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Computational and Theoretical Chemistry 985 (2012) 72–79
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Theoretical investigation of the static (dynamic) polarizability and second hyperpolarizability of DAAD quadrupolar push–pull molecules. A comparison among HF (TD-HF), DFT (TD-B3LYP), and MP2 (TD-MP2) methods Emildo Marcano a,⇑, Emilio Squitieri a, Juan Murgich b, Humberto Soscún c a b c
Escuela de Química, Facultad de Ciencias, Universidad Central de Venezuela, Apartado 47102, Caracas 1020-A, Venezuela Centro de Química, Instituto Venezolano de Investigaciones Científicas (IVIC), Apartado 21827, Caracas 1020-A, Venezuela Centro Nacional de Tecnología Química, Complejo Tecnológico Simón Rodríguez, Base Aérea La Carlota, Caracas, Venezuela
a r t i c l e
i n f o
Article history: Received 28 August 2011 Received in revised form 1 February 2012 Accepted 2 February 2012 Available online 16 February 2012 Keywords: Polarizability Second hyperpolarizability Quadrupolar push–pull molecules BLA p⁄/p Ratio DFT
a b s t r a c t Calculations at HF, DFT (SVWN, PBE, B3LYP, PBE0, BHHLYP and CAM-B3LYP functionals) and MP2 level were carried out with 6-31+G(d,p) basis set for estimate the polarizability (a) and the second hyperpolarizability (c) of DAAD quadrupolar push–pull molecules. Our results show that DFT schemes overestimate a values relative to MP2 ones, being SVWN and PBE those functionals that give the most important deviation, whereas that HF and MP2 results are comparable between them. Moreover, the electron correlation effect is negative at MP2 level. For the static second hyperpolarizability, a significant agreement was obtained between CAM-B3LYP and MP2 methods. In relation to the frequency-dependence, only the dynamic second hyperpolarizability, calculated from TD-HF, TD-B3LYP, and TD-MP2 methods, revealed a large dependence on k. When hci (hai) are correlated to p⁄/p (the ratio between the occupation number of the antibonding orbital, p⁄, and the bonding orbital, p) one significant exponential relationship was obtained, similar those obtain previously using the bond length alternation (BLA) parameter. Finally, our results demonstrate that DAAD molecules are more effective for third-order NLO applications than the corresponding dipolar molecules. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Third-order nonlinear optical (NLO) properties of p-conjugated organic systems have attracted a great deal of attention during the last two decades because of their potential applications in photonic devices such as optical switching, three dimensional memories, optical limiting, and photodynamic therapy [1–3]. The search for highly efficient third-order NLO systems has yielded several structure–property relationships of the second molecular hyperpolarizability (c) with tuning molecular parameters, such as the nature and length of the p-conjugated chain, the electron donor (D) and acceptor (A) groups, and the atomic charge density [4–6]. Accurate quantum chemical calculations of molecular hyperpolarizabilities are currently either beyond, or at, the borderline of what is feasible for large p-conjugated molecules. In these calculations even if electron correlation (EC) can lead to substantial changes in magnitude it is difficult to take into account due to the required computational efforts. In fact, computational requirements to perform conventional Møller–Plesset (MP), interaction of configurations CI, coupled-cluster CC or multiconfigurational ⇑ Corresponding author. E-mail address:
[email protected] (E. Marcano). 2210-271X/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.comptc.2012.02.001
self-consistent field calculations MCSCF rapidly escalate with the molecular size. To overcome this problem, density functional theory (DFT) methods can be employed. However, several papers have pointed out that DFT fails drastically in considering the evaluation of field-response properties. For example, conventional exchange–correlation (XC) functionals have been employed to calculate dipole moments, polarizability and hyperpolarizabilities [7,8], where it has been shown that the functional of correlation (C) has a negligible effect on the calculated values, whereas the exchange (X) contribution is responsible for the large overestimations when the size of the system increases. These enhanced polarizability values have been associated to an overestimation of the electron charge transfer between D/A groups linked to the p-conjugate system of the molecule, whereas for the first (b) and second (c) hyperpolarizability, an incomplete screening of the external electric field is responsible for the large discrepancies with respect to accurate values [9]. Therefore, XC functionals incorrectly describe the polarization of p-conjugated molecules when the intra-molecular charge separation is due to D/A substitution and/or to the presence of external electric fields. Likewise, the performance of different hybrid functionals (for example, O3LYP, B3LYP, B97-1, B98, PBEO and BHHLYP) has been taken in account to evaluate geometric and electronic properties
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E. Marcano et al. / Computational and Theoretical Chemistry 985 (2012) 72–79
of a great variety of important p-organic molecules, including those referred as push–pull ones, where, the bond length alternation (BLA) parameter has become important to rationalize the response properties. In this context, it has been found that some of these functionals give BLA results that are consistent with those obtained from MP2 calculations. However, these results have to be considered cautiously because in some cases BLA calculated using DFT approach show large deviation with respect to MP2 results [10]. The effect of EC on the first hyperpolarizability of D/A-substituted benzenes and stilbenes has been studied at DFT (B3LYP, B97-2, PBEO and BMK) and MP2 levels [11]. It has been found that MP2 approach is more consistent with the experimental data than DFT ones. In fact, the results of hybrid DFT functionals overestimate in many cases the b values, being the BMK functional that give the highest deviation. The failure of DFT calculations has been associated with the incorrect asymptotic behavior of typical XC potentials. These approximate XC potentials do not correctly treat self-repulsion, which leads to the so-called self-interaction error (SIE) [12]. Consequently, the excited states of dyes, in particular the charge transfer (CT) of chromophores, are inadequately described [13] [14,15]. Since these sources of error are absent in the Hartree–Fock (HF) method, the fraction of HF exchange proportionally reduced but does not eliminate the inaccuracies of pure DFT. Based on these ideas, hybrid functionals as B3LYP [16–18], PBE0 [19,20] and BHHLYP [21] have been employed with the inclusion of the fraction of HF exchange in 20%, 25%, and 50%, respectively, which has resulted in some improvements in both the determination of hyperpolarizabilities as the prediction of excited states of organic compounds [22]. A new class of DFT functionals has been proposed recently, which includes a growing fraction of ‘‘exact’’ exchange as the distance increases. These functionals, referred as long-range corrected (LC) functionals (for example, CAM-B3LYP) are able to improve correlation between calculated and experimental optical properties of several dyes [23–25]. One purpose of this paper is the comparison between DFT (SVWN, PBE, B3LYP, PBE0, BHHLYP and CAM-B3LYP functionals) and MP2 methods based on the calculation of the static polarizability and second hyperpolarizability of DAAD quadrupolar push–pull molecules. These molecules (see Fig. 1a) are of special interest in the literature because exhibit large cross sections for two-photon absorption (TPA), process that is associated to the molecular second hyperpolarizability [26]. Moreover, recent studies on NLO properties of polydiacetylene chains with D/A (ANH2/ANO2) substitution blocks have established that the polymers with DAAD blocks have higher diagonal c components than the DDAA ones. Additionally, by properly tuning the electronic properties of the block it is possible to achieve larger c value in DAAD polymers [27]. In relation to these DAAD molecules, we have reported theoretical calculations of the polarizability and second hyperpolarizability at PM3, PM6, HF, and MP2 level of several DAAD structures [28]. In that report we have established an important dependence of a and c with the molecular architecture (or skeleton) of the studied molecules. For instance, we have shown that independently of the electron acceptor group A in DAAD molecules, the molecule in Fig. 1a show the highest a and c values. Additionally, an analysis of the HF, MP2 and DFT (B3LYP and BHHLYP) results indicate that EC effects on a are consistent between MP2 and DFT methods, while large differences have been found for DFT c calculations with respect to MP2 ones. It is worth to mention that this behavior is independent of the basis set extension [29]. In the present paper, the calculations of the electronic contribution to the static dipole polarizability a and second hyperpolarizability c were obtained with HF, DFT, MP2 methods. The DFT calculations were carried out with the following functionals: (i)
73
Fig. 1. (a) DAAD quadrupolar molecule. (b) DA dipolar molecule. Here A: ACN and D: AH, ACH3, AOCH3, AOH, ANH2 and AN(CH3)2.
SVWN fundamental local density (LDA) functional, which depend on the electron density, (ii) PBE generalized-gradient (GGA) approximation, which depend on the density gradient [30], (iii) different contributions of the degree of exchange (X) functional regarding HF such as B3LYP (20%), PBE0 (25%), and BHHLYP (50%) hybrid functional, and (iv) self-interaction error through CAM-B3LYP functional. Additionally, to estimate at first time the frequency-dependent of these NLO properties we have implemented TD-HF, TD-B3LYP, and TD-MP2 time-dependent (TD) calculations, which details are given in the Computational section. The BLA parameter has been traditionally used as a measure of the intra-molecular charge transference (ICT) induced by the effect of the D/A groups linked to the p-conjugated system in push–pull molecules [31,32]. Recently, the ratio between the occupation number of the antibonding orbital (p⁄) and the bonding orbital (p), denoted by p⁄/p ratio, and the stabilization energy (E2), both calculated from Natural Bond Orbital (NBO) analysis at MP2/631+G(d,p) level, have been employed to interpret ICT in dipolar push–pull molecules. In particular, for isophorone chromophores [33], a kind of dipolar push–pull alkene, it was found that p⁄/p parameter can be linearly correlated to the first hyperpolarizability, b. More recently, we have extended correlations between p⁄/ p ratio and second hyperpolarizability c (a) for a series of DAAD quadrupolar push–pull molecules [28]. The stabilization energy E2 is associated with the p-delocalization between i and j NBOs under consideration, and there is a connection between E2 and the orbital overlap. The analysis of E2 energies has been used by several authors [34] to recognize ICT in p-conjugated compounds. For example, in polyacetylene, NBO analysis show that for short oligomer chains the incorporation of ethynyl spacers increases the p-delocalization energy, but, on the other hand, reduces the efficiency with which the delocalization is promoted along the molecular backbone [35]. For acetoacetanilide crystal, the different intramolecular interactions that are responsible for the stabilization of the molecule and their connection to experimental NLO properties were also revealed through NBO studies [36]. Therefore, another important point of our work will be evaluating p⁄/p and E2 as molecular parameters similar to BLA one to correlate the results obtained for the polarizability and second hyperpolarizability of DAAD molecules. Finally, in this paper we will explore the possibility of some additive relationship between the magnitudes of c (and a) of the structures 1a and 1b, since the DA dipolar moiety is relative to DAAD ones (as can be easily observed in Fig. 1).
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2. Computational details A representation of the DAAD molecules with the orientation of the coordinate system chosen to carry out the calculations is displayed in Fig. 1a. These molecules are composed by two D/A pairs opposite connected to the p-conjugated system. Since DAAD is a model for centrosymmetric molecules (symmetry center is located in the middle C5 and C6 double bond) where the permanent dipole moment and first hyperpolarizability are both zero. The static polarizability and second hyperpolarizability can be theoretically calculated either from analytical (CP-HF) [37] or numerical (finite field, FF) [38,39] methods. The quantities of interest are hai and hci, i.e., the averaged values of the static polarizability and second hyperpolarizability, which are given by:
hai ¼
PnEC ðxÞ ¼
1 ðaxx þ ayy þ azz Þ 3
ð1Þ
1 ðc þ cyyyy þ czzzz þ 2cxxyy þ 2cxxzz þ 2cyyzz Þ 5 xxxx
ð2Þ
and
hci ¼
hyperpolarizability of molecules of Fig. 1a can be reasonably calculated at MP2/6-31+G(d,p) level. Additionally, we also recognized that the DFT deviations for a and c results when compared with MP2 ones can be associated to the convergence criteria of density matrix [29]. Therefore, in all the calculations of present work we have used the value of 108 as convergence criteria for the density matrix. With the purpose to evaluate the frequency-dependence of the polarizability and second hyperpolarizability of DAAD molecules, we have carried out TD-HF/6-31+G(d,p) calculations at 532 and 1064 nm. Moreover, in order to consider EC effect on the dynamic properties, calculations at TD-MP2 and TD-B3LYP levels were obtained through the following equation [44]:
where hci in Eq. (2) can be compared with EFISHG or THG experimental results [40]. We have chosen the A electron acceptor group as ACN, whereas AH, ACH3, AOH, AOCH3, ANH2 and AN(CH3)2 represent to the D donor electron group [41]. The geometric optimization of these structures was carried out at MP2 level using the 6-31+G(d,p) basis set and GAUSSIAN [42]. The static tensor components of a and c were calculated with the GAMESS software [43] using Field Finite (FF) methods with electric field intensities of E = ±0.001 au in each (x, y, z) axis direction. In this report, these static tensor components are included as Supplementary material (Tables S1 and S2). The behavior of NLO properties in organic compounds is associated to the electronic delocalization of the isolated molecule. In fact, these properties are the response of the charge transference across the molecular structure by effect of the interaction with external electromagnetic fields until the molecular relaxation is reached. To give an accurate quantum representation of these effects in the calculation of NLO properties at molecular level it is important to take in account electron correlation (EC) [52]. In previous article [28] we established that EC effects on the second
PTD HFn ðxÞ PnEC ðx ¼ 0Þ: PTD HFn ðx ¼ 0Þ
ð3Þ
In Eq. (3), PnEC ðxi Þ is EC effect on the dynamic n-th order response (i.e., polarizability or second hyperpolarizability) of any post-HF level (in this paper, TD-MP2 and TD-B3LYP), and its frequency dependence x is separated in terms of the corresponding static response, PnEC ðx ¼ 0Þ, multiplied by the ratio of the frequency-dependency that arise using the simple TD-HF scheme [44]. 3. Results and discussion 3.1. NBO analysis for DAAD molecules Since that the ICT is a essential point in the analysis of the values a and c as function of the molecular structure, i.e., D/A groups, we have calculated p⁄/p on C5@C6 and E2 for p(C1@C2) ? p⁄(C3@C4) delocalization from NBO study at MP2/6-31+G(d,p) level. The results obtained are presented in Table 1. In this table we have also included values of molecular total energy (E), the distance of the double bond between C5 and C6 carbon atoms (RC5@C6), the corresponding BLA, and the rest of relevant values of E2. Analysis of results for molecules of structure 1a, allowed us to propose correlations between the RC5@C6 and E2 values and the corresponding p⁄/p ones, where good linear relationships were found (see Fig. 2). Additionally, an exponential behavior was obtained for BLA versus p⁄/p and E2, respectively. These results show that these
Table 1 Molecular energy (E/au), RC5AC6 (Å), BLA (Å), p⁄/p on C5@C6 bond, NBO‘s, and E2 (kcal/mol) calculated at MP2/6-31+G(d,p) level. Donor AH
E 569.1
RC5AC6 1.3674
BLA 0.128
p⁄/p (C5@C6)
NBO (i)
NBO (j)
E2
0.065
p(C1AC2) p(C3AC4)
0.069
p(C1AC2) p(C3AC4)
0.072
p(C1AC2) p(C3AC4)
p⁄(C3AC4) p⁄(C11AN12) p⁄(C3AC11) p⁄(C3AC4) p⁄(C11AN12) p⁄(C3AC11) p⁄(C3AC4) p⁄(C11AN12) p⁄(C3AC11) p⁄(C1AC2) p⁄(C3AC4) p⁄(C11AN12) p⁄(C3AC11) p⁄(C1AC2) p⁄(C3AC4) p⁄(C11AN12) p⁄(C3AC11) p⁄(C1AC2) p⁄(C3AC4) p⁄(C11AN12) p⁄(C3AC11) p⁄(C1AC2)
19.2 25.0 14.3 21.8 25.4 14.3 23.6 18.8 14.7 38.6 24.4 25.5 14.3 40.8 28.2 25.4 14.7 64.6 29.4 25.5 14.3 76.1
LP (N12) ACH3
647.1
1.3683
0.127
LP (N12) AOH
718.8
1.3691
0.124
AOCH3
796.4
1.3700
0.126
0.074
ANH2
679.2
1.3715
0.119
0.081
AN(CH3)2
835.3
1.3736
0.114
0.085
LP (N12) LP (O17) p(C1AC2) p(C3AC4) LP (N12) LP (O17) p(C1AC2) p(C3AC4) LP (N12) LP (N16) p(C1AC2) p(C3AC4) LP (N12) LP (N16)
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75
Fig. 2. Relationship between: (2a) RC5@C6, (b) E2, and (c) BLA with p⁄/p. (d) BLA versus E2 at MP2/6-31+G(d,p) level.
330 310 290 270 250 230 210 190 170 150 0.060
< α > (au)
(a)
3.2. Static polarizability of DAAD molecules
-N(CH3)2 -OCH3
-NH2
-OH -CH3
R² = 0.9344
-H
0.065
0.070
0.075
0.080
0.085
0.090
π*/π 1200 1100 1000 900 800 700 600 500 400 300 0.060
< γ >.103 (au)
(b)
-N(CH3)2
-NH2
-OCH3 -CH3 -OH
-H 0.065
0.070
0.075
R² = 0.9208
0.080
0.085
0.090
π*/π Fig. 3. Plot of (a) hai and (b) hci as function of p⁄/p at MP2/6-31+G(d,p) level.
molecular ‘‘parameters’’ can be used to correlate the polarizability and second hyperpolarizability of DAAD quadrupolar push–pull molecules (see Fig. 3). In all these figures and Table 1, we can recognize that p⁄/p ratio increases according to the sequence: AH < ACH3 < AOH < AOCH3 < ANH2 < AN(CH3)2, i.e., p⁄/p enhance when atoms with electron lone pair (LP) are linked to p-system. In fact, from Table 1 we can observe that the main p ? p⁄ transfer occurs for AN(CH3)2 substituent in the structure 1a, since the value of E2 is incremented by 12 kcal/mol as result of the overlap between LPN16 and p⁄C1AC2 orbitals.
The results of the static polarizability hai of DAAD Fig. 1a molecules calculated with the different computational chemistry methods are displayed in Table 2, where the highest ha i value is found for AN(CH3)2 donor group. With respect to the performance of DFT functionals, hai results are significantly overestimated if HF and MP2 values are taken into account (see Fig. S1 in Supplementary material). In fact, Table 4 show that the percentage error of hai at DFT (SVWN, PBE, B3LYP, PBE0, BHHLYP and CAM-B3LYP)/631+G(d,p) level with respect to MP2 ones is between 24% and 45%, being SVWN and PBE functionals those that give the main deviation. However, HF and MP2 ha i values are comparable between them. In fact, the corresponding absolute percentage error is less of 8% in some cases. Additionally, it is observed that EC effect on hai is negative at MP2 level [45]. In general, Table 2 shows the results of DFT functionals for hai in terms of the electron donor group. It is known that the quality of the DFT calculations depends on the choice of XC functional [46]. We have chosen SVWN and PBE exchange correlation functionals within LDA and GGA approach, respectively. The SVWN functional is composed by the local Slater exchange functional (S) [47] plus the uniform electron gas local correlation functional due to Vosko, Wilk, and Nusair (VWN) [48], whereas the description of PBE GGA has been reported elsewhere [49]. The results of hai for SVWN and PBE functionals have similar magnitude between them with percentage error of approximately 40% with respect to MP2 ones. Therefore, the consideration of gradient corrections in DFT does not improve the polarizability calculation with respect to LDA approach. These results are in agreement with different studies that have shown that the variation of conventional LDA schemes with respect to HF is primarily determined by the exchange (X) contribution, where the role of nonlocal corrections is very small [9]. In order to assess the HF exchange (X) contribution, B3LYP (20%), PBE0 (25%) and BHHLYP (50%) hybrid functionals have also been used to calculate hai. The results of Table 2 indicate that despite the hybrid functionals overestimate the static polarizability;
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Table 2 hai/au calculated with HF, DFT, and MP2 methods and 6-31+G(d,p) basis set. The results in parentheses are the q/d factors. DONOR
HF
SVWN
PBE
B3LYP
AH ACH3 AOH AOCH3 ANH2 AN(CH3)2
200.6 263.8 215.5 250.4 239.4 357.3
256.9 311.1 299.1 360.1 340.4 453.6
256.2 308.7 297.9 357.4 338.6 448.8
246.8 293.7 282.9 336.5 323.5 424.2
(1.4) (1.8) (1.5) (1.6) (1.6) (1.7)
the increment of HF exchange contribution is able to improve at qualitative level the tendencies of the calculated values with respect to MP2 ones. It is important also to observe the excellent agreement between BHHLYP and CAM-B3LYP functionals with deviation less of 2% between both schemes. CAM-B3LYP takes into account the long-range correction, which is able to give an improvement in the asymptotic behavior and avoids the self-interaction error [23]. 3.3. Static second hyperpolarizability Results for the second hyperpolarizability of Fig. 1a DAAD molecules are shown in Table 3 at several theory levels. These results shows that similarly to the polarizability values of Table 2, the highest hc i calculated value is obtained for molecule with AN(CH3)2 group [31,28]. However, it is worth to note that DFT EC effects with respect to MP2 hc i results are positive [45], on the contrary to the above behavior of the dipole polarizability (see Fig. S1 in Supplementary material). Additionally, no agreement there exists between the magnitude of the HF and MP2 results. In fact, results of Table 4 shows that a significant increment of almost 60% was obtained with MP2 results when compared to HF ones, while the tendency of hc i as increasing function of D was preserved. On the other hand, Table 4 display the results of HF and DFT hci deviation with respect to MP2 values, which in general are in good agreement for AH and ACH3 groups, i.e., the absolute percentage error is less of 11%. However, when using electron donor groups, such as ANH2 and AN(CH3)2, a considerable difference between
PBE0
BHHLYP
CAM-B3LYP
MP2
242.0 288.4 277.8 330.0 318.0 416.8
234.1 275.7 263.5 310.3 301.5 390.2
233.8 276.1 264.9 313.5 303.3 396.1
186.1 218.1 208.7 247.0 239.1 314.3
(1.4) (1.8) (1.5) (1.6) (1.6) (1.7)
(1.4) (1.8) (1.7) (1.7) (1.7) (1.1)
the magnitude of DFT and MP2 hci results is observed. In particular, SVWN and PBE functionals show similar hci deviation when compared to MP2 ones, where percentage error is between 26% (AOH) and 70% (ANH2). Therefore, LDA approximation and gradient corrections do not improve the prediction of second hyperpolarizability of the molecule 1a with D groups with electron lone pair. When PBE and PBE0 results are compared to MP2 ones, an important increment of hci is obtained, which confirm the importance of HF exchange (X) contribution for the prediction of this property. In fact, the mean absolute percentage error between PBE0 and MP2 is 19.6%. When B3LYP and BHHLYP functionals were employed, the results of hc i for most donor groups remain essentially unchanged, except for ANH2 and AN(CH3)2, where the inclusion of 50% of HF exchange (X) lead to an improvement of hc i. A significant agreement was obtained for the values of hci between CAM-B3LYP and MP2 methods (see Table S1); where the mean absolute percentage error was 7.2%. This result can be explained by the fact that CAMB3LYP combines the hybrid qualities of B3LYP (20% of HF exchange) and the long-range correction from Coulomb-attenuating method. Therefore, both effects are important to take properly in account EC effect for hci at low computational expenses. Recently, we found that there is an important dependence between the values of hc i and ha i with the molecular arrangement (isomers) of quadrupolar molecules [28]. In this work, we have extended that observation to include the dipolar moiety of Fig 1b. Therefore, we have also calculated the static polarizability and second hyperpolarizability of the DA dipolar push–pull molecule, which are shown in Tables 2 and 3 in the form of haiDAAD/haiDA
Table 3 hci 103 au Calculated with HF, DFT and MP2 methods and 6-31+G(d,p) basis set. The results in parentheses are the q/d factors. DONOR
HF
SVWN
PBE
B3LYP
AH ACH3 AOH AOCH3 ANH2 AN(CH3)2
158.5 293.8 191.2 266.7 281.3 710.7
306.1 467.4 269.4 471.0 210.7 485.7
309.2 455.7 270.8 466.7 223.1 496.7
341.2 490.1 365.0 565.6 379.8 736.1
(2.2) (2.1) (2.3) (2.6) (2.4) (3.2)
PBE0
BHHLYP
CAMB3LYP
MP2
327.2 471.7 368.7 573.8 397.6 753.1
308.1 429.7 383.2 565.9 487.7 901.6
337.0 481.5 452.8 651.3 570.6 1017.9
345.5 (2.3) 456.6 (2.2) 462.1 (2.3) 637.9 (2.7) 697.2 (2.7) 1170.7 (3.1)
(2.2) (2.2) (2.3) (2.9) (2.7) (3.4)
Table 4 Percentage error of hai and hci relative to MP2/6-31+G(d,p) calculations obtained from the values of the Tables 2 and 3. Property
hai
hci
DONOR
HF
SVWN
PBE
B3LYP
PBE0
BHHLYP
CAM-B3LYP
AH ACH3 AOH AOCH3 ANH2 AN(CH3)2 AH ACH3 AOH AOCH3 ANH2 AN(CH3)2
7.8 20.9 1.4 3.2 0.1 13.7 54.1 35.7 58.2 58.6 59.6 39.3
38.1 42.6 45.8 43.3 42.3 44.3 11.4 2.4 26.2 41.7 69.8 58.5
37.7 41.5 44.7 42.7 41.6 42.8 10.5 0.2 26.8 41.4 68.0 57.6
32.6 34.7 36.3 35.5 35.3 35.0 1.2 7.3 11.3 21.0 45.5 37.1
30.1 32.2 33.6 33.1 33.0 32.6 5.3 3.3 10.1 20.2 43.0 35.7
25.8 26.4 25.7 26.2 26.1 24.2 10.8 5.9 11.3 17.1 30.0 23.0
25.7 26.6 26.9 26.9 26.8 26.0 2.5 5.5 2.1 2.0 18.2 13.0
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and hciDAAD/hciDA ratios (referred as q/d factor in parenthesis), respectively, at B3LYP, BHHLYP and MP2 levels. Polarizability results included in Table 2 reveal that DAAD molecules are up to 1.6 times more polarizable than DA ones. For the second hyperpolarizability, we found the following approximate relationship: hciDAAD 2.6 hciDA. In consequence, from these results it is possible estimate hci and hai for DAAD molecules from DA moieties calculations. Additionally, these results also indicate that the quadrupolar 1a molecules are more effective for third-order NLO applications than the corresponding dipolar 1b. As was mentioned above, the molecules of Fig. 1a are of special attention due to exhibit large cross section for TPA, which is associated to the second hyperpolarizability. A reduced perturbation theory expression, named three-level model, can be used to describe qualitatively magnitude and sign variations of c. For a centrosymmetric molecule only two terms are important to estimate c [50]:
c ¼ ce þ ctp ¼ Kl401 D11 þ K0 l201 l212 D121 :
ð4Þ
In Eq. (4), ce and cp are the one and two photon terms, l01 is the transition moment from ground state to the first excited state (one photon state), l12 is the transition moment between first and higher excited states (two photon state), K and K0 are constants associated to the optical process under consideration and D is a dispersion term. Thus, c depends on the competition between the two terms with opposite signs, more specifically, between l01 and l12. Strong D/A groups favor the stabilization of the first excited state and the charge transfer from donor to acceptor ones, resulting in a larger l01 than l12, and as a consequence ce contributes mainly to c. This kind of molecules often exhibits negative value of this property. For the cases with weak D/A groups, the larger energy difference between the ground and first excited states leads to a smaller l01, making ctp dominant and c positive [51]. For all molecules studied in this work, both, the tensor components and hci have positive values. Therefore, the two donor groups at both ends of molecular backbone of the Fig. 1a would tend to halt the ICT, resulting in an even smaller l01 and ctp, which lead to a positive c value. Consequently, the results obtained in Table 3 confirm the applicability of these molecules for design of TPA devices [26]. 3.4. Basis set effects on static polarizability and second hyperpolarizability The behavior of polarizability and second hyperpolarizability values in terms of the basis set extension is another important aspect in the calculation of optical properties in organic molecules [53]. In this context, we have included ayy and cyyyy calculations at HF and DFT levels by using 6-31G, 6-31G(d), 6-31+G(d), 631+G(d,p,f), and 6-31+G(2d,2p) basis sets for the molecule with the highest values in the optical properties. In this case, the calculations were carried out on MP2/6-31+G(d,p) optimized geometries of the 1a structure with ACN/AN(CH3)2 substituted groups. In order to compare the DFT basis set effects results, we included ayy and cyyyy calculations until CAM-B3LYP and MP2/6-31+G (2d,2p) level. The results are shown in Table 7. At HF level, the inclusion of d polarization and sp diffuse functions increased the values of cyyyy with respect to those obtained with the 6-31G sets. Passing from 6-31G to 6-31G+(d), increased the cyyyy values by 5%. On the other hand, passing from 6-31+G(d) to 6-31G+(d,p), a decreased of cyyyy were found. The addition of one function f increased cyyyy value with respect to those obtained with 6-31G+(d,p). In the case of ayy the inclusion of d and p polarization and sp diffuse functions increased the values by 2%. Therefore, at HF level, the ayy magnitude was less sensitive to the basis set effects than the cyyyy property, how is expected. Within the DFT
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approach, the ayy values are not significantly influenced with the introduction of polarization and diffuse functions, and the qualitative tendencies were similar with respect to those obtained at HF levels. On the other hand, basis set effects on cyyyy show a dependency of the employed functional. For all basis sets, the B3LYP functional overestimated the cyyyy magnitude with respect to HF results when d and p polarization and sp diffuse functions were included. When the EC was calculated with BHHLYP functional, we also observed an overestimation of cyyyy values with respect to HF results for all basis sets, showing different tendencies to B3LYP functional results. For instance, for the 6-31+G(2d,2p) basis set, the differences was of 23%. When we compared the tendencies between DFT and MP2 in Table 7, we observed that in general, the addition of polarization d and p functions at 6-31+G(d,p) basis set lead to an increasing in cyyyy values (MP2 level). Despite CAMB3LYP increased the cyyyy values with respect to the others functional, the increased with the basis set extension is not sufficient for comparing with the MP2 level. Finally, the results in Table 7 shown that MP2/6-31+G(d) and MP2/6-31+G(d,p) levels can lead a similar values to ayy and cyyyy, calculated at MP2/6-31+G(d,p,f) and MP2/6-31+G(2p,2d) ones. Therefore, for NLO calculation in molecules such as 1a, with ACN/AN(CH3)2 substituted groups, a level MP2/6-31+G(d,p) can be used when not accuracy results are necessary at much lower computational cost. 3.5. Some structure–property relationships As has been mentioned previously, from an NBO analysis we have obtained the results shown in Table 1. In Fig. 3 we have plotted hai and hci as function of p⁄/p. As can be observed in this figure, as one goes from AH to AN(CH3)2 an exponential increment in the calculated values of hai and hci is obtained [31]. This increment can be explained in terms of the difference of the stabilization energy (DE2) values displayed in Table 1. For example, as one goes from AOH to AN(CH3)2 the magnitude of DE2 is 37.5 kcal/mol. This value of DE2 is associated to the stabilization energy between LPN16 ? p⁄C1C2 and LPO17 ? p⁄C1C2 delocalization, and represent the enhancement in the stabilization energy due to the delocalization of the electron lone pair of N in AN(CH3)2 to C1@C2 bond. Additionally, such delocalization leads to an increment in the magnitude of p⁄/p on C5@C6 bond. For the second hyperpolarizability, the exponential behavior observed in Fig. 2b has been reported previously [31,47] in terms of BLA parameter. We can switch p⁄/p values to the corresponding BLA, RC5@C6, and E2 by means of Fig. 2. 3.6. Dynamic polarizability second hyperpolarizability of DAAD molecules In Tables 5 and 6 we shown the values of hai and hci obtained from TD-HF, TD-B3LYP and TD-MP2 methods at k = 1, 1064, and 532 nm using 6-31+G(d,p) basis set from Eq (3). From these results, we observe that TD-HF hai and hc i calculations reproduce almost the static results obtained with the FF approach of Tables 2 and 3, respectively. In general, the TD-HF hai values show a magnitude increasing by frequency with k = 1064 nm and 532 nm. It is worth noticing that the effect of frequency-dependence decreases (or increase) when the electronic correlation is included at TD-MP2 (TD-B3LYP) level. For the dynamic second hyperpolarizability we observe that hci has a large dependence on k. For example, the value of hcik = 532 nm for AN(CH3)2 at TD-HF level is 12 times the static case, while at 1064 nm it is 1.6 times the value of hcik?1. Finally, let us stress that according to the Table 6 the dispersion effects on hci are enhanced when EC is included at TD-B3LYP and
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E. Marcano et al. / Computational and Theoretical Chemistry 985 (2012) 72–79
Table 5 Values of hai au with TD-HF, TD-B3LYP and TD-MP2 methods and 6-31+G(d,p) basis set. DONOR
k=1
AH ACH3 AOH AOCH3 ANH2 AN(CH3)2
k = 1064 nm
k = 532 nm
TD-HF
TD-HF
TD-B3LYP
TD-MP2
TD-HF
TD-3LYP
TD-MP2
200.6 231.7 215.5 250.4 239.4 303.3
209.1 241.6 225.1 261.8 251.8 319.4
257.2 268.9 295.5 351.8 340.3 379.2
193.9 199.7 218.1 258.2 251.6 281.0
244.9 283.7 266.7 311.9 308.5 394.8
301.3 315.9 350.1 419.1 416.9 468.7
227.1 234.6 258.3 307.6 308.2 347.3
Table 6 Values of hci 103 au with TD-HF, TD-B3LYP and TD-MP2 methods and 6-31+G(d,p) basis set. DONOR
k=1
AH ACH3 AOH AOCH3 ANH2 AN(CH3)2
k = 1064 nm
k = 532 nm
TD-HF
TD-HF
TD-B3LYP
TD-MP2
TD-HF
TD-B3LYP
TD-MP2
158.1 203.1 190.7 259.7 280.8 470.7
222.1 290.5 275.7 381.2 425.4 741.3
478.0 484.7 526.4 808.6 574.3 767.9
484.0 451.5 666.4 911.9 1054.2 1221.1
836.3 1200.5 1213.5 1831.4 2524.8 5647.5
1799.8 2002.9 2316.9 3884.6 3408.4 5849.9
1822.6 1866.0 2933.2 4381.0 6256.8 9302.9
Table 7 Basis set effects on the ayy /au and cyyyy /103 au properties at different theory levels for 1a structure with ACN/AN(CH3)2 substituted groups. Basis set
6-31G 6-31G(d) 6-31+G(d) 6-31+G(d.p) 6-31+G(d,p,f) 6-31+G(2d,2p)
HF
CAM-B3LYP
MP2
ayy
cyyyy
B3LYP
ayy
cyyyy
ayy
PBE0
cyyyy
ayy
cyyyy
ayy
cyyyy
ayy
cyyyy
642.8 640.6 670.1 673.3 676.0 679.2
1416.2 1274.7 1503.1 1323.5 1449.0 1455.7
732.6 736.3 780.3 784.0 782.3 789.2
1186.4 1101.0 1453.2 1482.4 1456.4 1501.0
724.4 729.1 767.4 771.2 770.9 776.6
1328.2 1216.9 1514.4 1557.3 1543.7 1581.7
683.2 685.8 721.9 725.6 725.4 731.1
1773.9 1609.6 1936.9 1957.7 1919.7 1933.0
678.7 684.7 724.8 737.9 739.4 742.3
2216.8 2020.2 2422.9 2428.9 2581.0 2657.0
534.5 571.8 613.8 529.9 613.4 633.5
3196.0 4651.9 5547.0 5390.4 5110.8 5340.4
TD-MP2 levels. However, note that for ANH2 and AN(CH3)2 the calculations at TD-B3LYP level underestimates hci in relation to MP2. 4. Conclusions We have calculated the static and dynamic polarizability and second hyperpolarizability of DAAD quadrupolar molecules at HF, DFT (SVWN, PBE, B3LYP, PBE0, BHHLYP, and CAM-B3LYP) and MP2 levels with 6-31+G(d,p) basis set as function of D/A groups, in terms of p⁄/p, BLA, RC5@C6, and E2 (see Table 3). The electron correlation effects using SVWN, PBE, B3LYP, PBE0, BHHLYP, and CAM-B3LYP functionals induce significant increasing on hai, while decrease the values of hci in relation to MP2 calculations (see Tables 3 and 4). The results also indicate that LDA (SVWN) and GGA (PBE) approximations are unsuitable for ha i and hc i calculations of the studied molecules. The inclusion of 50% of HF exchange (BHHLYP) and the long-range correction (CAM-B3LYP) are important to evaluate hci as MP2 level. The failures of some DFT approaches in giving good performance for NLO properties of DAAD quadrupolar molecules can be associated to the overestimation of charge transfer between the donor and acceptor groups. With respect to frequency-dependence results, the polarizability increases with the k at TD-HF level, but decreased when the electronic correlation is taken in account at TD-MP2 level. Unlike hai values, a significant dispersion effect on the second hyperpolarizability was found at k = 1064 and 532 nm, which is enhanced when EC is included at TD-B3LYP and TD-MP2 levels. Finally, Polarizability results included in Table 2 reveal that DAAD molecules are up to 1.6 times more polarizable than DA
BHHLYP
ones, while for the second hyperpolarizability, we found that hciDAAD 2.6 hciDA. In consequence, it is possible estimate hc i and hai for DAAD molecules from DA moieties calculations. We results indicate that the quadrupolar 1a molecules are more effective for third-order NLO applications than the corresponding dipolar 1b. Acknowledgments This work was supported by the ‘‘Consejo de Desarrollo Científico y Humanístico de la Universidad Central de Venezuela’’ (CDCH-UCV, Grant No. -PG 03-00-6710-2007). E. M. acknowledges to the ‘‘Fondo Nacional para el Avance de la Ciencia y la Tecnología’’ (FONACYT, Fellowship No. 200400436). Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.comptc.2012.02.001. References [1] D.A. Pathenopoulos, P.M. Rentzepis, Three-dimensional optical storage memory, Science 245 (1989) 843. [2] W. Zhou, S.M. Kuebler, K.L. Braun, T. Yu, J.K. Cammack, C.K. Ober, J.W. Perry, S.R. Marder, An efficient two-photon-generated photoacid applied to positivetone 3D microfabrication, Science 296 (2002) 1106. [3] H.S. Nalwa, S. Miyata (Eds.), Nonlinear Optics of Organic Molecules and Polymers, CRC Press, Inc., New York, 1997. [4] P.K. Frederiksen, M. Jølgensen, P.R. Ogilby, Two-photon photosensitized production of singlet oxygen, J. Am. Chem. Soc. 123 (2001) 1215–1221.
Author's personal copy
E. Marcano et al. / Computational and Theoretical Chemistry 985 (2012) 72–79 [5] D. Burland, Optical nonlinearities in chemistry: introduction, Chem. Rev. 94 (1994) 1–2. [6] S. Forrest, M. Thompsom, Introduction: organic electronics and optoelectronics, Chem. Rev. 107 (2007) 923–925. [7] Benoît Champagne, Eric A. Perpète, Stan J.A. van Gisbergen, Evert-Jan Baerends, Jaap G. Snijders, Chirine Soubra-Ghaoui, Kathleen A. Robins, Bernard Kirtman, Assessment of conventional density functional schemes for computing the polarizabilities and hyperpolarizabilities of conjugated oligomers: an ab initio investigation of polyacetylene chains, J. Chem. Phys. 109 (1998) 10489–10499. [8] S.J.A. van Gisbergen, P.R.T. Schipper, O.V. Gritsenko, E.J. Baerends, J.G. Snijders, B. Champagne, B. Kirtman, Phys. Rev. Lett. 83 (1999) 694–697. [9] Benoît. Champagne, Eric.A. Perpète, Denis. Jacquemin, Assessment of conventional density functional schemes for computing the dipole moment and (hyper)polarizabilities of push–pull p-conjugated systems, J. Phys. Chem. A. 104 (2000) 4755–4763. [10] (a) M. Del Zoppo, C. Castiglioni, V. Gerola, P. Zuliani, G. Zerbi, Effect of bond length alternation and of bond length alternation oscillations on the molecular nonlinear optical response of push pull polyenes, J. Opt. Soc. Am. B. 15 (1998) 308; (b) Denis Jacquemin, Antoine Femenias, Henry Chermette, Ilaria Ciofini, Carlo Adamo, Jean-Marie Andr´e, Eric A. Perpete, Assessment of conventional density functional schemes for computing the dipole moment and (hyper)polarizabilities of push–pull p-conjugated systems, J. Phys. Chem. A. 110 (2006) 5952–5959. [11] Kyrill Yu, Suponitsky, Sergio Tafur, Artem E. Masunov, Applicability of hybrid density functional theory methods to calculation of molecular hyperpolarizability, J. Chem. Phys. 129 (2008) 044109–044120. [12] M. Gruning, O.V. Gritsenko, S.J.A. van Gisbergen, E.J. Baerends, On the required shape corrections to the local density and generalized gradient approximations to the Kohn–Sham potentials for molecular response calculations of (hyper) polarizabilities and excitation energies, J. Chem. Phys. 116 (2002) 9591. [13] R.J. Magyar, S. Tretiak, Dependence of spurious charge-transfer excited states on orbital exchange in TDDFT: large molecules and clusters, J. Chem. Theory Comput. 3 (2007) 976. [14] A. Dreuw, M. Head-Gordon, Failure of time-dependent density functional theory for long-range charge-transfer excited states: the zincbacteriochlorin bacteriochlorin and bacteriochlorophyllspheroidene complexes, J. Am. Chem. Soc. 126 (2004) 4007–4016. [15] D.J. Tozer, Relationship between long-range charge-transfer excitation energy error and integer discontinuity in Kohn–Sham theory, J. Chem. Phys. 119 (2003) 12697–12700. [16] C.T. Lee, W.T. Yang, R.G. Parr, Development of the Colle–Salvetti correlationenergy formula into a functional of the electron density, Phys. Rev. B 37 (1988) 785–789. [17] A.D. Becke, Density-functional thermochemistry. III. The role of exact Exchange, J. Chem. Phys. 98 (1993) 5648–5653. [18] P.J. Stephens, F.J. Devlin, C.F. Chabalowski, M.J. Frisch, Ab initio calculation of vibrational absorption and circular dichroism spectra using density functional force fields, J. Phys. Chem. 98 (1994) 11623–11627. [19] M. Ernzerhof, G.E. Scuseria, Assessment of the Perdew–Burke–Ernzerhof exchange-correlation functional, J. Chem. Phys. 110 (1999) 5029–5037. [20] C. Adamo, V. Barone, Toward reliable density functional methods without adjustable parameters: the PBE0 model, J. Chem. Phys. 110 (1999) 6158– 6171. [21] A. Becke, A new mixing of Hartree–Fock and local density-functional theories, J. Chem. Phys. 98 (1993) 1372–1378. [22] Denis. Jacquenim, Eric.A. Perpete, Gustavo.E. Scuseria, Ilaria. Ciofini, Carlo. Adamo, TD-DFT performance for the visible absorption spectra of organic dyes: conventional versus long-range hybrids, J. Chem. Theory Comput. 4 (2008) 123–135. [23] T. Yanai, D.P. Tew, N.C. Handy, A new hybrid exchange–correlation functional using the Coulomb-attenuating method (CAM-B3LYP), Chem. Phys. Lett. 393 (2004) 51–57. [24] R. Baer, D. Neuhauser, Density functional theory with correct long-range asymptotic behavior, Phys. Rev. Lett. 94 (2005) 043002–043006. [25] E. Livshits, R. Baer, A well-tempered density functional theory of electrons in molecules, Phys. Chem. Chem. Phys. 9 (2007) 2932–2941. [26] M. Barzoukas, M. Blanchard-Desce, Molecular engineering of push–pull dipolar and quadrupolar molecules for two-photon absorption: a multivalence-bond states approach, J. Chem. Phys. 113 (2000) 3951–3960. [27] S. Ohnishi, Y. Orimoto, F. Gu, Y. Aoki, Nonlinear optical properties of polydiacetylene with donor-acceptor substitution block, J. Chem. Phys. 127 (2007) 084702–084713.
79
[28] E. Marcano, E. Squitieri, J. Murgich, H. Soscún, Conformational dependence of the second hyperpolarizability of quadrupolar molecules, J. Mol. Struct. (THEOCHEM) 911 (2009) 81–87. [29] E. Marcano, E. Squitieri, J. Murgich, H. Soscún, Second hyperpolarizability: MP2 vs. DFT methods, J. Comput. Methods Sci. Eng. 9 (2009) 1. [30] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev Lett. 77 (1996) 3865–3868. [31] S. Hahn, D. Kim, M. Cho, Nonlinear optical properties of the linear quadrupolar molecule: structurefunction relationship based on a three-state model, J. Phys. Chem. B 103 (1999) 8221–8229. [32] M. Del Zoppo, C. Castiglioni, V. Gerola, P. Zuliani, G. Zerbi, Effect of bond length alternation and of bond length alternation oscillations on the molecular nonlinear optical response of push pull polyenes, J. Opt. Soc. Am. B 15 (1998) 308–317. [33] E. Kleinpeter, A. Koch, B. Mikhova, B.A. Stambolyska, T.M. Kolev, Quantification of the push–pull character of the isophorone chromophore as a measure of molecular hyperpolarizability for NLO applications, Tetrahedron Lett. 49 (2008) 1323–1327. [34] C. Ravikumar, I. Hubert Joe, V.S. Jayakumar, Charge transfer interactions and nonlinear optical properties of push–pull chromophore benzaldehyde phenylhydrazone: a vibrational approach, Chem. Phys. Lett. 460 (2008) 552–558. [35] Davood. Nori-Shargh, Neda. Hassanzadeh, Meisam. Kosari, Sasan. Sharifi, Stereoelectronic interaction effects (associated with the anomeric effects) on the stability of the stereoisomers of 1,4,5,8-tetraoxadecalin, 1,4,5,8tetrathiadecalin and 1,4,5,8-tetraselenadecalin. An ab initio study and NBO analysis, J. Mol. Struct.: THEOCHEM 940 (2010) 129–134. [36] C. Ravikumar, I. Hubert Joe, D. Sajan, Vibrational contributions to the secondorder nonlinear optical properties of p-conjugated structure acetoacetanilide, Chem. Phys. 369 (2010) 1–7. [37] J.L. Brédas, C. Adant, P. Tackx, A. Persoons, Third-order nonlinear optical response in organic materials: theoretical and experimental aspects, Chem. Rev. 94 (1994) 243–278. [38] H.A. Kurtz, J.J.P. Stewart, K.M. Dieter, Calculation of the nonlinear optical properties of molecules, J. Comp. Chem. 11 (1990) 82–87. [39] K. Kamada, M. Ueda, H. Nagao, K. Tawa, T. Sugino, Y. Shmizu, K. Ohta, Molecular design for organic nonlinear optics: polarizability and hyperpolarizabilities of furan homologues investigated by ab initio molecular orbital method, J. Phys. Chem. A 104 (2000) 4723–4734. [40] D.P. Shelton, J.E. Rice, Measurements and calculations of the hyperpolarizabilities of atoms and small molecules in the gas phase, Chem. Rev. 94 (1994) 3–29. [41] E. Kleinpeter, A. Schulenburg, Quantification of the push–pull effect in substituted alkenes, Tetrahedron Lett. 46 (2006) 5995–5997. [42] Gaussian 03M. Revision B.04. M.J. Frisch et al., Gaussian, Inc., Pittsburgh PA, 2003. [43] GAMESS, M.W. Schmidt, K.K. Baldridge, A. Boatz, et al., J Comput Chem 14 (1993) 1347. [44] D.R. Kanis, M.A. Ratner, T.J. Marks, Design and construction of molecular assemblies with large second-order optical nonlinearities. Quantum chemical aspects, Chem Rev. 94 (1994) 195–242. [45] Qingxu. Li, Liping. Chen, Qikai. Li, Zhigang. Shuai, Electron correlation effects on the nonlinear optical properties of conjugated polyenes, Chem. Phys. Lett 457 (2008) 276–278. [46] R.G. Parr, W. Wang, Density Functional Theory of Atoms and Molecules, Oxford University Press, Oxford, UK, 1989. [47] J.C. Slater, A simplification of the Hartree–Fock method, Phys. Rev. 81 (1951) 385–390. [48] S.J. Vosko, L. Wilk, M. Nusair, Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis, Can. J. Phys. 58 (1980) 1200–1211. [49] J.L. Bredas, C. Adant, P. Tackx, A. Persoons, B.M. Pierce, Third-order nonlinear optical response in organic materials: theoretical and experimental aspects, Chem. Rev. 94 (1994) 243–278. [50] C.W. Dirk, W.C. Herndon, F. Cervantes-Lee, H. Selnau, S. Martinez, P. Kalamegham, A. Tan, G. Campos, M. Velez, Squarylium dyes: structural factors pertaining to the negative third-order nonlinear optical response, J. Am. Chem. Soc. 117 (1995) 2214–2225. [51] M. Yang, S. Li, J. Ma, Y. Jiang, An MP2 study of linear polarizabilities and second-order hyperpolarizabilities for centrosymmetric squaraines, Chem. Phys. Lett. 354 (2002) 316–323. [52] G. Maroulis, On the accurate theoretical determination of the static hyperpolarizability of trans-butadiene, J. Chem. Phys. 111 (1999) 583–592. [53] P. Karamanis, G. Maroulis, Single (CAC) and triple (CC) bond-length dependence of the static electric polarizability and hyperpolarizability of HAC@CAC@CAH, Chem. Phys. Lett. 376 (2003) 403–410.
