The hyperpolarizability of the tricyanomethanide molecular ion in solution

19 April 1996

OHEMIOAL PHYSIOS LETTERS

ELSEVIER

ChemicalPhysicsLetters252 (1996)389-397

The hyperpolarizability of the tricyanomethanide molecular ion in solution o

Yi Luo, Amary Cesar, Hans Agren Institute of Physics and Measurement Technology, Linki~ping University, S-58183 Linki~ping, Sweden

Received7 December1995;in final form8 February1996

Abstract

Considering the tricyanomethanide ion [C(Cbl)3]- as a prototype for octopole-like molecules we explore the solvent, correlation and frequency dependences of its polarizability and first hyperpolarizability using a reaction feld response model with single- and multiconfigurational self-consistent wavefunctions. The convention problem of relating the hyperpolarizability obtained from the hyper-Rayleigh scattering technique and from computations is disentangled. The validity of the simplifying two- and three-state models for the hyperpolarizability are examined.

1. Introduction

It has been proposed that octopolar molecules form a new class of molecules with potentially useful properties for nonlinear optics (NLO) [1]. Compared to dipolar molecules, which have been much studied both theoretically and experimentally, octopolar molecules possess certain practical advantages, for instance, easier noncentrosymmetric crystallization with lack of dipolar aggregate interaction, better ratio of the off-diagonal versus diagonal /3 tensor components and an improved efficiency-transparency balance [2]. These properties have now inspired experimental efforts similar to those already devoted to dipolar molecules, including the application of the hyper-Rayleigh scattering technique (HRS) [2]. One important aspect in the determination of hyperpolarizabilities is the contribution of the solvent effects since, generally, experiments are carried out in solutions. Indeed, a sizeable solvent effect contribution has been verified for the corresponding

cases of aromatic dipolar and extended xr-conjugated molecules [3-8]. The purpose of this work is to investigate the [C(CN)3]- molecule as a prototype for octopolar molecules, and compare with similar studies of dipolar molecules with respect to the solvent and other contributions to the hyperpolarizability. We point out some difficulties in the use of different conventions to derive the hyperpolarizability and the related HRS values, and problems in this respect for the analysis of fl of [C(CN) 3]- and other octopolar species. We explore the simplifying two- and three-state models and their sensitivity to the chosen parametrizations for the evaluation of hyperpolarizabilities.

2. Method and calculations

The frequency-dependent polarizabilities and first hyperpolarizabilities of the tricyanomethanide anion

0009-2614/96/$12.00 © 1996 ElsevierScienceB.V. All rights reserved PI1 S0009-2614(96)00167-4

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Y.Luo et al. / Chemical Physics Letters 252 (1996) 389-397

[C(CN)3]- in vacuum have been computed by means of linear and quadratic response theory methods [9-11]. The frequency dependent first hyperpolarizabilities of [C(CN)3]- in solution were obtained using a finite field approach to the frequency-dependent polarizabilities using the reaction field response method [5,12]. In this method the time-dependent response equations are solved for the solvated molecule perturbed by an external perturbation, such as an electromagnetic field. All properties are calculated for self-consistent field (SCF) and multiconfigurational self-consistent field (MCSCF) wavefunctions using the response theory methods implemented in the Dalton program package [13]. The MCSCF wavefunction is of full "rr CAS (complete active space) type. A perturbation series expansion, the so-called B convention [14], is used for defining the polarizabilities and hyperpolarizabilities. An atomic basis set including double zeta and polarization functions is used for the calculations. Such a basis set has been proven to be adequate for describing the linear and nonlinear optical properties of molecules of moderate size [5,15,16]. The geometry of [C(CN)3]- in the gas phase was optimized at the SCF level and found to be planar with D3%symmetry. The optimized bond lengths are 1.403 A for the C - C bond and 1.159 A for the C - N bond, which agree with previous experimental and theoretical resuits [17]. The optimized geometry of [C(CN)3]- in water was found to be almost the same as that in the gas phase. The same coordinate system and orientation as in Ref. [17] was used, i.e. the negative x axis coinciding with one of the dihedral axes, see Fig. 1; in this case, one could expect that /3xxx = - / 3 x y y = --/3yxy = --/3yyx [ 1 8 ] . Therefore, only the component along the x axis will be discussed throughout this Letter. The dependency of /3 on the different solvent environment was investigated within the multipole

C

I

x

Fig. 1. Symmetryaxes for [C(CN)3]-.

expanded polarized continuum reaction field model for solvents with dielectric constants that model 1,4dioxane (Cop = 2.023), ethylacetate (Est = 6.02), acetone (est = 20.7), methanol (~st = 32.63) and water (est = 78.54). The dielectric constants Eop and % refer to the optical and static components of the dielectric constant of a given solvent. A cavity of spherical shape has been employed for the solute [C(CN)3]- ando the radius of this cavity has been fixed at 3.96 A, corresponding to the sum of the C - C - N bond length plus one Van der Waals radius of 1.40 ,~ for the nitrogen atom. The importance of the solute electronic charge multipole expansion for describing the solute-solvent interaction contribution to /3 is emphasized by the fact that the dipole moment is zero for the [C(CN)3]- ground state of D3h symmetry. It should be noticed that, strictly, a pure octopole molecule is defined as a molecule in which all multipoles of lower order are zero. The [C(CN)3]molecule and those molecules suggested by Zyss and Ledoux [2] are not octopolar in this sense. However, the octopolar character defined by Zyss [1] was not based on the traditional charge distribution, but from the spherical representation of the /3 tensor [1]: the 'octopolar' molecules are the ones for which only a J = 3 spherical tensor component of the /3 tensor is excited. Generally, such molecules lack dipole moment in the ground state.

3. Results and discussion 3.1. P o l a r i z a b i l i t i e s a n d h y p e r p o l a r i z a b i l i t i e s

The SCF and MCSCF values of the static and frequency-dependent polarizabilities a and hyperpolarizabilities /3 of [C(CN)3]- in vacuum and the different solvent environments are shown in Fig. 2. The frequency-dependent polarizability a ( - t o ; to) and the first hyperpolarizabilities applicable to the Kerr effect /3(-to; to, 0) have been been calculated at to = 1.17 eV, the frequency for which the experimental values were obtained [17]. The dispersion effect at this frequency is observed to be small at both the SCF and MCSCF levels, which reflects the fact that this frequency is far from the calculated first resonance.

Y. Luo et al. / Chemical Physics Letters 252 (1996) 389-397 16

(a)

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Fig. 2. The dependence on dielectric constants ( e ) for calculated frequency-dependent (a) polarizabilities and (b) first hyperpolarizabilities for [C(CN)3]-. SCF and CASSCF results for frequencies 0.00 and 0.043 au are labeled as SCF-0.00, SCF-0.043, CAS-0.00 and CAS-0.043 in the box, respectively.

The correlation effect is found to be almost the same for the molecule in different environments, and has a small effect on the polarizabilities, increasing it

391

by about 2%. For the hyperpolarizabilities a significant effect is obtained; the full "rr MCSCF result is about 1.7 times that of SCF. Thus the [C(CN)3]example indicates the importance of correlation effects for describing the hyperpolarizabilities of octopolar molecules. The solvent dependence of the polarizability and hyperpolarizability can be seen from Fig. 2. The static polarizability shows an increase of 12-36% at both the SCF and MCSCF levels. This figure is larger than for the p-nitroaniline (PNA) molecule, which represents a dipolar molecule with a large solvent contribution, about 7-18% [5]. It was shown in previous studies that the solvent effect is more pronounced for the first hyperpolarizabilities than for the polarizabilities [4,5,19]. For instance, the first hyperpolarizability of PNA is increased in the range 45-150% for the dielectric constants covered [5]; for the water molecule it was found that even the sign of the hyperpolarizability is changed upon liquefaction [19]. Similar conclusions can also be drawn for the present case; the increase of the first hyperpolarizability of [C(CN) 3]- for the dielectric constants covered is in the range 54-221%. The frequency dependent polarizability a ( - to; to) of [C(CN)3]- in various environments is reported in Fig. 2. Octopolar molecules, here represented by the [C(CN) 3]- ion, are excellent systems to be analyzed within the reaction field model with an arbitrary multipolar expansion of the solute charge density. Being of D3h symmetry the electronic ground state ~1 has no permanent dipole moment, a dipolar model is thus insufficient to account for the ground state solute-solvent interaction, especially, no total energy shift could be predicted by this model. On the other hand, molecular properties such as polarizabilities and hyperpolarizabilities are dependent on the electronic (dipole) transition moments and, the latter, on the permanent dipole moments of the excited states, as well. Therefore, still keeping a D3h molecular arrangement, we may expect considerable reaction field solvent effects on these properties already at the lowest level of theory, with the dipole solutesolvent interaction originating from the solute charge distribution of the electronic E' and E" excited states, e.g. the permanent dipole moments of those excited states are non-zero, as shown in Table 1. Fig. 3 presents the convergence characteristics of

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Y.Luo et al. / Chemical Physics Letters 252 (1996) 389-397

the multipolar expansion for the ground state total energy, polarizability and hyperpolarizability of [C(CN)3]- in acetone calculated at the full ~ MCSCF level. Being a molecular ion, the L = 0 Born term gives a large contribution to the total energy but gives no contribution at all to the molecular properties, and, as expected, the dipole L = 1 does not contribute to the total energy. The convergence of the expansion for the total energy is quite rapid due to the dominance of the Born energy. Although the dipole approximation gives sizeable contributions to the solvent effect for the polarizability and the hyperpolarizability of [C(CN)a]-, about 58 and 34%, respectively, it is evident from Fig. 3 that quite a large multipolar expansion must be considered for modelling the solvent effects on these properties. It is plausible that this is the case for octopolar molecules in general. The convergence of the hyperpolarizability fl on the multipole expansion of the solute-solvent interaction has been tested for polar solute molecules such as PNA [5,8], water [19] and 1,1-dicyano-6-(dibutylamino)hexatriene (DCH) [8] where it was found to be quite fast, with sufficient inclusion of terms up to about L = 8. The difference between the dipolar and multipolar expansions is, however, only 10% for para-nitroaniline [5] and 20% for water [19]; a more drastic result was obtained for the DCH molecule where the dipole approximation fails to reproduce the qualitative trend of the hyperpolarizability fl in the range = 1-40 for the dielectric constant of the solvent. For elongated molecules a more crucial parameter entering the reaction field theory of the solute-solvent interaction is the shape of the cavity enclosing the solute molecule, a fact well illustrated for the DCH molecule in Ref. [8] and expected to be

true for polyenes in general. Given its moderate size and the excess of a unit electronic charge delocalized over the molecule, no major corrections for the octopolar ion [C(CN)3]- of optimized equilibrium geometry, polarizability cr and hyperpolarizability fl are expected to be introduced by restricting the solvent cavity to a spherical shape.

3.2. Three-state approximation

For one-dimensional charge transfer compounds, such as PNA, the first hyperpolarizabilities fl can be adequately described in terms of a two-state model in which only ground and charge transfer states are considered [3,20]. For [C(CN)3]- there are two doubly degenerate states dominating the absorption spectra [17], which is related to its octopolar nature and D3h symmetry (another example is triaminotrinitrobenzene [21]). A three-state approximation which includes the ground state and the two doubly degenerate excited states would thus be necessary [21]. These doubly degenerate states are the analogues of the charge-transfer states in PNA-like molecules. Since the ground state dipole moment is absent, the xxx component of the first hyperpolarizability fl can be expressed in terms of a sum-over-state expression as

,o2, ,o3)= E

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m,?/

(1) i.e. flxxx constitutes the summation of all contributions from all possible transition channels. The matrix element titan represents the contribution of a

Table 1 The excitationenergies,dipole transitionmomentsrelated to the ground state, 1E' and 2E' states of [C(CN)3]SCF MCSCF 1~ 1E' 2E' 1/g 1E' to0i 1/¢ 0 0.255 0.352 0 0.250

2E' 0.342

/t~j 1E' 2E'

0.756 0.276

0.840 1.918

0.267 0.746

0.746 0.463

1.091 1.674

0.965 0.757

Y. Luo et al. / Chemical Physics Letters 252 (1996) 389-397 I

2.$

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It should be noted that this expression for/3ff'~ is the same as the traditional two-state model for chargetransfer molecules. The two doubly degenerate states of [C(CN)3]-, 1E' and 2E', have relatively strong dipole transition moments as predicted by both the SCF and full ~r MCSCF response calculations. The three-state approximation is thus examined at both theoretical levels. With this approximation, only four possible transition routes,

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are included. Table 1 lists all necessary data for in vacuum calculated at both the SCF and MCSCF levels using the three-state model. Similar results have also been found for [C(CN) 3]- in the different solvents. The matrix /3c is found to be

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L Fig. 3. The dependence on multipole expansion for A P = Psolvent -Pg~s for [C(CN)3]- in acetone solution. P is (a) the ground state total energy, (b) the polarizability and the hyperpolarizability. The solid line is used as a guide for the eye.

where all values are in units of 10 -30 esu 1. We note that the relative contributions of the excited states 1E' and 2E' are different at the two different theoretical levels. At the SCF level, 1311 is much smaller than /315. However, the full 1r MCSCF calculation leads to the opposite prediction, with /311 giving the maximum contribution among the four channels. This can be understood by inspecting the results shown in Table 1. For both SCF and MCSCF calculations, the

single channel, such as 1,~ 1 ~ mE' ~ nE' ~ 1,~ 1 for [C(CN)3]-

/3Y(o,s, ,o3) =

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(2)

1The conversion factors for ot are 1 a u = 1 . 6 4 8 8 × 1 0 -41 a u = 3 . 2 0 6 3 6 1 × 1 0 -53 C 3 m 2 j - e = 8.639418× 10 -53 esu. C 2 m 2 j -1 = 1 . 4 8 1 8 × 1 0 -25 cm 3 / 3 : 1

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Y. Luo et al. / ChemicalPhysics Letters 252 (1996) 389-397

excitation energies as well as the transition moments are found to be almost the same for the two states. However, it is clearly shown that correlation has large effects on the dipole moments of those two excited states; the dipole moment of the 1E' state is increased almost 4 times, while the dipole moment of the 2E' state is decreased by a factor of 2. This indicates once more the importance of correlation for describing the hyperpolarizabilities of octopolar molecules. Based on the three-state approximation, we obtain fl~'SCF(0; 0, 0 ) = 1.13 ( 1 0 -30 esu) and •Th-MCSCF (0; 0, 0 ) = 1.49 ( 1 0 -3o esu), where the direct response calculations, in which the contributions from all states are included, give the results fl R'SCF(0; 0, 0) = 0.65 ( 1 0 - 30 eSU) and flR'MCSCF(0; 0, 0 ) = 1.13 (10 -30 esu). It is seen that the first hyperpolarizability of [C(CN) 3] is overestimated by the simplified three-state approximation at both the SCF and MCSCF levels. However, the three-state approximation could still be considered as a reasonable approximation at the MCSCF level. It can also be seen that the traditional two-state model is simply not adequate for the hyperpolarizability of [C(CN)3]-. The largest two-state contribution stems from the fl)1 term, in which only the ground and the 1E' states are included. It accounts only for one third of the total value. Table 2 includes the results of frequency-dependent fl calculations from both response theory and the three-state level model. The dispersion effect is small as it is calculated at a frequency far from the resonance. It has been noticed that ,rr electron theories have difficulties in describing the energy spectrum of octopolar molecules properly [21]. In our previous studies for PNA, a similar observation was made

[3,5]; the full -rr MCSCF result for the energy of the charge transfer state was actually worse than the SCF result. The transparency of [C(CN)3]- was found to be down to at least 300 nm, about 4.13 eV, in solution. There are no experimental results for the absorption spectrum of [C(CN) 3]- in the gas phase. The I N D O / S C I results in Ref. [17] indicate that the two excited states, 1E' and 2E', are located at 4.6 and 7.0 eV [17] in the gas phase. It was concluded that for such energy values a negligible resonance enhancement of the ESHG ( f l x x x ( - 2 t o ; to, to)) value at the frequency to = 1.17 eV was expected [17]. We have put the I N D O / S C I energy results together with the dipole moments obtained from our MCSCF calculation into the three-state model, see the results labeled as ThSM-E in Table 2. With these energy values, the static hyperpolarizability increases about a factor of two from the MCSCF three-state approximation result (see Table 2). Especially, the dispersion for the second-harmonic generation (SHG) at o9 = 1.17 eV is about 63%. Therefore, the resonance enhancement for SHG at to = 1.17 eV can not be negligible. 3.3. Comparison with experiments Usually, the first hyperpolarizabilities of organic molecules in solution are measured through electricfield-induced second-harmonic generation (ESHG). However, application of the ESHG method to octopolar molecules is impossible because the ground state dipole moment is absent. An alternative technique, the hyper-Rayleigh scattering (HRS) technique, has been developed and can be used for measuring the first hyperpolarizabilities of nonpolar and even ionic species [22]. For HRS, no orienting

Table 2 Comparison of the first hyperpolarizability flxaxx(- tol - to2; to1, to2) ( 10- 30 esu) for [C(CN) 3] - obtained from response calculations and three-state models (ThSM) SCF MCSCF RESP ThSM RESP ThSM ThSM-E a fl(0; 0, 0) fl(-to; 0, to) b fl(-2to; to, to) b

0.65 0.68 0.74

1.13 1.28 1.43

a The excitation energies reported in Ref. [17] are used. b to = 0.043 au = 1.17 eV (correspondingto 1064 mm).

1.13 1.18 1.29

1.49 1.70 1.95

2.99 3.75 4.87

Y. Luo et al. / Chemical Physics Letters 252 (1996) 389-397

dc electric field is needed to obtain the (HRS) signal [22]. It is known that special care should be exercised when one compares the theoretical results with the experimental counterparts for the hyperpolarizabilities, since there are different conventions involved in the measurements, especially for the ESHG technique [14]. A special convention has been used for ESHG measurements of molecules in solution, the so-called B* convention [14]. The relation among different conventions is: 6/3 BZ' = 2/3 a = fiT, where B and T refer to the perturbation and Taylor conventions, respectively. The convention B* is only defined for ESHG experiments but could be used indirectly for HRS measurements as well. In order to explain this we first discuss the procedure used in the HRS experiment for extracting the values of the first hyperpolarizabilities for molecules in solution. The intensity of the second-harmonic light, I 2,o, is proportional to the square of the intensity of the incident harmonic light, Io~ in the HRS experiment. It gives 12,o = GB2I 2, where G is a geometric factor indicating the amount of signal captured by the experimental geometry, and B E = Nsolvent fl21ven t + 2 Nsolute/3solute for a two-component system [22]. A linear dependence of the quadratic coefficient GB 2 on the number density Nsolute is always found [22]; from the intercept a and the slope b one obtains the relation

/3solute =

[ /3so,v~,,t I,

(9)

i.e. /3solute can be calculated when /3solvont is known or vice versa. Therefore, the convention for the measured value from HRS is dependent on the convention used for the reference data. If the reference data is obtained from ESHG measurements with the B * convention, the result measured from HRS should be determined by the B* convention as well. In the report of the first HRS measurements for the first hyperpolarizabilities of molecules in solution [22], the /3 value for PNA in chloroform was obtained using the earlier result of /3 for CHC13 measured from an ESHG experiment [23] as reference. Since the value for CHC13 is defined in the B * convention, the result for PNA in chloroform obtained from the HRS experiment in Ref. [22] must also be considered

395

as obtained in convention B*. By comparing the HRS result for PNA with other ESHG results, close agreement has been found [22]. The value of t8 for [C(CN)3]- was determined by using methanol as Ref. [17]. The hyperpolarizability of the latter species was obtained from earlier HRS measurements [22] using the value of /3 for PNA in methanol [20], which is known to be in convention B*, as reference. Therefore, the measured result for [C(CN)3]- reported in Ref. [17] must be in convention B*, i.e. /3~x'~(-2to; to, to) = 7 _ 1.5 (10 -30 esu) [17]. Within convention B one thus has /3xax~(-2to; to, to)--- 21 _ 4.5 (10 -30 esu) where to = 1.17 eV. It is found that the experimental values for [C(CN)3]- in three solvents ( n 2 0 , MeOH and EtOH) are identical within experimental error. There is only one theoretical investigation, an INDO//SCI calculation, available so far for the first hyperpolarizability /3axx for [C(CN)3]- in the gas phase, giving the static and SHG values 2.5 (10 -30 esu) and 3.4 (10 -30 esu), respectively, at the same frequency as that for the experiment [17]. Without considering the convention problem the I N D O / S C I result was claimed to be in excellent agreement with experiment [17]. With proper consideration of the convention problem, the result is, however, about 6 times smaller than the experimental value. Such a deviation can partly be traced to the solvent effect. Our results given by SCF and MCSCF response theory are also found to be too small compared with experiment. At the full MCSCF level, the static hyperpolarizability of [C(CN)3]- in water is about 3.38 (10 -30 esu). Since quite a small solvent shift for the excitation energies has been found, the same dispersion for the SHG signal at frequency to = 1.17 eV as in the gas phase can be expected. Therefore, the SHG value at to = 1.17 eV for [C(CN)3]- in water is 3.89 (10 -30 esu). As discussed in the previous section, the full q'r MCSCF excitation energies obtained from response theory are too large. As shown in Table 2, the SHG value would increase about 2.5 times if the excitation energies proposed in Ref. [17] are used (the result labeled with ThSM-E in Table 2). By considering this factor, the estimated SHG value at to = 1.17 eV for [C(CN)3]- in water would be raised to about 10 (10 -3o esu). Most ESHG experiments are referred directly or indirectly to the second harmonic generation coeffi-

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Y. Luo et al. / Chemical Physics Letters 252 (1996) 389-397

cient, dl~, for quartz. However, two absolute values have been reported for quartz [24,25]. They differ by a factor of 0.6. Using the recent (but not necessarily more accurate) value for quartz, the experimental data obtained from ESHG measurements should be scaled by a factor of 0.6, the/3 value for [C(CN)3]could become /3xBxx(-2o.~; oJ, og)(quartz I I ) = 12.6 + 2.7 (10 -30 esu) at o~= 1.17 eV. Our estimated results are much closer to this experimental value. Like fluorescence, hyper-Rayleigh scattering is an incoherent process. If the molecules fluoresce at the second-harmonic wavelength, the accuracy of the HRS measurements for the hyperpolarizability becomes doubtful. It has recently been shown experimentally that the values measured by HRS are considerably larger than their corresponding values from ESHG for molecules which fluoresce in the frequency-doubled wavelength region [26]. Such an effect has not been discussed in the measurements for [ C ( C N ) 3 ] - [17], and the influence of the molecular fluorescence at the second-harmonic wavelength on the /3 value remains unknown.

3.4. Summary We have chosen [ C ( C N ) 3 ] - as a test case for theoretical calculations on solvent-induced frequency-dependent polarizabilities ( a ) and first hyperpolarizabilities (/3) for octoplar molecules. Apart from the solvent effect we have studied the effect of electron correlation, the validity of few-state models and the convention problem of relating computed hyperpolarizabilities with hyper-Rayleigh scattering data. Our results can be summarized as follows. As for dipolar molecules, in general, the correlation effect for the [C(CN)3]- octopolar molecular is found to be important both for describing the hyperpolarizability and for relating the mechanism of optical nonlinearity in terms of few-state models. A three-state model including correlation was found to be a good approximation for the first hyperpolarizability, while the traditional two-state model is inadequate at any computational level. The multipolar expansion of the solute charge density used in the reaction field model is particularly important for the octopolar molecules, the ground state dipole moment of which is absent. It was shown that a dipolar model only provides about 58% and 34% of the total

solvent effect for the polarizability and the hyperpolarizability, respectively. The convergence of the multipolar expansion for the hyperpolarizability was found to be relatively slow. The calculated gas phase hyperpolarizabilities can not be directly compared with the experimental data measured in solution. However, even for the theoretical solution values, special care must be exercised when comparing with their experimental counterparts, both concerning the convention problem and the fact that there are two "standard" quartz values available. The special convention B * is only defined for ESHG measurements in solution, but could also be associated for HRS experiments when the internal reference value is obtained by the ESHG measurement. Using the smaller standard value for quartz, our estimated result for the hyperpolarizability of [C(CN) 3]- in water can be brought into good agreement with the experimental value.

Acknowledgement This work was supported by the Swedish Natural Science Research Council (NFR). We thank Dr. Mikael Lindgren for valuable discussions.

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