interactions that have an important effect. Zyss and Berthier [438] have modeled the latter using point charges located on the atoms involved in the H-bonds. On the basis of a self-consistent Hartree-Fock treatment, they found that the combination of intramolecular charge transfer and intermolecular bonding increases the magnitude of j8^ primarily in the direction of the parallel in-plane H-bonds. A subsequent AMI treatment [439] on a pair of linear coplanar ;?-nitroaniline molecules with their donor and accceptor groups in a hydrogen bonding arrangement also showed an enhancement by a maximum of 50% at the van der Waals contact distance. In none of these cases has the corresponding vibrational response been considered. At first blush it would appear that solid state interactions are much less critical for third-order, than for second-order, NLO materials. However, recent investigations indicate that second hyperpolarizabilities may be affected in a major way. Initial studies have been carried out on a pair of butadiene and hexatriene molecules

117

THEORETICAL APPROACH TO DESIGN

a=4.18A

Fig. 37. Projection of the all-trans PAfiberstructure in the plane perpendicular to the chain direction as determined in [444, 445]. Chains I and II are in the parallel configuration; chains I and III are in the quasi-perpendicular configuration. The bond length alternation is in-phase within the set of parallel chains and within the set of quasi-perpendicular chains. However, the two different sets are out-of-phase with one another, which results in a P21 In space group. Reprinted from B. Kirtman, C. E. Dykstra, and B. Champagne, Chem. Phys. Lett. 305,132 (1999). Copyright 1999, with permission from Elsevier Science.

in the nearest neighbor geometry appropriate to a stretched polyacetylene fiber (see Fig. 37) except with the single and double bonds in-phase [440]. It was found that the longitudinal static electronic second hyperpolarizability per molecule [ylJfi)/N\ is reduced by the pair interaction to about 2/3 of the isolated molecule value. As seen in the figure, there are two different nearest neighbor configurations, parallel and quasi-perpendicular. The 2/3 reduction holds for both configurations and for correlated (MP2/631G) as well as uncorrelated (RHF/6-31G) treatments. Larger basis sets containing diffuse functions significantly alter y^ but not the relative effect of the interchain interaction (Table XXXII). Furthermore, as additional nearest neighbors are included there is a monotonic decrease in Ji^/N for a total reduction of 76% when the first shell of six nearest neighbors is complete [441]. (In this case the experimental structure of the fiber was preserved in all respects.) Although a large superposition error is possible with the small 6-31G basis, tests reveal that is not the case. Prior to this work a semiempirical (DVB-PPP) treatment of hexatriene chains [442] had given quite different results. Although the pair interaction was found to yield a small decrease in Ji^/N this was more than overcome when the remaining nearest neighbors were included. In contrast, a semiempirical TDHF/AMl treatment carried out by Chen and Kurtz [443] for parallel pairs of cis and trans polyene chains containing up to six double bonds showed a marked decrease of yj^(0)/N upon dimerization. The longitudinal static vibrational hyperpolarizability was also studied [446], at the double harmonic RHF/6-31G level of approximation, for several representative polyene clusters containing up to four nearest neighbors. In this case the geometry was optimized and the corresponding electronic hyperpolarizability was compared at the optimized geometry. For short polyene chains low-frequency collective and interchain modes play an important role. As the chains are lengthened, however, the higher-frequency vibrations most significant for an isolated chain become dominant. The interchain interactions between the longer chains reduce the vibrational hyperpolarizability just as they did the corresponding

Table XXXII. Basis Set Effects upon the Static Electronic Polarizability and Second Hyperpolarizability of All-Trans Hexatriene Dimers^

6-31G 6-31G+PD TZ TZ-Ks TZ-Hsp TZ-l-spd

4(0) (au)

r£(0) (103 au)

294.0 (90.6) 319.6 (90.1) 309.9 (90.4) 313.7 (90.6) 322.3 (89.9) 323.3 (89.7)

82.4 (59.0) 182.3 (64.3) 111.3 (59.9) 123.0 (62.9) 179.5 (64.5) 169.6 (64.9)

^The ratio (in percent) between the (hyper)polarizability per chain in the dimer and the isolated monomeric value is given in parentheses.

electronic property. That is to say, yJ^/'^L ^^ essentially unaltered by the presence of surrounding chains, lliese results for the electronic and vibrational hyperpolarizabiUty are consistent with the hypothesis that the interaction between chains arises from mutual electrostatic polarization of the electron density. If the preceding hypothesis is correct, then a classical electrostatic reaction field cavity model should be able to describe the solid state medium effect on the electronic hyperpolarizability. A treatment has been carried out by Hamada [324] using a spherical cavity with local field factors based on the anisotropic Lorentz approximation. Because a spherical (rather than ellipsoidal) cavity was employed, along with the hyperpolarizability of the isolated molecule (rather than the solute molecule in its reaction field) it may not be surprising that an increase in y£ was found contrary to ab initio cluster calculations on the first shell of nearest neighbors. Indeed, the expression for a needle-shaped cavity is quite different than for a spherical cavity. In the former case, the cavity field tends to be small, leaving the reaction field dominant. The hexatriene cluster calculations also do not correspond to a solute

118

CHAMPAGNE AND KIRTMAN

Table XXXIII. Ratio between the (Hyper)polarizability per Chain in the Dimer and the Isolated Monomeric Value as a Function of the Number of CC Double Bonds in Polyacetylene Chains"

The only assumptions that need be made are (1) classical electrostatic polarization is the dominant effect and (2) the electrical properties of individual hexatriene (or longer) chains can be adequately described by a set of additive point polarizability and A^ a[(0) r£(0) hyperpolarizability tensors [448]. This is more sophisticated than 5 0.907 0.620 the earlier procedure of Dykstra et al. [437] in which these prop6 0.907 0.622 erties were concentrated at a single point. Given the distributed (hyper)polarizability tensors of the individual chains one can solve 7 0.909 0.629 the nonlinear classical electrostatic equations iteratively for the 8 0.912 0.640 interaction energy of an arbitrary cluster in the presence of a fi9 0.916 0.653 nite external field in the longitudinal direction [449]. Then, follow10 0.920 0.667 ing the Romberg procedure of Section 2.6, the longitudinal (hy11 0.924 0.682 per)polarizability per chain may be obtained by numerical differ12 0.928 0.697 entiation. 13 0.932 0.712 Although assumption (2) in the preceding paragraph can al14 0.936 0.726 ways be satisfied by taking a sufficient number of points, in reality the ab initio data set is limited and one wants to make the model as simple as possible. Thus, on the basis of previous experience the ab "CPHF/6-31G results on RHF/6-31G geometry. initio polarizability tensor for the hexatriene chain was distributed over the six carbon centers, which were taken to be equivalent exTable XXXIV. Comparison of yf^iO) of All-Trans Hexatriene Clusters cept at the chain ends where local C2 symmetry was invoked. The with y£(0) of the Oligomer Containing the Same Number of Carbon hydrogen centers were not utilized because they contribute only Atoms^ a small part of the polarization response. Because centers on the same chain are not allowed to interact the ab initio polarizability H-C6ArH6^-H A^ N-C6H8 bundle N-C6H8 needle tensor is exactly duplicated by this procedure. It turns out that the same prescription also reproduces the ab initio longitudinal polar2 1,317 69.5 165.5 izability of the seven chain first nearest neighbors cluster to 1%. 294.7 6,263 73.7 3 In the case of the hyperpolarizability tensor several distribution 430.2 77.2 4 15,677 schemes were tried before the ab initio result for the seven chain cluster of first nearest neighbors was satisfactorily (within 1.5%) "The values are given in 10^ au. The monomeric CgHg value at the RHF/6- duplicated. In the end, the longitudinal component was divided in31G level amounts to 54.4 x 10^ au. The bundles are built from paral- equivalently between the carbon atoms at the chain ends and the lel and quasi-perpendicular chains numbered as shown in Figure 37: for remaining interior carbon atoms. The chain end values were choN = 2, chains I and II; for N = 3, chains III, IV, and V; for N = 4, chains sen to fit butadiene whereas the interior values were taken to make III, IV, V, and VI. The needles are built from coUinear CgHg molecules up the difference between butadiene and hexatriene. Obviously, separated by a distance of 2 A. this distinction will become less significant for longer polyenes. All remaining elements of the hyperpolarizability tensor were located at the center of the chain and it was verified that their distribution immersed in a three-dimensional solvent because, for the cluster, is unimportant. More complicated schemes could certainly be dethere is no medium in the longitudinal direction. There are some vised but this one was sufficient to yield very good agreement for preliminary cluster results that bear on this point and on the effect all the subshell clusters on which it was checked as well as for the of increasing the chain length. For a parallel pair of chains ab initio effect of uniformly scaling all the interchain distances, despite the RHF/6-31G calculations [447] show (Table XXXIII) a slow reduc- large changes in yf^/N that occur in the latter test. Values of the static longitudinal electronic polarizability and hytion (in percent) of the interchain effect on yf^/N (and af^/N) for polyenes containing up to 28 carbons. On the other hand, for perpolarizability for successive complete shells of nearest neighcollinear arrays of 2 to 5 all-trans hexatriene molecules (separated bors are shown in Figure 38. The figure also contains a plot of the by 2 A) y^ P^^* chain increases markedly in comparison with an in- total a^ or TL V^^S^S th^ number of chains. Both curves are nearly dividual hexatriene chain (Table XXXIV). From these and earlier linear and, using the slope determined by the two points atN = 37 studies four conclusions emerge: (i) the effect of the surround- and 61 (i.e., the third and fourth shells), one obtains an immediate ings on 7£ are substantial, (ii) the sign and magnitude of the in- estimate of the infinite A^ limit for the property per chain. These terchain interactions depend upon the geometric arrangement of estimates are within 0.5% of those determined by a more sophisthe chains, (iii) the assessment of solid state effects requires ex- ticated extrapolation procedure [441]. Clearly the extrapolation is treme care, and (iv) in the cluster approach one must consider long highly accurate in this case. The calculated values show a decrease chains and large clusters. The cluster approach does have the ad- by over 50% in af^/N due to the solid state medium whereas yf^/N vantage, however, that problems associated with determining the is reduced by more than a factor of 20. The electrostatic treatment just discussed can be extended to appropriate cavity, as well as cavity and reaction field factors, are deal with long or infinite chains. This extension would be straightentirely avoided. In practice, it is not feasible to extend the ab initio cluster treat- forward were it not for the fact that the iterative solution of the ment much beyond the first shell. One can, nonetheless, utilize the electrostatic equations does not converge because of the increased ab initio results to parametrize an electrostatic cluster model in- strength of the interaction. To circumvent this difficulty one can cluding enough shells to permit the desired extrapolation [441]. solve the equations with an artificially reduced interaction and.

THEORETICAL APPROACH TO DESIGN 5000.0

94%) with retention of sign. Besides this sensitivity to y^, the sensitivity to details of the model remains to be probed particularly taking into account the reduced interchain effect observed (computationally) when the length of a parallel pair of polyene chains is increased, the enlargement of the cluster in the third dimension, and the presence of structural defects.

-•-7^/100 4000.0

3000.0

'2

119

2000.0

ACKNOWLEDGMENTS This work has benefited from long-standing collaborations with Jean-Marie Andre, David M. Bishop, Denis Jacquemin, Josep M. Luis, Eric A. Perpete, Kathleen A. Robins, and Joseph L. 0.0 Toto; from fruitful collaborations with Evert-Jan Baerends, David 20 30 40 50 70 Beljonne, Roberto Cammi, Celso P. de Melo, Erik Deumens, ClifNumber of chains (N) ford E. Dykstra, Christof Hattig, Ivo Kanev, Benedetta Mennucci, Fig. 38. Variation of ctL ^^d TL ^S successive shells of nearest neighbors David H. Mosley, Yngve Ohrn, Olivier Quinet, Milena Spassova, are added to the cluster. The properties were calculated by the electroJacopo Tomasi, Teressa T Toto, Stan J. A. van Gisbergen, and static method and are given in atomic units. Reprinted from B. Kirtman, C. E. Dykstra, and B. Champagne, Chem. Phys. Lett. 305,132 (1999). Copy- Ernst-Joachim Weniger; as well as from enlightening discussions with Rodney J. Bartlett, Yi Luo, Masayoshi Nakano, Gustavo right 1999, with permission from Elsevier Science. Scuseria, Zhigang Shuai, and Guiseppe Zerbi. B.C. thanks the Belgian National Fund for Scientific Research for his Research Associate position. We thank, for financial support and computer then, extrapolate to the desired value. A simple physical way of time, the Namur Scientific Computing Facility, the FNRS-FRFC, doing that is to uniformly multiply all the interchain distances by the "Loterie Nationale" for convention N° 2.4519.97, the Belgian a scaling parameter A' ^ 1- However, the values of x that yield a National Interuniversity Research Program P3-49 and P4-10, the stable solution of the electrostatic equations are too large to perMaui High Performance Computer Center, and the Donors of the mit a reUable extrapolation to unity using conventional techniques. Petroleum Research Fund administered by the American ChemiA viable alternative approach was found [450] by further explocal Society. ration of the model through calculations on all clusters that can be constructed from chains containing N = 10,14,18,..., 42 carbon atoms and 3-6 nearest neighbor shells (containing 37-127 chains), subject to a limit of 2366 electrical sites. Because the goal is to sim- REFERENCES ulate the infinite PAfiberthe following simplifications were made: 1. J. Kerr, Philos. Mag. 50, 337 and 446 (1875). (i) all carbons were parametrized equivalently; (ii) only the lon2. F. ?ocke\s, Abh. Gott. 39,1 (1884). gitudinal components of a^ and y^ were included at the atomic 3. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, Phys. Rev. centers; and (iii) the contribution from the central carbon atom of Lett. 7,118 (1961). the cluster was rigorously isolated as being most representative of 4. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, the infinite system. For this central carbon atom (indicated by a Phys. Rev. 127,1918 (1962). subscript c) it is useful to define K. Kamada, T. Sugino, M. Ueda, K. Tawa, Y. Shimizu, and K, Ohta, 1000.0

A / = [y^(cluster) - ^^(chain)]/^^(chain)

(104)

and similarly for Aa^. Equation (104) gives the fractional change in the property due to the interchain interactions. It turns out that A7^/7c(chain) depends only on a^Cchain) and on x- Plots of Ay^/y^ (chain) versus x for different values of a^ (chain) reveal a family of curves with an almost identical shape that closely resembles a Morse function. There also exists a simple quadratic relationship between the value of x at the minimum, /tmin» and a^(chain). The end result is that one can accurately fit ^r^/rc(chain) to a polynomial in ;^ - ;^inin and thereby extrapolate reliably to ;^ = 1. It is easy to see that the interchain effect depends very sensitively on the ab initio y^ (chain). Using a rough correlated value of y^ (chain) = 18 x 10^ au and an estimated a^ (chain) = 70 au it is found that Ay^ = -1.06. For these input parameters there is over a 100% decrease in y^—in other words the second hyperpolarizability changes sign and its magnitude is reduced to 6% of the isolated chain value. This result varies very weakly with a^ (chain), but decreasing y^(chain) by 33%, for example, changes Ay^ to -0.71, which corresponds to a 71% reduction in magnitude (rather than

Chem. Phys. Lett. 302, 615 (1999). 6. S. Van Elshocht, Th. Verbiest, M. Kauranen, A. Persoons, B. M. W. Langeveld-Voss, and E. W. Meijer,/. Chem. Phys. 107, 8201 (1997). 7. A. Willetts, J. E. Rice, D. M. Burland, and D. R Shelton, /. Chem. Phys. 97, 7590 (1992). S. R. Marder, C. B. Gorman, F. Meyers, J. W. Perry, G. Bourhill, J. L. Bredas, and B. M. Pierce, Science 265, 632 (1994). 9. B. Kirtman, B. Champagne, and D. M. Bishop, /. Am. Chem. Soc. 122, 8007 (2000). 10. B. J. Orr and J. E Ward, Mol. Phys. 20, 513 (1971); D. M. Bishop, /. Chem. Phys. 100, 6535 (1994). 11 D. M. Bishop and P. Norman, in "Handbook of Advanced Electronic and Photonic Materials and Devices" (H. S. Nalwa, Ed.), Vol. 9, Chap. 1, pp. 1-62. Academic Press, San Diego, 2001. 12. D. M. Bishop and B. Kirtman,/. Chem. Phys. 95, 2646 (1991). 13. D. M. Bishop, B. Kirtman, and B. Champagne, /. Chem. Phys. 107, 5780 (1997). 14. R O. Astrand, K Ruud, and D. Sundholm, /. Am. Chem. Soc. 103, 365 (2000). 15. A. J. Russell and M. A. Spackman, Mol. Phys. 84, 1239 (1995); 88, 1109 (1996); 90, 251 (1997). 16. J. M. Luis, M. Duran, J. L. Andres, B. Champagne, and B. Kirtman, /. Chem. Phys. Ill, 875 (1999).

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