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The wave functions of the linear quadrupolar molecule are assumed to be expressed by the linear combinations of the three basis functions, a VB config...
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J. Phys. Chem. B 1999, 103, 8221-8229

8221

Nonlinear Optical Properties of the Linear Quadrupolar Molecule: Structure-Function Relationship Based on a Three-State Model Sangjoon Hahn and Dongho Kim Spectroscopy Laboratory, Korea Research Institutue of Standards and Science, Taedok Science Town, Taejon 305-600, Korea

Minhaeng Cho* Department of Chemistry and Center for Electro and Photo ResponsiVe Molecules, Korea UniVersity, Seoul 136-701, Korea ReceiVed: September 11, 1998; In Final Form: June 8, 1999

Theoretical descriptions of the molecular polarizability and second hyperpolarizability (R and γ) of the linear quadrupolar molecule are presented. By using a valence-bond and two charge-transfer states, three eigenfunctions of a quadrupolar molecule are obtained as linear combinations of these basis states. The analytic expressions for R and γ are obtained, and the physical picture on the nonlinear optical (NLO) response of the quadrupolar molecule is presented. By introducing the bond-length-alternation coordinates into the model Hamiltonian, the relationships of various vibrational characteristics, such as IR, Raman, and hyper-Raman intensities and vibrational frequencies, with respect to the molecular structure are established. The vibrational second hyperpolarizability is also compared with the electronic one, and the two quantities show similar trends as the charge-transfer character changes. A number of distinctive features are discussed by comparing the results with those of the linear push-pull polyene. The NLO properties of a series of quadrupolar molecules are calculated by using the ab initio method, and also a few more examples of quadrupolar molecules are suggested.

1. Introduction Nonlinear optical (NLO) properties of various organic molecules have been studied extensively over the past decade, since those materials with large hyperpolarizabilities have a strong potential of being used for optoelectronics and a variety of optical devices.1,2 Particularly, the conjugated push-pull polyenes have received a lot of attention experimentally and theoretically.3-25 It has been suggested that optimizing the nonlinear optical property requires a specific donor-acceptor strength for a given conjugated bridge. In these molecules, the NLO properties arise from a charge-transfer electronic state in which electron density is transferred from the electron donor to the electron acceptor. To describe the NLO properties of a linear push-pull polyene, a two-state model, the so-called valence-bond (VB) charge-transfer (CT) model, was presented by Lu et al.,12 where the model Hamiltonian is given as

H)

(

EVB + 1/2k(Q - Q0VB)2 -t ECT + 1/2k(Q - Q0CT)2 -t

)

(1)

The electronic energies of the CT and VB states are denoted as ECT and EVB, respectively. t is the transfer integral. The bondlength-alternation coordinate is denoted as Q, and the two equilibrium positions associated with the VB and CT states are Q0VB and Q0CT, respectively. As discussed in ref 12, the BLA coordinate changes from Q0VB ) -0.12 Å to Q0CT ) +0.12 Å based on the experimental observation of the average bond length of trans-1,3,5,7-octatetraene.26 For a given push-pull

polyene, the BLA is determined by minimizing the lower eigenvalue of eq 1 with respect to the BLA coordinate, Q. By using this model Hamiltonian, the formal expressions of the molecular polarizability and hyperpolarizabilities were obtained.12 It was found that the general trends of the NLO properties, the solvation-induced effects on them,16 and various vibrational characteristics could be successfully described by eq 1.27 Despite that the linear push-pull polyene generally has large first and second hyperpolarizabilities, β and γ, the electronic structural asymmetry that gives rise to the charge-transfer state may present problems from a materials standpoint.28 That is to say, on crystallization, the dipoles tend to oppose one another, leading to unit cells with a centrosymmetry and therefore a bulk material with a comparatively small NLO property, χ(2). To circumvent this crystallization problem, a new type of nonlinear optical molecule having C3 symmetry, so-called octupolar molecule, has been studied recently.29-31 It was found that the nonlinear optical properties such as β and γ of this type of molecules are relatively large. Furthermore, since the permanent dipole moments of these octupolar molecules, e.g., TATB (1,3,5triamino-2,4,6-trinitrobenzene) and CV (crystal violet), vanish, a new crystallization engineering scheme can be used to produce the macroscopic assembly of these molecules. Because of the symmetry of an octupolar molecule, it was found that at least three electronic states are to be involved in the nonlinear optical response and two of them are degenerate excited states.31 Although there exist a few attempts to elucidate the nonlinear optical response of the octupolar molecules by using the semiempirical calculation method or PPP (Pariser-Parr-Pople)

10.1021/jp9836933 CCC: $18.00 © 1999 American Chemical Society Published on Web 09/11/1999

8222 J. Phys. Chem. B, Vol. 103, No. 39, 1999

Hahn et al. energies of two CT configurations are identical. There exist two transfer integrals, t and T, where the former represents a chargetransfer coupling between the VB configuration and one of the CT configurations and the latter is that between two CT configurations. In general, these two transfer integrals may not be identical. However from the analogy of this model with the VB-CT model for a linear push-pull polyene,12 it is conjectured that the two transfer integrals are likely to be similar. The eigenvalues of the model Hamiltonian can be found easily as

Figure 1. Two types of linear quadrupolar molecules. The type A molecule contains a central acceptor (A), whereas the type B molecule has a central donor (D). The wave functions of the linear quadrupolar molecule are assumed to be expressed by the linear combinations of the three basis functions, a VB configuration, and two CT configurations.

Hamiltonian, there is no simple model for the general structureproperty relationship. Recently, Cho et al. presented a fourstate model, where one valence-bond (VB) state and three charge-transfer (CT) states were used as a set of electronic basis functions.32 They found that both R and β monotonically increase as the CT character in the ground state increases. These patterns are in strong contrast with those of the linear pushpull polyene, where R and β of the latter type of molecules exhibit nonmonotonic patterns as the CT character of the ground state increases. It is believed that the difference in the symmetries of dipolar and octupolar molecules causes this striking dissimilarity. In this paper we shall also show that the quadrupolar molecule with an inversion center exhibits quite different NLO properties in comparison to those of the push-pull polyene. By extension of the model Hamiltonian, eq 1, the NLO properties of a quadrupolar molecule, which can be pictorially represented by Figure 1, are theoretically investigated in this paper. We shall mainly focus on two types of linear quadrupolar molecules denoted as types A and B, where the former contains a central acceptor and the latter has a central donor. Because of the centrosymmetry of these types of molecules, their β’s vanish. Therefore, the lowest-order optical nonlinearity is γ. In this paper, a simple three-state model for the NLO properties of quadrupolar molecules is presented. In section 2, the theoretical derivations of R and γ based on our three-state model are presented with brief discussions on the results. In sections 3 and 4, two effective conjugation coordinates are introduced in the model Hamiltonian and we shall discuss various vibrational features predicted by this model. Discussion and a few concluding remarks are given in section 5. 2. Electronic Polarizability and Hyperpolarizabilities As can be seen in Figure 1, a single VB configuration and two CT configurations are introduced to describe the electronic structure of a given linear quadrupolar molecule. These three configurations constitute a minimum number of basis functions. Hereafter the corresponding wave functions are denoted as φVB and φCT,j (with j)1,2), respectively. By using this basis set, one can construct the Hamiltonian as

(

EVB -t -t H ) -t ECT -T -t -T ECT

)

1 1 Eg ) (EVB + ECT - T) - [(∆E - T)2 + 8t2]1/2 2 2 Ee ) ECT + T 1 1 Ef ) (EVB + ECT - T) + [(∆E - T)2 + 8t2]1/2 2 2

(3)

where ∆E ≡ ECT - EVB. Here the subscript g denotes the ground state, and e and f are the two excited states. From the eigenvector elements, the corresponding eigenfunctions are given by the linear combinations of the three basis states, i.e.,

Ψg ) x1 - 2lφVB +

2

xlφCT,j ∑ j)1

Ψe ) -(φCT,1 - φCT,2)/x2 Ψf ) x1 - 2mφVB -

2

xmφCT,j ∑ j)1

(4)

Here l and m represent the relative CT characters in the ground (|g〉) and the second excited (|f〉) states, respectively,

1 l≡ 4 1 m≡ + 4

∆E - T 4x(∆E - T)2 + 8t2 ∆E - T 4x(∆E - T)2 + 8t2

(5)

The first excited state |e〉 is completely determined by the wave functions of the two CT configurations. If the electronic energy of the CT configuration is much lower than that of the VB configuration, that is to say, ∆E , 0, the CT character of the electronic ground state, l, approaches 1/2. In this limit, the ground state is given by a linear combination of the two CT configurations, e.g., Ψg ) (φCT,1 + φCT,2)/x2. On the other hand, as ∆E increases, l approaches 0 so that the ground state becomes close to the VB configuration. This trend will be confirmed in section 2-C by the numerical calculations for a few sets of parameters. 2-A. Permanent and Transition Dipole Moments. The approximations, that the transition dipole matrix elements among the three basis functions are negligible,12 are as usual invoked, i.e.,

〈φVB|µˆ |φCT,j〉 ) 0 〈φCT,j|µˆ |φCT,k〉 ) 0

for

j*k

(6)

(2)

where the electronic energies of VB and CT configurations are denoted as EVB and ECT, respectively. Since the linear quadrupolar molecule contains an inversion symmetry, the electronic

where µˆ is the electric dipole operator. Then, the permanent dipole moments of the three eigenstates all vanish, i.e., 〈Ψg|µˆ |Ψg〉 ) 〈Ψe|µˆ |Ψe〉 ) 〈Ψf|µˆ |Ψf〉 ) 0. The transition dipole matrix elements between two different eigenstates are obtained as

NLO Properties of the Linear Quadrupolar Molecule

x2l (µ ≡ 〈Ψ |µˆ |Ψ 〉 ) xm2 (µ

µg,e ≡ 〈Ψg|µˆ |Ψe〉 ) µe,f

e

f

CT,1

CT,1

J. Phys. Chem. B, Vol. 103, No. 39, 1999 8223

- µCT,2)

- µCT,2)

µg,f ) 〈Ψg|µˆ |Ψf〉 ) 0

(7)

The transition dipole matrix element between the ground state |g〉 and the second excited state |f〉 equals zero and the three eigenstates have zero permanent dipole moment. Thus, the first hyperpolarizability, βe, vanishes, as it should. It is interesting to note that the electric dipole matrix element µg,e is proportional to xl so that if the CT character of the ground electronic state increases, the transition dipole matrix elements µg,e and µe,f increase and decrease, respectively. Now let µ denote the absolute magnitude of the permanent dipole moment of a CT configuration. Then for a type A quadrupolar molecule, the permanent dipole moments of each CT configuration, µCT,1 and µCT,2, are (µ, 0, 0) and (-µ, 0, 0), respectively. The transition dipole moments given in eqs 7 can be rewritten in terms of µ, as µg,e ) -x2lµ(1,0,0) and µe,f ) x2mµ(1,0,0), respectively. Here it is assumed that the molecular principal axis is lying on the x-axis. Denoting q and d as the effective charge and the distance between the donor and the acceptor, the effective dipole moment of a CT configuration is given as qd. It is found that the transition dipole moments are linearly proportional to the permanent dipole moment of the CT configuration. 2-B. Molecular Polarizability and Second Hyperpolarizability. Now, it is straightforward to calculate R, β, and γ by using the sum-over-state expressions. First, the molecular polarizability along the principal axis is found to be

Rxx )

2µxg,e µxe,g 4lµ2 ) Eeg Eeg

(8)

Note that R is linearly proportional to the CT character and to µ2. In eq 8, the factor 2 is due to the number of permutation of indices. As briefly mentioned above, due to the centrosymmetry of the quadrupolar molecule the electronic contribution to βxxx, βexxx, vanishes. This can be confirmed by noting that the transition dipole moment between |g〉 and |f〉 and the permanent dipole moments of the three eigenstates vanish within the VB2CT model. The second hyperpolarizability is found to be

γexxxx

[

) 24 -

{

) 24 -

µxg,e µxe,g µxg,e µxe,g Eeg3 4l2µ4 4lmµ4 + Eeg3 Eeg2 Efg

+

}

µxg,e µxe,f µxf,e µxe,g Eeg2Efg

] (9)

The factor 24 is the number of permutation of indices in the tensor product. The first term in eq 9 is associated with the optical transition processes pictorially shown in Figure 2a, and the second is with that of Figure 2b. Note that the second term in eq 9 involves a product of l and m. Thus, in the region where the CT character l is small, the second contribution, associated with three electronic states, to γexxxx is dominant in comparison to the two-state contribution.

Figure 2. (a) Optical transition process contributing to the second hyperpolarizability. Note that only two low-lying electronic states are involved in this optical transition process. (b) Another contribution to γ. This term involves all three electronic states.

Figure 3. (a) (Solid and dashed curves) CT character, l (m), as a function of the energy gap between the CT and VB configurations, respectively. For the numerical calculation of this figure it is assumed that T ) t ) 1.0 eV. Here the absolute magnitude of the permanent dipole moment of a CT configuration, denoted as µ, is assumed to be unity throughout this paper. (b) Molecular polarizability plotted with respect to l. t is assumed to be 1 eV, and T varies as 0.5 eV (solid curve), 0.75 eV (long dashed curve), 1.0 eV (short dashed curve), and 1.25 eV (dash dotted curve). (c) Second hyperpolarizability plotted with respect to l. The two transfer integrals are the same as in Figure 3b. (d) Enlarged version of the small l region of Figure 3c. Note that γ increases within this range of l.

2-C. Model Calculations of rexx and γexxxx. In this section, a series of numerical calculations for a few sets of parameters will be presented to show the general trends of Rexx and γexxxx. First, the CT character, l, is plotted as a function of ∆E ()ECT - EVB) in Figure 3a for the case when the two transfer integrals equal 1 eV, i.e., T ) t ) 1 eV. As ∆E decreases, the CT character of the electronic ground state, l, increases. In contrast, the CT character of the second excited state, m, decreases as the energy of the CT configuration decreases in comparison to that of the VB configuration. In practice, ∆E can be lowered by introducing stronger donors and acceptors in a given quadrupolar molecule. Since the magnitude of the transition dipole matrix element, µg,e (µe,f), is proportional to xl (xm), one can selectively make µg,e (µe,f) large (small) with stronger donors and acceptors. This statement is valid regardless of the magnitudes of the two transfer integrals. Next, Rexx and γexxxx are calculated for different values of the two transfer integrals ranging from 0.5 to 1.25 eV. The calculated Rexx and γexxxx when t is fixed to be 1.0 eV and when T varies from 0.5 to 1.25 eV are plotted in Figure 3b,c, respectively. The small l region of Figure 3c is shown in Figure 3d. Note that the general behaviors of Rexx and γexxxx with respect to l are more or less the same in a wide range of T values for a fixed t ) 1.0 eV. In Figure 4a,b, for a fixed T )

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Hahn et al.

Figure 5. Two types of stretching modes: (a) asymmetric and (b) symmetric.

Figure 4. (a) Molecular polarizability plotted with respect to l when T is fixed to be 1.0 eV and t varies as 0.5 eV (solid curve), 0.75 eV (long dashed curve), 1.0 eV (short dashed curve), and 1.25 eV (dash dotted curve). (b) Second hyperpolarizability plotted with respect to l. The two transfer integrals are the same as in Figure 4a.

1.0 eV, the calculated Rexx and γexxxx by varying t from 0.5 to 1.25 eV are plotted. Again, the increasing patterns with respect to l are not altered by the magnitude of t. Thus, the general trends discussed in this paper are not strongly affected by changes in the magnitudes of the two transfer integrals. As the ground-state CT character increases, Rexx increases monotonically. This is a marked contrast to that of a linear push-pull polyene, where γexxxx increases and approaches a maximum value and then decreases to zero, as the CT character of the ground state increases. γexxxx in Figures 3c and 4b changes nonmonotonically with respect to l. However, as will be demonstrated in section 5, the CT characters of the realistic quadrupolar molecules are found to be less than 0.2. Thus, the increasing pattern shown in Figure 3d in the range 0 e l e 0.2 is likely to be observed in the experimental measurements. 2-D. Chain-Length Dependencies of rexx and γexxxx. As mentioned at the end of section 2-A above, the effective dipole moment of a CT configuration equals qd. Therefore, from eqs 8 and 9, it is expected that R and γ depend on the conjugation bridge length as

R∝d

2

and

γ∝d

4

(10)

Since our model does not take into account a variety of electron correlation effects at all, it should not be surprising if one finds that the chain-length dependencies of R and γ deviate from the simple predictions, eq 10. However, the general increasing patterns of the chain-length dependencies should be observed experimentally. 3. Vibrational Characteristics In this section, the vibrational properties of a quadrupolar molecule are considered by introducing two effective conjugation coordinates strongly coupled to the electronic degrees of freedom. It should be noted that an effective conjugation coordinate, also the so-called bond-length-alternation coordinate, was included in the VB-CT model for a linear push-pull polyene by Lu et al. (see eq 1).12 Assuming that this nuclear degree of freedom is a harmonic oscillator and extending the theoretical investigation of Zerbi and co-workers,14 Kim et al.15 showed that the vibrational contributions to R, β, and γ of a linear push-pull polyene are quantitatively similar to the

electronic ones. Furthermore, Cho combined this model with a Born-type solvation theory to show that a variety of vibrational characteristics such as vibrational frequency, IR and Raman intensities, and vibrational β and γ are strongly dependent on the solvent polarity.27 The concept of the effective conjugation coordinate that is applicable to a variety of linear polyenes was clearly elaborated by Zerbi and co-workers.17 This coordinate is associated with the CdC bond stretching motion with the simultaneous shrinking of CsC bonds in a given polyene. Often this coordinate is called as the dimerization coordinate.33 The linear quadrupolar polyene considered in this paper is likely to have a similar vibrational degree of freedom strongly coupled to the delocalization of π-electrons. In contrast to the linear push-pull polyene, the linear quadrupolar molecule has an inversion center so that one should consider a slightly different model to incorporate the vibrational degrees of freedom into the model Hamiltonian. From the geometry shown in Figure 1, the quadrupolar molecule has approximately D∞h symmetry, similar to a CO2 moleculesfor instance, the symmetry of a squaraine molecule is precisely D∞h. Then, in analogy with the CO2 molecule, one can consider two stretching local modes pictorially shown in Figure 5a,b. The corresponding symmetry species are ∑+ u (asymmetric stretching mode) and ∑+ (totally symmetric stretching mode), reg spectively. The former is IR-active, whereas the latter is Ramanactive. The two local modes are the bond-length-alternation vibrations of the two branches, and they will be denoted as QL and QR. Then, these two modes are coupled to each other so that one should consider the following vibrational potential

1 1 V(QL,QR) ) k0(QL - Q0L)2 + k0(QR - Q0R)2 - k′QLQR 2 2 (11) where the corresponding force constant is denoted as k0 and the coupling force constant is denoted as k′. k0 for a linear pushpull polyene was estimated to be 33.55 eV/Å2.12 Q0L and Q0R are the equilibrium BLA coordinates at the left and right branches, respectively. The coupling force constant, k′, should depend on the structure of the central acceptor (donor) in the type A (B) molecule. As discussed in ref 12, the effective BLA coordinate Q varies from -0.12 to +0.12 Å. The two normal modes obtained by diagonalizing eq 11 will be denoted as xA and xS, which are defined as xA ≡ 1/x2(QL - QR) and xS ≡ 1/x2(QL + QR), respectively. Here the subscripts A and S emphasize the asymmetric and the symmetric modes, respectively. The corresponding force constants associated with these two normal modes are obtained as kA ) k0 + k′ and kS ≡ k0 - k′. Thus, eq 11 can be rewritten as

1 1 V(xA,xS) ) kA(xA - x0A)2 + kS(xS - x0S)2 + C 2 2

(12)

NLO Properties of the Linear Quadrupolar Molecule

J. Phys. Chem. B, Vol. 103, No. 39, 1999 8225

where C is a constant and x0A ≡ (k0/x2kA)(Q0L - Q0R) and x0S ≡ (k0/x2kS)(Q0L + Q0R). In the case of the VB configuration, Q0L ) -0.12 Å and Q0R VB ) -0.12 Å so that xVB A,0 ) 0 and xS,0 ) -0.24k0/x2kS Å. Second, for the first CT configuration we have Q0L ) 0.12 Å, CT1 Q0R ) -0.12 Å, xCT1 A,0 ) 0.24k0/x2kA Å, and xS,0 ) 0. For the 0 second CT configuration, we have QL ) -0.12 Å, Q0R ) 0.12 CT2 Å, xCT2 A,0 ) -0.24k0/x2kA Å, and xS,0 ) 0. The extended model Hamiltonian including these vibrational degrees of freedom is, then,

[

]

H) EVB + VVB(xA,xS) -t -t ECT1 + VCT1(xA,xS) -T -t ECT2 + VCT2(xA,xS) -t -T (13) Figure 6. (a) Force constant ratios, KA/kA and KS/kS, plotted as solid and dashed curves, respectively. It is assumed that t ) T ) 1.0 eV. (b) Square root of IR intensity, |(∂µxg/∂xA)eq|, plotted with respect to l. (c) Square root of Raman intensity, |(∂Rxx/∂xS)eq|. (d) Square root of hyperRaman intensity, |(∂βxxx/∂xA)eq|.

where

1 1 2 2 VVB ≡ kAxA + kS(xS - xVB S,0 ) 2 2 1 1 2 2 VCT1 ≡ kA(xA - xCT1 A,0 ) + kSxS 2 2 1 1 2 2 VCT2 ) kA(xA - xCT2 A,0 ) + kSxS 2 2

(14)

To solve the model Hamiltonian in eq 13, the nuclear motions are treated adiabatically so that the diagonalization of eq 13 to find the ground- and excited-state eigenfunctions is straightforward. Although it is possible to obtain the analytic expressions of eigenvalues and eigenfunctions of eq 13, they are quite lengthy and complicated. The ground-state energy, Eg(xA,xS), is now a function of the two normal coordinates. Therefore, minimizing the ground-state energy gives us the equilibrium geometry. Because of the centrosymmetry, for any arbitrary quadrupolar molecule, Qeq R and Qeq L should be identical. Thus the antisymmetric stretching coordinate xA at the equilibrium geometry should be zero, xeq A ) 0. In contrast, the symmetric stretching mode coordinate at the equilibrium geometry is not zero. From the model Hamiltonian in eq 13, one can verify that xeq S varies from 0 to -0.24k0/x2kS Å. In analogy with the linear push-pull polyene, where the BLA is proportional to the CT character of the ground state, we find that the CT character, l, of the quadrupolar molecule is also directly proportional to the equilibrium value of the symmetric stretching mode coordinate. This observation can be formally proven by solving eq 13 to obtain Eg(xA,xS) and calculating the equilibrium coordinate at the potential energy minimum. Then, the relationship between l and xeq S is obtained as VB xeq S ) xS,0 (1 - 2l) ) 0.24k0(2l - 1)/x2kS Å

(15)

Furthermore, if the BLA coordinate is defined as Qg ≡ Qeq L ) at the equilibrium geometry, Q is related to l as Qeq g R

Qg ) (0.24l - 0.12)k0/kS Å

(16)

In general, the coupling force constant k′ is likely to be much smaller than k0.34 In this case, when k0/kS = 1, from eq 16 the equilibrium BLA coordinate varies from -0.12 to 0 Å, as l changes from 0 to 1/2. It is interesting to note that eq 16 is precisely identical to the result obtained by Lu et al. for a push-

pull polyene (see eq 16b of ref 12). Therefore, we believe that the relationship in eq 16 is an important result, showing that the conVentional BLA coordinate can also play an important role in establishing the structure-NLO-property relationship of the quadrupolar molecule in general. We shall present a series of quantum chemistry calculations to demonstrate the validity of these results in section 5. To obtain various quantities, such as force constants and IR and Raman intensities, we find it useful to separate the total Hamiltonian (13) into two parts as H ) H0 + H′ where H′ is defined as

[

0 H′ ) 0 0

0 -xCT1 A,0 xA 0

0 0 -xCT2 A,0 xA

]

(17)

Then, the eigenvalues and eigenfunctions of H0 are given as eqs 3 and 4 with the following replacements of EVB and ECT as

1 1 2 2 EVB f EVB + kS(xS - xVB S,0 ) + kAxA 2 2 1 1 ECT f ECT + kSxS2 + kAxA2 2 2

(18)

By using the standard perturbation theory, it is possible to obtain exact analytic expressions for the two force constants,

KS ≡

( ) ( ) ∂2Eg ∂x2S

KA ≡

eq

8t2 1 2 ) kS - kS2(xVB S,0 ) 2 [(∆Eeq - T)2 + 8t2]

∂ 2E g ∂xA2

eq

) kA -

CT2 2 leqkS2(xCT1 A,0 - xA,0 ) eq Eeq e - Eg

(19)

These values are plotted as a function of l (see Figure 6a).35 It is observed that, as the CT configuration becomes more stable than the VB configuration, the force constant, KA, of the asymmetric stretching mode decreases. Hence it is expected that the IR peak position of the xA mode is red shifted as the strength of the donor (acceptor) of the type A (B) molecule is increased. That is to say, the more the π-electrons are delocalized, the

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Hahn et al.

smaller the force constant of the IR-active mode is. This pattern is in good agreement with the observation in the linear pushpull polyene, where the effective force constant of the BLA coordinate decreases as the equilibrium BLA coordinate approaches zero (see Figure 1a in ref 27). In comparison to the IR-active asymmetric stretching mode, the force constant of the Raman-active symmetric stretching mode shows a nonmonotonic pattern with respect to the CT character. Therefore, as the donor (acceptor) strength of the type A (B) molecule is increased, the Raman mode frequency would be red-shifted and followed by a blue shift. Next, the IR and Raman intensities of xA and xS modes, respectively, are calculated. The IR peak intensity of the xA mode is defined as

IR intensity ∝

|( ) | ∂µxg ∂xA

2

(20)

eq

where µxg is the permanent dipole moment of the ground state along the molecular axis. Although the permanent dipole moment of the ground state at the equilibrium geometry, µxg( eq xeq A ,xS ), is zero, as expected from the molecular symmetry, the first derivative of µxg(xA,xS) at the equilibrium geometry is not necessarily zero. As the CT character increases (as more the electrons are delocalized over the molecule), the square root of IR intensity increases monotonically (see Figure 6b). Next, the Raman intensity is calculated by using the definition

Raman intensity ∝

|( ) | ∂Rxx ∂xS

v Figure 7. (a) Vibrational second hyperpolarizability, γxxxx . (b) Same as Figure 3c for the case when t ) T ) 1.0 eV. This is just for comparison with (a).

mode increases, (ii) that of the Raman-active symmetric stretching mode increases to a maximum value and decreases to zero, and (iii) the hyper-Raman peak intensity increases initially and decreases to zero and then increases again. It is hoped that these theoretical predictions are confirmed experimentally as well as theoretically by using the ab initio method in the near future. 4. Vibrational Contribution to the Second Hyperpolarizability Various vibrational characteristics associated with the effective conjugation coordinates, xA and xS, were discussed above. We next consider the vibrational contributions to the hyperpolarizabilities. As discussed in refs 36 and 37 (see also eqs 6 and 11 in ref 15), the vibrational first hyperpolarizability is given as

2

(21)

eq

where Rxx is the [x,x] component of the polarizability tensor. In Figure 6c, the square root of the Raman intensity is plotted. It reaches a maximum around l ) 0.3 and then decreases to zero. It is interesting to note that the electronic structure with maximum electron delocalization, that is when l ) 0.5, does not coincide with that associated with the maximum Raman intensity. This pattern is also in good agreement with that found in the linear push-pull polyene. As discussed in ref 27, the maximum Raman intensity of the linear push-pull polyene appears when the BLA coordinate is between -0.12 and 0 Å as well as between 0 and 0.12 Å. Likewise, the maximum Raman intensity of the symmetric stretching mode in a quadrupolar molecule is found in the middle of the two limiting cases, between l ) 0 and l ) 0.5 or, equivalently, between Qg ) -0.12 and 0 Å.27 It is also straightforward to calculate the hyper-Raman intensity, that is

|( ) |

∂βxxx hyper-Raman intensity ∝ ∂xA

2

(22)

eq

Note that the symmetric stretching mode is not hyper-Ramanactive. The square root of the hyper-Raman intensity is shown in Figure 6d. Although the electronic βxxx is zero for a centrosymmetric quadrupolar molecule, the first derivative of βxxx with respect to the xA coordinate is not zero in general so that the hyper-Raman scattering signal can be observed. We find that the hyper-Raman intensity shows a nonmonotonic behavior as the CT character increases. On the basis of these numerical results, it is concluded that, as the donor (acceptor) strength in the type A (B) molecule increases or equivalently as the CT character of the ground state increases, (i) the IR peak intensity of the asymmetric stretching

βvxxx ≡

( )( )

∂µxg 3 4π2c2νA2 ∂xA

eq

∂Rxx ∂xA

+

eq

( )( )

∂µxg 3 4π2c2νS2 ∂xS

eq

∂axx ∂xS

eq

(23) In the case of the quadrupolar molecule, both terms in eq 23 vanish. Thus β0xxx ) 0. More specifically, this is because (i) the asymmetric stretching mode is Raman-inactive, i.e., (∂Rxx/∂xA)eq ) 0 in the first term of eq 23 and (ii) the symmetric stretching mode is IR-inactive, i.e., (∂µg,x/∂xS)eq ) 0, in the second term of eq 23. Although one has to sum over the contributions from every normal mode including the two stretching modes to evaluate βvxxx in general, only the contributions from the two BLA modes are considered in eq 23, since these two modes are likely to be most strongly coupled to the electronic degrees of freedom. The vibrational second hyperpolarizability is given as14,15,36,37

γvxxxx ≡

∑

n)A,S4π

{( )( ) ( ) }

1 c Vn2

2 2

∂µg,x

∂βxxx

+3

4

∂xn

eq

∂xn

eq

2

∂Rxx ∂xn

eq

(24) When n ) A, the second contribution in the bracket of eq 24 vanishes, whereas the first contribution is zero when n ) S. In any case, one of the two terms is nonzero so that γvxxxx does not vanish. In Figure 7a, numerically calculated γvxxxx is presented, and it is compared with the electronic correspondence, γexxxx, which was given in eq 9. The absolute magnitude of γvxxxx changes nonmonotonically as the CT character increases, which reflects the trend of hyper-Raman intensity change. Both γexxxx and γvxxxx show a similar pattern with respect to the CT character l, though the two values differ quantitatively. It is interesting to note that the quadrupolar molecule exhibits a few different patterns in comparison to the linear push-pull polyene. In the latter case, only a single BLA coordinate is needed to

NLO Properties of the Linear Quadrupolar Molecule

J. Phys. Chem. B, Vol. 103, No. 39, 1999 8227

Figure 8. Linear quadrupole molecules. Their NLO properties are studied by using the ab initio (3-21G) calculation method.

describe the vibrational contributions to β and γ. Furthermore, the vibrational contributions, βVxxx and γvxxxx, of the linear pushpull polyene are quantitatiVely similar to the electronic contributions so that it was suggested that βVxxx and γvxxxx can be used to roughly estimate the magnitudes of the electronic correspondences, βexxx and γexxxx. In relation to our study, Yaron and Silbey carried out an investigation of the vibrational contribution to γ for a centrosymmetric polyene chain without donor or acceptor substitution.38 Unlike the case of the linear push-pull polyene, they found that the magnitude of γvxxxx is approximately 10% of γexxxx. In the case of the quadrupolar molecule with only two effective vibrational coordinates, we also found that γvxxxx is much smaller than γexxxx (see the difference in the y-axis scales in the two Figure 7a,b). Therefore, the idea that the vibrational contributions to NLO properties can be used to quantitatively estimate the electronic ones may not be applicable to the quadrupolar molecule. Nevertheless, a more detailed investigation for a wide range of parameter values is desired to draw a definite conclusion on the quantitative relationship between γvxxxx and γexxxx. 5. Ab Initio Calculations of the Electronic NLO Properties of a Series of Quadrupolar Molecules To confirm the theoretical predictions based on the VB-2CT model presented in section 2, Rexx and γexxxx of a series of quadrupolar molecules shown in Figure 8 are calculated by using the ab initio method. The central bezene ring with two nitro groups, which are electron-accepting substituents, can be considered as a central acceptor, and the two donors are connected to the central acceptor via conjugated polyenes. As the electron-donating group changes from H to N,Ndimethylamine, the electron-donating ability increases. Consequently, one can expect that the CT character increases by this substitution of a stronger donor. By using the 3-21G basis set for the ab initio calculation at the Hartree-Fock level with the GAMESS program,39 the molecular structures are optimized. Both R and γ tensor elements are evaluated by the finite-field method for a given optimized geometry. The calculated Rexx and γexxxx are plotted in Figure 9a,b with respect to the BLA defined as Qg ) R2 - (R1 + R3)/2 (see Figure 8A for the definition of the bond lengths). It is clearly observed that both Rexx and γexxxx increase as the BLA increases or, equivalently as the CT character increases. From the estimated BLA values with eq 16, it is found that the corresponding CT characters are all less than 0.2, with the rough approximation of k0/kS = 1. Now the general trend of γexxxx with respect to the CT character

e Figure 9. Rexxand γxxxx plotted with respect to the equilibrium BLA coordinate calculated by using the 3-21G basis set. The open circles in e (a) and (b) represent Rexx and γxxxx of the molecule shown in Figure 8B. (c) is a plot of the total charge of the central acceptor vs BLA (see the text for a more detailed description).

shown in Figure 3d is clearly confirmed by the ab initio results in Figure 9b. As discussed before, if the CT character increases, the mobile π-electrons are more widely delocalized over the conjugated molecular system. Thus, the electronic linear and nonlinear responses, represented by the magnitudes of Rexx and γexxxx, increase with the CT character for the molecules in Figure 8A. Note that as the donor changes from -H to -N(CH3)2, γexxxx increases almost 7-fold, whereas Rexx increases just about 1.4 times. Unfortunately, it was not possible to see the turnover behavior observed in Figure 3d. It will be interesting to experimentally and theoretically investigate along this line with the molecules covering a wider range of BLA values. Also it was predicted that the CT character is linearly proportional to the BLA, as can be seen in eq 16. To confirm this relationship, the total charge of the central acceptor, which is defined as a sum of the atomic charges of the dinitrobenzene group, is plotted with respect to the BLA in Figure 9c. Indeed, the extent of the charge separation is in linear relation to the BLA. In the series of calculations presented above, the quadrupolar molecules considered contain a common central acceptor, parasubstituted dinitrobenzene. In section 2, it was predicted that, as the central acceptor strength increases in the type A quadrupolar molecule, the NLO properties should increase, since

8228 J. Phys. Chem. B, Vol. 103, No. 39, 1999

Hahn et al.

Figure 11. Transition-metal-complex-type quadrupolar molecules.

Figure 10. Squaraine derivatives.

this change preferentially stabilizes the CT configuration rather than the VB configuration. To examine this possibility, we deliberately consider the molecule shown in Figure 8B. Note that the central part of this molecule is just a benzene ring without any electron-accepting groups substituted. Rexx and γexxxx of this molecule are calculated and presented in Figure 9a,b as an open circle. Indeed, it is found that Rexx and γexxxx of this weak-acceptor molecule is much smaller than those of the compound having a stronger central acceptor. Thus, the general prediction based on the VB-2CT model is again confirmed by this comparison. Particularly, γxxxx of this molecule (B) is about half of that of the compound (A) with the same donor, -NH2. Therefore, the general strategy of maximizing the NLO property, γxxxx, of the linear quadrupolar molecule should be clear by now. 6. Discussion, Summary, and a Few Concluding Remarks To make a comparison of the NLO properties of the quadrupolar molecule with those of the linear push-pull polyene, we first consider the electronic structure in terms of the BLA. In the case of a linear push-pull polyene, as the strengths of the donor and the acceptor increase, the BLA increases from negative (-0.12 Å) to positive (0.12 Å) values. On the other hand, in the case of a quadrupolar molecule, the BLA in each branch varies from -0.12k0/kS to 0 Å. Due to this limited range of the allowed BLA, the quadrupolar molecule only exhibits the half-side of the NLO responses of the linear push-pull polyene. In relation to the present investigation, the squaraine derivatives can serve as a series of the quadrupolar molecules that can be experimentally studied to confirm the theoretical predictions presented in this paper (see Figure 10). Recently, Ashwell and co-workers studied the second harmonic generation from the centrosymmetric molecule, squaraines.40 It was found that the crystal structure of an anilinium squaraine has a noncentrosymmetric “T-shaped” configuration with moderately close intermolecular contacts between acceptor and donor groups of adjacent molecules. Therefore, the intermolecular chargetransfer process becomes feasible, so that the molecular centrosymmetry is broken by the interaction with the neighboring molecules. Now let us focus on the molecular nonlinearity instead of the bulk property. For squaraines, the central acceptor is the -C4O2- group, and the corresponding donor can be an N,N-dialkyl group. Therefore squaraine derivatives belong to the type A quadrupolar molecule in Figure 1. By substituting different donors, one can change the electronic energy difference between the CT and VB configurations. This will in turn modify the CT character of the electronic states and will affect both R and γ. On the basis of the model presented in this paper, it is expected that both R and γ would increase as the donor strength

Figure 12. Two resonant structures of a cyanine molecule.

is increased from substituent i to ix in Figure 10. The ab initio studies of this series of quadrupolar molecules will be presented elsewhere. Recently, Schougaard et al. investigated the second hyperpolarizability of compounds i-iv in Figure 11, which belong to the type B quadrupolar molecules by using the third-harmonic generation method.41 The signal intensity of compound iv was found to be the largest among these compounds, where the optical field with a wavelength of 1064 nm was used to create a three-photon electronic coherence state. Since they compared the third-harmonic nonlinear responses of these molecules at finite frequency, it is not possible to directly apply our model to their systems because our result rather provides a static value. We believe that the optical resonance effect might play a crucial role in determining the third-harmonic field intensity in their experimental measurement of the second hyperpolarizabilities of these molecular systems. Still it is desirable to study this type of transition metal complex with centrosymmetry in more detail to elucidate the origin of the large optical nonlinearitys the extension of our results to the dynamical cases is a straightforward exercise and will be discussed elsewhere. Before we close this section, it should be noted that the cyanine-type push-pull polyene, shown in Figure 12, has a structure similar to the type A quadrupolar molecule shown in Figure 1 when the donors are N,N-dialkyl groups. However, there are two distinctive differences: (i) the cyanine-type pushpull polyene does not have the central acceptor, and (ii) the cyanine-type polyene does not have centrosymmetry. Therefore, one should not directly make a comparison of our result with that of the cyanine molecule. However, the resonance structures between the two CT configurations shown in Figure 1 closely resemble the resonance structures in the cyanine molecule. This resemblance is a clue explaining why the NLO property increases as the CT character increases in the quadrupolar molecule. Finally, we summarize the main results presented in this paper with a few concluding remarks. In this paper, a theoretical description of the NLO properties of the linear quadrupolar molecule is presented by introducing a three-state model Hamiltonian. Also the vibrational characteristics are discussed by considering two vibrational degrees of freedom that are associated with the symmetric and asymmetric stretching motions. A variety of vibrational properties, such as the frequencies of the IR- and Raman-active modes, IR, Raman, and hyper-Raman peak intensities, and vibrational contributions to β and γ are discussed. To confirm the general trends predicted by the model Hamiltonian, a series of quantum chemistry calculations of the NLO properties of the model quadrupolar molecules (Figure 8) were presented. On the basis of the results,

NLO Properties of the Linear Quadrupolar Molecule we believe that the general structure-function relationships for R and γ of the linear quadrupolar molecule can be described by the model discussed in this paper. However, because our model Hamiltonian is too crude to provide quantitative information on the absolute magnitudes of R and γ, further investigations by means of an ab initio method as well as experimental measurement of R and γ for a series of quadrupolar molecules are certainly desired. Acknowledgment. M.C. is grateful for the financial support from the Center for Electro & Photo Responsive Molecules, the Center for Molecular Science, KOSEF, and the Basic Science Research Institute program, BSRI-97-3407. D.K. appreciates the financial support from the Creative Research Initiatives of the Ministry of Science and Technology of Korea. References and Notes (1) Prasad, P. N.; William, D. J. Introduction to Nonlinear Optical Effects in Molecules and Polymer; John Wiley: New York, 1991. (2) Chemla, D. S.; Zyss, J. Nonlinear Optical Properties of Organic Molecules and Crystals; Academic: New York, 1987. (3) Marder, S. R.; Perry, J. W.; Schaeffer, W. P. Science 1989, 245, 626. (4) Samuel, I. D. W.; Ledoux, I.; Dhenaut, C.; Zyss, J.; Fox, H. H.; Schrock, R. R.; Silbey, R. J. Science 1994, 265, 1070. (5) Mukamel, S.; Takahashi, A.; Wang, H. X.; Chen, G. Phys. ReV. Lett. 1992, 69, 65. (6) Spano, F. C.; Soos, Z. G. J. Chem. Phys. 1993, 99, 9265. (7) Kanis, D. R.; Ratner, M. A.; Marks, T. J. Chem. ReV. 1994, 94, 195. Marks, T. J.; Ratner, M. A. Angew. Chem., Int. Ed. Engl. 1995, 34, 155. (8) Meyers, F.; Marder, S. R.; Pierce, B. M.; Bredas, J. L. J. Am. Chem. Soc. 1994, 116, 10703. (9) Joffre, M.; Yaron, D.; Silbey, R. J.; Zyss, J. J. Chem. Phys. 1992, 97, 5607. (10) Marder, S. R.; Gorman, C. B.; Meyers, F.; Perry, J. W.; Bourhill, G.; Bredas, J. L.; Pierce, B. M. Science 1994, 265, 632. (11) Gorman, C. B.; Marder, S. R. Proc. Natl. Acad. Sci. U.S.A. 1993, 90, 11297. (12) Lu, D.; Chen, G.; Perry, J. W.; Goddard, W. A., III. J. Am. Chem. Soc. 1994, 116, 10679. (13) Chen, G.; Mukamel, S. J. Chem. Phys. 1995, 103, 9355. (14) Castiglioni, C.; Del Zoppo, M.; Zerbi, G. Phys. ReV. 1996, B53, 13319. (15) Kim, H.-S.; Cho, M.; Jeon, S.-J. J. Chem. Phys. 1997, 107, 1936. (16) Chen, G.; Lu, D.; Goddard, W. A., III. J. Chem. Phys. 1994, 101, 5860. (17) Castiglioni, C.; Del Zoppo, M.; Zerbi, G. J. Raman Spectrosc. 1993, 24, 485. (18) Fincher, C. R., Jr.; Ozaki, M.; Heeger, A. J.; MacDiarmid, A. G. Phys. ReV. 1979, B19, 4140. (19) Harada, I.; Furukawa, Y.; Tasumi, M.; Shirakawa, H.; Ikeda, S. J. Chem. Phys. 1980, 73, 4746. (20) Piaggio, P.; Dellepiane, D.; Piseri, L.; Tubino, R.; Taliani, C. Solid State Commun. 1984, 50, 947. (21) Lussier, L. S.; Sandorfy, C.; Le-Thanh, L.; Vocelle, D. J. Phys. Chem. 1987, 91, 2282. (22) Masuda, S.; Torii, H.; Tasumi, M. J. Phys. Chem. 1996, 100, 15335. (23) Yamabe, T.; Akagi, K.; Tanabe, Y.; Fukui, K.; Shirakawa, H. J. Phys. Chem. 1982, 86, 2359.

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