Article pubs.acs.org/jced
Polarizabilities from Long-Range Corrected DFT Calculations Shintaro Maekawa and Krzysztof Moorthi* Computational Science Group, Mitsui Chemicals, Inc., Sodegaura, 299-0265, Japan S Supporting Information *
ABSTRACT: The long-range corrected DFT functionals, LCBLYP, LC-PBE and CAM-B3LYP with the augmented Dunning-type triple-ζ basis sets represent dynamical polarizabilities at λ = 589.3 nm of 105 medium-sized organic compounds containing C, H, O, N, S, P, F, Cl, Br and I elements with the root mean squared deviations (RMSD) of about 0.34, 0.35, and 0.42 Å3, respectively. These errors do not change appreciably when the augmented double-ζ basis sets are used. The functionals with 100% of Hartree−Fock (HF) exchange at long-range perform best for aromatic compounds and the CAM-B3LYP or B3LYP functionals for fully saturated compounds. The degrees of HF exchange in mid- and long-range affect strongly the shape and location of the distributions of absolute errors in polarizability, P(Δα). The differences between functionals belonging to the BLYP and PBE families, and having the same degree of HF exchange, have much smaller effect on the P(Δα) distributions.
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INTRODUCTION The organic optical materials are technologically important because of their unique combination of physical properties: low density, excellent optical properties, ease of property tailoring and material processing. Since performance of optical devices is often sensitive to small variations of the refractive index, an accurate determination of the refractive index of organic materials is required. The refractive index of a material, n, is related to the trace of the polarizability tensor, ⟨α⟩, and density, d, of material by the Lorentz−Lorenz equation 4πNA ⟨α⟩d n2 − 1 = 2 3 M n +2
functional theory (DFT) due to the favorable balance between computational cost and accuracy. DFT approaches to polarizability are expected to have superior predictive power when appropriate functional is selected. However, the functionals employing the local density approximation (LDA) or the generalized gradient approximation (GGA) exhibit incorrect asymptotic behavior of the exchange correlation potential, which decays exponentially instead of decreasing as −1/r.3,13 This imperfection contributes to an underestimation of excitation energies, an overestimation of longitudinal polarizabilities in conjugated systems, poor description of intermolecular interactions, etc.3,14,17−19,41 One approach to correct the exchange potential is based on splitting the two-electron operator into a short- and a long-range part14,19
(1)
where M is the molecular weight of the substance and NA is the Avogadro number. Equation 1 is a good approximation at high frequencies.1 Simulations for nonpolar liquids indicate that the error in evaluating polarizability through the Lorentz−Lorenz equation is generally below 1%.2 Polarizability is also important, among others, for modeling intermolecular interactions in systems containing water and ionic species3,4 and solvation effects in nonreacting5 and reacting systems.6 Solubility models of gases in liquids,7 phase equilibria in fluorocarbons8 etc. also require consideration of the polarizability of components. Empirical models of molecular polarizability or refractive indices of organic substances include, among many others, the atomic additive method (AAM),11 group contribution methods,12 and descriptor-based QSPR approaches.9,10 The mean absolute relative error (MARE) on polarizability of moderately conjugated organic substances, predicted using AAM and its variants is rather low, and varies within 1 and 2%.11,33 However, for substances dissimilar to those in training sets, predictive ability of empirical methods usually deteriorates. Theoretical calculations of polarizability are often performed using density © 2014 American Chemical Society
1 − [A + B erf(μr12)] A + B erf(μr12) 1 = + r12 r12 r12
(2)
where r12 is an interelectronic distance, erf is the error function and μ, A and B are adjustable parameters. The A and A+B parameters, which satisfy 0 ≤ A ≤ 1, 0 ≤ B ≤ 1, 0 ≤ A+B ≤ 1, represent the HF exchange contributions in the short- and longrange limits, respectively. The short-range term is usually approximated as a local density exchange. The long-range term Special Issue: Modeling and Simulation of Real Systems Received: March 7, 2014 Accepted: June 19, 2014 Published: July 3, 2014 3160
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Table 1. Static Electronic, α(0), and Vibrational Polarizability, αv(0), of Benzene Calculated with HF and DFT Methods with augcc-pVDZ (aDZ) and aug-cc-pVTZ (aTZ) Basis Sets; Benzene Geometry Calculated Using B3LYP Method with cc-pVDZ Basis Sets (A); B3LYP Method with cc-pVTZ Basis Sets (B); Taken from Ref 25 (C); Relative Error, ε (eq 5) aDZ//C
basis set//geometry α(0)/Å HF LC-BLYP CAM-B3LYP B3LYP BLYP PBE PBE0 LC-PBE expt a
3
10.00 9.93 10.13 10.24 10.44 10.33 10.12 9.82 9.96 ± 0.1a
aTZ//C ε
α(0) /Å
0.4 −0.3 1.7 2.8 4.8 3.7 1.6 −1.4 −
10.04 9.96 10.15 10.27 10.47 10.37 10.15 9.83
3
aDZ//A
aTZ//B
aTZ//B
ε
ε
ε
αv(0)/Å3
0.8 −0.1 1.9 3.1 5.1 4.1 1.9 −1.3
2.2 1.5 3.4 4.5 6.6 5.5 3.3 0.3
0.9 0.0 2.0 3.1 5.2 4.2 2.0 −1.2
0.33 0.33 0.33 0.33 0.37 0.41 0.35 0.36 0.29a
Reference 26. x E LR − HF = −
1 2
∑ ∑ ∬ Ψ*iσ(r1)Ψ*jσ(r1) σ
× Ψjσ (r2) dr1 dr2
ij
adopted from ref 11 if not stated otherwise. In this set, methanol exhibits the lowest (3.26 Å3) and pentadecane the highest (28.55 Å3) polarizability. For electronic structure computations, for the C, H, N, O, P, S, F, Cl and Br atoms, correlation-consistent valence double-ζ, ccpVDZ, and triple-ζ, cc-pVTZ, Dunning basis sets20,21 have been employed, which are further abbreviated as DZ and TZ sets, respectively. For the I atom appropriate pseudopotentials aug-ccpVDZ-PP, aug-cc-pVTZ-PP have been employed.22 The isolated molecular structures have been fully optimized using B3LYP functional with DZ and TZ basis sets. Symmetry constraints were not imposed during optimization. Frequency analysis has been performed in order to confirm that the stationary points found correspond to minima. For calculation of static and dynamic electronic polarizabilities, the diffused functions have been added to the DZ and TZ basis sets, and the resulting basis sets are referred to as aDZ and aTZ, respectively. In the LC-BLYP and LC-PBE methods, the attenuation parameter, μ, (eq 2) is set to μa0 = 0.47 where a0 is the atomic unit of length. In the CAMB3LYP method μa0 = 0.33 is used. All electronic structure calculations have been performed using the Gaussian09 program.42 Several measures of discrepancies between the experimental and calculated values have been used: the absolute errors
erf(μr12) Ψiσ (r2) r12 (3)
is evaluated using the occupied spin orbitals ψiσ with the modified long-range operator. The method is generally applicable to the GGA functionals. The application of the above scheme with A = 0 and B = 1 (long-range correction, LC)14 to the BLYP15,16,23 functional yields LC-BLYP,17 and to PBE34 the LC-PBE functional. Analogous separation applied to the PBE037 hybrid functional with A = 0.25 and B = 0.75 yields CAM-PBE0 (or LCPBE0) functional.38 These modified functionals contain 100% of the HF exchange in the long-range limit. The separation based on eq 2 applied to the B3LYP15,23,39,40 functional with A = 0.19 and B = 0.46 yields the CAM-B3LYP19 functional, with 65% of HF exchange in the long-range limit. The modified functionals represent better the conjugated oligomer (hyper)polarizabilities17,41 and excitation energies of dyes.19,35,36 The theoretical studies of polarizability are often limited to small sets of molecules. Hence, information on the performance of the long-range corrected functionals based on larger sets of molecules is desirable. We calculate polarizabilities for more than a hundred diverse organic compounds using several long-range corrected (LC-BLYP, LC-PBE, and CAM-B3LYP), and conventional (BLYP, PBE, B3LYP, and PBE0) functionals, and Hartree−Fock theory. The results are compared to available experimental data in order to evaluate the performance of the long-range corrected functionals considered.
Δx = xc − xe
(4)
where xc is the calculated value and xe is the experimental value; relative percentage errors x − xe ε= c 100 xe (5)
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DATA AND COMPUTATIONAL METHODS The set of 105 organic substances (Table S-1 in the Supporting Information [SI]) has been adopted from Tables 2 and 6 of Bosque and Sales.11 The molecules include C, H, O, N, S, P, F, Cl, Br, and I elements. The substances include cyclic and acyclic nonaromatic hydrocarbons, aromatic hydrocarbons, aromatic and aliphatic alcohols, amines, nitriles, nitro-derivatives, amides, aldehydes, ketones, esters of carboxylic and inorganic acids, ethers, carboxylic acids, halogen derivatives, amines, and various sulfur compounds. One molecule (methyl linoleate) has been excluded due to long computational times. The set contains 42 compounds which possess only σ-bonds (σ-subset) and 63 compounds containing at least one π-bond (π-subset). The experimental polarizability values for these molecules (extracted from refractive indices of liquids measured at the D-sodium line λ = 589.3 nm and at 293 or 298 K, using eq 1) have been also
mean signed relative errors (MSRE) ε MSRE = ∑ i N i
(6)
where N is the number of data points; mean absolute relative errors (MARE) MARE =
∑ i
|εi| N
(7)
and root-mean-square deviations, ρ ρ= 3161
∑i (Δxi)2 N
(8)
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Table 2. Static Electronic Polarizability, α(0), of Naphthalene, Anthracene and Naphthacene Calculated with HF and DFT Methods and with aug-cc-pVDZ (aDZ), and aug-cc-pVTZ (aTZ) Basis Sets for Two Geometries: (I) Calculated at B3LYP/ccpVDZ level; (II) Calculated at B3LYP/cc-pVTZ level; Relative Error, ε (eq 5), in Parentheses naphthalene
a
anthracene
naphthacene
basis set//geometry
aDZ//I
aTZ//II
aDZ//I
aTZ//II
aDZ//I
aTZ//II
method
α(0)/Å3
α(0)/Å3
α(0)/Å3
α(0)/Å3
α(0)/Å3
α(0)/Å3
HF LC-BLYP CAM-B3LYP B3LYP BLYP PBE PBE0 LC-PBE expt
17.49 (0.6) 17.38 (0.0) 17.80 (2.4) 18.10 (4.1) 18.51 (6.5) 18.34 (5.5) 17.90 (3.0) 17.20 (−1.0) 17.38a 17.40b
17.24 (−0.8) 17.12 (−1.5) 17.54 (1.0) 17.84 (2.7) 18.24 (4.9) 18.09 (4.1) 17.65 (1.5) 16.94 (−2.5)
26.33 (3.7) 26.14 (2.9) 26.89 (5.9) 27.50 (8.3) 28.21 (11.1) 27.99 (10.2) 27.20 (7.1) 25.90 (2.0) 25.3a 25.4a
25.94 (2.1) 25.75 (1.4) 26.49 (4.3) 27.10 (6.7) 27.79 (9.4) 27.59 (8.6) 26.80 (5.5) 25.50 (0.4)
36.68 (6.0) 36.35 (5.1) 37.54 (8.5) 38.63 (11.7) 39.78 (15.0) 39.46 (14.0) 38.15 (10.3) 36.00 (4.1) 34.6b
36.12 (4.4) 35.80 (3.5) 36.98 (6.9) 38.05 (10) 39.17 (13.2) 38.93 (12.5) 37.63 (8.8) 35.48 (2.6)
Reference 28. bReference 26.
Table 3. Mean Signed Relative Error, MSRE (eq 6), Mean Absolute Relative Error, MARE (eq 7), Root Mean Squared Deviation, ρ (eq 8) and Maximum Relative Error, MAXRE, for Substances in Table S-1 in the SI protocol
a
X/aug-cc-pVDZ//B3LYP/cc-pVDZ
X
MSRE/%
MARE/%
HF LC-BLYP CAM-B3LYP B3LYP BLYP LC-PBE PBE0 PBE AAMa
−3.3 −0.4 2.2 4.2 8.3 −1.4 3.0 7.8 0.1
4.0 2.1 2.6 4.2 8.3 2.3 3.1 7.8 2.1
HF LC-BLYP CAM-B3LYP B3LYP BLYP LC-PBE PBE0 PBE AAMa
−5.3 −2.0 0.6 2.7 6.9 −2.9 1.5 6.4 −0.3
5.3 2.3 1.3 2.7 6.9 3.0 1.6 6.4 2.1
HF LC-BLYP CAM-B3LYP B3LYP BLYP LC-PBE PBE0 PBE AAMa
−1.9 0.6 3.2 5.2 9.3 −0.3 4.0 8.6 0.4
3.2 2.0 3.4 5.2 9.3 1.8 4.1 8.6 2.1
ρ/Å3
X/aug-cc-pVTZ//B3LYP/cc-pVTZ MAXRE/%
all compounds 10.9 5.6 9.0 11.8 16.3 5.1 10.2 15.2 10.6 σ-bonded compounds 0.60 10.4 0.26 4.2 0.31 4.4 0.52 6.8 1.02 12.4 0.33 5.1 0.41 5.5 0.98 12.1 0.32 6.9 π-bonded compounds 0.50 10.9 0.36 5.6 0.65 9.0 0.93 11.8 1.48 16.3 0.31 4.9 0.77 10.2 1.40 15.2 0.34 10.6 0.54 0.32 0.54 0.79 1.32 0.32 0.65 1.25 0.33
MSRE/%
MARE/%
ρ/Å3
MAXRE/%
−4.3 −1.4 1.2 3.2 7.4 −1.4 2.5 7.3
4.7 2.3 2.2 3.3 7.4 2.5 2.8 7.3
0.65 0.34 0.42 0.64 1.15 0.35 0.59 1.18
10.2 5.6 8.5 11.2 15.7 6.2 9.5 14.6
−6.1 −2.7 −0.1 2.0 6.3 −3.0 1.4 6.4
6.1 2.8 1.5 2.1 6.3 3.1 1.9 6.4
0.72 0.35 0.24 0.41 0.90 0.37 0.42 0.99
9.6 5.3 3.7 5.9 11.2 6.0 5.9 12.5
−3.1 −0.5 2.1 4.1 8.1 −1.0 3.3 8.0
3.7 2.0 2.7 4.1 8.1 2.0 3.5 8.0
0.61 0.33 0.50 0.75 1.29 0.34 0.68 1.30
10.2 5.6 8.5 11.2 15.7 6.2 9.5 14.6
AAM-based polarizabilities taken from ref 11.
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RESULTS AND DISCUSSION
calculated using pure BLYP and PBE functionals with aTZ basis sets, are about 5 and 4%, respectively. The hybrid B3LYP and PBE0 functionals, and the long-range corrected CAM-B3LYP yield lower ε, about 3.1, 1.9, and 1.9%, respectively. The polarizabilities calculated using HF and LC-BLYP methods are very close, within 0.8% and −0.1%, respectively, to the
Acenes. Tables 1 and 2 present calculated polarizabilities for benzene, naphthalene, anthracene, and naphthacene. For an equilibrium geometry of Gauss and Stanton25 (RC−C = 1.3914 ± 0.001 Å, RC−H = 1.0802 ± 0.002 Å) for benzene (Geometry C in Table 1), the relative percentage errors on polarizability, ε (eq 5), 3162
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experimental value. The LC-PBE method yields a more negative ε = −1.3%. The LC-BLYP, LC-PBE polarizability values (9.96, 9.83 Å3, respectively) are lower than the HF-based value (10.04 Å3), similarly as in the case of first oligomers of polyynes.17,18 In order to compare calculated static electronic contributions to experiment, α(0), the vibrational polarizability, αv(0), should be subtracted from experimental values.26 The experimental electronic contributions to benzene polarizability, 9.96 Å3 (ref 26), 10.00 ± 0.1 Å3 (ref 43) considered here are consistent with the total static polarizability (10.56 ± 0.1 Å3)24 when the calculated vibrational polarizability contribution (∼0.3 Å3, Table 1) is included. The benzene geometry calculated using the B3LYP method with TZ basis sets (RC−C = 1.3908 Å, RC−H = 1.0821 Å) (B3LYP/ TZ, Geometry B) approximates Geometry C very well. Hence, polarizabilities based on Geometry B are practically the same as these based on Geometry C (Table 1). The geometries computed at the B3LYP/TZ level are also very good approximations of the equilibrium geometries of higher acenes.28 In Table 2, we adopt therefore, geometries computed at the B3LYP/DZ and B3LYP/TZ levels. The conventional functionals, BLYP, PBE, B3LYP and PBE0 strongly overestimate polarizability values for higher acenes, and the relative errors increase with the number of fused rings (Table 2). For example, relative errors for naphthalene, anthracene, and naphthacene, calculated at the BLYP/aTZ//B3LYP/TZ level, increase as 4.9, 9.4, and 13.2%, respectively. The polarizabilities computed with the LC-PBE or LC-BLYP functionals exhibit the lowest relative errors (between −2.5 and 3.5%) among the methods studied. For all acenes considered, the calculated polarizability increases as LC-PBE < LC-BLYP < HF < CAM-B3LYP < PBE0 < B3LYP < PBE < BLYP (Tables 1 and 2). Also for polyynes, polarizabilities based on the long-range corrected methods, are generally lower than the Hartree−Fock polarizabilities.17,18 These results suggest importance of taking into account electronic correlations in calculating polarizabilities of acenes. The triple-ζ basis sets generally improve the agreement with experiment. In summary, the long-range corrected functionals, particularly LC-PBE and LC-BLYP, predict acene electronic polarizability values, which agree well with experiment at least up to naphthacene. Diverse Organic Compounds. Table 3 summarizes error measures (eqs 6−8) calculated for the set of 105 organic molecules in Table S-1 in the SI. The highest RMSDs are observed for pure BLYP and PBE functionals, 1.15 and 1.18 Å3, respectively, and the lowest ones for LC-BLYP and LC-PBE, 0.34 and 0.35 Å3, respectively. The latter values are close to the value resulting from the best fit of an empirical AAM model (0.33 Å3).11 Figure 1 shows representative plots of the calculated vs experimental polarizabilities, α(calcd) = f(α(exptl)), for compounds in Table S-1 in the SI. For clarity, we include only BLYP, CAM-B3LYP, and LC-BLYP results. For comparison with correlations, we plot also results for AAM.11 The CAM-B3LYP and LC-BLYP plots fall in vicinity of the α(calcd) = α(exptl) line, and cannot be discerned from the AAM plot. On the other hand, the slope coefficient for the BLYP plot is higher than unity, which suggests that the BLYP (and PBE, not shown) systematically overestimate polarizability. This has been also seen for acenes (Tables 1−2). In Figure 2 we plot cumulative distribution functions (CDF), P(Δα), of absolute errors, Δα, (eq 4) computed for each method
Figure 1. Representative plots of experimental vs calculated polarizabilities for a set of 105 organic molecules in Table S-1 in the SI. The solid line denotes α(calcd) = α(exptl).
Figure 2. Cumulative probability distribution of absolute error in polarizability, P(Δα), calculated using the X/aug-cc-pVTZ//B3LYP/ cc-pVTZ protocols for substances in Table S-1 in the SI. The solid black line denotes Gaussian distribution with zero median and standard deviation σ = 0.33 Å3.
with triple-ζ basis sets. For comparison, we include also Gaussian (normal) CDF (black solid line in Figure 2) P(Δα) =
⎛ Δα ⎞⎤ 1⎡ ⎟⎥ ⎢1 + erf⎜ ⎝ σ 2 ⎠⎦ 2⎣
(9)
with median set to zero and standard deviation, σ = 0.33 Å3, adopted from AAM.11 The medians in Figure 2 increase as HF < LC-PBE < LC-BLYP < CAM-B3LYP < PBE0 < B3LYP < PBE ≅ BLYP, which roughly coincides with the degree of HF exchange at long-range, 100, 100, 100, 65, 25, 20, 0, 0%, respectively. The CDF plots for pure functionals, BLYP and PBE, collapse on each other, which suggests that both functionals yield essentially equivalent statistics for this data set. The plots for these pure functionals are non-Gaussian; they are asymmetric with the upper median shifted to strongly positive Δα. The curves corresponding to hybrid functionals, B3LYP and PBE0, also fall close to each other, but show some differences presumably due to slightly different degrees of distance independent HF exchange, 20 and 25%, respectively. Among the conventional functionals studied, the PBE0 method yields the lowest RMSD, 0.59 Å3. The CAM-B3LYP functional performs better than the B3LYP and PBE0 methods: for about 40% of the compounds the absolute errors, Δα, are close to zero. However, for the remaining compounds, the errors rapidly increase (Figure 2). The CAMB3LYP, and particularly, LC-BLYP and LC-PBE distributions resemble better Gaussian CDF (Figure 2). The CDFs of the LCBLYP and LC-PBE functionals overlap to a large extent; 3163
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π-subsets, could reduce discrepancies Δα. Indeed, when CAMB3LYP functional is applied to the σ-subset, MARE and RMSD decrease to 1.5% and 0.24 Å3, respectively. The lowest maximum relative errors are observed for the LCBLYP and LC-PBE functionals, about 5.6 and 6.2% for N,Ndiisopropylcyanamide, respectively (Table 3). These errors are lower than the MAXRE calculated by the AAM correlation (10.6% for acetonitrile) and by conventional functionals, for example, B3LYP and BLYP, 11.2 (methyl phenyl sulfane) and 15.7% (benzoyl bromide), respectively. The polarizabilities calculated with augmented triple-ζ (aTZ) basis sets are expected to approximate better fully converged polarizabilities resulting from the use of complete basis sets. Nevertheless, the mean absolute errors on polarizabilities calculated with smaller augmented double-ζ basis sets are practically the same as those with aTZ basis sets (Table 3), which may be useful in practice. The essential features of the cumulative distributions, P(Δα), calculated with aDZ basis sets remain the same as in Figure 2, and therefore are not shown. Wavelength Dependence of Polarizability of Selected Substances. The Abbe number, ν, is a simple measure of the refractive index dispersion
therefore, the performance of both functionals is very similar (Figure 2, Table 3). The overlap of the LC-BLYP and LC-PBE CDFs is consistent with the observed overlap of BLYP and PBE CDFs, and the same way of implementing the HF exchange (A = 0, B = 1 and μa0 = 0.47 in eq 2). The HF method yields negative Δα for about 90% of the compounds. Table 4 presents the list of substances that are common to the first (last) quantile of the LC-BLYP/aTZ and CAM-B3LYP/aTZ Table 4. Substances Common to the First and the Last Quantile of the LC-BLYP and CAM-B3LYP Distributions Depicted in Figure 2 first quantile
last quantile
N,N-diisopropylcyanamide cis-decalin 2,6-di-tert-butylpyridine methylcyclohexane 2,4-dimethyl-3-pentanol N,N-diethylcyanamide piperidine-1-carbonitrile cyclohexane pentamethylene-sulfide 4-methyl-2-pentanone N,N-diethylacetamide 2-methyltetrahydrofuran chlorocyclohexane n-hexane 2-methoxy-1,3-dioxolane tetrahydrofuran 3-pentanol
benzoyl bromide diphenyl ether dibenzylether methyl phenyl sulfane 1,2-dibromobenzene 2-cyanopyridine ethyl salicylate 3-phenyl-1-propanol tetrachloroethylene 2,4-dimethylphenol diallylamine iodobenzene benzaldehyde dodecanenitrile 3-methylphenol
ν=
nD − 1 nF − nC
(10)
where nD, nF and nC are the refractive indices of the material at wavelengths of 589.3, 486.1, and 656.3 nm, respectively. Table 5 presents the Abbe numbers calculated using the LC-BLYP, CAM-B3LYP, and B3LYP functionals with the aTZ basis sets. Generally, the relative errors in Abbe numbers of aromatic compounds (benzene, toluene, 1-methylnaphthalene, pyrrole, pyridine, and 2-methylpyridine) calculated using the LC-BLYP functional are very low, 1, −1.3, −2, 4, 2.3, 1%, respectively. The CAM-B3LYP functional tends slightly, and B3LYP more strongly, to underestimate Abbe numbers of the π-bonded compounds (Table 5). Figure 3 presents the wavelength dependence of the refractive indices for selected substances. The LC-BLYP-based refractive indices for benzene, agree exceptionally well with experimental values not only within the visible wavelength region but also in IR region. Nevertheless, we did not include the dynamical vibrational contribution to
distributions. There are 17 such substances in the first quantile, out of which 11 substances are fully saturated. In the last quantiles of the LC-BLYP and CAM-B3LYP distributions, there are 15 common substances, out of which 12 are aromatic and the remaining three substances contain π-bonds. The results in Table 4 and Figure 2 suggest that polarizabilities of compounds showing extreme deviations and calculated by these two methods are likely to differ by about 0.2 to 0.4 Å3. Thus, dividing compounds according to the bond type they possess, into σ- and
Table 5. Abbe Numbers, ν, (eq 10) and Refractive Indices at λ = 589.3 nm, nD, for Some Solvents Calculated Using the LC-BLYP, CAM-B3LYP, B3LYP Methods with aug-cc-pVTZ Basis Sets exptlf,g
LC-BLYP
CAM-B3LYP
B3LYP
cmpd
T/K
ν
nD
ν
nD
ν
nD
ν
nD
benzenea,e toluenea,e 1-methylnaphthaleneb,c pyrroled,e pyridined,e 2-methylpyridined carbon disulfidea,e trans-decalineb,e methylcyclohexaneb,e 2,2,4-trimethyl pentaneb,e pyrrolidined chloroforma,e
293 293 298 293 293 293 293 293 298
30.1 30.9 20.5 32.3 30.8 31.5 18.3 55.4 56.2
1.501 33 1.496 71 1.615 12 1.510 15 1.510 16 1.501 02 1.627 72 1.469 32 1.420 58
30.4 30.5 20.1 33.6 31.5 31.8 20.1 63.6 62.5
1.500 1.507 1.642 1.501 1.509 1.503 1.646 1.440 1.398
28.8 29.4 18.3 30.8 30.0 30.0 20.1 58.5 57.6
1.513 1.511 1.666 1.518 1.522 1.517 1.650 1.453 1.410
27.8 28.0 16.7 28.9 29.0 28.6 20.2 54.2 53.5
1.520 1.520 1.684 1.530 1.530 1.527 1.650 1.463 1.419
293 293 293
56.4 52.8 49.8
1.391 45 1.442 83 1.446 16
62.9 58.4 54.2
1.370 1.421 1.442
57.7 51.6 49.5
1.381 1.436 1.457
53.3 45.6 45.7
1.389 1.448 1.468
a
Reference 29, Tables V−VIII. bReference 27. cDensity taken from ref 30. dReference 31. eDensity taken from ref 32. fLiquid phase optical measurements. gUncertainty on refractive index 10−4 to 10−5 and on Abbe number 0.1. 3164
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B3LYP functionals represent better the polarizabilities, refractive indices and Abbe numbers of fully saturated compounds. The change from the triple-ζ to the double-ζ basis sets has small effect on error averages.
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ASSOCIATED CONTENT
S Supporting Information *
Polarizability values for 105 organic compounds calculated using HF, BLYP, B3LYP, LC-BLYP, CAM-B3LYP, PBE, PBE0, and LC-PBE methods with aug-cc-pVTZ basis sets and molecular geometries optimized at the B3LYP/cc-pVTZ level. This material is available free of charge via the Internet at http:// pubs.acs.org.
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Figure 3. Experimental refractive indices, n, for benzene (solid squares, ref 29, 293 K), carbon disulfide (solid diamonds, ref 29, 293 K) and trans-decaline (solid circles, ref 27, 298 K) as a function of wavelength, λ, compared to the values calculated using eq 1 with polarizabilities based on the X/aug-cc-pVTZ//B3LYP/cc-pVTZ protocols where X is LCBLYP (open circles), CAM-B3LYP (open squares), and B3LYP (crosses).
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected] Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We are indebted to Professor Doros N. Theodorou (NTU of Athens) for reading the manuscript and critical comments, to Dr T. Nakano for discussions, and to Director Dr T. Kakigano for encouragement and support.
polarizability, which could be important for longer wavelengths. For alkanes and cycloalkanes the LC-BLYP functional underestimates refractive index by about 2% and overestimates the Abbe number by 10−15% (Table 5). For these compounds, the CAM-B3LYP and B3LYP functionals yield Abbe numbers that agree better with experiment (Table 5, Figure 3). Also, for chloroform the CAM-B3LYP functional yields Abbe number (49.5) that agrees exceptionally well with experiment (49.8), with refractive index well represented (Table 5). For carbon disulfide, the Abbe numbers calculated by these three functionals are higher than the experimental values by about 10%, and refractive indices by about 1 to 2% (Table 5, Figure 3). In general, the LC-BLYP functional is the most successful for aromatic compounds, while CAM-B3LYP or B3LYP functionals represent best the polarizabilities of fully saturated compounds. However, all three methods tend to overestimate the refractive index for carbon disulfide and 1-methylnaphthalene.
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REFERENCES
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CONCLUSIONS The degree of HF exchange in mid- and long-range is a major factor determining shape and location of the distributions of absolute errors in polarizability, P(Δα) in Figure 2. The differences between functionals belonging to the BLYP and PBE families play smaller role: for example, for pure BLYP and PBE methods the P(Δα) distributions collapse on each other, thus providing approximately equivalent statistics for the data set studied. This pertains also to the LC-BLYP and LC-PBE methods with 100% of HF exchange in the long-range.The latter methods represent polarizabilities of 105 medium-sized organic compounds containing C, H, N, O, S, P, F, Cl, Br and I with the mean absolute relative error of about 2.5% and RMSD of 0.35 Å3, which are similar to the errors resulting from a good quality empirical method.11 Figure 2 suggests that fine-tuning of LC functionals is possible. The electronic contributions to static polarizabilities of acenes predicted using LC-BLYP and LC-PBE methods are slightly lower than polarizabilities calculated using Hartree−Fock theory, similarly as in the case of polyynes.17,18 The functionals with 100% HF exchange in the long-range limit, are most successful in predicting polarizabilities, and related properties for aromatic compounds. The CAM-B3LYP or 3165
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