Linear and second-order nonlinear optical properties of ionic organic crystals Tomasz Seidler, Katarzyna Stadnicka, and Benoît Champagne Citation: The Journal of Chemical Physics 141, 104109 (2014); doi: 10.1063/1.4894483 View online: http://dx.doi.org/10.1063/1.4894483 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Investigation of the linear and second-order nonlinear optical properties of molecular crystals within the local field theory J. Chem. Phys. 139, 114105 (2013); 10.1063/1.4819769 Large second-order optical nonlinearity in a ferroelectric molecular crystal of croconic acid with strong intermolecular hydrogen bonds Appl. Phys. Lett. 102, 162901 (2013); 10.1063/1.4802727 Theoretical investigation on the linear and nonlinear susceptibilities of urea crystal J. Chem. Phys. 128, 244713 (2008); 10.1063/1.2938376 First-principles calculations of band structures and dynamic optical properties of CsCdBr 3 and RbCdI 3 ∙ H 2 O crystals J. Appl. Phys. 99, 013516 (2006); 10.1063/1.2159086 The mechanism of linear and nonlinear optical effects in fluoride crystals J. Appl. Phys. 98, 033504 (2005); 10.1063/1.1977199

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THE JOURNAL OF CHEMICAL PHYSICS 141, 104109 (2014)

Linear and second-order nonlinear optical properties of ionic organic crystals Tomasz Seidler,1,2,a) Katarzyna Stadnicka,1 and Benoît Champagne2,a) 1 2

Faculty of Chemistry, Jagiellonian University, ul, Ingardena 3, 30-060 Kraków, Poland Laboratoire de Chimie Théorique, University of Namur, rue de Bruxelles, 61, B-5000 Namur, Belgium

(Received 7 July 2014; accepted 21 August 2014; published online 11 September 2014) The linear and second-order nonlinear optical susceptibilities of three ionic organic crystals, 4-N,N-dimethylamino-4 -N -methyl-stilbazolium tosylate (DAST), 4-N,N-dimethylamino-4 -N methyl-stilbazolium 2,4,6-trimethylbenzenesulfonate (DSTMS), and 4-N,N-dimethylamino-4 -N phenyl-stilbazolium hexafluorophosphate (DAPSH), have been calculated by adopting a two-step multi-scale procedure, which consists in calculating: (i) the ion properties using ab initio or density functional theory methods and then (ii) in accounting for the crystal environment effects using classical electrostatic models. Provided that the ionic properties are evaluated at the second-order Møller-Plesset level and that the dressing field effects using point charges are accounted for, the agreement with experiment is excellent and enables to explain the origin of the larger χ (2) response of DAPSH with respect to DAST and DSTMS. The study has also demonstrated that including the dressing field leads to a decrease of the χ (2) response of ionic crystals whereas its effect is opposite for molecular crystals. Moreover, the results have also demonstrated that this multi-scale approach can be used to interpret the impact of the nature and position of the counterion on the linear and nonlinear optical susceptibilities of ionic crystals. Finally, it has been shown that the use of a conventional exchange-correlation functional like B3LYP leads to severe overestimations of χ (1) but large underestimations of χ (2) whereas the use of homogeneous dipole field is not recommended because it usually leads to overestimations of the linear and nonlinear optical susceptibilities. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4894483] I. INTRODUCTION

Organic nonlinear crystals are subject to extensive investigations because of their potential applications in modern telecommunication and opto-electronics owing to the electronic origin of their fast responses.1 One of the great advantages of this class of materials is the flexibility in their design. Following a special set of rules one is able to maximize the molecular second-order nonlinear response of the chromophore, the first hyperpolarizability, β. However, the demand for molecular asymmetry usually leads to large ground state dipole moments of the chromophores, which therefore usually crystallize in a centrosymmetric fashion, thus canceling out the macroscopic second-order nonlinearity, χ (2) . One of the strategies to overcome this tendency is to make use of the strong Coulomb interactions,2 which has led to the preparation, among others, of 4-N,N-dimethylamino-4 -N -methyl-stilbazolium tosylate (DAST), 4-N,N-dimethylamino-4 -N -methylstilbazolium 2,4,6-trimethylbenzenesulfonate (DSTMS), and 4-N,N-dimethylamino-4 -N -phenyl-stilbazolium hexafluorophosphate (DAPSH). DAST was described for the first time by Marder et al.3 in 1989 and it is still considered the most promising organic nonlinear optical (NLO) crystal, especially for electrooptic applications and broadband THz wave generation.4 Still, the experimentally determined a) Electronic

addresses: [email protected].

[email protected]

0021-9606/2014/141(10)/104109/10/$30.00

and

non-resonant SHG susceptibilities of DAPSH were shown to be even superior to those of DAST.5 The aim of this study is to demonstrate how the Local Field Theory can successfully calculate and interpret the χ (1) and χ (2) tensors of ionic organic crystals. Indeed, this approach is usually employed to characterize χ (1) and χ (2) of molecular crystals, as illustrated by the recent investigation on 2-methyl-4-nitroaniline (MNA) and 4-(N,N-dimethylamino)3-acetamidonitrobenzene (DAN), of which the unit cells are built from one type of neutral molecules.6 Therefore, in this paper, we concentrate on the aspects of the calculations that are specific to the ionic crystals. The work is organized as follows: (i) Sec. II summarizes the methodology for determining the χ (1) and χ (2) tensors as well as the properties of the isolated ions, (ii) the results are discussed in Sec. III; first, on the isolated cations properties by emphasizing on the influence of the chromophore conformation and of the in-crystal surrounding on the NLO-phore properties, (iii) then, the macroscopic electric properties are presented and analyzed, and (iv) finally the conclusions are drawn.

II. THEORETICAL METHODS AND COMPUTATIONAL DETAILS

Starting from X-ray diffraction data,7–9 the geometry was optimized with CRYSTAL0910, 11 at the B3LYP/631G(d,p) level of theory. Only the fractional coordinates were

141, 104109-1

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optimized while the cell parameters were kept frozen to their experimental values. A. The electrostatic interaction scheme

The Local Field Theory was introduced by Munn12 and Hurst and Munn13, 14 for describing the macroscopic properties of molecular crystals. Since then it has been successfully used for many molecular crystals providing refractive indices and χ (2) tensor components in good agreement with experiment.6, 15–18 The equations relating the molecular electric properties, the polarizability α and the first hyperpolarizability β, to their macroscopic counterparts, the electric susceptibilities χ (1) and χ (2) follow the well-known formulas19 χ (1) (−ω; ω) =

1  T d (ω) · α (−ω; ω), k ε0 V k k

(1)

T DDF T /B3LY P

1  T d (ω + ω2 ) 2ε0 V k k 1 k

k

(2)

k

izability tensors of the kth (sub)ion/molecule in the unit cell, V is the unit cell volume, and ε0 is the dielectric permittivity of the vacuum. d (ω) is the local field tensor relating the k macroscopic E and microscopic (local) F k electric fields F k = d · E. k

(3)

In the dipole approximation, d is evaluated using the dimenk sionless Lorentz factor tensor L,  D kk (ω) (4) d (ω) = k

and

CP KS/B3LY P αij (ω

T DDF T /B3LY P

where α and β are the polarizability and first hyperpolark

αij

2 MP 2 βijMP k (ω) = βij k (ω = 0)

·β (−(ω1 + ω2 ); ω1 , ω2 ) : d (ω1 )d (ω2 ), k

The electric responses of the isolated ions were calculated at two levels of approximation, with density functional theory (DFT) and the B3LYP exchange-correlation (XC) functional as well as with second-order Møller-Plesset (MP2) perturbation theory. The MP2 method is selected because it can closely reproduce the α and β values obtained using higher-level ab initio methods20 like coupled clusters scheme whereas the B3LYP XC functional is chosen to illustrate the ability of density functional theory with conventional XC functionals. More details about the impact of the choice of XC functional can be found in Ref. 21. At the MP2 level, the α and β calculations were performed using the finite field (FF) method.22 The frequency dispersion on the static MP2 properties was included via a modified multiplicative scheme23 employing the time-dependent DFT (TDDFT) and the B3LYP XC functional, αijMP 2 (ω) = αijMP 2 (ω = 0)

χ (2) (−(ω1 + ω2 ); ω1 , ω2 ) =

B. Ion properties

k

  1 D −1 (ω) = 1 − L · α(−ω; ω) . V ε0

Here, α(ω) denotes the diagonal supermatrix of the ion or molecular polarizabilities. In addition to accounting for local field effects, the crystal properties also result from the modifications of the electronic wavefunctions and properties due to the surrounding molecules. These are described by the incrystal polarizing field determined using point-charges. This approach for determining χ (1) and χ (2) of molecular crystals works well because it introduces inhomogeneity of the dressing electric field.6 The atomic charges follow the definition of Mulliken and are obtained from periodic boundary conditions (PBC) B3LYP/6-31G(d,p) calculations using the CRYSTAL09 package.10, 11 The ions are surrounded spherically with charges up to an arbitrary radius (Rcut ) chosen on the basis of the convergence of the electric field calculated at the center of nuclear charge of the ion of interest.

CP KS/B3LY P βtot (ω

= 0)

,

(6)

.

(7)

(ω) = 0)

α ij and β ijk denote the (hyper)polarizability tensor components and β tot is the norm of the hyperpolarizability vector of which the components are defined in Eq. (8). CPKS stands for the coupled-perturbed Kohn-Sham method, which is employed to obtain the static DFT responses. The eigenbasis of the static MP2 polarizability is used as reference frame in Eq. (6). Within the discussion, we also refer to additional  quantities, i.e., (i) the isotropic polarizability αiso = 13 i αii and (ii) β tot the norm of the vectorial part of the first hyperpolarizability, of which the components are defined according to the formula βi =

(5)

βtot

(ω)

1 (β + βj ij + βjj i ). 5 j ijj

(8)

The T convention is used to define the first hyperpolarizabilities. All the calculations of ion properties were performed with the Gaussian 09 package24 and the 6-311++G(d,p) basis set. The molecular (hyper)polarizabilities were distributed evenly over the defined sub-ions (the submolecule analogues). This scheme is known as the RLFTn (Rigorous Local Field Theory n) where n is the number of the heavy atoms. Any subion was defined as the centroid of nuclear charges of a heavy atom with the hydrogen atoms attached to it. The Lorentz factor tensor was calculated for the lattice of sub-ions, which means that no ionic pair was privileged. The χ (1) tensor components are reported in the abc* orthogonal reference frame while the χ (2) (−2ω; ω, ω) tensor elements are reported in the eigenbasis of the calculated dielectric tensor ε(ω) = 1 + χ (1) (ω).

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FIG. 1. DAST – schematic drawing of the ionic pair in the abc* reference axes system with the atomic numbering scheme together with the electric dressing field (left) and its unit cell (right).

III. RESULTS AND DISCUSSION A. Crystal structures

Since 1989 and the study by Marder et al.,3 DAST crystal has been the subject of numerous studies about linear and nonlinear optical properties, as reviewed recently by Jazbinšek et al.4 DAST crystallizes in the Cc space group with cell parameters a = 10.365 Å, b = 11.322 Å, c = 17.893 Å, β = 92.24◦ , Z = 4.7 The ionic pair in the abc* axes system and the unit cell of DAST is shown in Figure 1. Optimized fractional coordinates and Mulliken charges are summarized in Table S1 in the supplementary material.25 DSTMS is very similar to DAST and has been first described in 2007.26 It also crystallizes in the Cc space group with cell parameters a = 10.2665 Å, b = 12.2788 Å, c = 17.9626 Å, β = 93.039◦ , Z = 4.8 The ionic pair in the abc* axes system and the unit cell of DSTMS is shown in Figure 2 while optimized fractional coordinates and Mulliken charges are summarized in Table S2 in the supplementary material.25 DAPSH is one of the DAST analogues. DAPSH in its a polymorphic form crystallizes in the Cc space group with cell parameters a = 19.384 Å, b = 10.636 Å, c = 11.784 Å, β = 125.93◦ , Z = 4.9 The first ion pair of the unit cell (P F6− is at x + 1/2, y, z + 1/2 with respect to the coordinates found in the crystallographic information file (CIF)) and the unit cell are presented in Figure 3. The optimized fractional coordinates together with the Mulliken populations are reported in Table S3 in the supplementary material.25 Selected bond lengths and torsion angles of the NLOphore cations are summarized in Table I. The differences be-

tween the DAST and DSTMS are marginal while DAPSH differs slightly more, owing to the substitution of the methyl by a phenyl group. So, the bond length alternation (BLA) of the C3–C7–C8–C9 segment goes from 0.077±0.001 Å in DAST and DSTMS to 0.070 Å in DAPSH, evidencing a better charge-transfer π -conjugation for the latter. Note however that DAPSH is slightly less planar as indicated by the larger deviations of the torsion angles from 0◦ (or 180◦ ). B. Polarizing field effects on the NLO-phore (hyper)polarizabilities

The static (λ = ∞) electric properties of the cations are presented in Tables S4, S5, and S6 for DAST, DSTMS, and DAPSH, respectively,25 while the dynamic electric properties for λ = 1907 nm in Tables S7 – S9.25 To allow comparison between their molecular properties, the DSTMS and DAPSH cations orientations were adjusted to fit the core of the DAST chromophore in the least square sense. The differences of polarizability and first hyperpolarizability between DAST and DSTMS cations attain at most 1% and reflect their similar geometries. As a result of the change of the N-bearing substituent and the reduced BLA, the (hyper)polarizabilities of DAPSH are larger than those of DAST and DSTMS, by 20%–40%. The dressing electric fields, originating from the surrounding point charges are compared in Table II. For simplicity, we report the fields averaged over the heavy atoms, after reorienting the geometries to fit the DAST cation core. The surrounding electric fields are similar for DAST and DSTMS,

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FIG. 2. DSTMS – schematic drawing of the ionic pair in the abc* reference axes system with the atomic numbering scheme together with the electric dressing field (left) and its unit cell (right).

and about 20% larger than for DAPSH. The calculated polarizing electrostatic potentials ( = ϕ crystal − ϕ cation ) compared in Figure 4 reflect the differences of the field strength in the plane containing the C3C7C8C9 atoms. Note also that the nature and the relative spatial position of the counterion has

an impact on the orientation of the dressing field (Table II), and, as shown below, on the dressed (hyper)polarizabilities. Indeed, the dressing field orientation deviates by 9.3◦ with respect to the charge transfer direction of DSTMS whereas this deviation attains 24.6◦ and 40.6◦ for DAST and DAPSH,

FIG. 3. DAPSH – schematic drawing of the ionic pair in the abc* reference axes system with the atomic numbering scheme together with the electric dressing field (top) and its unit cell (bottom).

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TABLE I. Comparison of selected PBC/B3LYP optimized bond lengths (in Å) and torsion angles (in ◦ ) of the DAST, DSTMS, and DAPSH cations (see Figures 1–3 for the atom labels).

N1C6 N1C1 C1C2 C2C3 C3C4 C4C5 N1C5 C3C7 C7C8 C8C9 C9C10 C10C11 C11C12 C12C13 C13C14 C9C14 C12N2 C2C3C7C8 C3C7C8C9 C7C8C9C10 C1N1C6C17

DAST

DSTMS

DAPSH

1.469 1.358 1.372 1.418 1.415 1.376 1.357 1.440 1.363 1.442 1.413 1.384 1.421 1.427 1.380 1.414 1.364 − 1.6 − 178.2 − 176.3 ...

1.469 1.358 1.374 1.418 1.415 1.373 1.357 1.439 1.364 1.442 1.413 1.384 1.421 1.425 1.380 1.415 1.363 − 1.3 − 177.6 − 177.6 ...

1.446 1.364 1.369 1.419 1.421 1.374 1.365 1.437 1.368 1.439 1.413 1.384 1.419 1.423 1.377 1.418 1.365 0.6 179.9 165.1 33.9

respectively. Moreover, the angle between the dressing electric fields for DAST and DAPSH is ∼45◦ . The dressing field effects on the cation properties are comparable in the case of DAST and DSTMS and larger than for DAPSH. For DAST and DSTMS, the decrease of the static polarizability for the dressed cations with respect to the isolated one is of the order of 6% and 11% for the static α iso at the B3LYP and MP2 levels while at λ = 1907 nm it is of the order of 7% and 12%, re-

TABLE II. Dressing electric fields (in GV/m), averaged over the heavy nuclei, of the DAST, DSTMS, and DAPSH cation cores and the angle between the electric field and the charge-transfer N1N2 axis (the DSTMS and DAPSH vectors are transformed to fit the orientation of the DAST cation core).

DAST DSTMS DAPSH

Fx

Fy

Fz

|F |

− 1.12 − 1.03 − 0.65

0.02 − 0.32 0.16

− 0.06 0.08 0.52

1.12 1.08 0.85



(N 1N 2, F ) 155.4 170.7 139.4

spectively. For DAPSH, these dressing effects are smaller by 33% and 3% at the B3LYP and MP2 levels, respectively. The situation gets much different for the first hyperpolarizability. So, at the MP2 level, the static β tot of DAST and DSTMS decrease by about 31% and the effect is slightly amplified at λ = 1907 nm. In the case of DAPSH, as a result of the smaller and differently oriented dressing field, the decrease of β tot is about 50% smaller at the MP2 level (and only marginally different when using B3LYP). Note that the situation is quite different at the B3LYP level, where the β tot decreases are of the order of 5%–8% for the two first crystals and where β tot of DAPSH is mostly unchanged. In addition, after inclusion of the dressing effects, the MP2 β values remain larger than the B3LYP ones by 45%–55%. The polarizability and first hyperpolarizability of the anions in the static limit with and without taking into account the in-crystal polarizing electric field are summarized in Table S10 in the supplementary material.25 The crystal dressing field reduces strongly the first hyperpolarizability of the tosylate and 2,4,6-trimethylbenzenesulfonate anions. So, at the MP2 level, their β tot decrease by 87% and 82%, respectively. The small nonzero hyperpolarizability of hexafluorophosphate anion results from its geometry, that deviates slightly from ideal octahedral ones.

FIG. 4. PBC/B3LYP/6-31G(d,p) polarizing electrostatic potential = ϕ crystal − ϕ cation in the cationic chromophore plane (the increment is 0.004 a.u.).

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C. Electric susceptibilities

1. Linear optical properties

The calculated χ (1) tensor components are listed in Table III. The results for DAST agree well with experiment, especially when using MP2 molecular properties. The (1) tensor comonly difference concerns the sign of the χ13 ponent. The experimentally determined angle to rotate the abc* axes system into the indicatrix axes was reported to be −5.4◦ .27 The calculations lead to a similar absolute value with frequency dispersion smaller than 0.5◦ but the sign is opposite (+5.5 − 6◦ ). From structure analysis, it is found that, after projection on the (010) crystallographic plane, the charge-transfer axis (considered approximately to be parallel to the N–N direction) and the a crystallographic axis form an angle of ∼3.2◦ while the angle between the projection of the orthogonal direction (i.e., the direction normal to the aromatic plane) and the c crystallographic axis is ∼5.2◦ . As a matter of fact, the MP2-based calculated refractive indices of DAST also nicely match the experimental values over the whole frequency range (Figure 5). Note that this agreement would not have been met without accounting for the in-crystal polarizing field effect, which decreases the cation polarizability. When employing B3LYP molecular properties, the calculated χ (1) tensor components are typically 10% too large and the largest refractive index is overestimated by 0.1. A similar analysis can be performed for DSTMS (Table III and Figure 6). Note that the experimentally determined angle of inclination of the x indicatrix with respect to the a crystallographic axes 3.6(3)◦ is slightly smaller than that obtained in our calculations: 5◦ –6◦ (with frequency dispersion of ∼1◦ ). The agreement between

FIG. 5. Experimental and calculated frequency dispersion of the refractive indices of DAST.

experimental and MP2-calculated refractive indices is again almost perfect for both the static values and for the frequency dispersion in the λ = ∞ to λ = 600 nm range. For DAPSH, again a better reproduction of the experimental χ (1) results is achieved at MP2 level of theory. The wavelength dispersion of the refractive indices, shown in Figure 7 fits very well the experimental one with only slight underestimation of nx and overestimations of ny and nz . The small discrepancies between experiment and our calculations are partially due to slightly different orientations of the optical indicatrix with respect to the crystallographic frame. The experimentally determined angle between the x indicatrix and the a crystallographic axes is 20(2)◦5 while calculations give ∼17.5◦ .

TABLE III. χ (1) tensor components of DAST, DSTMS, and DAPSH at different wavelengths as calculated using Eq. (1) and different methods in comparison with experiment. The molecular properties are evaluated at the B3LYP and MP2 levels while accounting for the charge field. Using the MP2 molecular properties, additional calculations were performed setting the charge field to zero or by ignoring the counterion contribution to the charge field. The experimental data are taken from Refs. 4, 26, and 5 respectively. DAST (1)

(1)

DSTMS (1)

Method

λ/nm

χ11

χ22

χ33

Exp

∞ 1907 1542 1318

3.303 3.440 3.523 3.621

1.513 1.547 1.568 1.593

1.466 1.484 1.495 1.508

B3LYP Charge field

∞ 1907 1542 1318

3.684 3.856 3.958 4.076

1.553 1.579 1.593 1.610

MP2 No field

∞ 1907

4.231 4.498

MP2 Charge field

∞ 1907 1542 1318 ∞ 1907 1542 1318

MP2 Charge field (anion ignored)

(1)

χ13

(1)

(1)

(1)

DAPSH (1)

(1)

(1)

(1)

(1)

χ11

χ22

χ33

χ13

χ11

χ22

χ33

χ13

0.175 0.187 0.193 0.202

3.098 3.220 3.293 3.379

1.647 1.674 1.690 1.708

1.500 1.504 1.507 1.510

−0.101 −0.108 −0.113 −0.118

3.554 3.726 3.834 3.965

1.455 1.495 1.519 1.546

1.415 1.438 1.452 1.469

−0.889 −0.951 −0.990 −1.038

1.443 1.451 1.456 1.461

− 0.223 − 0.237 − 0.244 − 0.254

3.441 3.592 3.682 3.786

1.683 1.712 1.729 1.749

1.472 1.480 1.484 1.489

−0.200 −0.208 −0.214 −0.219

3.753 3.949 4.066 4.204

1.700 1.732 1.751 1.772

1.465 1.491 1.506 1.523

−0.792 −0.848 −0.881 −0.920

1.632 1.669

1.451 1.460

− 0.265 − 0.288

3.899 4.130

1.769 1.811

1.475 1.484

−0.222 −0.236

4.310 4.587

1.735 1.776

1.519 1.555

−0.994 −1.077

3.382 3.532 3.621 3.724

1.514 1.537 1.550 1.565

1.442 1.450 1.455 1.460

− 0.204 − 0.217 − 0.225 − 0.234

3.190 3.323 3.403 3.494

1.625 1.651 1.667 1.684

1.467 1.475 1.480 1.485

−0.188 −0.197 −0.202 −0.208

3.487 3.663 3.768 3.890

1.666 1.694 1.711 1.730

1.425 1.450 1.464 1.481

−0.713 −0.765 −0.797 −0.833

2.539 2.671 2.750 2.841

0.892 0.910 0.921 0.933

0.626 0.629 0.630 0.632

− 0.122 − 0.130 − 0.135 − 0.140

2.304 2.424 2.495 2.577

0.876 0.896 0.908 0.922

0.566 0.568 0.569 0.570

−0.091 −0.096 −0.098 −0.102

3.186 3.351 3.450 3.565

1.438 1.465 1.481 1.500

1.278 1.302 1.316 1.333

−0.700 −0.751 −0.781 −0.817

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FIG. 6. Experimental and calculated frequency dispersion of the refractive indices of DSTMS.

2. Nonlinear optical properties

The χ (2) tensor components are presented in Table IV. For DAST, the MP2-based results reproduce the experimen(2) values and their frequency dispersion to a very good tal χ111 extent up to λ = 1542 nm whereas at λ = 1318 nm they underestimate it by about 50%. This is associated with the difficulty of the method to describe the NLO responses close to the resonance regime. On the other hand, the B3LYP XC functional, as employed to evaluate the dressed molecular (hyper)polarizabilities, leads to large underestimations. The (2) (2) and χ212 compoKleinman symmetry conditions for the χ122

FIG. 7. Experimental and calculated dispersion of refractive indices of DAPSH.

nents are satisfied by the calculated values, at all the wavelengths, which contrasts with experiment. The calculated val(2) ]. For ues are closer to the smallest experimental value [χ212 DSTMS, the MP2 dressed (hyper)polarizability values lead (2) responses slightly below those of DAST (15% at to χ111 λ = 1907 nm) whereas no clear trend could be estimated from experiment. The MP2-based results for DSTMS are within one error bar of the experimental values. Like for DAST, contrary to experiment, no deviation from Kleinman symmetry (2) (2) and χ212 conditions is observed in the calculations for χ122 and the best match is obtained with the smallest experimen(2) ]. Again, the B3LYP values are systematically tal value [χ122

TABLE IV. χ (2) tensor components (in pm/V) of DAST, DSTMS, and DAPSH at different wavelengths as calculated using Eq. (2) and different methods in comparison with experiment. The molecular properties are evaluated at the B3LYP and MP2 levels while accounting for the charge field. Using the MP2 molecular properties, additional calculations were performed setting the charge field to zero or by ignoring the counterion contribution to the charge field. The experimental data are taken from Refs. 4, 26, and 5 respectively. DAST (2)

(2)

DSTMS (2)

(2)

(2)

DAPSH (2)

(2)

(2)

(2)

(2)

(2)

(2)

(2)

Method

λ/nm

χ111

χ122

χ212

χ111

χ122

χ212

χ111

Exp

∞ 1907

420(110)

64(8)

50(6)

428(40)

62(8)

70(8)

490(80) 580(80)a

1542 1318

580(30) 2020(220)

82(6) 192(18)

78(4) 106(24)

B3LYP Charge field

∞ 1907 1542 1318

215.2 355.7 509.0 927.0

25.7 41.2 57.2 95.6

25.7 41.1 57.1 95.2

176.4 291.3 415.9 734.4

30.0 47.7 65.8 106.9

30.0 47.6 65.8 106.8

239.0 404.6 602.9 ...

17.3 30.0 44.6 ...

17.3 33.9 55.8 ...

2.0 8.6 23.7 ...

2.0 6.0 14.3 ...

1.8 2.9 3.6 ...

1.8 3.9 7.4 ...

MP2 No field

∞ 1907

565.8 1095.5

66.6 121.5

66.6 120.8

458.4 875.4

75.2 135.7

75.2 135.3

515.2 1002.1

43.7 83.0

43.7 91.1

9.9 34.0

10.4 46.6

6.0 9.6

6.0 14.6

MP2 Charge field

∞ 1907 1542 1318

275.5 447.5 626.6 1035.7

32.7 52.0 71.4 112.0

32.7 51.9 71.2 111.5

233.0 378.0 528.8 869.2

38.5 60.9 83.4 130.4

38.5 60.9 83.4 130.2

334.3 558.1 809.4 ...

32.2 53.3 76.2 ...

32.2 54.6 81.3 ...

3.5 10.7 24.6 ...

3.5 9.0 18.5 ...

3.4 5.5 7.4 ...

3.4 6.4 10.4 ...

∞ 1907 1542 1318

242.7 392.3 547.9 905.2

28.8 46.1 63.7 102.4

28.8 46.0 63.6 102.6

214.6 347.3 485.3 795.9

33.6 53.7 74.1 117.6

33.6 53.7 74.0 117.3

304.6 507.1 731.9 ...

30.3 49.7 70.5 ...

30.3 51.0 75.1 ...

4.0 8.8 16.4 ...

3.4 6.1 9.5 ...

4.0 8.8 16.4 ...

4.0 9.9 20.5 ...

MP2 Charge field (anion ignored) a

χ122

χ212

30(4)

χ113

χ311

12(10) 10.2a

χ133

χ313

10(8) 16.0a

Second experimental estimate.

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smaller than the MP2 ones and in worse agreement with experiment. In the case of DAPSH, too small coherence length and crystal surface imperfections made difficult the measurement of the χ (2) tensor components. In fact, two sets are available and they differ by slightly more than one standard deviation.5 Still, both suggest a stronger NLO activity of DAPSH with respect to DAST and DSTMS. This conclusion is supported by (2) value both the MP2 and B3LYP calculations. The MP2 χ111 is in good agreement with the largest experimental value [580 χ (2)

pm/V] whereas the 111 (2) ratio (10.5), predicted at the same χ122 level, is in better agreement with the smaller experimental es(2) value at λ = 1542 nm timate (13 ± 0.9). The calculated χ111 attains 809 pm/V, which places DAPSH among the most efficient organic NLO crystals. Note that λ = 1318 nm is beyond the first resonance and the corresponding χ (2) responses are therefore not reported. The neglect of the polarizing inhomogeneous electric field leads to very strong overestimations of the χ (2) tensor (2) , the largest tensor comcomponents. By considering χ111 ponent, this overestimation attains, in the static limit, about 100% for DAST and DSTMS and only 55% for DAPSH. At λ = 1907 nm, this overestimation gets even larger and amounts to 103%–145% and 80%, respectively. So, contrary to MNA and DAN where it enhances the macroscopic nonlinear optical susceptibilities,6 the dressing field reduces substantially those of ionic crystals and the larger the field, the larger this reduction as illustrated by comparing DAST and DSTMS to DAPSH.

IV. FURTHER DISCUSSIONS, CONCLUSIONS, AND OUTLOOK

The linear and nonlinear optical susceptibilities of three ionic crystals have been calculated by adopting a two-step multi-scale procedure, which consists in calculating first the ion/molecular properties using ab initio or DFT methods and then in accounting for the crystal environment effects using classical electrostatic models. Owing to the good performance of this method to reproduce and analyze experimental results over a large range of wavelengths, this study extends the range of applicability of this multi-scale approach from molecular crystals as discussed in a previous paper6 to ionic crystals and opens the way for designing organic crystals for NLO applications.28 So, the method confirms the larger χ (2) response of DAPSH with respect to DAST and DSTMS and attributes the difference to (i) a larger molecular first hyperpolarizability and to (ii) smaller crystal polarizing field effects. Like in Ref. 6, the present study has also substantiated the key role played by the crystal dressing field on the linear and nonlinear optical responses and has evidenced a crucial difference between molecular and ionic crystals. Indeed, for the former, the crystal dressing field enhances the macroscopic responses whereas for the latter it leads to a strong reduction, typically by factor of 2. This difference between the ionic and molecular crystals originates from the direction of the incrystal dressing field, which reduces or enhances the charge transfer between the donor and acceptor groups, respectively.

FIG. 8. DAST cation together with five different choices of anion and their positions that can be used for determining the dipole electric field.

These results have also demonstrated that this multi-scale approach can be used to interpret the impact of the nature and position of the counterion on the linear and nonlinear optical susceptibilities of ionic crystals. Moreover, the good agreement between theory and experiment is achieved provided the molecular properties are evaluated at the MP2 level whereas a conventional XC functional like B3LYP leads to large overestimations of χ (1) but large underestimations of χ (2) . In these calculations, the dressing field has been determined by using point charges surrounding the chromophores, which enables to account for its inhomogeneities. These were calculated using periodic boundary conditions. Following previous works,6, 15 it was however interesting to assess the performance of the alternative scheme that is based on the determination of a homogeneous dipole electric field. However, the charged nature of the components of the unit cell complicates the situation. One variant of that scheme consists in carrying out the usual self-consistent field procedure for the cation and the anion separately and in obtaining a dipole field for evaluating the ion properties of each of them. In that case, to overcome the origin-dependency problem, the dipole moment is calculated at the barycenters of the ions. Another variant consists in determining a unique dipole electric field for the cation/anion pair, which raises the question of the definition of several cation/anion pairs as shown in Figure 8. These different dipole field approaches were then compared to the charge field approach in the case of DAST crystal and static property calculations performed with the B3LYP/6311++G(d,p) method (Table V). In comparison to the charge field results, when adopting the scheme with a different dipole field for the cation and the anion, (i) the dressing field is substantially increased and is differently oriented with respect to the unit cell components, (ii) the χ (1) tensor components are overestimated, in particular the largest one (by 34%), lead(2) ing to large discrepancies with experiment, and (iii) the χ111 tensor component increases by about 60% with respect to the charge field case. In the latter case, this overestimation

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J. Chem. Phys. 141, 104109 (2014)

TABLE V. Results from calculations performed using the uniform lattice dipole polarizing electric field (λ = ∞; the electric field F is given in GV/m while χ (2) in pm/V) for different ways of defining the electric field in comparison with the charge field approach. Results are provided at the 0th cycle of the selfconsistent field procedure as well as for the last cycle corresponding to convergence of the dipole field within 0.05 GV/m. In the case of the procedure with a different dressing field for the cation and anion, the electric field components are reported in the format cation anion .

Charge field Cation and anion Ion pair 1 Ion pair 2 Ion pair 3 Ion pair 4 Ion pair 5

Cycle

χ11

(1)

χ22

(1)

χ33

(1)

χ13

(1)

χ111

(2)

χ122

Fx

... 0 2 0 1 0 1 0

3.684 4.209 4.941 4.977 4.377 4.240 3.859 − 31.214

1.553 1.636 1.724 1.915 1.918 1.810 1.752 0.394

1.443 1.449 1.441 1.610 1.612 1.553 1.549 − 6.130

− 0.223 − 0.255 − 0.299 − 0.466 − 0.322 − 0.221 − 0.288 16.285

25.7 34.4 39.9 64.3 47.0 48.2 30.5 − 358.9

− 1.12

0 2 0 2

6.080 3.767 3.890 4.344

2.145 1.779 1.741 1.807

1.603 1.596 1.598 1.560

− 0.459 − 0.287 − 0.193 − 0.110

215.2 281.3 342.3 554.6 412.0 353.5 234.5 − 127 205.7 Divergence 1078.1 219.6 202.8 337.0

originating from the dipole field is partially compensated by the underestimation of the first hyperpolarizability evaluated at the B3LYP level. The results obtained for the ion pairs are even more problematic because they exhibit large variations as a function of the choice of ion pair. Nevertheless, in some cases (anions in positions 2 and 4) the agreement with the charge field results is better than with the first variant but there is no justification to prefer a priori one ion pair than another. These results also show that in some cases, the procedure for obtaining the dressing field is not convergent. Without extending this analysis to calculation using MP2 molecular properties, these dipole field results and their comparisons with the charge field results demonstrate the performance of the latter and the strong limitations of the former. The impact of the anion properties on the macroscopic electric properties was also assessed at the MP2 level. This is achieved by setting to zero the (hyper)polarizabilities of the anions in the unit cell. In DAST and DSTMS, the anion ac(1) (1) )–60% (χ33 ) of the χ (1) tensor compocounts for 25% (χ11 nents, resulting in a decrease of n1 and n2 by 10%–15% and of n3 by 18%–20% (Table III). In DAPSH, the influence of the hexafluorophosphate anion is less prominent because its polarizability is smaller than for the other counterions. Indeed, (1) (1) (1) (1) , χ22 , and χ33 by 8%–15% and χ13 removing it impacts χ11 by only 2%, and therefore the refractive indices by less than 5%. The anion contributions to the χ (2) tensor components (Table IV) attain 8%–13% for DAST and DSTMS. In the case of DAPSH, it is responsible for up to 10% of the dominant (2) χ (2) tensor components while the smaller components, χ133 (2) and χ313 , are reduced. The DAPSH case where the anion first hyperpolarizability is negligible illustrates nicely that the impact of the anion on χ (2) mostly originates from the local field tensors. ACKNOWLEDGMENTS

This research was supported in part by PL-Grid Infrastructure as well as by the Belgium government (IUAP N◦

(2)

242.6 28.1 22.1 41.5

2.0 −3.1 2.1 −3.1

Fy 0.02 0.7 1.1 0.7 1.1

Fz − 0.06 1.4 3.0 1.4 2.9

− 0.1 − 0.1 − 2.2 − 2.2 19.8

− 5.1 − 5.2 2.5 2.5 37.9

− 6.1 − 6.1 5.7 5.7 − 32.6

− 5.3 − 3.5 0.0 0.1

− 5.4 − 4.3 2.9 2.9

6.4 6.3 − 5.3 − 5.2

|F | 1.12 2.5 4.5 2.7 4.4

7.9 8.0 6.6 6.6 53.8 9.9 8.4 6.0 6.0

P7/05, Functional Supramolecular Systems). T.S. acknowledges the support from a Project operated within the Foundation for Polish Science MPD Programme co-financed by the European Union’s (EU) European Regional Development Fund as well as the financial support from IUAP N◦ P7/05. The calculations were performed on the Technological Platform of High-Performance Computing of the Consortium des Equipements de Calcul Intensif, for which we gratefully acknowledge the financial support of the FNRS-FRFC (Convention Nos. 2.4.617.07.F and 2.5020.11), and of the University of Namur. 1 Non-linear

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J. Chem. Phys. 141, 104109 (2014) 25 See

supplementary material at http://dx.doi.org/10.1063/1.4894483 for investigation of linear and second-order nonlinear optical properties of ionic organic crystals within the local field theory. 26 L. Mutter, F. D. J. Brunner, Z. Yang, M. Jazbinšek, and P. Günter, J. Opt. Soc. Am. B 24, 2556 (2007). 27 G. Knöpfle, R. Schlesser, R. Ducret, and P. Günter, Nonlinear Opt. 9, 143 (1995). 28 T. Seidler, K. Stadnicka, and B. Champagne, “Second-order nonlinear optical susceptibilities and refractive indices of organic crystals from a multiscale numerical simulation approach,” Adv. Opt. Mater. (published online 2014). Note that in the chemical structure of DAPSH in Table I, a N atom has been misplaced. The calculations were performed on the correct structure.

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