Analysis of nonlinear optical properties in donor-acceptor materials.

Analysis of nonlinear optical properties in donor–acceptor materials Paul N. Day, Ruth Pachter, and Kiet A. Nguyen Citation: The Journal of Chemical Physics 140, 184308 (2014); doi: 10.1063/1.4874267 View online: http://dx.doi.org/10.1063/1.4874267 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/18?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Ultrafast optical nonlinearities and figures of merit in acceptor-substituted 3,4,5-trimethoxy chalcone derivatives: Structure-property relationships J. Appl. Phys. 103, 103511 (2008); 10.1063/1.2924419 Extended conjugation and donor-acceptor substitution to improve the third-order optical nonlinearity of small molecules Appl. Phys. Lett. 90, 251106 (2007); 10.1063/1.2750396 Quantum-chemical investigation of second-order nonlinear optical chromophores: Comparison of strong nitrilebased acceptor end groups and role of auxiliary donors and acceptors J. Chem. Phys. 124, 044510 (2006); 10.1063/1.2155385 Nonlinear optical property calculations by the long-range-corrected coupled-perturbed Kohn–Sham method J. Chem. Phys. 122, 234111 (2005); 10.1063/1.1935514 Nonlinear optical properties of tetrahedral donor–acceptor octupolar molecules: Effective five-state model approach J. Chem. Phys. 116, 9165 (2002); 10.1063/1.1473818

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THE JOURNAL OF CHEMICAL PHYSICS 140, 184308 (2014)

Analysis of nonlinear optical properties in donor–acceptor materials Paul N. Day,1,2 Ruth Pachter,1 and Kiet A. Nguyen1,3 1

Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio 45433, USA General Dynamics Information Technology, Inc., Dayton, Ohio 45431, USA 3 UES, Inc., Dayton, Ohio 45432, USA 2

(Received 4 February 2014; accepted 18 April 2014; published online 9 May 2014) Time-dependent density functional theory has been used to calculate nonlinear optical (NLO) properties, including the first and second hyperpolarizabilities as well as the two-photon absorption crosssection, for the donor-acceptor molecules p-nitroaniline and dimethylamino nitrostilbene, and for respective materials attached to a gold dimer. The CAMB3LYP, B3LYP, PBE0, and PBE exchangecorrelation functionals all had fair but variable performance when compared to higher-level theory and to experiment. The CAMB3LYP functional had the best performance on these compounds of the functionals tested. However, our comprehensive analysis has shown that quantitative prediction of hyperpolarizabilities is still a challenge, hampered by inadequate functionals, basis sets, and solvation models, requiring further experimental characterization. Attachment of the Au2 S group to molecules already known for their relatively large NLO properties was found to further enhance the response. While our calculations show a modest enhancement for the first hyperpolarizability, the enhancement of the second hyperpolarizability is predicted to be more than an order of magnitude. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4874267] I. INTRODUCTION

Conjugated organic molecules have long been of interest for a variety of nonlinear optical (NLO) applications.1–4 Typical are so-called donor-π -acceptor (D−π -A) molecules, where a conjugated aromatic or polyene system makes up the π -electron core, and an electron donor, e.g., an amino group, is bonded at one end, and an electron acceptor group, such as a nitro group, is attached at the opposite end. However, although interest in NLO properties for such compounds has resulted in a relatively large number of theoretical studies at various levels of approximation,5–23 even some investigating open-shell singlet systems with diradical character,24–27 as well as of experimental work,5, 28–33 a consistent comparison with experiment for validation remains elusive. This is important for application to newly developed materials, such as those that could be attached to gold clusters.34 Measurements of the first hyperpolarizability (β) by second-harmonic generation (SHG) and of the second hyperpolarizability (γ ) by third-harmonic generation (THG) and by electric-field induced second harmonic generation (EFISH, or dc-SHG) were carried out for dimethylamino nitrostilbene (DANS), p-nitroaniline (pNA), nitrobenzene, and other molecules at wavelength 1908 nm by Cheng et al.28 As these were carried out in solution (DANS in chloroform, pNA in acetone, and nitrobenzene neat), comparison between measured and calculated values requires inclusion of a solvation model in the calculations, and thus one is included in this work. Measured values of β for pNA at 1064 nm in six different solvents have been reported by Stahelin et al.,29 while Teng et al.5 have reported measured values of β for pNA in dioxane solvent at five different photon wavelengths. Gas-phase EFISH was carried out for pNA and nitrobenzene by Kaatz et al.,31 at 1064 nm. Note that comparison with these gas-phase values 0021-9606/2014/140(18)/184308/13/$30.00

allows for direct comparison of theory with experiment without the complication of including a solvation model. In earlier theoretical work, Champagne et al.7 evaluated exchange-correlation (X-C) functionals with time-dependent density functional theory (TDDFT), and MP2, for prediction of static (hyper)polarizabilities for pNA. The basis sets used in that study are small, although a few of their results have surprisingly good agreement with higher-level calculations. Lu et al.21 evaluated X-C functionals for calculating β using the experimental results of Cheng et al.28 as a reference, but reported only the mean absolute errors (MAE) and not their actual calculated β values. They concluded that X-C functionals with a high percentage of exact exchange, such as LC, M06-HF, and ωB97, performed best for calculating the first hyperpolarizability. Sim et al.6 reported MP2 values of β and γ for pNA, estimating frequency-dependent values by scaling the static MP2 values using static and frequency-dependent Hartree-Fock (HF) values. That work illustrated the importance of including electron correlation in the calculation of hyperpolarizabilities. The scaling method was also used by Suponitsky et al.13 to calculate β at 1064 nm for MP2, CCSD, and DFT. In that study, four hybrid functionals were tested against CCSD using modest basis sets, and the conclusions were that the 6-31+G* basis is adequate at least for obtaining reasonable relative values of β, and that the most accurate values were obtained with the functional that is a metahybrid, BMK (9% larger than CCSD), followed closely by PBE0 (10%), while B97-2 and B3LYP yielded values that were too large by 16% and 22%, respectively. Soscun et al.9 studied pNA in the gas-phase with MP2 and B3LYP, reporting static β and γ values at the HF, MP2, and B3LYP levels of theory. The comparison to experiment is only qualitative since the experimental values are in solution and have been scaled to zero frequency using a two-state approximation.

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Day, Pachter, and Nguyen

Garza et al.19 reported gas-phase values of β for pNA calculated with TDDFT, with comparison to the measured values of Kaatz.31 That study concluded that long-range corrected (LC) functionals, in particular ones with a high percentage of exact exchange in the asymptotic regions such as ωB97X, are best for calculating hyperpolarizabilities in D−π -A molecules. In an evaluation of X-C functionals for the calculation of excitation energies in charge transfer (CT) systems,35 we found that the LC functionals, in particular ωB97X and LC-PBE, performed best for D-π -A molecules with a high degree of CT, such as DANS. In a theoretical study of several other donor-acceptor molecules, de Wergifosse and Champagne22 evaluated ab initio methods and X-C functionals for calculating the first hyperpolarizability using CCSD(T) results as a reference. Some of the conclusions from that study were that the 6-31+G(d) basis set is adequate, that MP2 performs well, and that none of the X-C functionals tested performed well consistently. Hammond and Kowalski14 calculated (hyper)polarizabilities of several small molecules and of pNA using the high-level CCSD method and a variety of basis sets, and used the results to evaluate X-C functionals for the calculation of (hyper)polarizabilities. They concluded that the CAMB3LYP functional is generally superior to GGA and hybrid functionals, but that it is still prone to yield significant error in certain cases. Comparison of theoretical (hyper)polarizabilities with experiment has the added complication that in the system measured, the molecules have a distribution of geometries, both because of the existence of multiple low-lying local minimum-energy configurations and because of nuclear vibration around each of these minima. A complete theoretical evaluation might involve sampling the different geometries, calculating the properties for each geometry, and performing a thermal average. Eriksen et al.23 addressed this problem by carrying out TDDFT on pNA for 100 different geometrical configurations. However, as pointed out,23 the main contribution to inaccuracy in prediction is still due to the X-C functional employed in TDDFT. In this work, we report a comprehensive study of calculated (hyper)polarizabilities for several organic NLO compounds using TDDFT and various X-C functionals and basis sets, comparing the results with experimental data as is available (including frequency-dependent measurements), as well as with higher-level theoretical results. In addition, we report values for the static (hyper)polarizabilities calculated at the MP2/aug-cc-pVTZ level as an accurate reference. Willetts et al.30 and Reis32 provided valuable information on the different conventions for reporting experimental values of β and γ , which we have taken into account. The computed values of various frequency-dependent hyperpolarizabilities are of greater interest than the static hyperpolarizabilities, due to the interest in comparison to experiment (and eventually to the prediction of properties); however, for establishing a theoretical method, i.e., an X-C functional and basis set that yield accurate results, benchmarking for the static case could be directly transferable to the frequency-dependent hyperpolarizabilities. While the accuracy may be lower for frequencies close to resonance, this will be the case for calculations at any level of theory. As resonance effects depend on the computed excited state energies, a theoretical method that yields accu-

J. Chem. Phys. 140, 184308 (2014)

rate excited state energies and static polarizabilities should indeed yield accurate frequency-dependent polarizabilities. Following our benchmarking, we aimed to understand variation of the NLO properties upon bonding to gold metal atoms and report on the results for pNA with a gold dimer attached, (Au2 S-pNA)−1 . We previously investigated34 the onephoton absorption (OPA) and two-photon absorption (TPA) for the gold dimer and its thiolated analogue, as well as for the larger thiolated gold clusters, [Au25 (SR)18 ]−1 and [Au12 (SR)9 ]+1 , and large values for the TPA cross-section were predicted for each of these systems, as well as verified by experiment36, 37 for the case of [Au25 (SR)18 ]−1 . A recent study38 has highlighted the unique NLO properties of gold clusters that are too small for plasmonic effects. For comparison, the chemical effect of covalent attachment of gold atoms for TPA, which is proportional to the imaginary part of the second hyperpolarizability, was considered. The TPA of conjugated organic compounds has been studied for some time both experimentally39–45 and theoretically,46–49 including measurements50–52 and calculations17, 53–55 on DANS. In our previously reported calculations of the TPA of DANS using TDDFT,55 we found good agreement with experiment50 using the CAMB3LYP functional with the polarizable continuum model (PCM) solvation model and a Lorentzian linewidth function derived from the experimental results. Muragan et al.17 also used the CAMB3LYP functional, but compared the PCM solvation model to several QM/MM derived solvation models. By combining these solvation models with a variable linewidth, they were able to produce a more complicated picture of the solvent dependence of the TPA. In this study, we investigate the effect on TPA of the covalent attachment of gold atoms. We focus on gas-phase calculations of TPA and compare DANS with (Au2 S- DANS)−1 , as an example of possible chromophore attachment to gold nanoparticles. II. COMPUTATIONAL METHODS

Geometry optimizations were carried out with the Gaussian 09 program (Revision C.01)56 using the hybrid XC functional, B3LYP.57–61 For nitrobenzene, pNA, and DANS, the basis set 6-311G** was used, while for (Au2 SpNA)−1 and (Au2 S-DANS)−1 the LANL2DZ basis set and ECP was used. The X-C functionals tested in the calculations of the (hyper)polarizabilities include PBE,62 PBE0,63 B3LYP,57–61 BHandHLYP, CAMB3LYP,64 and a modified version described previously and labeled mCAMB3LYP.65 The mCAMB3LYP functional, like CAMB3LYP, uses an Ewald split of the exact (Hartree-Fock) exchange and the density functional exchange, but with different α and β parameters. The CAMB3LYP functional uses the parameters α = 0.19 and β = 0.46, where α is the fraction of exact change at small electron separations, while the fraction of exact exchange in the asymptotic regions is α + β = 0.65. In the mCAMB3LYP functional, the parameter β is reduced to 0.19, thus reducing α + β to 0.38, making mCAMB3LYP a less extreme version of this functional, with its fraction of exact exchange in the asymptotic region falling between that of CAMB3LYP and B3LYP. The BHandHLYP functional as im-


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plemented in Gaussian 09,56 which is similar to that proposed by Becke66 but has 50% exact exchange and 50% Becke58 exchange and LYP59 correlation, was included to study the effect of such a large fraction of exact exchange for the full range rather than just in the long-range region. Static MP2 and DFT (hyper)polarizabilities were calculated using the finite field method with the GAMESS program.67 The default applied field of 0.001 a.u. was used in all calculations except the MP2/aug-cc-pVTZ calculation on nitrobenzene, where 0.005 a.u. was used in order to obtain sufficient precision for extracting the second hyperpolarizabilities in the energy expansion. Gaussian 09 (Revision C.01)56 and Dalton68, 69 were used to calculate frequency-dependent (hyper)polarizabilities using TDDFT. To model solvent effects, we have used the PCM of Tomasi et al.70 In the PCM calculations with Gaussian 09, the United Atom Topological Model (UA0) was used for consistency with the Dalton results. Data analysis of experimental macroscopic observable measurements to extract the microscopic hyperpolarizabilies includes the consideration of factors such as the number density of the moiety, local field factors, and refractive indices,

and analysis may differ between groups. Relationships between the different conventions that have been previously applied for the analysis of dc EFISH experiments are given in Table I of Willetts et al.30 The hyperpolarizabilities reported in our work use the “T” convention, defined by expanding the field-induced dipole in a Taylor series, 1 T 2 1 β F + γT F3 (1) 2! 3! where μ0 is the permanent dipole moment, F is the local electric field, and we have truncated the series after the third-order term. In the “B” convention, the 1/n! factor is absorbed in the hyperpolarizability, yielding a simple perturbation expansion, μind = μ0 + α T F +

μind = μ0 + α B F + β B F 2 + γ B F 3 .

The local electric field can be written in terms of its static and time-dependent components, F = F0 + Fω cos(ωt).

αxx + αyy + αzz . (5) 3 The first and second hyperpolarizabilities are third and fourth order tensors, respectively, with two and three independent frequency parameters, respectively. The values reported here are the component of the first hyperpolarizability along the permanent molecular dipole moment, given by α T = α B = α X = αiso =

1 (6) βzT = βzzz + (βzxx + 2βxzx + βzyy + 2βyzy ), 3 and the orientationally averaged second hyperpolarizability, γ¯ T =

x,y,z 1 γiijj + γij ij + γijj i . 15 i,j

(7)

(3)

When Eq. (3) is substituted in Eq. (1), and some trigonometric identities are used, the result is

1 μind = μ0 + α0T F0 + α t (−ω; ω)Fω cos(ωt) + β0T F02 + β T (−ω; ω, 0)F0 Fω cos(ωt) 2 1 1 + β T (0; ω, −ω)Fω2 + β T (−2ω; ω, ω)Fω2 cos(ωt) 4 4 1 1 1 × γ0T F03 + γ T (−ω; ω, 0, 0)F02 cos(ωt) + γ T (0; ω, −ω, 0)Fω2 F0 6 2 4 1 T 1 1 × γ (−2ω; ω, ω, 0)Fω2 F0 cos(2ωt) + γ T (−3ω; ω, ω, ω)Fω3 cos(3ωt) + γ T (−ω; ω, ω, −ω)Fω3 cos(ωt). 4 24 8

In Eq. (4), each numerical coefficient is a product of the Taylor series coefficient and the field-expansion coefficient. In the “X” convention, both of these factors are absorbed in the hyperpolarizability. The linear polarizability is a frequency-dependent second-order tensor. All linear polarizabilities reported here correspond to the isotropic polarizability, given by

(2)

(4)

For the evaluation of Eq. (6), the assumption is that the z-axis has been chosen to align with the permanent dipole moment of the molecule. Since convention “B” differs from convention “T” only in that the Taylor series coefficients are implicitly included, the value of βzB is 12 that of βzT , and the value of γ¯ B is 1/6 that of γ¯ T . A fourth convention labeled “B*” is defined only for the EFISH experiment and is used in the work of Cheng et al.28 It differs from the B convention in that an additional factor of 1/3 is implicitly included in the first hyperpolarizability. Thus, in order to convert the reported EFISH results of Cheng et al.28 to the T convention, both the first and second hyperpolarizabilities were multiplied by 6. A factor of 6 was also used to convert their THG second hyperpolarizabilities to the T convention. As was pointed out by Reis,32 the experimental results of Teng et al.5 and Stahelin et al.29 use the “X” convention in reporting measured values of β(−2ω;ω,ω) for pNA. Since in the X convention the coefficient of 14 on this term (see Eq. (4)) is absorbed into β, these values were multiplied by 4 to convert to the T convention. Another complication is that while these quantities with the z subscript are


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FIG. 1. Schematic structures for (1) nitrobenzene, (2) pNA, (3) DANS, (4) (Au2 S-pNA)−1 , (5) (Au2 S-DANS)−1 .

widely used for expressing the component of β in the direction of the dipole, another quantity for each convention uses the subscript, where the relationship is β|| = 35 βz , which was also considered where needed. Values for β in atomic units were converted to esu × 10−30 by multiplying by 8.6391 × 10−3 , while γ in atomic units was converted to esu × 10−36 by multiplying by 5.03657 × 10−4 . Throughout this paper, linear polarizabilities, first hyperpolarizabilities, and second hyperpolarizabilities are reported in 10−23 , 10−30 , 10−36 esu, respectively. TPA calculations were carried out using the Dalton program in the same manner as we described previously.55 The linewidth, EFWHM , used is 0.52 eV, and the damping factor, , was set to 0.05 eV.

and the B3LYP functional. We used the 6-311++G** basis, and evaluate the six X-C functionals mentioned previously. The RMS error is found to be lowest for the BHandHLYP (0.47) and CAMB3LYP (0.54) functionals, and largest for PBE (1.70), the only functional with no exact exchange. Additionally, the calculated excitation energy to the primary charge transfer state (2A2 ) is closest to the CCSD value when the BHandHLYP and CAMB3LYP functionals are used.

TABLE I. Measured values of the dipole (μ, in Debye), the linear polarizability (α in 10−23 esu), and the first two hyperpolarizabilities (β in 10−30 esu and γ in 10−36 esu) for compounds 1, 2, and 3, including γ values for THG and EFISH. Nitrobenzene

III. RESULTS AND DISCUSSION

Figure 1 shows schematic structures for the NLO compounds considered in the calculations, namely, nitrobenzene (1), pNA (2), DANS (3), (Au2 S-pNA)−1 (4), and (Au2 SDANS)−1 (5). Table I lists the measured hyperpolarizability data for 1, 2, and 3, which was used for comparison. Because pNA has been extensively characterized, in the gas phase and in solution, we discuss results for this test problem first. Next, results for nitrobenzene and DANS are compared to experiment followed by predictions for the compounds with the Au2 S group bonded to the electron-donor group.

A. pNA in the gas phase

Mikhailov et al.,71 reported excited state energies for pNA calculated using EOM-CCSD and TDDFT. As a first screening of X-C functionals for NLO calculations on pNA, we use their CCSD state energy values as a reference in Table II. They used the SVP and SVP+ basis sets with CCSD, and we use the results using the larger basis as a reference, and they also carry out TDDFT calculations using the SVP basis

Measured31 Measured28

Solventa λ (nm) μ (D) None Neat

1064 1908

α

4.22 4.00 1.40

βzT 2.84 11.40

γ¯TTH G

T γ¯EF I SH

9.86 34.20 244.80

p-Nitroaniline Measured28 Measured5 Measured31 Measured5 Measured29 Measured29 Measured29 Measured29 Measured29 Measured29 Measured5 Measured5 Measured5

Acetone DX None DX DX CHCl3 DCM ACN Acetone CH3 OH DX DX DX

1908 1907 1064 1064 1064 1064 1064 1064 1064 1064 1370 909 830

6.20 1.70 6.20 4.22 6.20 7.00 6.40 6.20 6.20 7.30 6.10 6.20 6.20 6.20

55.20 38.40 15.44 67.60 65.20 67.20 67.60 116.80 103.60 128.00 47.20 100.00 160.00

90.00 29.15

DANS Measured28

CHCl3 Measuredb ,10, 28 CHCl3

1908 1908

6.60 3.40 438.00 1350.00 6.60 4.05 1008.28 2725.38

a

DX: 1,4-dioxane, DCM: dichloromethane, ACN: acetonitrile. (Hyper)polarizabilities have been re-calculated from the experimental values of Cheng28 using the local-field factors of Tu.10 b


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TABLE II. Excited state energies (eV) for pNA calculated with six functionals and compared to the EOM-CCSD values reported by Mikhailov et al.71 State

CCSDa

B3LYPb

CTDAc

CAMd

B3LYPd

mCAMd

PBE0d

BH&Hd

PBEd

2A1 3A1 4A1 5A1 6A1

4.65 6.43 7.24 7.77 8.49

4.09 6.21 6.98 6.62 6.52

5.21 6.16 6.71 7.25 7.85

4.30 6.20 6.92 7.35 7.43

3.88 5.92

4.05 6.03 6.76 6.79

4.03 6.08 6.67 6.77 6.99

4.42 6.35 7.02 7.57 7.58

3.50 5.52 5.60 5.84 5.87

1B2 2B2 3B2 4B2 5B2

4.67 5.98 6.49 8.40 7.81

4.59 5.12 6.27 7.23 8.00

5.48 6.77 6.02 7.46 7.67

4.79 5.53 6.37

4.50 4.97 6.06 6.72

4.63 5.18 6.20

4.64 5.14 6.19

4.96 5.70 6.45

4.14 4.53 5.25 5.54

1B1 2B1 3B1 4B1 5B1

4.74 5.21 6.28 8.06 ...

4.44 5.96 6.92 5.99 7.91

4.40 6.17 7.06 6.79 7.82

4.52 5.27 6.24 6.90 7.32 7.41

4.40 4.91 5.82 6.07 6.58

4.45 5.03 5.96 6.60 6.69 7.03

4.47 5.16 6.07 6.37 6.81

4.72 5.35 6.30 6.99 7.38 7.51

4.14 4.62 4.66 5.51

1A2 2A2 3A2 4A2 5A2

4.21 6.08 ... 8.15 7.79

3.96 6.89 6.35 7.03 8.09

3.96 7.00 7.05 7.50 8.21

3.97 6.03 6.97 7.03 7.35

3.84 5.65 6.30 6.52 6.57 6.77

3.89 5.77 6.56 6.72 6.98 7.04

3.91 5.90 6.63 6.78 6.82 6.91

4.22 6.10 6.95 7.07 7.61

3.53 5.21 5.38 5.62 5.83 5.95

RMS

...

0.92

0.69

0.54

1.10

0.69

0.82

0.47

1.70

6.57 6.42

a

EOM-CCSD/SVP+ results of Mikhailov,71 used as reference for calculation of RMS error. TDDFT results of Mikhailov71 using SVP basis. c CTDA-DFT results of Mikailov71 using B3LYP/SVP. d Our calculated TDDFT values using the 6-311++G** basis and the CAMB3LYP, B3LYP, mCAMB3LYP, PBE0, BHandHLYP, and PBE X-C functionals. b

The computed static (hyper)polarizabilities for pNA in the gas-phase are listed in Table III. The finite field method in the GAMESS program has been used to compute the dipole (μ), linear polarizability (α), and the first- and second-

hyperpolarizability (β and γ ) at the MP2/aug-cc-pVTZ level of theory. The quantities are calculated with the finite field in two different ways, using both an expansion in the energy and an expansion in the dipole moment. The latter gives the more

TABLE III. Calculated static gas-phase (hyper)polarizability data for pNA, and the percent error (%) relative to the CCSD/ d-aug-CC-pVTZ values for α and β, and relative to the MP2/aug-cc-pVTZ (dipole expansion) value for γ . The units are as in Table I. p-Nitroaniline CCSDa,14 CCSDb,14 CCSDc,14 CCSDd,14 CCSDe,14 CCSDf,14 HFa,14 HFb,14 CCSa,14 CC2a,14 MP4g,7 MP2g,7 MP2/6-31+G**7 MP2/DZP6 MP2(energy)b MP2(dipole)b MP2(energy)h MP2(dipole)h MP2/A9

μ

α

%

βzT

%

γ¯ T

%

6.92 6.95 6.91 6.95 6.91 7.19 8.28 7.61 7.78 6.82 7.07 7.07 7.14 6.87 6.85 6.85 6.92 6.92 6.99

1.55 1.54 1.56 1.55 1.56 1.44 1.46 1.42 1.54 1.66 1.89 1.91 2.23 1.58 1.56 1.56 1.47 1.47 1.48

0.0 − 0.1 0.7 ... 0.7 − 6.6 − 5.9 − 8.5 − 0.5 7.1 22.3 23.2 44.5 2.1 1.1 1.1 − 4.7 − 4.7 − 4.4

14.23 13.71 13.91 13.62 13.91 15.94 8.09 6.11 8.28 18.44 16.79 16.68 20.07 17.10 14.09 14.06 15.36 15.39 17.13

4.5 0.7 2.1 ... 2.1 17.0 − 40.6 − 55.1 − 39.2 35.4 23.2 22.5 47.3 25.5 3.4 3.2 12.8 13.0 25.8

14.32 13.47 19.82 19.26 26.58 19.08 20.60 18.11 19.73

− 24.9 − 29.4 3.9 0.9 39.3 ... 8.0 − 5.1 3.4


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TABLE III. (Continued.) p-Nitroaniline SVWN/cc-pVTZ SVWN/6-31+G**7 BLYP/6-31+G**7 LDA:LDA/DZ7 LB94:LDA/DZ7 LDA:LDA/TZ+P7 LB94:LDA/TZ+P7 LDA:LDA/TZ+2P7 LB94:LDA/TZ+2P7 BP86:LDA/TZ+2P7 SAOP:LDA/TZ+2P7 LDA:LDA/TZ+2P+Diffuse7 PBEa,14 PBEb,14 RT-PBE11 PBEb PBE0h PBE0a,14 PBE0b,14 PBE0b B3LYP (80%)g,7 B3LYP (50%)g,7 B3LYP(20%)g,7 B3LYP(20%)i,7 B3LYP/Sadlej8 B3LYPa,8 B3LYPa,14 B3LYPb,14 B3LYP(energy)b B3LYP(dipole)b B3LYPb B3LYPh mCAMB3LYPh BHandHLYPh CAMB3LYPh CAMB3LYPb CAMB3LYP (energy)b CAMB3LYP(dipole)b CAMB3LYPa,14 CAMB3LYPb,14

μ

α

%

βzT

%

7.72 8.39 8.21 9.05 9.56 8.08 8.67 8.08 8.67 7.93 8.77 7.93 7.69 7.47

1.48 2.49 2.50 2.17 2.27 2.39 2.48 2.39 2.48 2.35 2.38 2.52 1.64 1.62

− 4.5 61.0 61.8 40.1 46.5 54.8 60.4 54.6 60.3 52.2 54.0 62.9 5.7 4.9

7.63 7.76 7.84 7.40 7.55 8.18 8.06 8.01 8.13

1.64 1.49 1.56 1.54 1.56 1.81 1.91 2.03 2.35

6.0 − 3.9 1.0 − 0.4 1.0 17.2 23.4 31.2 52.2

7.87 7.48 7.60 7.60 7.61 7.81 7.72 7.79 7.63 7.43 7.43 7.43 7.51 7.33

1.58 1.57 1.59 1.59 1.58 1.51 1.49 1.44 1.46 1.54 1.54 1.54 1.54 1.52

2.4 1.3 2.5 2.5 2.5 − 2.4 − 3.7 − 7.1 − 5.5 − 0.5 − 0.5 − 0.5 − 0.6 − 2.0

11.76 17.32 17.74 15.11 15.65 15.39 16.51 15.22 16.44 14.99 15.33 14.86 14.09 14.78 14.02 13.75 13.95 13.02 12.89 12.58 12.28 13.77 14.67 17.80 13.70 13.44 13.68 13.73 13.32 13.32 13.33 14.68 14.07 12.34 13.00 11.78 11.78 11.78 12.19 11.61

− 13.7 27.2 30.2 10.9 14.9 13.0 21.2 11.8 20.7 10.0 12.5 9.1 3.5 8.5 2.9 0.9 2.4 − 4.4 − 5.3 − 7.6 − 9.8 1.1 7.7 30.7 0.6 − 1.3 0.5 0.8 − 2.2 − 2.2 − 2.2 7.8 3.3 − 9.4 − 4.5 − 13.5 − 13.5 − 13.5 − 10.5 − 14.8

γ¯ T

%

5.83 6.80 8.02 3.00 0.11 4.67 1.77 4.89 1.94 5.11 3.00 7.80

− 69.5 − 64.4 − 58.0 − 84.3 − 99.4 − 75.5 − 90.7 − 74.4 − 89.8 − 73.2 − 84.3 − 59.1

17.01 14.12

− 10.9 − 26.0

15.56 7.81 7.84 6.16 10.62

− 18.5 − 59.1 − 58.9 − 67.7 − 44.4

15.88 16.64 16.53 14.68 14.84 13.55 14.41 15.50 15.82 15.48

− 16.8 − 12.8 − 13.4 − 23.1 − 22.3 − 29.0 − 24.5 − 18.8 − 17.1 − 18.9

a

aug-CC-pVDZ basis. aug-CC-pVTZ basis. c d-aug-CC-pVDZ basis. d d-aug-CC-pVTZ basis. e t-aug-CC-pVDZ basis. f 6-31+G*. g 6-31G basis. h 6-311++G** basis. i 6-31+G* basis. b

accurate results, but the similarity of the results from the two expansions noted is an indicator of the reliability of the calculated quantities. Hammond and Kowalski14 used the B3LYP functional but a larger basis set, cc-pVDZ, for the geometry optimization of pNA, and also warn that different geometries can yield significantly different values for hyperpolarizabilities. However, our calculated hyperpolarizabilities that correspond to the same functional and basis set are in excellent agreement with their results, differing by less than 3% in each case except for with PBE, where the difference is 7%. Their

values for μ, α, and β calculated14 at the CCSD level of theory and with their best basis set, d-aug-cc-pVTZ, are used as a reference. The error analysis shows that the MP2 results are very similar to the CCSD results with the large basis sets, so the MP2 value is used as a reference for γ . Percent errors for α, β, and γ are included in Table III. When the aug-cc-pVTZ basis set is used, PBE, B3LYP, PBE0, BHandHLYP, and CAMB3LYP results are in fairly good agreement with the reference values, although PBE shows the best agreement followed by B3LYP. Use of the


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J. Chem. Phys. 140, 184308 (2014)

smaller 6-311++G** basis set results in only a small deterioration in accuracy, but use of the much smaller 6-31G and 6-31+G** basis sets used with B3LYP, reported by Champagne et al.7 yielded results with substantially larger errors. However, since in some cases a larger basis set results in a larger error, these results may involve some cancellation of errors. The CAMB3LYP functional did not perform as well as expected, having error values for β and γ of 14% and 19%, respectively. These results demonstrate the criticality of the parameters that are used in long range-separated functionals, as we previously pointed out.35 Next, the gas-phase data for pNA at finite wavelengths are summarized in Table IV. Hyperpolarizabilities at 9398 cm−1 (1064 nm) and 5241 cm−1 (1908 nm) were previously reported using MP26 (scaled from the zero frequency MP2 values using CPHF results) and also for the X-C functionals PBE,11 BLYP,8 and B3LYP.8, 20 We used CAMB3LYP, mCAMB3LYP, B3LYP, PBE0, and PBE using larger basis sets than in the previous studies, and also scaled MP2 using the aug-cc-pVTZ basis. The static hyperpolarizability results of Table III indicate that the MP2/aug-cc-pVTZ values are a significant improvement over the MP2/6-31+G* values, but the CPHF method may underestimate the frequency dependence (dispersion), as was implied by Suponitsky et al.13 Experimentally, the only gas-phase result is that of Kaatz et al.,31 at a wavelength of 1064 nm. At 1908 nm, the MP2 values are used as a reference, while at 1064 nm, the error is evaluated both using MP2 as the reference and using the measured values. While Suponitsky et al.13 reported a CCSD value for β (scaled to 1064 nm from zero frequency), the basis set used was only 6-31+G*, and considering that the CCSD/6-31+G* static value for β reported in Table III has a 17% error from

the large basis CCSD, while the MP2/aug-cc-pVTZ value only has a 3.2% error, the latter is considered the preferable choice as a reference. At 1908 nm, the B3LYP, PBE0, BHandHLYP, CAMB3LYP and mCAMB3LYP functionals using either the aug-cc-pVTZ or the 6-311++G** basis demonstrate fair agreement with the MP2 results for the first hyperpolarizability. For the second hyperpolarizability, the γ EFISH values of 20.3 and 19.4 calculated using the PBE and B3LYP functionals, respectively, are in good agreement with the reference value of 20.8, while the PBE0, CAMB3LYP, mCAMB3LYP, and BHandHLYP values of 18.1, 17.7, 17.2, and 15.5, respectively, are only in fair agreement. At 1064 nm, the CAMB3LYP results are in excellent agreement with the MP2 reference, while BHandHLYP is in good agreement for β but only in fair agreement for γ , and PBE0 is in good agreement for γ but only in fair agreement for β. Since the reported measured value31 is 21% lower than the MP2 reference, the agreement with experiment is not as good for most of the calculated values. The difference may be partly because the calculated values do not include any vibrational effects, but previous work by Quinet et al.72 indicates that, in this frequency range, we might expect the inclusion of vibrational effects to reduce the calculated β by only 3% to 5%. In a previous study by Suponitsky et al.,13 good agreement with the experimental β was obtained using the BMK, PBE0, and B97-2 functionals with the 6-311+G(2df,pd) basis by optimizing the geometry at the same level of theory with C2v symmetry constraint, and scaling from static values using static and dynamic Hartree-Fock values. As vibrational effects were not included in that study, the good agreement with experiment is likely due to some cancellation of errors

TABLE IV. Gas-phase dipole and (hyper)polarizability data for pNA at 1908 nm, 1064 nm, and 1300 nm. The percent error (%) is relative to the scaled MP2/aug-cc-pVTZ. For β at 1064 nm, the error relative to the measured value (%m ) is also listed. The units are as in Table I. (a) Photon energy of 5241 cm−1 (wavelength = 1908 nm) p-Nitroaniline MP2a,b MP2/DZPb,6 LDA/TZP12 LDA/TZ2P+12 SVWN/cc-pVTZ LDA/cc-pVTZ RT-PBE/DZP11 PBEc PBEa PBE0-LDA0/TZP12 PBE0-LDA0/TZ2P+12 PBE0c PBE0a B3LYP8 B3LYPa B3LYPc BHandHLYPc mCAMB3LYPc CAMB3LYPc CAMB3LYPa

α

βzT

%

1.57 1.59

15.49 19.20 37.54 36.58 14.26 14.34 17.08 18.25 16.77 30.76 29.40 16.23 14.65 16.23 15.71 17.27 14.00 16.40 14.92 13.51

... 24 142 136 −8 −7 10.3 17.9 8.3 99 90 5 −5 5 1 12 − 9.6 6 −4 − 13

1.49

1.58 1.65

1.50 1.57 1.60 1.52 1.45 1.50 1.47 1.55

γ¯TTH G

T γ¯EF I SH

%

20.75

...

7.35 7.35

− 65 − 65

γ¯ITDRI

24.00 11.46 22.66 9.43

16.93 20.32

− 18.4 − 2.1

16.60 18.11

− 20 − 13

19.42 17.39 15.46 17.25 16.52 17.71

−6 − 16 − 25.5 − 17 − 20 − 15

10.78 17.18

20.76 20.44 19.22

16.39 15.77


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J. Chem. Phys. 140, 184308 (2014)

TABLE IV. (Continued.) (b) Photon energy of 9398 cm−1 (wavelength = 1064 nm) p-Nitroaniline Measured1 MP2a,b MP2/DZPb,6 MP2b,d,13 MP2b,e,13 CCSDb,e,13 RT-PBE/DZP11 PBEa BLYPf,8 BLYP/Sadlej8 B3LYPf,8 B3LYP/Sadlej8 RT-B3LYP/6-31G*20 B3LYP/6-31G*20 RT-B3LYPe,20 B3LYPe,20 B3LYPa,19 B3LYPb,d,13 B3LYPc B3LYPa B97-2b,d,20 BMKb,d,20 PBE0b,d,20 PBE0c PBE0a M06-HFc,19 ωB97c,19 LC-OLYPc,19 LC-BLYPc,19 ωB97Xc,19 LC-wPBEc,19 BHandHLYPc mCAMB3LYPc CAMB3LYPc CAMB3LYPa

α

1.59 1.59

1.69

1.30 1.30 1.55 1.55

1.55 1.62

1.52 1.60

1.47 1.53 1.49 1.57

βzT 15.44 19.54 24.00 16.76 19.82 16.43 33.48 29.61 30.60 31.72 24.66 25.88 24.07 22.77 26.67 26.66 25.99 17.79 26.79 24.50 16.30 15.70 15.85 24.29 22.05 12.24 13.87 13.56 14.36 14.79 14.12 19.32 24.38 21.09 19.21

%

%m

− 21 ... 23 − 14 1 − 16 71 52 57 62 26 32 23 17 36 36 33 −9 37 25 − 17 − 20 − 19 24 13 − 37 − 29 − 31 − 27 − 24 − 28 −1 25 8 −2

... 27 55 9 28 6 117 92 98 106 60 68 56 48 73 73 68 15 74 59 6 2 3 57 43 − 21 − 10 − 12 −7 −4 −8 25 58 37 24

γ¯TTH G

T γ¯EF I SH

%

%m

29.15 25.57 27.60

14 ... 8

... − 12 −5

34.10

33

17

γ¯ITDRI

43.17

11.79

96.31

19.18

103.0

76.03 54.56

27.96 29.98

9 17

−4 3

25.62 26.96

0 5

− 12 −8

21.78 26.56 24.03 25.07

− 15 4 −6 −2

− 25 −9 − 18 − 14

21.59 20.03

(c) Photon energy of 7692 cm−1 (wavelength = 1300 nm) p-Nitroaniline PBE/TZP(STO)16 CAMB3LYP/TZP(STO)16 CAMB3LYPc mCAMB3LYPc

α

βzT

γ¯TTH G

T γ¯EF I SH

17.02 1.48 1.51

17.72 19.94

29.88 34.36

19.79 21.16

γ¯ITDRI 11.73 14.61 17.72 18.70

a

aug-CC-pVTZ basis. Scaled from static MP2 using static and frequency-dependent HF. Note that although frequency-dependent β and γ at the CCSD level could be calculated by the Dalton68 program, this cannot be done in a parallel calculation, making it a challenge for molecules the size of pNA with large basis sets. c 6-311++G** basis. d 6-311+G(2df,pd) basis. e 6-31+G* basis. f aug-CC-pVDZ basis. b

of opposite sign. The good agreement our B3LYP value for γ has with experiment is also likely due to some cancellation of errors. Calculations at 7692 cm−1 (1300 nm) were also performed in order to compare to the results of Rinkevicius et al.16 Clearly, diffuse functions in the basis set are important in the calculation of hyperpolarizabilities, as the value of 17.7 obtained for γ IDRI at 1300 nm using CAMB3LYP/6-

311++G** is 20% larger than the value reported previously16 using the same functional but only a TZP basis. For further analysis, the gas-phase values for β z T , γ EFISH , and γ THG are plotted in Figure 2. As stated above, the tested X-C functionals have only fair agreement with the MP2 reference values, with none clearly superior. While CAMB3LYP agrees best at the highest frequency for both β and γ EFISH , at the lower frequency the best agreement for β is with the


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J. Chem. Phys. 140, 184308 (2014)

culated values of γ THG are much larger, particularly for the B3LYP functional. These results demonstrate that application of TDDFT has to be carefully benchmarked for specific molecular systems, and claims of the adequacy of long-range corrected functionals may not be appropriate.19 B. pNA in solution

Table S1 in the supplementary material73 lists the measured and calculated values for pNA in solution, and a comparison for β and γ in acetone and dioxane is given in Figure S1. While the calculated results do not include corrections for vibrational effects, a previous study72 indicates that at these frequencies this effect on β is only 3%–5%, while the experimental uncertainty is estimated to be about 15%. While the value for μ obtained at 1908 nm in acetone using CAMB3LYP with PCM is larger than the reported measured value, the calculated β and γ THG are in good agreement with experiment, having about a 10% error, as also shown in Fig. S1 of the supplementary material,73 and the value for α is also in fair agreement with experiment. The errors are larger when the B3LYP, PBE, and PBE0 functionals are used. Table S1 of the supplementary material73 also lists the β measurements of Stahelin et al.29 in six different solvents at 1064 nm along with our corresponding calculated values of the (hyper)polarizabilities with five X-C functionals. These data are used to generate Figure 3, which shows the calculated and measured β values for pNA at 1064 nm as a function of the solvent dielectric constant. While none of the X-C functionals give excellent agreement with experiment for every solvent, the CAMB3LYP results have the best agreement. The CAMB3LYP values are in particularly good agreement with experiment when the solvent is chloroform or acetone, and it is also the best in acetonitrile, while only the BHandHLYP functional has lower error in dichloromethane. The only exception to the good performance of the CAMB3LYP functional in Figure 3 is for the solvent dioxane, where CAMB3LYP has the largest error compared to experiment of all the functionals except BHandHLYP. In a solvent with low polarity and a low static

FIG. 2. Hyperpolarizabilities of gas-phase pNA as a function of field frequency. Green dashed line: DZP basis, red dashed line: 6-311++G** basis, black solid line: aug-cc-pVTZ basis. Star: MP2, triangle: B3LYP, circle: PBE0, square: CAMB3LYP, diamond: mCAMB3LYP, X:BHandHLYP, dashed line: PBE, plus sign: measured value. (a) β z T in 10−30 esu, (b) γ EFISH in 10−36 esu, (c) γ THG in 10−36 esu.

B3LYP functional, while for γ EFISH the PBE functional has the best agreement. At zero frequency, the TDDFT γ EFISH values are consistently smaller than the MP2 value. The THG values plotted in Figure 2(c) are similar for all the functionals and only slightly below MP2 for the static case and for 1908 nm (5241 cm−1 .) At 1064 nm (9398 cm−1 ), the cal-

FIG. 3. Measured and calculated first hyperpolarizability of pNA at 1064 nm as a function of solvent dielectric constant.


184308-10


dielectric constant, such as dioxane, the increase in β due to solvation is smaller than in the in the more polar and more easily polarizable solvents, and thus other sources of error, such as the neglect of vibrational effects and experimental error, are more significant. While for most of the solvents tested here the pure DFT functional, PBE, overestimates the effect of solvation on β, for dioxane these errors tend to cancel and PBE has the lowest error. The PBE0 functional performs fairly well and has the lowest error when the solvent is methanol. Not only do the CAMB3LYP results have the lowest average error of any of the X-C functionals, but good agreement is found over a wide range of dielectric constants. The γ EFISH data, plotted in Figure S1(b) of the supplementary material73 , show qualitative agreement between all functionals considered, but no experimental data are available for comparison. The THG data in Figure S1(c) of the supplementary material73 demonstrate fair agreement with experiment when the CAMB3LYP functional is used, as the calculated γ is only about 10% larger than the measured value at 1064 nm, while the B3LYP result is significantly higher. Figures S1(d) and S1(e) of the supplementary material73 report the β and γ EFISH values, respectively, for pNA in 1,4-dioxane. The TDDFT calculations underestimate β compared to experiment, although at the lower frequencies the difference is small. For this solvent, CAMB3LYP has the poorest agreement with experiment of the tested functionals, while PBE shows the best agreement, except at the higher frequencies. The large increase in β using PBE at 830 nm (12 048 cm−1 ) is likely due to a resonance effect, as the excitation energy of the CT state at this level of theory (26 445 cm−1 ) is fairly close to twice this photon frequency (24 096 cm−1 ). The resonance effect at 830 nm is even more pronounced in the EFISH data in Figure S1(e) of the supplementary material73 , where the value of γ EFISH using PBE is 3 to 4 times the values from the other functionals. C. Nitrobenzene and DANS

The data for nitrobenzene are given in Table S2 of the supplementary material73 . In the finite field calculation at the MP2/aug-cc-pVTZ level with the GAMESS program, we found that when the default field of 0.001 a.u. was used, the energies obtained were not precise enough to evaluate the second hyperpolarizability components with the energy expansion, so for this calculation we used a field of 0.005 a.u. The good agreement between the values obtained for the energy and dipole expansions when this field is used indicates that the error is low. When the MP2/aug-cc-pVTZ results are used as a reference for the static gas-phase calculations, the calculated CAMB3LYP/aug-cc-pVTZ results are found to have a low error. The error is slightly increased when the smaller 6–311++G** basis is used, although when mCAMB3LYP is used with this basis set the error is surprisingly low. In comparing to the gas-phase measured values31 of β and γ at 1064 nm, we note that while the excellent agreement with experiment of the calculated γ using CAMB3LYP/6-311++G** is possibly partially serendipitous, the results are encourag-

J. Chem. Phys. 140, 184308 (2014)

ing. The calculated value of β at 1908 nm for neat nitrobenzene at the CAMB3LYP/6-311++G** level of theory is in fair agreement with experiment. The experimental value for γ in the condensed-phase at 1908 nm with EFISH is an order of magnitude larger than all the calculated values, and is to be investigated in future work. The hyperpolarizability data for DANS are listed in Table V. Again, the MP2 static gas-phase results are used as reference. CAMB3LYP/6-311++G** results are in excellent agreement with the MP2 values. While the results of Krawczyk15 are limited to β, they are in good agreement with our results. The experimental values reported by Cheng et al.28 in chloroform at 1908 nm have been recalculated from the raw experimental data using the localfield factors of Tu, Luo, and Agren.10 When these experimental values are used as a reference, the β value calculated with PBE0 is in good agreement, while the β calculated with CAMB3LYP is much lower. However, the γ value calculated with CAMB3LYP is in good agreement with experiment.

D. Molecular-Au attachment

In (Au2 S-pNA)−1 , a sulfur atom is bonded to one of the atoms in a gold dimer and also to the nitrogen atom in the amino group of pNA, replacing one of the hydrogen atoms. (Hyper)polarizability data for (Au2 S-pNA)−1 are given in Table VI. While the results indicate that the attached gold dimer enhances the NLO properties of pNA, the values for the hyperpolarizabilities vary significantly over the four XC functionals tested. At the highest frequency of 9398 cm−1 (λ = 1064 nm), resonance effects result in unphysical values when the B3LYP, PBE, and PBE0 functionals are used. With the gold dimer attached, the molecule has a higher density of excited states, and the first excited state is lower in energy, making resonance effects more likely. In particular, at the B3LYP level of theory, which has the largest resonance effect, the excitation energy to the first excited state is 18 712 cm−1 , which is nearly exactly twice the photon energy, resulting in large resonance effects for second-order and higher processes. The excitation energy at the CAMB3LYP level of theory is 24 563 cm−1 , which not only is likely closer to the correct value, but also is large enough to avoid resonance problems for second-order processes at this field frequency. From the results with the CAMB3LYP functional, attachment of the Au2 S group increases β by over 100% and γ EFISH by over an order of magnitude in the results for static frequency and for ν = 5241 cm−1 (λ = 1908 nm). Even larger enhancement factors of 6 for β and over 30 for γ EFISH are predicted at a frequency of 9398 cm−1 (λ = 1064 nm). We have previously reported55 a TPA spectrum for DANS that was calculated using quadratic response TDDFT in the Dalton program, and here we have also calculated the TPA for (Au2 S-DANS)−1 . As we described previously,54, 55 the TPA of a polar molecule can often be described by the two-state approximation, in which the TPA cross-section is proportional to the square of the ground-to-excited state transition dipole moment and to the square of the difference in the dipole moment between the two states. This is the case for


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J. Chem. Phys. 140, 184308 (2014)

TABLE V. Dipole and (hyper)polarizability data for DANS. In the static, gas-phase case, the percent (%) errors are relative to the MP2/6-311++G** values, while for the data in chloroform at 1908 nm, they are relative to the experimental values. The units are as in Table I. DANS

Solvent

λ (nm)

μ (D)

α

None None None None None None None None None

Inf. Inf. Inf. Inf. Inf. Inf. Inf. Inf. Inf.

9.04 9.04

3.99 3.99

Solvent

MP2(energy)a MP2(dipole)a MP2a,15 B3LYPa,15 CAMB3LYPa,15 CAMB3LYPa CAMB3LYPb CAMB3LYP/SDDall LC-B3LYPa

11.2

βzT 146.8 147.6 144.73 253.6 144.1 147.2 126.6 168.6 97.71

% −1 ... −2 72 −2 0 − 14 14 − 34

γ¯ T

%

410.8 410.8

0 ...

407.1 327.0 439.5 247.5

−1 − 20 7 − 40

10.5 9.96 11.2 10.0

4.11 3.80 3.69 3.77

λ (nm)

μ (D)

α

βzT

γ¯TTH G

None None None None None None

1908 1908 1908 1908 1908 1908

10.5 11.2 10.1 11.1 9.96 10.0

4.17 3.75

190.9 217.6

837.4

3.86 3.81

162.1 118.3

DANS

Solvent

λ (nm)

μ (D)

α

βzT

γ¯ T

CAMB3LYPa CAMB3LYPb PBE0a LC-PBEa

CHCl3 CHCl3 CHCl3 CHCl3

Inf. Inf. Inf. Inf.

12.1 11.4 13.4 11.5

5.02 4.59 5.78 4.58

322.8 261.7 649.6 189.8

1213 902.5 2730 613.3

DANS

Solvent

λ (nm)

μ (D)

α

βzT

%

Measured28 Measuredd,10, 28 CAMB3LYPa CAMB3LYPb PBE0a LC-PBEa

CHCl3 CHCl3 CHCl3 CHCl3 CHCl3 CHCl3

1908 1908 1908 1908 1908 1908

6.60 6.60 12.1 11.4 13.4 11.5

3.40 4.05 5.13 4.67 5.98 4.64

438 1008. 442.5 350.1 1080 236.4

... − 56 − 65 7 − 77

DANS CAMB3LYPa CAMB3LYP/SDDall CAMB3LYP/SDD CAMB3LYPc CAMB3LYPb LC-B3LYPa

642.0 926.7 644.5 544.7

T γ¯EF I SH

γ¯ITDRI

567.5 623.9 449.8 617.0 447.2 312.1

504.3

γ¯TTH G

%

1350 2725 2958

... 9%

404.3 547.5 401.3

T γ¯EF I SH

1857 1331 5801 810.0

a

6-311++G** basis set. 6-311G** basis set. c Stuttgart_rlc basis set and ECP. d (Hyper)polarizabilities have been re-calculated from the experimental values of Cheng28 using the local-field factors of Tu.10 b

the first TPA peak in these two molecules, which is the result of the charge transfer excited state. The two spectra are compared in Figure 4 where we see that attachment of the gold dimer causes a red-shift of about 5500 cm−1 in the first peak TABLE VI. Calculated dipoles, polarizabilities, and hyperpolarizabilities for (Au2 S-pNA)−1 . The units are as in Table I. Au2 S-p-Nitroaniline Solvent λ (nm) μ (D) CAMB3LYP/SDD B3LYP/SDD PBE/SDD PBE0/SDD CAMB3LYP/SDD B3LYP/SDD PBE/SDD PBE0/SDD CAMB3LYP/SDD B3LYP/SDD PBE/SDD PBE0/SDD

None None None None None None None None None None None None

Inf. Inf. Inf. Inf. 1908 1908 1908 1908 1064 1064 1064 1064

7.27 7.56 7.51 7.75 7.27 7.56 7.51 7.75 7.27 7.56 7.51 7.75

α

βzT

3.57 27 3.84 47 4.07 67 3.76 47 3.63 38 3.93 95 4.21 261 3.85 86 3.80 118 4.18 − 17 437 4.66 − 510 4.07 1334

T γ¯EF I SH

208 396 613 354 289 771 2738 637 827 − 89 470 10 321 39 206

and more than doubles its height to 268 GM. The addition of the gold dimer destabilizes both the HOMO and the LUMO, but destabilizes the HOMO to a larger extent, decreasing the H-L gap and causing this red-shift. The addition of the gold atoms also increases the density of the occupied orbitals near the HOMO, and the H-1->L and the H-2->l transitions make a significant contribution to this excited state along with H->L. However, the transition dipole moment is nearly the same for these two systems, and an increase in the dipole difference is responsible for the increase in TPA cross-section. With the gold atoms, the system has a higher density of states and thus many more terms contributing in the sum-over-states description of the TPA cross-section, and thus the spectrum is more complicated at higher energies. For the system with the gold-atoms, excited states near 44 000 cm−1 are close to resonant with the first excited state at 22 000 cm−1 , causing the unphysically large TPA peak in this region. As with pNA, the addition of Au2 S to the electron donor group reduces the excitation energy as well as increasing the density of states, resulting in the red-shift of the spectrum as well as in an increase in magnitude.


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FIG. 4. TPA spectra for DANS (solid blue line) and [Au2 S-DANS]−1 (dashed green line), calculated using the CAMB3LYP functional.

IV. CONCLUSIONS

In the calculation of hyperpolarizabilities using TDDFT, several functionals have shown the ability to quantitatively model NLO effects in organic D-π −A compounds, including GGA functionals, e.g., PBE, hybrid functionals, e.g., PBE0 and B3LYP, and long range-separated hybrids such as CAMB3LYP. While the CAMB3LYP functional is considered the best functional to use for D-π −A compounds due to its more accurate exchange in asymptotic regions, results were mixed for these four functionals. While the CAMB3LYP hyperpolarizabilities agreed with higher-level theory better than the other functionals at the higher frequency, the other functionals frequently showed better agreement in the static and low-frequency calculations. When the calculated β for pNA in solution is compared to experiment, CAMB3LYP has the best agreement for the solvents chloroform, acetone, and acetonitrile, but in dichloromethane, methanol, and dioxane solvents, the best agreement was for B3LYP, PBE0, and PBE, respectively. We conclude that quantitative prediction of hyperpolarizabilities is still a challenge hampered by inadequate functionals, basis sets, and solvation models, requiring further experimental characterization for classes of NLO compounds. Attachment of gold atoms to D−π -A molecules can be a valuable method to enhance and tailor NLO properties. The predicted enhancement of the second hyperpolarizability (γ ) of pNA by the attachment of Au2 S to the donor group is more than an order of magnitude at each of the studied frequencies, applying the CAMB3LYP functional. A significant enhancement in the first hyperpolarizability (β) is also predicted. Attachment of Au2 S to DANS more than doubles the magnitude of the TPA and red-shifts the peak by about 5500 cm−1 .

ACKNOWLEDGMENTS

Support for this work was provided by the Air Force Office of Scientific Research. The AFRL DoD Supercomputing Resource Center is acknowledged for providing computational resources and helpful support.


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