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Calculation of macroscopic linear and nonlinear optical susceptibilities for the naphthalene, anthracene and meta-nitroaniline crystals
Chemical Physics 261 (2000) 359±371
www.elsevier.nl/locate/chemphys
Calculation of macroscopic linear and nonlinear optical susceptibilities for the naphthalene, anthracene and meta-nitroaniline crystals H. Reis a,1, M.G. Papadopoulos a,*, P. Calaminici b,2, K. Jug b, A.M. K oster b,2 a
Institute of Organic and Pharmaceutical Chemistry, National Hellenic Research Foundation, Vasileos Constantinou 48, GR-11635 Athens, Greece b Theoretische Chemie, Universit at Hannover, Am Kleinen Felde 30, 30167 Hannover, Germany Received 31 May 2000; in ®nal form 13 September 2000
Abstract The macroscopic ®rst- to third-order susceptibilities of naphthalene, anthracene and meta-nitroaniline (mNA) are calculated using a rigorous local ®eld approach. Molecular (hyper)polarizabilities used as input are determined by density functional theory calculations with specially designed basis sets and for mNA also by MP2 calculations with the 6-31++G** basis set. In the case of mNA, the permanent electric local ®eld due to the surrounding dipoles in the crystal is taken into account for the ®rst- and second-order susceptibility by a self-consistent approach. The molecular dipole moment and ®rst hyperpolarizability of mNA are drastically changed by the permanent local ®eld. In all cases the calculated ®rst-order susceptibility compares very favorably with experimental data, if the molecular response is distributed over all heavy atoms in the molecules. Similarly, the calculated second-order susceptibility for mNA is in good agreement with available experimental data, if the same distribution scheme is used and the permanent local ®eld is taken into account properly. This implies that accurate values for the molecular second-order hyperpolarizability c have to be available. The anisotropic Lorentz ®eld factor approximation yields results that are only slightly worse than the best ones of the rigorous local ®eld theory for the ®rst-order susceptibilities, but fails for the second-order susceptibility of mNA, due to its incapability to describe the large eect of the permanent local ®eld on the ®rst-order hyperpolarizability b. Ó 2000 Elsevier Science B.V. All rights reserved.
1. Introduction
*
Corresponding author. E-mail address: [email protected] (M.G. Papadopoulos). 1 Also corresponding author. Fax: +30-1727-3831. 2 Present address: Departamento de Quõmica, CINVESTAV, Centro de Investigaci on y de Estudios Avanzados del I.P.N., Av. Instituto Politecnico Nacional, 2508 A.P. 14-740 Mexico D.F. 07000, Mexico.
Calculations of electrical properties of organic molecules by the methods of theoretical chemistry have been proven to be a very useful tool in the search for new nonlinear optical (NLO) materials. However, most of these calculations have focused on the nonlinear responses of isolated molecules, while the great majority of NLO experiments are conducted on the condensed phase and primarily
0301-0104/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 0 ) 0 0 3 0 5 - 0
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H. Reis et al. / Chemical Physics 261 (2000) 359±371
determine the macroscopic optical susceptibilities, which are also the properties of interest for most applications. In order to compare experimental and calculated (hyper)polarizabilities, quantitative relationships between the molecular and macroscopic properties need to be established. In molecular materials composed of organic molecules, where the molecules retain essentially their identities, the anisotropic Lorentz ®eld factor approximation (ALFFA) is often used to extract molecular (hyper)polarizabilities from the macroscopic properties [1], although the lack of a theoretically sound basis limits the general application of this approach [2]. The Lorentz approximation has also been used to calculate the NLO susceptibilities of molecular crystals using the electric properties of whole unit cells calculated at the semiempirical INDO level [3], and of crystals composed of polyacetylene-like polymers, with the properties of the oligomers calculated at the coupled-perturbed Hartree±Fock level [4]. For molecular materials the macroscopic properties can be expressed in terms of the molecular properties, intermolecular interactions and the molecular packing in the bulk. For molecular crystals in particular, a theory has been developed [5,6] that rigorously calculates the local ®eld at a molecule in the crystal as a sum of the external ®elds and the ®elds arising from all the other molecules in the crystal and allows thereby to express the macroscopic linear and nonlinear susceptibilities in terms of the molecular (hyper)polarizabilities and the permanent electric ®eld in the crystal. In order to overcome known problems of the point multipole expansion concerning size and shape eects of larger, especially elongated molecules, a simple ad hoc partitioning scheme for the molecular electrical response has been proposed. In the ®rst applications of this theory [7±14] the use of semiempirical or low-level ab initio input parameters made it necessary to replace the theoretically calculated polarizabilities by eective polarizabilities calculated from experimental refractive indices in order to get reasonable agreement with measured results. Eects arising from the permanent crystal ®eld were not taken into account. In two recent studies of the benzene [15] and urea [16] crystal it was shown that accurate
ab initio polarizabilities are able to reproduce the experimental ®rst-order susceptibility accurately. In the case of urea it was further shown that the permanent dipolar local ®eld has a large eect on the molecular dipole moment and ®rst hyperpolarizability, although the in¯uence on the calculated second-order susceptibility is small, due to the symmetric arrangement of the molecules in the unit cell. Both in urea and benzene only small size extension eects on the calculated susceptibilities were found using the ad hoc distributing approach mentioned above. We present here a study of the macroscopic linear and nonlinear susceptibilities of the larger molecules naphthalene, anthracene and meta-nitroaniline (mNA), using molecular properties from density functional theory (DFT) calculations with basis sets especially designed for the calculation of molecular (hyper)polarizabilities [17]. In order to explore the suitability of more standard methods for the prediction of macroscopic susceptibilities we also use the molecular properties calculated at the MP2 level with the standard 6-31++G** basis set as input for the mNA susceptibility calculations. We further explore the eect of using different partitioning schemes of the molecular response on the predicted susceptibilities and in the case of mNA, we take into account eects of the permanent dipolar local ®eld. The results are compared with the predictions of the ALFFA and with available experimental susceptibilities.
2. Methods If a crystal is placed into an external electric ®eld E it develops a macroscopic polarization P , which can be expanded in powers of E as: . P =0 v 1 E v 2 : EE v 3 .. EEE ;
1
where v n are the macroscopic electric nth order susceptibilities. In a molecular crystal the molecules retain essentially their identities and the susceptibilities can be considered as arising from the molecular response, described by the molecular polarizability a and the ®rst and second hyperpo-
H. Reis et al. / Chemical Physics 261 (2000) 359±371
larizability b and c. The induced molecular dipole moment pk at a site rk is given by: . pk ak F k 1=2b : F k F k 1=6c .. F k F k F k ; k
k
2
where F k is the local electric ®eld at rk , which, in dipole approximation, is the sum of the macroscopic ®eld E and the ®eld arising from the surrounding induced dipoles. The Taylor expansion convention for the de®nition of the hyperpolarizabilities has been used in Eq. (2) [18]. A local ®eld theory which takes the dipolar interactions rigorously into account and connects the susceptibilities with molecular properties has been developed by Hurst and Munn [5,6]. With static ®elds, the resulting expressions are: X ÿ1 v 1 0 v ak d k ; 3 k
v 2
i 1 X .. h b . dkdkdk ; 20 v k k
v 3 v 3 direct
v 3
cascading
h i : c : dkdk k
1 2 0 v2
: b Dkk0 Lk0 k00 b k
k
00
4
1 X h T Ti dk dk 60 v k i Xh d Tk d Tk
kk 0 k
h
00
i : d k00 d k00 ;
5
where superscript T denotes matrix transposition, the summations are over all Z molecules constituting the unit cell of volume v, Lkk0 are Lorentzfactor tensors, i.e. dipole lattice sums which give the ®eld at a molecule at site k due to the dipole moments of all molecules at the translationally invariant sites k 0 [19,20]. They are calculated using Ewald-summations, as described in [20,21]. Dkk0 and d k are local ®eld tensors connected with each other and with molecular properties by: iÿ1 X Xh dk Dkk 0 I ÿ L a= 0 v 0 : 6 k0
k0
kk
The bold-faced tensors L, I, a are of order 3Z, I is a generalized unit tensor with 3 3 subtensors 1dkk0 ,
361
while L and a have subtensors Lkk0 and ak dkk0 , respectively. Eqs. (2)±(6) assume that the molecules are treated like point dipoles and take no account of size and shape eects. It has been shown that using this approximation to calculate molecular polarizabilities from experimental ®rst-order susceptibilities leads to seriously wrong results for elongated molecules [22]. In a more realistic and yet easily tractable model [19,23] the molecules are divided into sets of sub-molecules, each treated as a point dipole. If it is assumed that every sub-molecule gives the same contribution to the susceptibilities, the point-dipole formalism can be retained and the quantities occurring in the foregoing equations are appropriate averages over these sub-molecules [8]. In the dipole approximation, nonpolar molecules do not create electric ®elds in the absence of a polarizing external ®eld and the molecular properties a, b, and c appearing in Eqs. (3)±(6) are therefore those of the free molecule, if the geometry in the crystalline phase and that of the free molecule do not dier considerably and eects like orbital con®nement and valence compression are neglected. In a crystal composed of polar molecules, the permanent dipoles lk create a permanent electric local ®eld F k0 at the site rk of a molecule and the molecular properties in Eqs. (3)±(6) are those in the presence of the additional ®eld. In the rigorous local ®eld theory, F k0 is given by [9]: F k0
1 X D L 00 lk00 : 0 v 0 00 kk 0 k0 k
7
kk
Here, lk00 is the permanent dipole moment at zero ®eld, while ak occurring in Dkk0 is the polarizability in the presence of the permanent crystal ®eld. The ®elds F k0 are of the order of several GV/m for polar molecules [24] and it has been shown for urea that its in¯uence on the (hyper)polarizabilities can be quite large [16]. The eects of the permanent crystal ®eld can be approximately calculated using the following series expansions for a and b with respect to the permanent local ®eld F k0 [16]:
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H. Reis et al. / Chemical Physics 261 (2000) 359±371
ak F k0 ak 0 b 0 F k0 12c 0 : F k0 F k0 ; 8 k
b F k0 b 0 c 0 F k0 ; k
k
k
9
k
where the quantities at zero ®eld are denoted by an argument 0. Eqs. (6)±(9) are repeated until selfconsistency is achieved. The ®eld eect on the second hyperpolarizability and thus on v 3 cannot be calculated in this approximation. Due to the rather semi-quantitative nature of the ad hoc distribution scheme, quantitative agreement with experimental data cannot be expected. Therefore, it is not justi®ed to calculate molecular properties including all possible eects at the highest levels possible, as long as the data used capture the essential features relevant for a successful prediction of the right order of magnitude and the correct ordering of the macroscopic susceptibilities. Consequently, several eects that can be expected to be of smaller importance, but whose inclusion would have led to a substantial increase of the computational cost, such as frequency dependence and vibrational contributions were ignored. Although it is known that static vibrational contributions can sometimes be of comparable magnitude as pure electronic contributions, all the experimental values with which we can compare were obtained using only optical frequencies. Therefore, only a usually small part of the static vibrational contribution, the contribution due to zero point vibrational averaging (ZPVA) would be nonnegligible, while the socalled pure vibrational contribution can be neglected [25]. Neglecting ZPVA introduces an maximum error of 10±20% [25] and we estimate an upper error limit of the same order to arise from the neglect of the frequency dependence, depending on the property considered and the frequency used. As in previous work, we want to compare the results of the rigorous local ®eld theory with the ALFFA, which is commonly used to calculate molecular (hyper)polarizabilities from experimental determined susceptibilities [1], although it lacks a sound theoretical foundation for crystal classes other than cubic. Here the local ®eld tensors d are given by:
h i. d 2I 3;
10
where is the relative permittivity tensor and I is the unit tensor. The principal components of at optical frequencies are the squares of the refractive indices ni , if the relative permeability can be approximated by unity. Usually, in applications of ALFFA, the refractive indices are taken from experiments where the principal values of the indicatrix are determined [26]. The molecular properties l, a, b and c were calculated in the framework of DFT via the ®nite ®eld method developed by Kurtz et al. [27] which was recently implemented in A L L CH E M [17]. The density functional program A L L CH E M [28] works with Gaussian basis sets and Gaussian-type auxiliary functions in order to avoid the calculation of four-center Coulomb integrals [29,30]. An ecient integral algorithm for the calculation of three-center Coulomb integrals was applied [31] which exploits some special features of the auxiliary function used. For the calculations of the molecular electronic properties of naphthalene, anthracene and mNA the exchange-correlation contributions proposed by Vosko, Wilk and Nusair (VWN) [32] were used. The exchange-correlation potential was numerically integrated on an adaptive grid [33]. It is well known that a general characteristic required for basis sets to perform well for polarizability calculations is that they should contain diuse functions [34]. An economical strategy for constructing these kinds of basis sets is to augment valence basis sets of reasonable good quality with additional polarization functions [35±38]. We have chosen as valence basis a triple zeta basis set (TZVP) which was optimized for local DFT calculations [39]. The basis set was then augmented with ®eld-induced polarization (FIP) functions due to Zeiss et al. [36]. They derived the FIP function exponents from an analytic analysis of the ®eld-induced charges in hydrogen orbitals. The ®rst-order FIP functions are used for the calculation of a and b. For the calculation of c second-order FIP functions, which are f-type Gaussians for main-row elements, are additionally added to the basis set. In order to avoid the con-
H. Reis et al. / Chemical Physics 261 (2000) 359±371
Fig. 1. Molecular coordinate system used for anthracene and naphthalene.
tamination of the valence basis set with the diuse s- and p-type Gaussians of the FIP functions, spherical basis functions are used in all calculations. We have named the resulting basis sets TZVP-FIP1 and TZVP-FIP2 [17,40±42]. The molecular structures of naphthalene and anthracene were optimized. For the optimization the VWN functional [32] and a local density functional optimized double zeta basis set (DZVP) [39] was used. The optimized structure parameters of naphthalene and anthracene, given in Ref. [17], are in good agreement with the published crystal structure parameters [43,44]. The molecular coordinate systems used for naphthalene and anthracene are shown in Fig. 1. For the calculation of the Lorentz tensors in Eqs. (5) and (6) a knowledge of the crystal structure is required. They were taken from Ref. [43] for naphthalene and from Ref. [44] for anthracene. Both molecules crystallize in the monoclinic space group P21 =a with Z 2. For naphthalene the unit cell geometry parameters are, at T 295 K: b 6:003 A, c 8:658 A, and a 8:235 A, b 122:92° and for anthracene, at T 290 K: b 6:038 A, c 11:184 A, and a 8:562 A, b 124:7°. In order to explore the eect of the spatial distribution of the (hyper)polarizabilities on the calculated susceptibilities, three distribution schemes were used: in the ®rst scheme, abbreviated RLFT1, every molecule was represented by a point in the center of mass (point dipole model), in the second, abbreviated by RLFT2 for naphthalene and RLFT3 for anthracene, one point submolecule was placed at the center of each ring, and in the third scheme one submolecule was placed at each C atom, (RLFT10 for naphthalene and RLFT14 for anthracene). For the calculation of mNA we used the geometry from the experimentally determined crystal
363
Fig. 2. Molecular coordinate system used for mNA.
structure of Ref. [45]. For this system we have also calculated static (hyper)polarizabilities at the SCF and MP2 level with GAMESS [46] using a 6-31++G** basis set employing the ®nite ®eld technique. The corresponding molecular coordinate system is shown in Fig. 2. It has been chosen so as to diagonalize the polarizability tensor at the MP2/6-31++G** level. The nondiagonal components of the polarizability tensor calculated with DFT in this coordinate system are negligible. The crystal structure of mNA was taken from Ref. [45]. The molecule crystallizes in the orthorhombic space group Pbc21 with Z 4 and with b 19:330 A the unit cell parameters a 6:501 A, Three distribution models have and c 5:082 A. been used for the calculation of macroscopic susceptibilities: the point dipole model RLFT1, a second model with three submolecules, one at the center of the nitro group, one at the nitrogen atom of the amino group, and one in the center of the phenyl ring (RLFT3) and ®nally a model with one submolecule on each C, N, and O atom, abbreviated by RLFT10. 3. Results and discussion 3.1. Naphthalene and anthracene Table 1 presents our calculated static molecular (hyper)polarizabilities of naphthalene and anthracene. Deviations of nondiagonal components of c from Kleinman symmetry due to numerical inaccuracies of the ®nite ®eld method have been averaged. The reported mean polarizabilities a and second hyperpolarizabilities c in Table 1 are calculated as: ÿ 11 a 13 axx ayy azz ;
364
H. Reis et al. / Chemical Physics 261 (2000) 359±371
Table 1 Calculated molecular polarizabilities aii =10ÿ40 (C2 m2 Jÿ1 ) and second hyperpolarizabilities ciijj =10ÿ64 (C4 m4 Jÿ3 ) of naphthalene and anthracene Naphthalene
Anthracene
axx ayy azz
28.59 21.03 10.87
49.98 28.30 14.26
a
20.17
30.85
cxxxx cyyyy czzzz cxxyy cxxzz cyyzz
41 600 24 280 10 150 12 660 8770 6350
129 740 26 120 13 910 15 630 17 890 7530
c
26 320
50 380
c
1X c : 5 i;j iijj
12
Comparison of the static a and c with available experimental data showed that a is about 10% too large, while the computed c is about 30% too small [17]. It should be noted, however, that the experimental values were determined at ®nite frequencies and that these values from dierent NLO processes can dier as much as 40%. In Table 2 the calculated ®rst-order susceptibility for dierent local ®eld models are compared with experimental results determined at k 546 nm. For the ALFFA model, the local ®eld factors dij were calculated from experimental refractive indices [26]. In the case of naphthalene and anthracene the axes of the indicatrix do not coincide with the crystal axes: the middle axis of the indicatrix is parallel to the b axis, while the angle between the small axis and the crystallographic a axis is 23.4° and 26.9° for naphthalene and anthracene, respectively [26]. 3 The values show clearly the inadequacy of the RLFT1 model, i.e. the point dipole approximation, for both naphthalene and anthracene: ®rst, the values strongly dier from the experimental ones, with some of the values for anthracene being 3 Due to a misprint the value for naphthalene given in Ref. [26] is 42.3°.
Table 2 Comparison of experimental [26] ®rst-order susceptibilities v 1 (at k 546 nm) of the naphthalene and anthracene crystals with those calculated for dierent partitioning schemes 1 vaa
vbb
1
vc c
1
vac
1
Naphthalene RLFT1 RLFT2 RLFT10 ALFFA
2.17 1.64 1.55 1.60
2.79 2.39 2.01 2.04
1.65 1.99 2.31 2.39
ÿ0.73 ÿ0.50 ÿ0.49 ÿ0.54
Experimental
1.56
1.97
2.55
ÿ0.53
Anthracene RLFT1 RLFT3 RLFT14 ALFFA
4.94 1.94 1.87 1.86
4.25 2.60 2.14 2.38
1.81 2.41 2.84 3.66
ÿ1.85 ÿ0.83 ÿ0.90 ÿ1.39
Experimental
1.72
2.19
2.54
ÿ0.58
very large, and further they also show a wrong order in magnitude compared to the experimental 1 1 1 results: RLFT1 gives vaa > vbb > vcc for anthracene, while the order of the experimental values is 1 1 1 > vbb > vaa , and for naphthalene the order is vcc 1 1 1 1 1 1 > vcc for RLFT1 and vcc > vbb > vaa vbb > vaa for the experimental values. Distributing the polarizability over the number of rings yields values for the components which are much more in line with the experimental ones, but the diagonal values are still partially wrongly ordered in both cases: RLFT2 for naphthalene gives 1 1 1 > vaa , and RLFT3 for anthracene yields vbb > vcc 1 1 1 . vbb > vcc > vaa Similar results were reported some years ago by Bounds and Munn [22], who calculated the molecular in-crystal polarizabilities of naphthalene and anthracene from the measured ®rst-order susceptibility, using both the point dipole approximation and the RLFT2 and RLFT3 schemes for naphthalene and anthracene, respectively. The set of equations giving the polarizabilities aij as a 1 function of the susceptibilities vij is underdetermined and consequently only lower and upper limits for aij can be calculated. Nevertheless, unacceptably large anisotropies for the polarizabilities were found for the RLFT1 model, while physically more realistic values were found using the more distributed schemes.
H. Reis et al. / Chemical Physics 261 (2000) 359±371
365
RLFT14, especially with respect to the components involving the c crystal axis. In Table 3 the values for the third-order susceptibility v 3 for the naphthalene and anthra-
From the models that use only computed input values, the correct order of the components is only achieved in the most distributed models RLFT10 and RLFT14 for naphthalene and anthracene, respectively. Both models yield further values that are surprisingly close to the experimental results, considering the various approximations used in the theoretical model. In the case of nonpolar molecules as naphthalene and anthracene one may expect that the actual distributions of the polarizability densities over a molecule embedded in the crystal do not dier very much from those represented by the RLFT10/RLFT14 schemes, provided that the electric ®elds caused by multipole moments higher than dipole are not strong enough to change the free molecular polarizabilities considerably. For naphthalene, the semiempirical ALFFA yields results that are comparable with those of the RLFT10 model. It should be kept in mind that in the case of ALFFA the local ®eld tensor d is calculated from the experimental refractive indices, thereby inherently incorporating the frequency dependence in d, while in the case of the RLFT schemes, d is calculated from the static polarizabilities, which would introduce an additional source of discrepancy when compared with frequency dependent experimental data. In the case of anthracene the ALFFA model yields results which are considerably worse than those of
cene crystal, respectively, are shown, calculated from Eq. (5). As the ®rst hyperpolarizability b is zero for centrosymmetric molecules, v 3 v 3 direct
for naphthalene and anthracene. The prediction of the ALFFA model has not been calculated here as, to our knowledge, the dispersion of the refractive indices for the naphthalene and anthracene crystal has not been published yet and it would therefore not be possible to compare the results of the RLFT schemes with the predictions of the ALFFA model in the way it is generally used. The trends observed in v 1 concerning the in¯uence of the dierent distribution schemes on the components is repeated and enhanced for v 3 : the RLFT1 scheme gives very large values for the components involving both the a and b axis compared to the distributed schemes. The order in magnitude of the diagonal components is again changed with increasing spatial distribution of the (hyper)polarizabilities: for naphthalene we have 3
3
3 > vc c c c for RLFT1 and RLFT2 and vbbbb > vaaaa 3 3 3 for the RLFT10 model, vc c c c > vbbbb > vaaaa 3
3
while for anthracene it is v 3 aaaa > vbbbb > vc c c c ,
Table 3 Calculated third-order susceptibilities v 3 =10ÿ24 (m2 Vÿ2 ) of the naphthalene and anthracene crystals for dierent partitioning schemes 3 vaaaa
3
vbbbb
vc c c c
3
vaabb
3
vaac c
3
Naphthalene
RLFT1 RLFT2 RLFT10
4880 1630 1320
6890 3690 1920
460 880 1780
1940 820 520
770 430 500
Anthracene
RLFT1 RLFT3 RLFT14
12 450 800 1070
96 620 2350 2030
29 450 4160 1950
2150 1770 3900
16 170 920 570
3
vbbc c
3
vaaac
3
vac c c
3
vabbc
Naphthalene
RLFT1 RLFT2 RLFT10
670 640 620
ÿ1420 ÿ340 ÿ280
ÿ390 ÿ170 ÿ210
ÿ620 ÿ200 ÿ140
Anthracene
RLFT1 RLFT3 RLFT14
2280 750 770
ÿ33 240 ÿ770 ÿ810
ÿ4960 ÿ620 ÿ1110
ÿ5300 ÿ230 ÿ200
366 3
H. Reis et al. / Chemical Physics 261 (2000) 359±371 3
3
3
3 vbbbb > vaaaa > vc c c c , and vc c c c > v 3 aaaa > vbbbb ,
for the RLFT1, RLFT2, and RLFT10 model, respectively. Unfortunately, we are not aware of any published experimental values for v 3 with which
Table 4 Calculated molecular dipole moments li =10ÿ30 (C m), polarizabilities aii =10ÿ40 (C2 m2 Jÿ1 ), ®rst bijj =10ÿ52 (C3 m3 Jÿ2 ) and second hyperpolarizabilities ciijj =10ÿ64 (C4 m4 Jÿ3 ) of mNA
we could compare our calculated results. The results for naphthalene and anthracene given here may be compared with those of benzene given in Ref. [15]. In this case the point dipole model RLFT1 and the partitioned scheme RLFT6, where the (hyper)polarizabilities were distributed over the C atoms did not yield mark 1 3 edly dierent values for both the vii and viijj components, while the results of the ALFFA 1 model were similar to the RLFT models for vii , but gave a dierent order for the diagonal com 3 ponents viiii . This shows that crystalline benzene is a rather small molecule in comparison with the intermolecular distances in the crystal and may well be described by the point dipole approximation, in contrast with the results for naphthalene and anthracene found here. 3.2. meta-Nitroaniline Table 4 shows the static molecular (hyper)polarizabilities of mNA obtained from DFT calculations with the TZVP-FIP basis sets and from SCF and MP2 calculations with the 6-31++G** basis set. The reported mean ®rst hyperpolarizability b was calculated as: b
3 X bijj li : 5 i;j jlj
13
Comparison of the SCF and MP2 calculations with the 6-31++G** basis set shows that the eect of electronic correlation is substantial for mNA: while l and a change only moderately, the values of b and c increase by about 80% and 50%, respectively. The DFT calculations yield similar values, although generally larger in absolute magnitude compared to MP2/6-31++G** for the components li , aii and bijj . For the components ciijj , on the other hand, the DFT calculation yields considerably larger values than the MP2 with the 6-31++G** basis set. The reason for this eect is the use of second-order f-type FIP functions
SCFa
MP2a
DFTb
lx ly lz
ÿ20.44 ÿ6.80 ÿ0.47
ÿ18.57 ÿ4.84 ÿ0.48
ÿ20.43 ÿ4.99 ÿ0.46
jlj
21.54
19.19
21.03
axx ayy azz
18.61 17.24 8.06
20.36 17.60 8.39
23.00 18.91 9.12
a
14.64
15.44
17.01
bxxx byyy bzzz bxxy bxxz bxyy bxzz byyz byzz
ÿ94.5 34.5 ÿ2.5 ÿ20.0 ÿ2.0 ÿ53.3 9.7 ÿ2.3 14.1
ÿ174.5 5.4 ÿ1.7 ÿ7.4 ÿ2.9 ÿ70.6 16.5 ÿ2.3 14.3
ÿ170.7 10.6 0.0 ÿ71.5 ÿ2.0 ÿ77.9 18.0 ÿ2.7 6.1
b
73.3
130.9
142.2
cxxxx cyyyy czzzz cxxyy cxxzz cyyzz
11 800 5400 2700 6300 2400 6100
18 300 16 100 7600 8100 3800 3300
52 900 19 800 9200 10 400 5800 6300
c
9900
14 500
25 300
a
6-31++G** basis. b TZVP-FIP basis.
(TZVP-FIP2 basis set) in the DFT calculation, which are absent in the 6-31++G** basis set. It was already shown [17] that these functions considerably improve the quality of calculated c tensors. Table 5 shows the calculated values of the permanent electric dipole local ®eld for the dierent distribution models. The components of the ®elds refer to the coordinate system of the reference molecule in the unit cell. In the case of RLFT3 and RLFT10 the ®eld was averaged over the number of submolecules. In the case of RLFT1 with the electric properties calculated at the DFT level the procedure did not converge, due to very large changes of the ®eld-dependent polarizability with every step. These in turn were caused by very large
H. Reis et al. / Chemical Physics 261 (2000) 359±371 Table 5 Average permanent electric local ®eld F (GV mÿ1 ) at the mNA reference molecule (in the crystal reference system) Fa MP2/6-31++G** RLFT1 2.33 RLFT3 2.81 RLFT10 2.39 DFT RLFT1 RLFT3 RLFT10
Fb
Fc
1.31 ÿ0.55 0.02
9.84 5.95 2.93
No convergence 3.55 ÿ0.60 2.85 ÿ0.01
7.94 3.41
Table 6 The molecular properties l=10ÿ30 (C m), a=10ÿ40 (C2 m2 Jÿ1 ), and b=10ÿ52 (C3 m3 Jÿ2 ) of mNA in presence of the permanent local ®eld calculated for the RLFT10 partitioning model, and their changes D 100A F ÿ A 0=A 0 compared to the quantities without permanent local ®eld l a b
values of the electric ®eld, which invalidated the truncation of the series in Eqs. (8) and (9) after the term with the second hyperpolarizability. The effect of the spatial distribution of the molecular response on the permanent electric dipole ®eld is substantial and similar for both sets of ab initio input values: the component Fb virtually vanishes, Fc strongly decreases, and Fa is nearly unaected by increasing the spatial distribution. Table 6 shows the values of the molecular properties l F k0 , a F k0 and b F k0 , using the ®eld calculated for the RLFT10 model. The quantities l and b are strongly enhanced by the permanent local ®eld, while a is only weakly aected. Similar results have been found for the urea crystal [16], where the permanent local ®eld even led to a re-
367
MP2
D (%)
DFT
D (%)
26.74 15.84 197.5
39 2.5 51
31.28 17.65 327.9
49 3.8 131
versal of sign for the quantity b. In this case, a weak ®eld eect on the second hyperpolarizability c has been reported, too. As mentioned in Section 2 the ®eld eect on c cannot be calculated by the approximate method employed here. The calculated macroscopic ®rst-order suscep 1 tibilities vii of mNA for the dierent partitioning schemes are given in Table 7, both with and without taking into consideration the permanent local ®eld, and are compared with experimental values [47,48] at wavelengths k P 1064 nm. Again, the smaller partitioning models RLFT1 and RLFT3 give the wrong ordering of magnitude for the components, for both ab initio levels and irrespective of the permanent local ®eld. Generally, the permanent local ®eld leads just to a small to 1 moderate increase of the values of vii .
Table 7 1 Comparison of the calculated ®rst macroscopic susceptibility vii of mNA, obtained using dierent partition models with and without the respective permanent local ®eld jF 0 j (GV mÿ1 ) with experimental values MP2/6-31++G** RLFT1 RLFT1 RLFT3 RLFT3 RLFT10 RLFT10 ALFFA k (nm) 1540 1064 1064
DFT/TZVP-FIP 1
jF 0 j
v 1 aa
vbb
v 1 cc
jF 0 j
0 10.2 0 6.6 0 3.8 ±
1.55 1.78 1.67 1.82 1.69 1.77 1.98
1.78 1.91 1.47 1.52 1.64 1.68 1.72
1.95 2.32 1.72 1.88 1.41 1.46 1.53
0 1.85 No convergence 0 1.99 8.7 2.40 0 1.99 4.4 2.15 ± 2.22
Experiment 1
v 1 aa
vbb
v 1 cc
Reference
1.89 1.95 1.96
1.78 1.82 1.83
1.61 1.66 1.66
[47] [47] [48]
For ALFFA, the experimental refractive indices at k 1540 nm [47] have been used.
v 1 aa
1
vbb
v 1 cc
2.01
2.48
1.61 1.71 1.82 1.86 1.83
2.10 2.68 1.66 1.79 1.70
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H. Reis et al. / Chemical Physics 261 (2000) 359±371
Only the RLFT10 scheme is able to reproduce the right ordering for the components and gives values which do not dier very much from the experimental values. The properties calculated from the molecular DFT polarizabilities are slightly too large compared with the experimental values, even without the permanent local ®eld. This is probably due to a slight overestimate of the molecular polarizability for mNA in the DFT calculation using the local VWN functional. The 1 vii calculated from the MP2 polarizabilities with the 6-31++G** basis set are slightly too small, which may be due to an underestimate of the permanent local ®eld, caused at least partially by too small values for the second hyperpolarizability tensor c. In conclusion both approaches, DFT and MP2, in combination with the RLFT10 scheme show good agreement with the experimental results for the ®rst order susceptibility. In Table 8 we present the calculated compo 2 nents of the static second-order susceptibility viij and for comparison some available experimental values [48±50], using the second harmonic generation (SHG) technique. The most signi®cant fea 2
2 vcbb ture of the experimental results, that is, vcaa 2
2 and vccc vcbb is reproduced by nearly all calcu-
lated sets of data. Comparison of static calculated values with frequency dependent experimental data is more problematic for v 2 than for v 1 , as the dependence of the ®rst hyperpolarizability b ÿx; x1 ; x2 on the frequencies involved is more complex than in the case of a ÿx; x (see e.g. the sum-over-states expressions for a and b in Ref. [51]). Moreover, in all the experimental results of v ÿx; x1 ; x2 cited at least one frequency involved is signi®cantly lower than in the case of v 1 , which may enhance the frequency dependent contribution to v 2 considerably. A small but measurable optical absorption of 4±6 cmÿ1 has been reported for the mNA crystal at k 532 nm [47], which corresponds to one of the frequencies involved in the case of SHG at k 1064 nm. The importance of the frequency dependent contribution to v 2 ÿ2x; x; x for SHG at k 2pc=x 1064 nm can be seen by comparing the results of Carenco et al. [48] at k 1319 and 1064 nm, reproduced in Table 8. Therefore, no quantitative agreement between the calculated static values and the experimental SHG values at k 1064 nm can be expected. On the other hand, comparing the calculated values with the less frequency dependent experimental SHG measurement at k 1319 nm
Table 8 2 Comparison of the second macroscopic susceptibility components viij =10ÿ12 (m Vÿ1 ) of mNA, calculated using dierent partition models with and without the respective permanent local ®eld jF 0 j (GV mÿ1 ) with experimental values from SHG MP2/6-31++G** RLFT1 RLFT1 RLFT3 RLFT3 RLFT10 RLFT10 ALFFAa
jF 0 j
2 vcbb
v 2 ccc
jF 0 j
0 10.2 0 6.6 0 3.8 ±
11 29 9 18 7 10 10
5 15 3 5 2 3 2
10 62 6 26 3 8 5
0 20 No convergence 0 16 8.7 65 0 11 4.4 22 ± 13
k (nm) SHG SHG SHG SHG a
DFT/TZVP-FIP
v 2 caa
Experiment 1319 1064 1064 1064
2
v 2 caa
vcbb
v 2 ccc
Reference
27 6 39 4 30 6 66 10
± 31 31 <1
25 5 41 4 35 7 29 4
[48] [48] [49] [50]
k 1319 nm, using the experimental refractive indices of Ref. [48].
v 2 caa
2
vcbb
v 2 ccc
6
28
2 14 0 4 0
17 115 7 21 9
H. Reis et al. / Chemical Physics 261 (2000) 359±371
we see that the RLFT10 scheme, using the DFT molecular properties with permanent local ®eld, is able to reproduce the experimental values nearly quantitatively, together with a reasonably small 2 value for vcbb . It should be stressed that the RLFT10 scheme in combination with the DFT (hyper)polarizabilities is able to reproduce the experimental SHG values only if the permanent local ®eld is taken into account properly. The failure of the RLFT10 scheme, using the 6-31++G** basis set input values is due to a strong underestimate of the permanent local ®eld eect on the ®rst hyperpolarizability b. This underlines the importance of second-order FIP functions for the calculation of the c tensor and thus for b F k0 in Eq. (9). Turning now to the predictions of ALFFA, we see from Table 8 that the Lorentz approximation fails to reproduce the experimental values and that its results are very close to those of the RLFT10 scheme without permanent local ®eld. Indeed the incapability to reproduce the permanent local ®eld eect is the main reason for the dierences between ALFFA and RLFT10: if the molecular properties given in Table 6, calculated at the DFT level and including the ®eld eect are used to calculate v 2 for ALFFA, instead of those of 2
2 2 ; vcbb ; vccc the ®eld-free molecule, we ®nd vcaa ÿ12 ÿ1 mV , values that are quite close 25; 3; 23 10 to those found for RLFT10 with ®eld. The Lorentz ®eld factor has originally been calculated for a cubic arrangement of molecules, with the molecule of interest in the center of a macroscopically small, but microscopically large sphere, embedded in a dielectric continuum. The ®elds due to the surrounding molecules in the sphere sum up to zero at the center of the sphere, and therefore only the macroscopic ®eld and the ®eld due to the polarisation on the surface of the sphere contribute to the local ®eld [52]. ALFFA is an ad hoc generalisation of this special case and is therefore inherently incapable to take the permanent local ®eld into account. The ®ndings reported here for crystals are in agreement with a theoretical investigation of the liquid state by Wortmann and Bishop [53]. They showed that the Lorentz model
369
can be regarded as a special case of the more general Onsager model for the ®rst-order susceptibility and an approximation to the second-order susceptibility, but in terms of what has been called ``solute'' molecular properties by the authors, that is, properties in the presence of the static reaction ®eld due to the permanent dipole moments of the surrounding molecules. But this reaction ®eld occurs only in the Onsager description, and the Lorentz model is unable to take it into account, similar as it is the case here for the permanent local ®eld in the crystal. Wortmann and Bishop note further that the Lorentz model applies erroneously an additional cavity ®eld factor for the secondorder susceptibility v 2 , which leads to values for v 2 that are roughly about 20% too large compared to the Onsager model. This and the partial incorporation of the frequency dependence by using experimental refractive indices for the calculation of the local ®eld factors d, may be the main reasons for the larger values found here for ALFFA compared to the RLFT10 scheme without the permanent local ®eld, if calculated with the same set of molecular properties. The good performance of the RLFT10 model for the ®rst-order and, if the permanent ®eld eect is properly taken into account, also for the secondorder susceptibilities indicates that, as in the case of the hydrocarbons, the simple distribution scheme used here does not dier dramatically from the actual polarizability and ®rst-order hyperpolarizability densities of the mNA molecule in the crystal. In meta-substituted benzene derivatives there are no large donor±acceptor charge transfer contributions to b [54]. It appears from our results that for such molecules the eect of distributing the molecular response functions homogeneously over the space occupied by the molecules is more important for the calculation of the susceptibilities than a realistic description of the spatial anisotropy of the (hyper)polarizability densities. It would be very interesting to investigate if this situation changes for molecules where large donor± acceptor charge transfer contributions to b lead to more strongly anisotropic (hyper)polarizability distributions, as for example the para-nitroaniline derivatives.
370
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4. Conclusions The macroscopic linear and nonlinear dipole susceptibilities of the naphthalene, anthracene and mNA crystals have been calculated in this work, using as far as possible theoretically determined input values. The molecular (hyper)polarizabilities have been calculated at the DFT level, using the TZVP-FIP1 and TZVP-FIP2 basis sets which are especially designed for calculations of these properties. For mNA, the permanent electric ®elds caused by the surrounding dipoles in the crystal have been taken into account in a self-consistent manner. In contrast to calculations on urea [16] and benzene [15] using the same distribution method, the spatial partitioning of the molecular response has been found to have a crucial in¯uence on the calculated susceptibilities. Distributing the properties over all heavy atoms yield susceptibilities that are in very good agreement with available experimental values. Such a good performance is rather surprising, considering the simple approach of distributing the molecular response equally over the submolecules. This indicates that a realistic description of the spatial anisotropy of these functions is not essential for a reliable calculation of the crystal susceptibilities, at least in the case of molecules without large donor±acceptor charge transfer contributions. Estimates of the susceptibilities based on the ALFFA were found to be in better agreement with experiment than those based on the less distributed schemes. However, the larger distribution models are generally in better agreement with experiment than the Lorentz approach, especially in the case 2 of viij of mNA. In this case, the Lorentz approach neglects completely the large eect of the permanent local ®eld on the ®rst hyperpolarizability of the free molecule. If the Lorentz approach were used in the usual way, i.e. to extract molecular hyperpolarizabilities from experimental susceptibilities of the mNA crystal, the resulting ®rst hyperpolarizabilities would therefore be approximately those of the free molecule in the presence of the permanent electrical local ®eld, rather than those of the isolated molecule. This is in line with the analysis of local ®eld corrections in liquids in
the framework of the Onsager reaction model by Wortmann and Bishop [53], who showed that the static reaction ®eld associated with the permanent dipole moments is the main reason for large dierences between quantum-mechanically calculated (hyper)polarizabilities of isolated molecules and the solute (hyper)polarizabilities derived from experiments by applying local ®eld corrections. The results reported here and those for the urea [16] and benzene [15] crystals suggest that for molecular crystals the permanent local ®eld eect on the linear polarizability is generally small and may be neglected, but its eect on the molecular dipole moment and ®rst hyperpolarizability can be very large and must be taken into account. It should be mentioned that the good agreement of the calculated ®rst-order susceptibility with experimental data for the hydrocarbon crystals does not exclude the possibility of large electric ®elds in these crystals, which may arise from multipole moments higher than dipole. As the ®rst hyperpolarizability is zero for nonpolar molecules, any ®elds would have to be very large in order to change the polarizabilities signi®cantly. Together with the results reported for the benzene [15] and urea [16] crystals, this work shows that the rigorous local ®eld approach is able to predict reliably the macroscopic optical susceptibilities of molecular crystals, nearly without recourse to experimental information. The presented combination of the local ®eld approach with density functional theory opens the possibility for a rational design of novel NLO materials.
Acknowledgements We acknowledge ®nancial support from the European Commission in the form of a TMR Network Grant (contract no. ERBFMRXCT960047).
References [1] D.S. Chemla, J. Zyss (Eds.), Nonlinear Optical Properties of Organic Molecules and Crystals, vols. 1 and 2, Academic Press, London, 1987.
H. Reis et al. / Chemical Physics 261 (2000) 359±371 [2] R.H.C. Janssen, J.-M. Bomont, D.N. Theodorou, S. Raptis, M.G. Papadopoulos, J. Chem. Phys. 110 (1999) 6463. [3] X.-L. Zhu, X.-Z. You, Y. Zhong, Z. Lu, S.-L. Guo, Chem. Phys. 253 (2000) 241. [4] T. Hamada, J. Chem. Soc., Faraday Trans. 92 (1996) 3165. [5] M. Hurst, R.W. Munn, J. Mol. Electron. 2 (1983) 35. [6] M. Hurst, R.W. Munn, J. Mol. Electron. 2 (1983) 43. [7] M. Hurst, R.W. Munn, J. Mol. Electron. 2 (1986) 139. [8] M. Hurst, R.W. Munn, J. Mol. Electron. 3 (1987) 75. [9] M. Hurst, R.W. Munn, in: R.A. Hann, D. Bloor (Eds.), Organic Materials for Non-linear Optics, Royal Society of Chemistry Special Publication no. 69, Royal Society of Chemistry, London, 1989, p. 3. [10] M. Hurst, R.W. Munn, J.A. Morley, J. Mol. Electron. 6 (1990) 15. [11] M. Hurst, R.W. Munn, in: A. Aviram (Ed.), Molecular Electronics ± Science and Technology, Engineering Foundation, New York, 1990, p. 267. [12] R.W. Munn, Int. J. Quant. Chem. 43 (1992) 159. [13] R.W. Munn, S.P.B. Smith, Adv. Mater. Opt. Electron. 1 (1992) 65. [14] M. Malagoli, R.W. Munn, J. Chem. Phys. 107 (1997) 7926. [15] H. Reis, S. Raptis, M.G. Papadopoulos, R.H.C. Janssen, D.N. Theodorou, R.W. Munn, Theor. Chem. Acc. 99 (1998) 384. [16] H. Reis, M.G. Papadopoulos, R.W. Munn, J. Chem. Phys. 109 (1998) 6828. [17] P. Calaminici, K. Jug, A.M. K oster, J. Chem. Phys. 109 (1998) 7756. [18] A. Willets, J.E. Rice, D.M. Burland, D.P. Shelton, J. Chem. Phys. 97 (1992) 7590. [19] R.W. Munn, Mol. Phys. 64 (1988) 1. [20] P.G. Cummins, D.A. Dunmur, R.W. Munn, R.J. Newham, Acta Cryst. A 32 (1976) 847. [21] H. Reis, M.G.P. Papadopoulos, C. H attig, J.G. Angy an, R.W. Munn, J. Chem. Phys. 112 (2000) 6161. [22] P.J. Bounds, R.W. Munn, Chem. Phys. 59 (1981) 47. [23] T. Luty, Chem. Phys. Lett. 44 (1976) 335. [24] R.W. Munn, M. Hurst, Chem. Phys. 147 (1990) 35. [25] D.M. Bishop, Adv. Chem. Phys. 104 (1998) 1. [26] J.W. Rohledder, R.W. Munn (Eds.), Magnetism and Optics of Molecular Crystals, Wiley, Chichester, 1992.
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[27] H.A. Kurtz, J.J.P. Stewart, K.M. Dieter, J. Comput. Chem. 11 (1990) 82. [28] A.M. K oster, M. Krack, M. Leboeuf, B. Zimmermann, A L L CH E M , Universit at Hannover, 1998. [29] B.I. Dunlap, J.W.D. Connolly, J.R. Sabin, J. Chem. Phys. 71 (1979) 4993. [30] J.W. Mintmire, B.I. Dunlap, Phys. Rev. A 25 (1982) 88. [31] A.M. K oster, J. Chem. Phys. 104 (1996) 4114. [32] S.H. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 58 (1980) 1200. [33] M. Krack, A.M. K oster, J. Chem. Phys. 108 (1998) 3226. [34] H.-J. Werner, W. Meyer, Mol. Phys. 31 (1976) 855. [35] A.J. Sadlej, Collection Czechoslovak Chem. Commun. 53 (1988) 1995. [36] G.D. Zeiss, W.R. Scott, N. Suzuki, D.P. Chong, S.R. Langho, Mol. Phys. 37 (1979) 1543. [37] M. Jaszu nski, B.O. Roos, Mol. Phys. 52 (1984) 1209. [38] B.O. Roos, A.J. Sadlej, Chem. Phys. 94 (1985) 43. [39] N. Godbout, D.R. Salahub, J. Andzelm, E. Wimmer, Can. J. Phys. 70 (1992) 560. [40] D.P. Chong, private communication. [41] P. Calaminici, K. Jug, A.M. K oster, J. Chem. Phys. 111 (1999) 4613. [43] D.W.J. Cruickshank, Acta Cryst. 10 (1957) 504. [44] R. Mason, Acta Cryst. 17 (1964) 547. [45] A.C. Skapski, J.L. Stevenson, J. Chem. Soc., Perkin Trans. 2 (1973) 1197. [46] M.W. Schmidt, K.K. Baldridge, J.A. Boatz, S.T. Elbert, M.S. Gordon, J.H. Jensen, S. Koseki, N. Matsunaga, K.A. Nguyen, S.J. Su, T.L. Windus, M. Dupuis, J.A. Montgomery, J. Comput. Chem. 14 (1993) 1347. [47] B.L. Davydov, V.F. Zolin, L.G. Korenewa, E.A. Lavrovsky, Opt. Spectrosc. (USSR) 39 (1975) 403. [48] A. Carenco, J. Jerphagnon, A. Perigaud, J. Chem. Phys. 66 (1977) 3806. [49] J.G. Bergman, G.R. Crane, J. Chem. Phys. 66 (1977) 3803. [50] P.D. Southgate, D.S. Hall, J. Appl. Phys. 43 (1972) 2765. [51] B.J. Orr, F.F. Ward, Mol. Phys. 20 (1971) 513. [52] C.J.F. Boettcher, Theory of Electric Polarization, vol. I, Dielectrics in Static Fields, second ed., Elsevier, Amsterdam, 1973. [53] R. Wortmann, D.M. Bishop, J. Chem. Phys. 108 (1998) 1001. [54] B.F. Levine, Chem. Phys. Lett. 37 (1976) 516.
www.elsevier.nl/locate/chemphys
Calculation of macroscopic linear and nonlinear optical susceptibilities for the naphthalene, anthracene and meta-nitroaniline crystals H. Reis a,1, M.G. Papadopoulos a,*, P. Calaminici b,2, K. Jug b, A.M. K oster b,2 a
Institute of Organic and Pharmaceutical Chemistry, National Hellenic Research Foundation, Vasileos Constantinou 48, GR-11635 Athens, Greece b Theoretische Chemie, Universit at Hannover, Am Kleinen Felde 30, 30167 Hannover, Germany Received 31 May 2000; in ®nal form 13 September 2000
Abstract The macroscopic ®rst- to third-order susceptibilities of naphthalene, anthracene and meta-nitroaniline (mNA) are calculated using a rigorous local ®eld approach. Molecular (hyper)polarizabilities used as input are determined by density functional theory calculations with specially designed basis sets and for mNA also by MP2 calculations with the 6-31++G** basis set. In the case of mNA, the permanent electric local ®eld due to the surrounding dipoles in the crystal is taken into account for the ®rst- and second-order susceptibility by a self-consistent approach. The molecular dipole moment and ®rst hyperpolarizability of mNA are drastically changed by the permanent local ®eld. In all cases the calculated ®rst-order susceptibility compares very favorably with experimental data, if the molecular response is distributed over all heavy atoms in the molecules. Similarly, the calculated second-order susceptibility for mNA is in good agreement with available experimental data, if the same distribution scheme is used and the permanent local ®eld is taken into account properly. This implies that accurate values for the molecular second-order hyperpolarizability c have to be available. The anisotropic Lorentz ®eld factor approximation yields results that are only slightly worse than the best ones of the rigorous local ®eld theory for the ®rst-order susceptibilities, but fails for the second-order susceptibility of mNA, due to its incapability to describe the large eect of the permanent local ®eld on the ®rst-order hyperpolarizability b. Ó 2000 Elsevier Science B.V. All rights reserved.
1. Introduction
*
Corresponding author. E-mail address: [email protected] (M.G. Papadopoulos). 1 Also corresponding author. Fax: +30-1727-3831. 2 Present address: Departamento de Quõmica, CINVESTAV, Centro de Investigaci on y de Estudios Avanzados del I.P.N., Av. Instituto Politecnico Nacional, 2508 A.P. 14-740 Mexico D.F. 07000, Mexico.
Calculations of electrical properties of organic molecules by the methods of theoretical chemistry have been proven to be a very useful tool in the search for new nonlinear optical (NLO) materials. However, most of these calculations have focused on the nonlinear responses of isolated molecules, while the great majority of NLO experiments are conducted on the condensed phase and primarily
0301-0104/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 0 ) 0 0 3 0 5 - 0
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H. Reis et al. / Chemical Physics 261 (2000) 359±371
determine the macroscopic optical susceptibilities, which are also the properties of interest for most applications. In order to compare experimental and calculated (hyper)polarizabilities, quantitative relationships between the molecular and macroscopic properties need to be established. In molecular materials composed of organic molecules, where the molecules retain essentially their identities, the anisotropic Lorentz ®eld factor approximation (ALFFA) is often used to extract molecular (hyper)polarizabilities from the macroscopic properties [1], although the lack of a theoretically sound basis limits the general application of this approach [2]. The Lorentz approximation has also been used to calculate the NLO susceptibilities of molecular crystals using the electric properties of whole unit cells calculated at the semiempirical INDO level [3], and of crystals composed of polyacetylene-like polymers, with the properties of the oligomers calculated at the coupled-perturbed Hartree±Fock level [4]. For molecular materials the macroscopic properties can be expressed in terms of the molecular properties, intermolecular interactions and the molecular packing in the bulk. For molecular crystals in particular, a theory has been developed [5,6] that rigorously calculates the local ®eld at a molecule in the crystal as a sum of the external ®elds and the ®elds arising from all the other molecules in the crystal and allows thereby to express the macroscopic linear and nonlinear susceptibilities in terms of the molecular (hyper)polarizabilities and the permanent electric ®eld in the crystal. In order to overcome known problems of the point multipole expansion concerning size and shape eects of larger, especially elongated molecules, a simple ad hoc partitioning scheme for the molecular electrical response has been proposed. In the ®rst applications of this theory [7±14] the use of semiempirical or low-level ab initio input parameters made it necessary to replace the theoretically calculated polarizabilities by eective polarizabilities calculated from experimental refractive indices in order to get reasonable agreement with measured results. Eects arising from the permanent crystal ®eld were not taken into account. In two recent studies of the benzene [15] and urea [16] crystal it was shown that accurate
ab initio polarizabilities are able to reproduce the experimental ®rst-order susceptibility accurately. In the case of urea it was further shown that the permanent dipolar local ®eld has a large eect on the molecular dipole moment and ®rst hyperpolarizability, although the in¯uence on the calculated second-order susceptibility is small, due to the symmetric arrangement of the molecules in the unit cell. Both in urea and benzene only small size extension eects on the calculated susceptibilities were found using the ad hoc distributing approach mentioned above. We present here a study of the macroscopic linear and nonlinear susceptibilities of the larger molecules naphthalene, anthracene and meta-nitroaniline (mNA), using molecular properties from density functional theory (DFT) calculations with basis sets especially designed for the calculation of molecular (hyper)polarizabilities [17]. In order to explore the suitability of more standard methods for the prediction of macroscopic susceptibilities we also use the molecular properties calculated at the MP2 level with the standard 6-31++G** basis set as input for the mNA susceptibility calculations. We further explore the eect of using different partitioning schemes of the molecular response on the predicted susceptibilities and in the case of mNA, we take into account eects of the permanent dipolar local ®eld. The results are compared with the predictions of the ALFFA and with available experimental susceptibilities.
2. Methods If a crystal is placed into an external electric ®eld E it develops a macroscopic polarization P , which can be expanded in powers of E as: . P =0 v 1 E v 2 : EE v 3 .. EEE ;
1
where v n are the macroscopic electric nth order susceptibilities. In a molecular crystal the molecules retain essentially their identities and the susceptibilities can be considered as arising from the molecular response, described by the molecular polarizability a and the ®rst and second hyperpo-
H. Reis et al. / Chemical Physics 261 (2000) 359±371
larizability b and c. The induced molecular dipole moment pk at a site rk is given by: . pk ak F k 1=2b : F k F k 1=6c .. F k F k F k ; k
k
2
where F k is the local electric ®eld at rk , which, in dipole approximation, is the sum of the macroscopic ®eld E and the ®eld arising from the surrounding induced dipoles. The Taylor expansion convention for the de®nition of the hyperpolarizabilities has been used in Eq. (2) [18]. A local ®eld theory which takes the dipolar interactions rigorously into account and connects the susceptibilities with molecular properties has been developed by Hurst and Munn [5,6]. With static ®elds, the resulting expressions are: X ÿ1 v 1 0 v ak d k ; 3 k
v 2
i 1 X .. h b . dkdkdk ; 20 v k k
v 3 v 3 direct
v 3
cascading
h i : c : dkdk k
1 2 0 v2
: b Dkk0 Lk0 k00 b k
k
00
4
1 X h T Ti dk dk 60 v k i Xh d Tk d Tk
kk 0 k
h
00
i : d k00 d k00 ;
5
where superscript T denotes matrix transposition, the summations are over all Z molecules constituting the unit cell of volume v, Lkk0 are Lorentzfactor tensors, i.e. dipole lattice sums which give the ®eld at a molecule at site k due to the dipole moments of all molecules at the translationally invariant sites k 0 [19,20]. They are calculated using Ewald-summations, as described in [20,21]. Dkk0 and d k are local ®eld tensors connected with each other and with molecular properties by: iÿ1 X Xh dk Dkk 0 I ÿ L a= 0 v 0 : 6 k0
k0
kk
The bold-faced tensors L, I, a are of order 3Z, I is a generalized unit tensor with 3 3 subtensors 1dkk0 ,
361
while L and a have subtensors Lkk0 and ak dkk0 , respectively. Eqs. (2)±(6) assume that the molecules are treated like point dipoles and take no account of size and shape eects. It has been shown that using this approximation to calculate molecular polarizabilities from experimental ®rst-order susceptibilities leads to seriously wrong results for elongated molecules [22]. In a more realistic and yet easily tractable model [19,23] the molecules are divided into sets of sub-molecules, each treated as a point dipole. If it is assumed that every sub-molecule gives the same contribution to the susceptibilities, the point-dipole formalism can be retained and the quantities occurring in the foregoing equations are appropriate averages over these sub-molecules [8]. In the dipole approximation, nonpolar molecules do not create electric ®elds in the absence of a polarizing external ®eld and the molecular properties a, b, and c appearing in Eqs. (3)±(6) are therefore those of the free molecule, if the geometry in the crystalline phase and that of the free molecule do not dier considerably and eects like orbital con®nement and valence compression are neglected. In a crystal composed of polar molecules, the permanent dipoles lk create a permanent electric local ®eld F k0 at the site rk of a molecule and the molecular properties in Eqs. (3)±(6) are those in the presence of the additional ®eld. In the rigorous local ®eld theory, F k0 is given by [9]: F k0
1 X D L 00 lk00 : 0 v 0 00 kk 0 k0 k
7
kk
Here, lk00 is the permanent dipole moment at zero ®eld, while ak occurring in Dkk0 is the polarizability in the presence of the permanent crystal ®eld. The ®elds F k0 are of the order of several GV/m for polar molecules [24] and it has been shown for urea that its in¯uence on the (hyper)polarizabilities can be quite large [16]. The eects of the permanent crystal ®eld can be approximately calculated using the following series expansions for a and b with respect to the permanent local ®eld F k0 [16]:
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H. Reis et al. / Chemical Physics 261 (2000) 359±371
ak F k0 ak 0 b 0 F k0 12c 0 : F k0 F k0 ; 8 k
b F k0 b 0 c 0 F k0 ; k
k
k
9
k
where the quantities at zero ®eld are denoted by an argument 0. Eqs. (6)±(9) are repeated until selfconsistency is achieved. The ®eld eect on the second hyperpolarizability and thus on v 3 cannot be calculated in this approximation. Due to the rather semi-quantitative nature of the ad hoc distribution scheme, quantitative agreement with experimental data cannot be expected. Therefore, it is not justi®ed to calculate molecular properties including all possible eects at the highest levels possible, as long as the data used capture the essential features relevant for a successful prediction of the right order of magnitude and the correct ordering of the macroscopic susceptibilities. Consequently, several eects that can be expected to be of smaller importance, but whose inclusion would have led to a substantial increase of the computational cost, such as frequency dependence and vibrational contributions were ignored. Although it is known that static vibrational contributions can sometimes be of comparable magnitude as pure electronic contributions, all the experimental values with which we can compare were obtained using only optical frequencies. Therefore, only a usually small part of the static vibrational contribution, the contribution due to zero point vibrational averaging (ZPVA) would be nonnegligible, while the socalled pure vibrational contribution can be neglected [25]. Neglecting ZPVA introduces an maximum error of 10±20% [25] and we estimate an upper error limit of the same order to arise from the neglect of the frequency dependence, depending on the property considered and the frequency used. As in previous work, we want to compare the results of the rigorous local ®eld theory with the ALFFA, which is commonly used to calculate molecular (hyper)polarizabilities from experimental determined susceptibilities [1], although it lacks a sound theoretical foundation for crystal classes other than cubic. Here the local ®eld tensors d are given by:
h i. d 2I 3;
10
where is the relative permittivity tensor and I is the unit tensor. The principal components of at optical frequencies are the squares of the refractive indices ni , if the relative permeability can be approximated by unity. Usually, in applications of ALFFA, the refractive indices are taken from experiments where the principal values of the indicatrix are determined [26]. The molecular properties l, a, b and c were calculated in the framework of DFT via the ®nite ®eld method developed by Kurtz et al. [27] which was recently implemented in A L L CH E M [17]. The density functional program A L L CH E M [28] works with Gaussian basis sets and Gaussian-type auxiliary functions in order to avoid the calculation of four-center Coulomb integrals [29,30]. An ecient integral algorithm for the calculation of three-center Coulomb integrals was applied [31] which exploits some special features of the auxiliary function used. For the calculations of the molecular electronic properties of naphthalene, anthracene and mNA the exchange-correlation contributions proposed by Vosko, Wilk and Nusair (VWN) [32] were used. The exchange-correlation potential was numerically integrated on an adaptive grid [33]. It is well known that a general characteristic required for basis sets to perform well for polarizability calculations is that they should contain diuse functions [34]. An economical strategy for constructing these kinds of basis sets is to augment valence basis sets of reasonable good quality with additional polarization functions [35±38]. We have chosen as valence basis a triple zeta basis set (TZVP) which was optimized for local DFT calculations [39]. The basis set was then augmented with ®eld-induced polarization (FIP) functions due to Zeiss et al. [36]. They derived the FIP function exponents from an analytic analysis of the ®eld-induced charges in hydrogen orbitals. The ®rst-order FIP functions are used for the calculation of a and b. For the calculation of c second-order FIP functions, which are f-type Gaussians for main-row elements, are additionally added to the basis set. In order to avoid the con-
H. Reis et al. / Chemical Physics 261 (2000) 359±371
Fig. 1. Molecular coordinate system used for anthracene and naphthalene.
tamination of the valence basis set with the diuse s- and p-type Gaussians of the FIP functions, spherical basis functions are used in all calculations. We have named the resulting basis sets TZVP-FIP1 and TZVP-FIP2 [17,40±42]. The molecular structures of naphthalene and anthracene were optimized. For the optimization the VWN functional [32] and a local density functional optimized double zeta basis set (DZVP) [39] was used. The optimized structure parameters of naphthalene and anthracene, given in Ref. [17], are in good agreement with the published crystal structure parameters [43,44]. The molecular coordinate systems used for naphthalene and anthracene are shown in Fig. 1. For the calculation of the Lorentz tensors in Eqs. (5) and (6) a knowledge of the crystal structure is required. They were taken from Ref. [43] for naphthalene and from Ref. [44] for anthracene. Both molecules crystallize in the monoclinic space group P21 =a with Z 2. For naphthalene the unit cell geometry parameters are, at T 295 K: b 6:003 A, c 8:658 A, and a 8:235 A, b 122:92° and for anthracene, at T 290 K: b 6:038 A, c 11:184 A, and a 8:562 A, b 124:7°. In order to explore the eect of the spatial distribution of the (hyper)polarizabilities on the calculated susceptibilities, three distribution schemes were used: in the ®rst scheme, abbreviated RLFT1, every molecule was represented by a point in the center of mass (point dipole model), in the second, abbreviated by RLFT2 for naphthalene and RLFT3 for anthracene, one point submolecule was placed at the center of each ring, and in the third scheme one submolecule was placed at each C atom, (RLFT10 for naphthalene and RLFT14 for anthracene). For the calculation of mNA we used the geometry from the experimentally determined crystal
363
Fig. 2. Molecular coordinate system used for mNA.
structure of Ref. [45]. For this system we have also calculated static (hyper)polarizabilities at the SCF and MP2 level with GAMESS [46] using a 6-31++G** basis set employing the ®nite ®eld technique. The corresponding molecular coordinate system is shown in Fig. 2. It has been chosen so as to diagonalize the polarizability tensor at the MP2/6-31++G** level. The nondiagonal components of the polarizability tensor calculated with DFT in this coordinate system are negligible. The crystal structure of mNA was taken from Ref. [45]. The molecule crystallizes in the orthorhombic space group Pbc21 with Z 4 and with b 19:330 A the unit cell parameters a 6:501 A, Three distribution models have and c 5:082 A. been used for the calculation of macroscopic susceptibilities: the point dipole model RLFT1, a second model with three submolecules, one at the center of the nitro group, one at the nitrogen atom of the amino group, and one in the center of the phenyl ring (RLFT3) and ®nally a model with one submolecule on each C, N, and O atom, abbreviated by RLFT10. 3. Results and discussion 3.1. Naphthalene and anthracene Table 1 presents our calculated static molecular (hyper)polarizabilities of naphthalene and anthracene. Deviations of nondiagonal components of c from Kleinman symmetry due to numerical inaccuracies of the ®nite ®eld method have been averaged. The reported mean polarizabilities a and second hyperpolarizabilities c in Table 1 are calculated as: ÿ 11 a 13 axx ayy azz ;
364
H. Reis et al. / Chemical Physics 261 (2000) 359±371
Table 1 Calculated molecular polarizabilities aii =10ÿ40 (C2 m2 Jÿ1 ) and second hyperpolarizabilities ciijj =10ÿ64 (C4 m4 Jÿ3 ) of naphthalene and anthracene Naphthalene
Anthracene
axx ayy azz
28.59 21.03 10.87
49.98 28.30 14.26
a
20.17
30.85
cxxxx cyyyy czzzz cxxyy cxxzz cyyzz
41 600 24 280 10 150 12 660 8770 6350
129 740 26 120 13 910 15 630 17 890 7530
c
26 320
50 380
c
1X c : 5 i;j iijj
12
Comparison of the static a and c with available experimental data showed that a is about 10% too large, while the computed c is about 30% too small [17]. It should be noted, however, that the experimental values were determined at ®nite frequencies and that these values from dierent NLO processes can dier as much as 40%. In Table 2 the calculated ®rst-order susceptibility for dierent local ®eld models are compared with experimental results determined at k 546 nm. For the ALFFA model, the local ®eld factors dij were calculated from experimental refractive indices [26]. In the case of naphthalene and anthracene the axes of the indicatrix do not coincide with the crystal axes: the middle axis of the indicatrix is parallel to the b axis, while the angle between the small axis and the crystallographic a axis is 23.4° and 26.9° for naphthalene and anthracene, respectively [26]. 3 The values show clearly the inadequacy of the RLFT1 model, i.e. the point dipole approximation, for both naphthalene and anthracene: ®rst, the values strongly dier from the experimental ones, with some of the values for anthracene being 3 Due to a misprint the value for naphthalene given in Ref. [26] is 42.3°.
Table 2 Comparison of experimental [26] ®rst-order susceptibilities v 1 (at k 546 nm) of the naphthalene and anthracene crystals with those calculated for dierent partitioning schemes 1 vaa
vbb
1
vc c
1
vac
1
Naphthalene RLFT1 RLFT2 RLFT10 ALFFA
2.17 1.64 1.55 1.60
2.79 2.39 2.01 2.04
1.65 1.99 2.31 2.39
ÿ0.73 ÿ0.50 ÿ0.49 ÿ0.54
Experimental
1.56
1.97
2.55
ÿ0.53
Anthracene RLFT1 RLFT3 RLFT14 ALFFA
4.94 1.94 1.87 1.86
4.25 2.60 2.14 2.38
1.81 2.41 2.84 3.66
ÿ1.85 ÿ0.83 ÿ0.90 ÿ1.39
Experimental
1.72
2.19
2.54
ÿ0.58
very large, and further they also show a wrong order in magnitude compared to the experimental 1 1 1 results: RLFT1 gives vaa > vbb > vcc for anthracene, while the order of the experimental values is 1 1 1 > vbb > vaa , and for naphthalene the order is vcc 1 1 1 1 1 1 > vcc for RLFT1 and vcc > vbb > vaa vbb > vaa for the experimental values. Distributing the polarizability over the number of rings yields values for the components which are much more in line with the experimental ones, but the diagonal values are still partially wrongly ordered in both cases: RLFT2 for naphthalene gives 1 1 1 > vaa , and RLFT3 for anthracene yields vbb > vcc 1 1 1 . vbb > vcc > vaa Similar results were reported some years ago by Bounds and Munn [22], who calculated the molecular in-crystal polarizabilities of naphthalene and anthracene from the measured ®rst-order susceptibility, using both the point dipole approximation and the RLFT2 and RLFT3 schemes for naphthalene and anthracene, respectively. The set of equations giving the polarizabilities aij as a 1 function of the susceptibilities vij is underdetermined and consequently only lower and upper limits for aij can be calculated. Nevertheless, unacceptably large anisotropies for the polarizabilities were found for the RLFT1 model, while physically more realistic values were found using the more distributed schemes.
H. Reis et al. / Chemical Physics 261 (2000) 359±371
365
RLFT14, especially with respect to the components involving the c crystal axis. In Table 3 the values for the third-order susceptibility v 3 for the naphthalene and anthra-
From the models that use only computed input values, the correct order of the components is only achieved in the most distributed models RLFT10 and RLFT14 for naphthalene and anthracene, respectively. Both models yield further values that are surprisingly close to the experimental results, considering the various approximations used in the theoretical model. In the case of nonpolar molecules as naphthalene and anthracene one may expect that the actual distributions of the polarizability densities over a molecule embedded in the crystal do not dier very much from those represented by the RLFT10/RLFT14 schemes, provided that the electric ®elds caused by multipole moments higher than dipole are not strong enough to change the free molecular polarizabilities considerably. For naphthalene, the semiempirical ALFFA yields results that are comparable with those of the RLFT10 model. It should be kept in mind that in the case of ALFFA the local ®eld tensor d is calculated from the experimental refractive indices, thereby inherently incorporating the frequency dependence in d, while in the case of the RLFT schemes, d is calculated from the static polarizabilities, which would introduce an additional source of discrepancy when compared with frequency dependent experimental data. In the case of anthracene the ALFFA model yields results which are considerably worse than those of
cene crystal, respectively, are shown, calculated from Eq. (5). As the ®rst hyperpolarizability b is zero for centrosymmetric molecules, v 3 v 3 direct
for naphthalene and anthracene. The prediction of the ALFFA model has not been calculated here as, to our knowledge, the dispersion of the refractive indices for the naphthalene and anthracene crystal has not been published yet and it would therefore not be possible to compare the results of the RLFT schemes with the predictions of the ALFFA model in the way it is generally used. The trends observed in v 1 concerning the in¯uence of the dierent distribution schemes on the components is repeated and enhanced for v 3 : the RLFT1 scheme gives very large values for the components involving both the a and b axis compared to the distributed schemes. The order in magnitude of the diagonal components is again changed with increasing spatial distribution of the (hyper)polarizabilities: for naphthalene we have 3
3
3 > vc c c c for RLFT1 and RLFT2 and vbbbb > vaaaa 3 3 3 for the RLFT10 model, vc c c c > vbbbb > vaaaa 3
3
while for anthracene it is v 3 aaaa > vbbbb > vc c c c ,
Table 3 Calculated third-order susceptibilities v 3 =10ÿ24 (m2 Vÿ2 ) of the naphthalene and anthracene crystals for dierent partitioning schemes 3 vaaaa
3
vbbbb
vc c c c
3
vaabb
3
vaac c
3
Naphthalene
RLFT1 RLFT2 RLFT10
4880 1630 1320
6890 3690 1920
460 880 1780
1940 820 520
770 430 500
Anthracene
RLFT1 RLFT3 RLFT14
12 450 800 1070
96 620 2350 2030
29 450 4160 1950
2150 1770 3900
16 170 920 570
3
vbbc c
3
vaaac
3
vac c c
3
vabbc
Naphthalene
RLFT1 RLFT2 RLFT10
670 640 620
ÿ1420 ÿ340 ÿ280
ÿ390 ÿ170 ÿ210
ÿ620 ÿ200 ÿ140
Anthracene
RLFT1 RLFT3 RLFT14
2280 750 770
ÿ33 240 ÿ770 ÿ810
ÿ4960 ÿ620 ÿ1110
ÿ5300 ÿ230 ÿ200
366 3
H. Reis et al. / Chemical Physics 261 (2000) 359±371 3
3
3
3 vbbbb > vaaaa > vc c c c , and vc c c c > v 3 aaaa > vbbbb ,
for the RLFT1, RLFT2, and RLFT10 model, respectively. Unfortunately, we are not aware of any published experimental values for v 3 with which
Table 4 Calculated molecular dipole moments li =10ÿ30 (C m), polarizabilities aii =10ÿ40 (C2 m2 Jÿ1 ), ®rst bijj =10ÿ52 (C3 m3 Jÿ2 ) and second hyperpolarizabilities ciijj =10ÿ64 (C4 m4 Jÿ3 ) of mNA
we could compare our calculated results. The results for naphthalene and anthracene given here may be compared with those of benzene given in Ref. [15]. In this case the point dipole model RLFT1 and the partitioned scheme RLFT6, where the (hyper)polarizabilities were distributed over the C atoms did not yield mark 1 3 edly dierent values for both the vii and viijj components, while the results of the ALFFA 1 model were similar to the RLFT models for vii , but gave a dierent order for the diagonal com 3 ponents viiii . This shows that crystalline benzene is a rather small molecule in comparison with the intermolecular distances in the crystal and may well be described by the point dipole approximation, in contrast with the results for naphthalene and anthracene found here. 3.2. meta-Nitroaniline Table 4 shows the static molecular (hyper)polarizabilities of mNA obtained from DFT calculations with the TZVP-FIP basis sets and from SCF and MP2 calculations with the 6-31++G** basis set. The reported mean ®rst hyperpolarizability b was calculated as: b
3 X bijj li : 5 i;j jlj
13
Comparison of the SCF and MP2 calculations with the 6-31++G** basis set shows that the eect of electronic correlation is substantial for mNA: while l and a change only moderately, the values of b and c increase by about 80% and 50%, respectively. The DFT calculations yield similar values, although generally larger in absolute magnitude compared to MP2/6-31++G** for the components li , aii and bijj . For the components ciijj , on the other hand, the DFT calculation yields considerably larger values than the MP2 with the 6-31++G** basis set. The reason for this eect is the use of second-order f-type FIP functions
SCFa
MP2a
DFTb
lx ly lz
ÿ20.44 ÿ6.80 ÿ0.47
ÿ18.57 ÿ4.84 ÿ0.48
ÿ20.43 ÿ4.99 ÿ0.46
jlj
21.54
19.19
21.03
axx ayy azz
18.61 17.24 8.06
20.36 17.60 8.39
23.00 18.91 9.12
a
14.64
15.44
17.01
bxxx byyy bzzz bxxy bxxz bxyy bxzz byyz byzz
ÿ94.5 34.5 ÿ2.5 ÿ20.0 ÿ2.0 ÿ53.3 9.7 ÿ2.3 14.1
ÿ174.5 5.4 ÿ1.7 ÿ7.4 ÿ2.9 ÿ70.6 16.5 ÿ2.3 14.3
ÿ170.7 10.6 0.0 ÿ71.5 ÿ2.0 ÿ77.9 18.0 ÿ2.7 6.1
b
73.3
130.9
142.2
cxxxx cyyyy czzzz cxxyy cxxzz cyyzz
11 800 5400 2700 6300 2400 6100
18 300 16 100 7600 8100 3800 3300
52 900 19 800 9200 10 400 5800 6300
c
9900
14 500
25 300
a
6-31++G** basis. b TZVP-FIP basis.
(TZVP-FIP2 basis set) in the DFT calculation, which are absent in the 6-31++G** basis set. It was already shown [17] that these functions considerably improve the quality of calculated c tensors. Table 5 shows the calculated values of the permanent electric dipole local ®eld for the dierent distribution models. The components of the ®elds refer to the coordinate system of the reference molecule in the unit cell. In the case of RLFT3 and RLFT10 the ®eld was averaged over the number of submolecules. In the case of RLFT1 with the electric properties calculated at the DFT level the procedure did not converge, due to very large changes of the ®eld-dependent polarizability with every step. These in turn were caused by very large
H. Reis et al. / Chemical Physics 261 (2000) 359±371 Table 5 Average permanent electric local ®eld F (GV mÿ1 ) at the mNA reference molecule (in the crystal reference system) Fa MP2/6-31++G** RLFT1 2.33 RLFT3 2.81 RLFT10 2.39 DFT RLFT1 RLFT3 RLFT10
Fb
Fc
1.31 ÿ0.55 0.02
9.84 5.95 2.93
No convergence 3.55 ÿ0.60 2.85 ÿ0.01
7.94 3.41
Table 6 The molecular properties l=10ÿ30 (C m), a=10ÿ40 (C2 m2 Jÿ1 ), and b=10ÿ52 (C3 m3 Jÿ2 ) of mNA in presence of the permanent local ®eld calculated for the RLFT10 partitioning model, and their changes D 100A F ÿ A 0=A 0 compared to the quantities without permanent local ®eld l a b
values of the electric ®eld, which invalidated the truncation of the series in Eqs. (8) and (9) after the term with the second hyperpolarizability. The effect of the spatial distribution of the molecular response on the permanent electric dipole ®eld is substantial and similar for both sets of ab initio input values: the component Fb virtually vanishes, Fc strongly decreases, and Fa is nearly unaected by increasing the spatial distribution. Table 6 shows the values of the molecular properties l F k0 , a F k0 and b F k0 , using the ®eld calculated for the RLFT10 model. The quantities l and b are strongly enhanced by the permanent local ®eld, while a is only weakly aected. Similar results have been found for the urea crystal [16], where the permanent local ®eld even led to a re-
367
MP2
D (%)
DFT
D (%)
26.74 15.84 197.5
39 2.5 51
31.28 17.65 327.9
49 3.8 131
versal of sign for the quantity b. In this case, a weak ®eld eect on the second hyperpolarizability c has been reported, too. As mentioned in Section 2 the ®eld eect on c cannot be calculated by the approximate method employed here. The calculated macroscopic ®rst-order suscep 1 tibilities vii of mNA for the dierent partitioning schemes are given in Table 7, both with and without taking into consideration the permanent local ®eld, and are compared with experimental values [47,48] at wavelengths k P 1064 nm. Again, the smaller partitioning models RLFT1 and RLFT3 give the wrong ordering of magnitude for the components, for both ab initio levels and irrespective of the permanent local ®eld. Generally, the permanent local ®eld leads just to a small to 1 moderate increase of the values of vii .
Table 7 1 Comparison of the calculated ®rst macroscopic susceptibility vii of mNA, obtained using dierent partition models with and without the respective permanent local ®eld jF 0 j (GV mÿ1 ) with experimental values MP2/6-31++G** RLFT1 RLFT1 RLFT3 RLFT3 RLFT10 RLFT10 ALFFA k (nm) 1540 1064 1064
DFT/TZVP-FIP 1
jF 0 j
v 1 aa
vbb
v 1 cc
jF 0 j
0 10.2 0 6.6 0 3.8 ±
1.55 1.78 1.67 1.82 1.69 1.77 1.98
1.78 1.91 1.47 1.52 1.64 1.68 1.72
1.95 2.32 1.72 1.88 1.41 1.46 1.53
0 1.85 No convergence 0 1.99 8.7 2.40 0 1.99 4.4 2.15 ± 2.22
Experiment 1
v 1 aa
vbb
v 1 cc
Reference
1.89 1.95 1.96
1.78 1.82 1.83
1.61 1.66 1.66
[47] [47] [48]
For ALFFA, the experimental refractive indices at k 1540 nm [47] have been used.
v 1 aa
1
vbb
v 1 cc
2.01
2.48
1.61 1.71 1.82 1.86 1.83
2.10 2.68 1.66 1.79 1.70
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H. Reis et al. / Chemical Physics 261 (2000) 359±371
Only the RLFT10 scheme is able to reproduce the right ordering for the components and gives values which do not dier very much from the experimental values. The properties calculated from the molecular DFT polarizabilities are slightly too large compared with the experimental values, even without the permanent local ®eld. This is probably due to a slight overestimate of the molecular polarizability for mNA in the DFT calculation using the local VWN functional. The 1 vii calculated from the MP2 polarizabilities with the 6-31++G** basis set are slightly too small, which may be due to an underestimate of the permanent local ®eld, caused at least partially by too small values for the second hyperpolarizability tensor c. In conclusion both approaches, DFT and MP2, in combination with the RLFT10 scheme show good agreement with the experimental results for the ®rst order susceptibility. In Table 8 we present the calculated compo 2 nents of the static second-order susceptibility viij and for comparison some available experimental values [48±50], using the second harmonic generation (SHG) technique. The most signi®cant fea 2
2 vcbb ture of the experimental results, that is, vcaa 2
2 and vccc vcbb is reproduced by nearly all calcu-
lated sets of data. Comparison of static calculated values with frequency dependent experimental data is more problematic for v 2 than for v 1 , as the dependence of the ®rst hyperpolarizability b ÿx; x1 ; x2 on the frequencies involved is more complex than in the case of a ÿx; x (see e.g. the sum-over-states expressions for a and b in Ref. [51]). Moreover, in all the experimental results of v ÿx; x1 ; x2 cited at least one frequency involved is signi®cantly lower than in the case of v 1 , which may enhance the frequency dependent contribution to v 2 considerably. A small but measurable optical absorption of 4±6 cmÿ1 has been reported for the mNA crystal at k 532 nm [47], which corresponds to one of the frequencies involved in the case of SHG at k 1064 nm. The importance of the frequency dependent contribution to v 2 ÿ2x; x; x for SHG at k 2pc=x 1064 nm can be seen by comparing the results of Carenco et al. [48] at k 1319 and 1064 nm, reproduced in Table 8. Therefore, no quantitative agreement between the calculated static values and the experimental SHG values at k 1064 nm can be expected. On the other hand, comparing the calculated values with the less frequency dependent experimental SHG measurement at k 1319 nm
Table 8 2 Comparison of the second macroscopic susceptibility components viij =10ÿ12 (m Vÿ1 ) of mNA, calculated using dierent partition models with and without the respective permanent local ®eld jF 0 j (GV mÿ1 ) with experimental values from SHG MP2/6-31++G** RLFT1 RLFT1 RLFT3 RLFT3 RLFT10 RLFT10 ALFFAa
jF 0 j
2 vcbb
v 2 ccc
jF 0 j
0 10.2 0 6.6 0 3.8 ±
11 29 9 18 7 10 10
5 15 3 5 2 3 2
10 62 6 26 3 8 5
0 20 No convergence 0 16 8.7 65 0 11 4.4 22 ± 13
k (nm) SHG SHG SHG SHG a
DFT/TZVP-FIP
v 2 caa
Experiment 1319 1064 1064 1064
2
v 2 caa
vcbb
v 2 ccc
Reference
27 6 39 4 30 6 66 10
± 31 31 <1
25 5 41 4 35 7 29 4
[48] [48] [49] [50]
k 1319 nm, using the experimental refractive indices of Ref. [48].
v 2 caa
2
vcbb
v 2 ccc
6
28
2 14 0 4 0
17 115 7 21 9
H. Reis et al. / Chemical Physics 261 (2000) 359±371
we see that the RLFT10 scheme, using the DFT molecular properties with permanent local ®eld, is able to reproduce the experimental values nearly quantitatively, together with a reasonably small 2 value for vcbb . It should be stressed that the RLFT10 scheme in combination with the DFT (hyper)polarizabilities is able to reproduce the experimental SHG values only if the permanent local ®eld is taken into account properly. The failure of the RLFT10 scheme, using the 6-31++G** basis set input values is due to a strong underestimate of the permanent local ®eld eect on the ®rst hyperpolarizability b. This underlines the importance of second-order FIP functions for the calculation of the c tensor and thus for b F k0 in Eq. (9). Turning now to the predictions of ALFFA, we see from Table 8 that the Lorentz approximation fails to reproduce the experimental values and that its results are very close to those of the RLFT10 scheme without permanent local ®eld. Indeed the incapability to reproduce the permanent local ®eld eect is the main reason for the dierences between ALFFA and RLFT10: if the molecular properties given in Table 6, calculated at the DFT level and including the ®eld eect are used to calculate v 2 for ALFFA, instead of those of 2
2 2 ; vcbb ; vccc the ®eld-free molecule, we ®nd vcaa ÿ12 ÿ1 mV , values that are quite close 25; 3; 23 10 to those found for RLFT10 with ®eld. The Lorentz ®eld factor has originally been calculated for a cubic arrangement of molecules, with the molecule of interest in the center of a macroscopically small, but microscopically large sphere, embedded in a dielectric continuum. The ®elds due to the surrounding molecules in the sphere sum up to zero at the center of the sphere, and therefore only the macroscopic ®eld and the ®eld due to the polarisation on the surface of the sphere contribute to the local ®eld [52]. ALFFA is an ad hoc generalisation of this special case and is therefore inherently incapable to take the permanent local ®eld into account. The ®ndings reported here for crystals are in agreement with a theoretical investigation of the liquid state by Wortmann and Bishop [53]. They showed that the Lorentz model
369
can be regarded as a special case of the more general Onsager model for the ®rst-order susceptibility and an approximation to the second-order susceptibility, but in terms of what has been called ``solute'' molecular properties by the authors, that is, properties in the presence of the static reaction ®eld due to the permanent dipole moments of the surrounding molecules. But this reaction ®eld occurs only in the Onsager description, and the Lorentz model is unable to take it into account, similar as it is the case here for the permanent local ®eld in the crystal. Wortmann and Bishop note further that the Lorentz model applies erroneously an additional cavity ®eld factor for the secondorder susceptibility v 2 , which leads to values for v 2 that are roughly about 20% too large compared to the Onsager model. This and the partial incorporation of the frequency dependence by using experimental refractive indices for the calculation of the local ®eld factors d, may be the main reasons for the larger values found here for ALFFA compared to the RLFT10 scheme without the permanent local ®eld, if calculated with the same set of molecular properties. The good performance of the RLFT10 model for the ®rst-order and, if the permanent ®eld eect is properly taken into account, also for the secondorder susceptibilities indicates that, as in the case of the hydrocarbons, the simple distribution scheme used here does not dier dramatically from the actual polarizability and ®rst-order hyperpolarizability densities of the mNA molecule in the crystal. In meta-substituted benzene derivatives there are no large donor±acceptor charge transfer contributions to b [54]. It appears from our results that for such molecules the eect of distributing the molecular response functions homogeneously over the space occupied by the molecules is more important for the calculation of the susceptibilities than a realistic description of the spatial anisotropy of the (hyper)polarizability densities. It would be very interesting to investigate if this situation changes for molecules where large donor± acceptor charge transfer contributions to b lead to more strongly anisotropic (hyper)polarizability distributions, as for example the para-nitroaniline derivatives.
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4. Conclusions The macroscopic linear and nonlinear dipole susceptibilities of the naphthalene, anthracene and mNA crystals have been calculated in this work, using as far as possible theoretically determined input values. The molecular (hyper)polarizabilities have been calculated at the DFT level, using the TZVP-FIP1 and TZVP-FIP2 basis sets which are especially designed for calculations of these properties. For mNA, the permanent electric ®elds caused by the surrounding dipoles in the crystal have been taken into account in a self-consistent manner. In contrast to calculations on urea [16] and benzene [15] using the same distribution method, the spatial partitioning of the molecular response has been found to have a crucial in¯uence on the calculated susceptibilities. Distributing the properties over all heavy atoms yield susceptibilities that are in very good agreement with available experimental values. Such a good performance is rather surprising, considering the simple approach of distributing the molecular response equally over the submolecules. This indicates that a realistic description of the spatial anisotropy of these functions is not essential for a reliable calculation of the crystal susceptibilities, at least in the case of molecules without large donor±acceptor charge transfer contributions. Estimates of the susceptibilities based on the ALFFA were found to be in better agreement with experiment than those based on the less distributed schemes. However, the larger distribution models are generally in better agreement with experiment than the Lorentz approach, especially in the case 2 of viij of mNA. In this case, the Lorentz approach neglects completely the large eect of the permanent local ®eld on the ®rst hyperpolarizability of the free molecule. If the Lorentz approach were used in the usual way, i.e. to extract molecular hyperpolarizabilities from experimental susceptibilities of the mNA crystal, the resulting ®rst hyperpolarizabilities would therefore be approximately those of the free molecule in the presence of the permanent electrical local ®eld, rather than those of the isolated molecule. This is in line with the analysis of local ®eld corrections in liquids in
the framework of the Onsager reaction model by Wortmann and Bishop [53], who showed that the static reaction ®eld associated with the permanent dipole moments is the main reason for large dierences between quantum-mechanically calculated (hyper)polarizabilities of isolated molecules and the solute (hyper)polarizabilities derived from experiments by applying local ®eld corrections. The results reported here and those for the urea [16] and benzene [15] crystals suggest that for molecular crystals the permanent local ®eld eect on the linear polarizability is generally small and may be neglected, but its eect on the molecular dipole moment and ®rst hyperpolarizability can be very large and must be taken into account. It should be mentioned that the good agreement of the calculated ®rst-order susceptibility with experimental data for the hydrocarbon crystals does not exclude the possibility of large electric ®elds in these crystals, which may arise from multipole moments higher than dipole. As the ®rst hyperpolarizability is zero for nonpolar molecules, any ®elds would have to be very large in order to change the polarizabilities signi®cantly. Together with the results reported for the benzene [15] and urea [16] crystals, this work shows that the rigorous local ®eld approach is able to predict reliably the macroscopic optical susceptibilities of molecular crystals, nearly without recourse to experimental information. The presented combination of the local ®eld approach with density functional theory opens the possibility for a rational design of novel NLO materials.
Acknowledgements We acknowledge ®nancial support from the European Commission in the form of a TMR Network Grant (contract no. ERBFMRXCT960047).
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