QTAIM-Based Scheme for Describing the Linear and Nonlinear

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QTAIM-Based Scheme for Describing the Linear and Nonlinear Optical Susceptibilities of Molecular Crystals Composed of Molecules With Complex Shapes Tomasz Seidler, Anna Krawczuk, Benoît Champagne, and Katarzyna Stadnicka J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b10026 • Publication Date (Web): 28 Jan 2016 Downloaded from http://pubs.acs.org on February 4, 2016

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The Journal of Physical Chemistry

QTAIM-Based Scheme for Describing the Linear and Nonlinear Optical Susceptibilities of Molecular Crystals Composed of Molecules with Complex Shapes Tomasz Seidler,∗,†,‡ Anna Krawczuk,‡ Benoˆıt Champagne,∗,† and Katarzyna Stadnicka‡ Laboratoire de Chimie Théorique, Unité de Chimie Physique Théorique et Structurale, University of Namur, rue de Bruxelles, 61, B-5000, Namur, Belgium, and Faculty of Chemistry, Jagiellonian University, ul, Ingardena 3, 30-060, Kraków, Poland E-mail: [email protected]; [email protected]

∗

To whom correspondence should be addressed University of Namur ‡ Jagiellonian University †

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Abstract The Quantum Theory of Atoms in Molecules (QTAIM) to distribute the molecular polarizability tensors over submolecular sites is employed to improve the prediction of local field tensors of molecular crystals and therefore of their linear and second-order nonlinear optical susceptibilities. This extension of the two-step multi-scale procedure is intended to better describe crystals built of molecules having complex shapes. When combined with a simple charge embedding approach to account for the crystal field effects on the molecular (hyper)polarizabilities this QTAIM local field theory (Q-LFT) approach is efficient to predict the χ(1) and χ(2) tensor components. Moreover, it does not require substantial computational resources because the largest calculations are performed on the individual molecules. This is illustrated by considering derivatives of (S )-2-(α-methylbenzylamino)-5nitropyridine (MBANP) as well as 2-methyl-4-nitroanline (MNA), which is a prototypical push-pull π-conjugated compound. In the later case, the use of the Q-LFT only leads to minor differences in the χ(1) and χ(2) tensor components with respect to the standard approach where the polarizability is equipartitioned. In the case of the MBANP derivatives, the Q-LFT scheme leads to systematic decrease of the linear and nonlinear optical responses. This generally improves the agreement between the calculated and experimental refractive indices and second-order nonlinear optical susceptibilities. Indeed, the standard partitioning scheme leads to an overestimation of the phenyl out-of-plane polarization component, and therefore of the refractive indices. The χ(2) variations among the MBANP derivatives have further been analyzed in terms of molecular geometry, crystal polarizing field, and crystal packing.

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Introduction Molecular shape and its dynamics are key parameters governing the properties and the reactivity of molecules, supramolecules, and polymers. This applies to the field of nonlinear optics where the shape and symmetry of the chromophore impact strongly their second- and third-order nonlinear optical (NLO) responses, not only at the molecular level but also at the macroscopic scale. 1,2 This is particularly the case of the molecular first hyperpolarizability (β) and its macroscopic counterpart, the second-order NLO susceptibility (χ(2) ), which vanishes for centrosymmetric objects. Additionally, tuning the shape of the NLO-phore and the in-crystal intermolecular interactions is a tool for controlling the crystal packing and therefore foe optimizing the crystal χ(2) responses. 3–7 As presented in this paper, the shape of the NLO-phore influences also the performance requirements on the theoretical approaches employed to predict these macroscopic properties, which raises the need for developing improved methods. This paper addresses the elaboration of such methods to predict the χ(1) and χ(2) responses of crystals built of molecules of any shape, ranging from conventional quasi linear push-pull π-conjugated compounds to molecules of complex shape. Predicting the linear and nonlinear optical properties of molecular crystals remains a challenge for quantum chemistry approaches because accurate determination of the (hyper)polarizabilities of molecular or ionic building blocks requires high-level quantum-chemical methods, 8–13 whereas the effects of the crystal surrounding are by definition long-range. As reviewed by one of us, 14 different approaches have been elaborated, from the simple oriented gas approximation, 15–17 to the cluster scheme, 17,18 and to periodic boundary conditions (PBC) calculations. 19–22 The strategy that we follow relies on a multi-scale approach, combining the evaluation of the molecular properties using first principles methods and the inclusion of the crystal field effects by employing an electrostatic interactions scheme. This method, which follows early developments by Munn, Dykstra, and co-workers, 23–28 has already been employed successfully to conventional push-pull π-conjugated systems like 2-methyl-4-nitroaniline (MNA) and 4-(N,N -dimethylamino)-3-acetamidonitrobenzene 3



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(DAN) 13,29 as well as to ionic organic crystals like 4-N,N -dimethylamino-4´-N´-methylstilbazolium tosylate (DAST). 30,31 Thus, in recent works, the impact of the geometry, the description of the dressing field, the amplitude of the zero-point vibrational average (ZPVA) contributions to the molecular (hyper)polarizabilities, and the choice of level of approximation [including the exchange-correlation functional in density functional theory (DFT) treatments] have been addressed. 13,29 In Ref. 13 it was demonstrated that the approach based on MP2 first hyperpolarizabilities give χ(1) and χ(2) values in very close agreement with those obtained with a higher-order method like CCSD. Furthermore, in Ref. 32 the MP2-based χ(1) and χ(2) values were demonstrated to provide very good correlation with experimental data. In the present study, the molecules we focus on are different from MNA, DAN, or DAST. They have a more complex shape, they are nonplanar and the above method is expected to present limitations. Indeed, in our previous studies, in order to describe the screening of the external electric field by the crystal environment, i.e. to evaluate the local field experienced by the NLO-phores in the crystal, the polarizability tensor of the molecule or ion is distributed equally on all the non-hydrogen atoms 33–35 which is an approach very close to the use of the Lorentz tensors averaged over the heavy atom sites. 36,37 Therefore, a more elaborate scheme is proposed here, which enables to describe the susceptibilities of crystals built of molecules of complex shape, like (S )-2-(α-methylbenzylamino)-5-nitropyridine (MBANP) and its derivatives (Scheme 1). 38 These crystals are chosen because they have been shown to be NLO material candidates, 39–42 while their NLO responses depend on the presence of one enantiomer or both in a racemate. Indeed, the MBANP compound crystallizes in monoclinic system (P 21 space group) as pure enantiomer and in orthorhombic (Aba2 space group) as racemate and these two forms have very different χ(2) responses. Two similar compounds are also included: 3-methyl and N -methyl MBANP derivatives, 38 further called 3MMBANP and NMMBANP, respectively. The polarizability distribution scheme that is adopted here is based on Bader’s Quantum Theory of Atoms in Molecules (QTAIM). 43 The QTAIM partitioning was shown to

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the QTAIM polarizability partitioning can be combined with the standard local field theory (LFT) to provide accurate values of the χ(1) and χ(2) crystal responses via an improved description of the local field tensor. This is the subject of this work. The paper is organized as follows. Next Section summarizes the basic equations and computational methods used to obtain the macroscopic optical susceptibilities of crystalline phases and the underlying molecular properties. The results of bulk properties are presented in the following Section. The effect of molecular packing on the χ(2) is then discussed and, finally, conclusions are drawn in the last Section.

Computational methods and models Linear and nonlinear optical susceptibilities The induced macroscopic polarization of a crystal P − P 0 , in the dipole approximation, is defined according to: 1 . [P − P 0 ] = χ(1) · E + χ(2) : EE + χ(3) ..EEE + . . . ε0

(1)

where P 0 is the permanent polarization vector, E is the macroscopic external electric field vector, χ(1) , χ(2) , and χ(3) are the successive electric susceptibility tensors. In insulating molecular crystals (in contrast to conducting ones), the macroscopic induced polarization P can be considered as a sum of molecular induced dipole moments pi whereas the interactions between the molecules only modify these microscopic properties. Within these assumptions, Refs. 23–28 provided the χ(1) and χ(2) expressions in terms of their molecular α and β counterparts:

χ(1) (−ω; ω) =

1 X T d (ω) · αk (−ω; ω) ε0 V k k

6


(2)

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χ(2) (− (ω1 + ω2 ) ; ω1 , ω2 ) =

1 X T d (ω1 + ω2 ) · β (− (ω1 + ω2 ) ; ω1 , ω1 ) : dk (ω1 ) dk (ω2 ) 2ε0 V k k k (3)

The sum over k runs over all (sub)molecules of the unit cell. αk and β are the k -th k

(sub)molecule polarizability and first hyperpolarizability tensors, ε0 is the dielectric permittivity of vacuum, V is the unit cell volume, and

dk (ω) =

X

Dkk′ (ω)

(4)

k′

D

−1

1 (ω) = 1 − L · α (−ω; ω) V ε0

(5)

where L is the Lorentz factor tensor, α is the supermatrix of (sub)molecular polarizabilities. Thus, dk interrelates the local F k electric field (on the k -th (sub)molecule) with the macroscopic electric field E: F k = dk · E

(6)

This is known as the local field theory (LFT) approach. The calculations of the molecular properties should also take into account the polarizing electric field originating from the surrounding molecules. This field can be approximated by the dipole field, defined homogenous at the (sub)molecular scale, originating from static dipole moments located at the nodes of the crystal lattice 33,51 or by an inhomogeneous field modeled by atomic point charges. 29 In the current work, the second approach is used and the point charges are evaluated within the Mulliken scheme from periodic boundary conditions (PBC) B3LYP/6-31G(d,p) calculations, performed with the CRYSTAL09 package. 52,53 The reliability of this choice has been already discussed in a recent paper. 54 This general approach, which includes both local and in-crystal polarizing field belongs, therefore to electrostatic interaction schemes. 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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Molecular responses and QTAIM distribution Following previous works, the molecular properties (α and β) were obtained at the MP2/6311++G(d,p) levels of approximation using Gaussian09 55 (for MBANP, additional MP2 calculations were performed with the 6-311++G(2d,2p) basis set). In order to evaluate Eqs. 2 - 6, the polarizability partitioning follows the method developed by Laidig and Bader. 50 Provided the electron density is divided between the “atoms” of a molecule, oneelectron molecular properties such as the dipole moment µ can be decomposed into atomic contributions µΩ following the definition µ=

Xh

µp (Ω) + q (Ω) X Ω

Ω

i

(7)

where

µp (Ω) = −

ˆ

Ω

rΩ ρ (r) dτ

(8)

is the first electric moment of atom Ω (rΩ = r − X Ω ), X Ω is the position vector of its nucleus, and

q (Ω) = ZΩ −

ˆ

ρ (r) dτ

(9)

Ω

is its atomic charge. The first term in the brackets of Eq. 7, µp (Ω), represents the polarization contribution to the dipole moment while the second is the origin-dependent intramolecular charge-transfer (also called charge-translation) 56 contribution µc (Ω) = q (Ω) X Ω . A somehow more sophisticated redefinition of the latter term lifts the origin-dependency: Nb (Ω)

µc (Ω) =

X

[X b (Ω|Λ) − X Ω ] Q ΛG Ω

Λ=1

8


(10)

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where Nb (Ω) is the number of bond critical points connected to the nucleus of atom Ω, X b (Ω|Λ) is the position of the bond critical point between atom Ω and Λ, and Q ΛG Ω is the 50 net charge of the group ΛG The differentiation of Ω that is bonded to atom Ω via atom Λ.

the atomic dipole moments, µ (Ω) = µp (Ω) + µc (Ω), with respect to external static electric fields yields the atomic contributions to the molecular static dipole polarizability tensor. This approach, initially used for simple molecules, has also been employed to treat complex molecules 57 as well as supramolecular systems involving weak electrostatic interactions like hydrogen bonds. 56,58 Note that the atomic polarizabilities sum up to the molecular polarizability. To reach a sufficient numerical accuracy on the distributed polarizabilities, the Romberg procedure with field strengths of 0.003 and 0.006 a.u. was used. The density integration within the Bader’s QTAIM methodology was performed with the AIMAll program package. 59 The accuracy of the whole procedure was monitored by comparing the sums of the atomic contributions to the static (MP2 and CPKS) molecular polarizabilities, showing maximum deviations of 0.5 a.u. Since this method does not account for the coupling between the atomic volumes and atomic charges, the atomic polarizability tensors obtained with this method are not symmetric. 58 The antisymmetric parts of the atomic polarizability tensors were then discarded since they do not have a clear physical meaning and they sum up to zero for the molecule. The frequency dispersion of the atomic polarizabilities was introduced via a multiplicative scheme: at at αij (ω) = αij (ω = 0)

αij (ω) αij (ω = 0)

(11)

at where αij and αij are ij -th components of the molecular and atomic polarizability tensors,

respectively. The atomic polarizabilities were then summed up to obtain group polarizabilities, which are then used within the LFT scheme. This procedure is referred to as Q-LFT (QTAIM - Local Field Theory). Contrary to the polarizability tensor, the first hyperpolarizability tensor is partitioned in the standard (simplified) way, i.e. equally on the groups formed by each heavy atom and the hydrogen atoms attached to them. 9



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Frequency dispersion of molecular properties The electric responses of the isolated molecules were calculated with the second-order MøllerPlesset (MP2) perturbation theory. Like in previous investigations, 13,29,30 the finite field (FF) method was used to calculate the static (hyper)polarizabilities from differentiating once (α) or twice (β) the dipole moment. The frequency dispersion of the MP2 properties is described by employing a modified multiplicative scheme, where the static MP2 values are combined with the static coupled-perturbed Kohn-Sham (CPKS) and dynamic time-dependent DFT (TDDFT) tensors evaluated using the B3LYP exchange-correlation (XC) functional. The static MP2 and CPKS/B3LYP polarizability tensors are diagonalized separately with the aid of the transformation matrices X ′ and X ′′ , respectively. The TDDFT/B3LYP polarizabilities are then transformed with the aid of the X ′′ matrix to the eigen-axes system of the CPKS/B3LYP polarizability tensor. Finally, the dynamic MP2 polarizability tensor components are obtained according to the formula: ′M P 2 ′M P 2 αij (ω) = αij (ω = 0)

α′′B3LY P (ω) ′′B3LY P δij + (1 − δij ) αij (ω) ′′B3LY P α (ω = 0)

(12)

The static hyperpolarizability tensors are transformed to a reference frame minimizing most of the off-diagonal components (using the transformation matrices Y ′ and Y ′′ for MP2 and B3LYP, respectively). These transformation matrices are built in such a way that 1°) the z axis is chosen parallel to β (ω = 0) (of which the components are given by βi = P 1 j (βijj + βjij + βjji )), 2°) the x axis is as much parallel as possible to the direction of the 5

lowest eigenvalue of the static polarizability tensor (the z-component of this direction vector is removed using the Gram-Schmidt orthogonalization method), and 3°) y is orthogonal to x and z (y = z × x). Then, the multiplicative scaling procedure is employed: ′M P 2 βijk

(ω) =

′M P 2 βijk

′′B3LY P βijk (ω) (ω = 0) ′′B3LY P βijk (ω = 0)

10


(13)

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′′B3LY P When a βijk (ω = 0) is very close to zero, it is however preferred to replace the multiplica-

tive approximation scheme by the additive one. At the end, the scaled MP2 polarizability and first hyperpolarizability tensors are transformed back to the starting axes system with the aid of the X ′T and Y ′T transformation matrices.

Results of calculations 2-methyl-4-nitroaniline The simple-shaped planar MNA chromophore is chosen as a “blank test” for the new methodology. MNA was already described by means of the standard LFT approach without taking into account the in-crystal polarizing electric field, 34,35 with the use of a uniform dipole electric field 29,60 as well as with more realistic point charge model. 13,29,54 The last and most evolved model of the polarizing electric field is used here. The effect of the geometry (PBCoptimized versus experimental neutron diffraction coordinates) on χ(1) and χ(2) has already been reported in Ref. 13 and here only the experimental neutron-derived structure is employed. The molecular properties are available from the Supplementary Information of Ref. 13. The static atomic/group polarizabilities obtained at the MP2 level are provided in Table S1. The reference axis system was set according to the principal components of the molecular polarizability tensor. The group polarizabilities are visualized in Figure 1. The representation of polarizabilities as ellipsoids was adapted following the paper by Krawczuk et al., 58 where the main axes of the ellipsoids are proportional to the eigenvalues of the polarizability tensors while the out-of-diagonal terms determine its relative orientation with respect to the reference frame. The most characteristic feature of the QTAIM polarizabilities is that the polarizability ellipsoids are strongly elongated along covalent bonds and/or the push-pull charge transfer direction. This is especially significant for the nitrogen atoms of the nitro and amino groups (see the αxx components in Table S1), which are responsible for the push-pull effect. Then, oxygen atoms are characterized by significantly smaller atomic 11




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for χ(2) , these differences are small and cannot conclude for the superiority of one or the other scheme. The situation gets however different for compounds with more complex shapes, as MBANP (vide infra). Table 1: Effect of the polarizability distribution scheme on χ(1) , n, and χ(2) (in pm/V) of MNA. Results are obtained using neutron diffraction coordinates, with inclusion of the charge polarizing field, and using the MP2/6-311++G(d,p) static α and β responses, whereas the frequency dispersion of the later is described using Eqs. 12 and 13).

exp 64–66

LFT

Q-LFT

(1)

(1)

λ/nm

χ11

χ22

(1)

χ33

(1)

χ13

nx

ny

nz

(2)

χ111

∞

2.453

1.956

1.422

0.709

1.953

1.719

1.436

1064

2.804

2.000

1.521

0.883

2.063

1.732

1.439

300(75)

∞

2.657

1.931

1.421

0.807

2.014

1.712

1.422

-134.3

1064

2.865

1.983

1.479

0.904

2.076

1.727

1.426

-363.3

∞

2.565

1.896

1.374

0.752

1.982

1.702

1.418

-119.4

1064

2.752

1.939

1.424

0.837

2.039

1.714

1.421

-306.1

(2)

χ122

(2)

χ212

-11.4 -29.4

-26.6

-11.0 -27.1

-25.4

(S )-2-(α-methylbenzylamino)-5-nitropyridine derivatives (S )-2-(α-methylbenzylamino)-5-nitropyridine crystallizes in the P 21 space group (a=5.321 ˚ A, b=6.293 ˚ A, c=17.650 ˚ A, β = 93.65o , Z=2) 67 as pure enantiomer. The content of the unit cell is shown in Figure 2. Two types of hydrogen-bonded chains of the C11 (8) type 68,69 could be distinguished in the structure, one along [110] and the second one parallel to [1¯10]. The crystal structure from neutron diffraction experiment was available only for this crystal and was used without reoptimization. Its racemate, (±)-MBANP, crystallizes in Aba2 space group (a=17.149 ˚ A, b=18.665 ˚ A, c=7.936 ˚ A, Z=8). 42 The crystal packing along c direction of (±)-MBANP is shown in Figure 3. The only hydrogen bond patterns found in this structure are R22 (8) rings between molecules related by twofold axis. The 3-methyl (3MMBANP) and N -methyl (NMMBANP) derivatives of MBANP are also considered. 3MMBANP crystallizes in P 21 21 21 space group (a=5.481 ˚ A, b=6.791 ˚ A, c=35.877 ˚ A, Z=4) 70 with a very similar hydrogen bonding system as in MBANP, whereas in the monoclinic NMMBANP structure (P 21 , a=8.979 ˚ A, b=15.977 ˚ A, c=9.410 ˚ A, β = 96.52o , Z=4) in the asymmetric unit a pair 13




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Selected geometrical parameters are summarized in Table 2. A common feature of all the molecules is that the additional phenyl group is not conjugated to the NLO-phore core. Thus, the contribution of the phenyl substituent to the polarizability (and, to a lower extent, the first hyperpolarizability) should be considered a priori as additive rather than in resonance with that of the 2A5NP moiety. The Q-LFT methodology allows for analysis of the microscopic origin of the crystal properties. Two scalar properties derived from the (hyper)polarizability tensors are used for this purpose - the isotropic polarizability (αiso ) and the norm of the first hyperpolarizability vector (βtot ) (Table 3). Additionally the pointcharge electric field averaged over the atoms of the 2A5NP moiety (Table 4) will be used for discussion throughout the paper. Table 3: In-crystal electric field effects on selected molecular properties as determined at the MP2/6-311++G(d,p) level (no field/charge field values). The “2A5NP ring” line reports the corresponding polarizability values for the NLO-phore only, as obtained from the QTAIM partitioning. For 3MMBANP, the CH3 group is not included into summation to get the NLO-phore core polarizability. moiety

property MBANP (±)-MBANP 3MMBANP NMMBANP (mol 1) NMMBANP (mol 2) αiso 187.5/191.5 192.2/195.8 201.5/206.7 210.5/215.8 210.9/218.7 molecule βtot 1281.4/1849.5 1510.2/2053.0 1318.1/2003.1 2156.4/3006.9 2179.6/3305.2 2A5NP ring αiso 100.0/103.9 103.4/106.9 100.9/106.3 108.4/113.4 108.7/116.3

Table 4: Components of the point charge electric field (in GV/m) averaged over the atoms of the 2A5NP ring (F ), norm of the projection of the F vector on the 2A5NP plane (Fk ), and angle between F and the charge-transfer axis (L) taken as C4-N3 axis (the vector components are transformed to fit the orientation of the 2A5NP ring of MBANP). system MBANP (±)-MBANP 3MMBANP NMMBANP (mol 1) NMMBANP (mol 2)

Fx Fy 2.90 3.22 1.17 1.51 3.12 2.95 1.63 2.18 1.01 2.95

17

Fz 2.30 0.89 0.30 0.17 -0.38

|F | Fk 4.90 4.67 2.10 1.96 4.30 4.30 2.73 2.69 3.14 3.04


∠ (L, F ) 53.4 49.3 34.6 29.4 16.3


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Monoclinic MBANP The MBANP molecule is presented in Figure 6. The angle between the normals to the 2A5NP and phenyl ring planes amounts to ca. 82.5o . Since the polarizabilities of the 2A5NP chromophore dominate the optical responses of the whole molecule, the use of the conventional partitioning scheme is expected to lead to an inadequate description of the properties of the remaining part of the molecule (the phenyl ring and its substituent), in particular owing to large out-of-plane polarizability components. The neutron diffraction coordinates are summarized together with the Mulliken population charges in Table S5. The calculated molecular static and dynamic (at λ = 1064 nm) properties are given in Tables S6 and S7, respectively. The static (λ = ∞) MP2/QTAIM group polarizabilities are shown in Table S8 while graphical representation of the QTAIM decomposition into group polarizability tensors is presented in Figure 6. As intuitively predicted, the polarizability of the 2A5NP core ring is very similar to that observed for MNA. Again the polarizabilities of the nitrogen atoms are the largest and are strongly enhanced in the charge-transfer direction. On the other hand, the polarizability tensors of the phenyl ring atoms are very close to that of the atoms of an isolated phenyl ring with small out-of-plane components. In the case of MBANP the effect of the polarizing electric field on the polarizability is negligible, whereas it accounts for enhancement of βtot of the order of 43%. The embedding charge field mainly leads to an enhancement of the 2A5NP ring polarizability and especially of its αyy component. The contribution of the 2A5NP ring to the total molecular polarizability (αiso ) attains 5354%, which decomposes into the diagonal components as follows: αxx ca. 51%, αyy ca. 62% and αzz ca. 48%.

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a

(1)

χ33 ny

nz


21

- results for MP2/6-311++G(d,p)

1.773 1.833 1.889 2.165

0.145 0.132 0.124 0.050

1.603 1.622 1.640 1.719

1.676 1.696 1.714 1.789

1.696 1.709 1.722 1.782

1.717 1.738 1.759 1.875

Q-LFT

1.808 1.876 1.936 2.201

1.784 1.811 1.845 2.019

1.671 1.717 1.764 1.967

1.942 0.030 1.677 2.018 0.008 1.696 2.094 -0.013 1.711 2.460 -0.146 1.756

∞a ∞ 1064 532

2.183 2.280 2.406 3.078

LFT

1.630 1.724 1.693

1.610 1.695 1.682

nx

1.819 1.877 1.928 2.141

0.068

0.110

(1)

χ13

∞a ∞ 1064 532

1.683 1.973 1.842

1.665 1.874 1.755

(1)

χ22

1.883 2.474 2.118 -0.032 1.697 1.864 1.767

1064

(1)

χ11

532

exp 39,41

∞

method λ/nm

35.2 33.1 31.6 13.4

12.8 3.3 -4.3 -21.2

-7.6

20.3

33.9

∠ (x, a)

24.0 23.9 51.1

40.1 40.9 102.6

50 72 77 73 41 39,40 72 74

(2)

χ222

(2)

(2)

(2)

(2)

χ123

(2)

χ213

(2)

χ312

-0.5 0.0 1.9 3.4

14.9 -5.7 14.6 -6.3 33.4 30.3 -14.7 -14.0 -15.0

7.9 10.8 -12.2 12.4 7.0 -12.3 39.3 41.2 12.1 10.2 -26.2 -26.5 -26.3

(2)

χ233 χ323 χ112 χ211

Table 5: Linear and second-order nonlinear optical susceptibility tensor components (in pm/V) and refractive indices of MBANP. The calculations were performed using neutronographic crystal coordinates at the MP2 level of approximation [the molecular properties were evaluated while taking into account the polarizing electric field; except when specified, all results were obtained with the 6-311++G(2d,2p) basis set].

Page 21 of 44 The Journal of Physical Chemistry


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Page 22 of 44

(2)

75 Taking as reference value for α-quartz dQ the χ222 experimental esti11 = 0.30(2) pm/V, 72 73 mates at 1064 nm are: 83 × dQ 126(20) × dQ 69 × dQ 11 = 50 pm/V, 11 = 77(12) pm/V, 11 =

41 pm/V 39,40 and 72 pm/V. 74 Despite rather scattered experimental values, the use of the (2)

standard LFT partitioning method gives too large χ222 values. This is further supported by the fact that the present calculations neglect the ZPVA correction, which might typically account for a 10-15% increase as shown in the case of MNA. 13 Switching from LFT to Q-LFT (2)

yields a drop of the χ222 value by up to 50%, whereas this drop was only ca. 16% for MNA. These contrasted results on MBANP and MNA demonstrate higher requirements for the description of the local field tensor when the NLO-phore is not a one-dimensional push-pull (2)

π-conjugated molecule. Besides the χ222 value, a general drop of the magnitude of all the χ(2) tensor components is observed in the abc∗ reference frame. Moreover, the different orientation of the optical indicatrix within the two approaches leads to drastic differences for the out-of-diagonal components: the 233/323 components reduce in favor of the 112/211 components when going from LFT to Q-LFT schemes. Provided accurate theoretical results are used for the local field tensor, the ordering of the amplitudes of the χ(2) tensor components (2)

(2)

(2)

(2)

is χ222 > χ211 > χ213 > χ233 . (±)-MBANP (racemate) The MBANP racemate is characterized by better physical and chemical stability than its enantiomer. 42 The conformation of the molecule is different from the one present in the enantiomorphic crystal (and the other MBANP derivatives). Indeed, the 2A5NP ring is rotated by approximately 180o with respect to the α-methylbenzyl substituent as compared to its orientation in the monoclinic MBANP (Table 2 and Figure 8). The angle between the normals to the 2A5NP and phenyl rings is ca. 79.5o after geometry optimization (from experimental data it is ca. 84.4o ). The full set of optimized fractional coordinates for the first molecule of the unit cell is listed in Table S9 together with the Mulliken atomic charges. The static and dynamic (at λ = 1064 nm) molecular properties are given in Tables S11 and S12,

22


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Page 24 of 44

is of the order of 0.03-0.04. The frequency dispersion of the refractive indices as predicted from MP2 molecular properties is drawn in Figure 9. Calculations reproduce the large decrease of the dominant χ(2) tensor components of (±)-MBANP with respect to those of the enantiomeric pure form. Still, they overestimate the experimental χ(2) tensor components, 42 though the Q-LFT method slightly improves the results with respect to the standard LFT approach (Table 6). The source of overestimations may be partially attributed to the use of PBC-optimized crystal geometry. 13 Table 6: Linear and second-order nonlinear optical susceptibility tensor components and refractive indices of (±)-MBANP. The calculations were performed at the MP2/6311++G(d,p) level of approximation, using PBC-optimized coordinates. (1)

(1)

(1)

(2)

(2)

(2)

(2)

(2)

method

λ/nm

χ11

χ22

χ33

nx

ny

nz

exp 42

1064 532

1.863 2.211

1.822 2.098

1.509 1.624

1.692 1.792

1.680 1.760

1.584 1.620

13.6

9.4

1.7

LFT

∞ 1064 532

1.954 2.043 2.498

1.811 1.889 2.267

1.732 1.785 2.005

1.719 1.744 1.870

1.676 1.700 1.807

1.653 1.669 1.734

10.2 24.7 21.9

7.8 18.1 17.7

0.1 2.6

Q-LFT

∞ 1064 532

1.764 1.820 2.090

1.623 1.684 1.992

1.475 1.510 1.649

1.662 1.679 1.758

1.620 1.638 1.730

1.573 1.584 1.628

9.1 19.5 18.9

7.5 16.7 16.7

0.8 3.4

24


χ113

χ311

χ223

χ322

χ333

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Page 26 of 44

The experimental linear optical properties of 3MMBANP were taken from Ref. 76 while the χ(2) tensor components from Ref. 74. The calculated χ(1) and χ(2) values are summarized in Table 7. Similarly to the previous examples the use of the Q-LFT method leads globally to a better description of the linear optical properties. Still, the overestimations of all the LFT refractive indices are generally overcorrected by the Q-LFT scheme. The wavelength dispersion of the refractive indices estimated with the two LFT approaches is compared to experiment in Figure 10. Using both distribution schemes the χ(2) values are in good agreement with experiment, though breaking of the Kleinman symmetry is underestimated and the ordering of the amplitudes of the three tensor components is different. Table 7: Linear and second-order nonlinear optical susceptibility tensor components and refractive indices of 3MMBANP. The calculations were performed using PBC-optimized coordinates at the MP2/6-311++G(d,p) level of approximation. (1)

(1)

(1)

(2)

(2)

(2)

method

λ/nm

χ11

χ22

χ33

nx

ny

nz

χ123

χ213

χ312

exp 74,76

∞ 1064 532

1.625 1.656 1.857

1.930 2.021 2.609

1.740 1.771 1.977

1.620 1.630 1.690

1.712 1.738 1.900

1.655 1.665 1.725

22.0

18.0

14.0

LFT

∞ 1064 532

1.677 1.726 1.939

2.095 2.216 2.899

1.761 1.819 2.09

1.636 1.651 1.714

1.759 1.793 1.975

1.662 1.679 1.758

-24.8

-10.9 -27.3

-26.7

Q-LFT

∞ 1064 532

1.558 1.606 1.844

1.822 1.892 2.220

1.637 1.683 1.906

1.600 1.614 1.687

1.680 1.701 1.794

1.624 1.638 1.705

-22.6

-9.8 -22.1

-23.5

26


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in-crystal polarizing field leads to enhancements of αiso and βtot of the order of 2.5%/3.7% and 39.5%/51.5% for molecule 1/2 (Table 3 and 4). Similarly, this better alignment of the polarizing field with the charge transfer axis of molecule 2 leads to a stronger αiso enhancement for the 2A5NP core, 7% in comparison to only 4.5% for molecule 1. A comparison between NMBBANP and 3MMBANP shows that a much stronger enhancement of the electric properties is achieved by the electron-donating effect of the methyl group attached to the amino nitrogen than to the pyridine ring in meta position with respect to the electron accepting nitro group. The experimental 38 and calculated linear and nonlinear electric properties of the NMMBANP crystal are summarized in Table 8 while the dispersion of refractive indices obtained from MP2 properties are shown in Figure 11. Similarly to the other crystals, the Q-LFT linear susceptibilities and refractive indices are systematically smaller than those obtained with the LFT scheme. This situation can again be considered as favorable since the ZPVA contributions are not taken into account. The experimental χ(2) tensor components given in Table 8 were taken from Ref. 38 but were rescaled accordingly to newer results from the (2)

same group 74 for χ222 of MBANP: 90 pm/V in the earlier reference versus 72 pm/V in the latter. Again, the Q-LFT χ(2) values are smaller than the LFT ones but the agreement with experiment remains rather poor. Indeed, the experimental values display a spectacular breaking of the Kleinman symmetry, which is not the case for the calculations, no matter which distribution scheme is used.

28


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30

(1)

(1)

χ33

(1)

χ13

nx

ny

nz

1.874 1.399 1.817 -0.557 1.845 1.549 1.513 1.968 1.427 1.898 -0.630 1.888 1.558 1.517 2.493 1.547 2.334 -1.080 2.120 1.596 1.527

∞ 1064 532

LFT

Q-LFT

2.112 1.514 1.802 -0.579 1.886 1.586 1.535 2.209 1.538 1.865 -0.641 1.924 1.593 1.541 2.890 1.686 2.297 -1.108 2.177 1.639 1.564

χ22

2.068 1.563 1.938 -0.569 1.891 1.601 1.559 2.189 1.605 2.036 -0.651 1.941 1.614 1.567 2.905 1.792 2.599 -1.190 2.225 1.671 1.597

∞ 1064 532

(1)

χ11

∞ 1064 532

exp 38

method λ/nm

43.5 43.4 42.9

41.7 41.6 41.3

37.5 37.5 37.5

∠ (x, a)

(2)

19.3 49.6 43.5

23.4 64.4 55.6

0.0

-0.9

0.0

(2)

χ211 χ123

-15.5 78.4

(2)

χ112

0.0 -0.5

-0.4 -1.7

-24.0

(2)

χ213

(2)

(2)

(2)

0.1

-0.6

9.4

3.8

-0.5 -0.9 -0.4

-0.4 -0.8 0.0

0.0

0.6 3.4

0.9 4.9

7.0

χ312 χ233 χ323 χ222

(2)

Table 8: Linear and second-order nonlinear optical susceptibility tensor components and refractive indices of NMMBANP. The calculations were performed using PBC-optimized coordinates at the MP2/6-311++G(d,p) level of approximation. The Journal of Physical Chemistry Page 30 of 44

Page 31 of 44

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approximation, be analyzed by resorting to the oriented gas model. 16 The crystal packing effect is here considered for the static limit (λ = ∞) with the use of the oriented gas model 16 which here consists simply in the tensorial sum of the molecular first hyperpolarizabilities χ(2) =

1 X β 2V ε0 k k

(14)

The symmetry operations in point group 2 reduce the number of non-zero χ(2) tensor components to 4 (112, 123, 222, 233), in point group mm2 to 3 (113, 223, 333), and in point group 222 to 1 (123). 1, 2 and 3 refer to the crystal axes in the case of orthorhombic systems. For monoclinic systems the abc∗ reference frame, used in the calculations, was further transformed to the optical indicatrix system based on the experimental angle between the a crystallographic axis and the x axis of the optical indicatrix (for λ = ∞). This angle is equal to +33.9o for MBANP and to 37.5o for NMMBANP. A comparison of the obtained results is presented in Table 9. The difference between MBANP and NMMBANP can be qualitatively explained by considering the angle between the push-pull charge transfer axis, approximated as N3-N2 direction, and the b crystallographic axis. For MBANP this angle is 32.2o while for NMMBANP it is larger and it is equal to 71.6o and 48.4o for the first and second molecule in the asymmetric unit, respectively. Thus, the worse alignment of the (2)

dominant β tensor component with respect to the b axis leads to a much smaller χ222 in the case of NMMBANP. (2)

Taking the χ222 of MBANP as reference value for all the compounds within a given method we see that the simple tensor sum provides results in qualitative agreement with the Q-LFT methodology. On the other hand, as discussed above, the LFT approach provides a wrong description of the χ(1) tensor for MBANP and thus the ordering of the χ(2) tensor components is qualitatively different. The comparison between the simple tensor sum and the Q-LFT method shows that the local field tensor accounts for a factor of 2-3.5. This 2 (nω ) +2 factor can be estimated from the simple Lorentz approximation, F = fIω fJω fK2ω = I 3 × 2 2 ) +2 (nωJ ) +2 (n2ω × K3 , using a weakly-frequency-dependent refractive index of about 1.5, leading 3 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


32

(2)

(2)

MBANP 2 (2)

(2)

(2)

(2)

(±)-MBANP mm2 (2)

component χ112 χ123 χ222 χ233 χ113 χ223 χ333 7.0 (17) -12.3 (-30) 40.9 (100) 12.4 (30) 10.2 (25) 7.8 (19) 0.1 (0) LFT Q-LFT 14.6 (61) -6.3 (-26) 23.9 (100) 0.0 (0) 9.1 (38) 7.5 (31) 0.8 (3) tensorial sum 5.9 (51) -2.4 (-21) 11.5 (100) 0.4 (3) 3.1 (27) 3.2 (28) 0.4 (3)

system point group

(2)

χ123 -10.9 (-27) -9.8 (-41) -3.0 (-26)

3MMBANP 222 (2)

(2)

(2)

(2)

χ112 χ123 χ222 χ233 23.4 (57) -0.4 (-1) 0.9 (2) -0.4 (-1) 19.3 (81) 0.0 (0) 0.6 (3) -0.5 (-2) 6.9 (60) -0.8 (-7) 0.2 (2) -0.1 (-1)

NMMBANP 2

Table 9: Comparison between successive approximate schemes for calculating the χ(2) tensor components (in pm/V); (values in (2) parentheses are the values relative to χ222 of MBANP in % within selected method).

The Journal of Physical Chemistry Page 32 of 44

Page 33 of 44

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to F ≈ 2.84.

Conclusions In our efforts towards developing methods to predict and interpret the linear and nonlinear optical properties of molecular crystals, the two-step multi-scale procedure, which has been shown to be reliable for push-pull π-conjugated systems, 32 has been extended with the aim of describing crystals built of molecules having complex shapes. This has been performed by adopting the Quantum Theory of Atoms in Molecules (QTAIM) to distribute the molecular polarizability tensor over the submolecular sites, leading therefore to an improved prediction of the local field tensors. When combined with a simple charge embedding approach, to account for the crystal field effects on the molecular (hyper)polarizabilities, this so-called Q-LFT method is efficient to predict the χ(1) and χ(2) tensor components. Moreover, it does not require substantial computational resources since the largest calculations are performed on the individual molecules. This is illustrated by considering derivatives of (S )-2-(α-methylbenzylamino)-5-nitropyridine (MBANP) as well as a prototypical push-pull π-conjugated compound, 2-methyl-4-nitroanline (MNA). In the later case, the use of Q-LFT only leads to minor differences in the χ(1) and χ(2) tensor components with respect to the standard approach where the polarizability is equipartitioned. Still, for χ(2) the difference goes in the right direction, when considering the contributions of the thermal motions (ZPVA correction). 13 In the case of the MBANP derivatives, the Q-LFT scheme leads to systematic decrease of the linear and nonlinear optical responses, improving generally the agreement with experiment. Indeed, the standard partitioning scheme leads to an overestimation of the phenyl out-of-plane polarization component, and therefore of the refractive indices. The χ(2) variations among the MBANP derivatives has been analyzed in terms of molecular geometry, crystal polarizing field and crystal packing. Further improvements of the methodology are expected to result from taking explicitly into account the hydrogen bonding, especially

33



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in the case of chain-type systems as observed in MBANP and 3MMBANP. These chains extending in the [110] and [1¯10] directions may in principle enhance the polarizability tensor components in the ab plane and reduce the underestimations of the refractive indices.

Acknowledgement This research was supported in part by PL-Grid Infrastructure, by the Francqui Foundation as well as by the Belgium government (IUAP N° P7/05, Functional Supramolecular Systems). T.S. acknowledges the financial support of IUAP N° P7/05 for his post-doctoral grant. The calculations were performed on the computers of the Consortium des Equipements de Calcul Intensif and mostly those of the Technological Platform of High-Performance Computing, for which we gratefully acknowledge the financial support of the FNRS-FRFC (Conventions No. 2.4.617.07.F and 2.5020.11) and of the University of Namur.

Supporting Information Available Components of the static (λ = ∞) and dynamic (λ =1064 nm) molecular polarizability α and first hyperpolarizability β tensors, QTAIM atomic/group polarizability tensor components, with and without in-crystal field, neutron diffraction and PBC optimized atomic fractional coordinates and PBC Mulliken charges of the crystals.

This material is available free of

charge via the Internet at http://pubs.acs.org/.

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perpolarizability Coefficients of Some Mono-and Disubstituted Stilbene Models for the Design of Nonlinear Optical Materials. J. Phys. Chem. 1988, 92, 2385–2390.

Graphical TOC Entry Local Field Tensor molecular polarizability

equi-partitioning

QTAIM partitioning

LFT

Q-LFT

scheme

scheme (1)

,

(2)

44


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QTAIM-Based Scheme for Describing the Linear and Nonlinear

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