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Ab initio calculations of static dipole polarizabilities and Cauchy moments for the halomethanes, CHmClnF4−m−n Yulia N. Kalugina Tomsk State University, 36 Lenin Av., Tomsk 634050, Russia Ajit J. Thakkar Department of Chemistry, University of New Brunswick, Fredericton, New Brunswick E3B 5A3, Canada Chem. Phys. Lett., 644, 20–24 (2016) http://dx.doi.org/10.1016/j.cplett.2015.11.044

Abstract Coupled-cluster calculations of the static electronic dipole polarizabilities and Cauchy moments are reported for all 15 halomethanes CHm Cln F4−m−n . Comparison with available experimental static polarizabilities is made. Excluding three experimental values which seem to be in error, the mean absolute deviation between the CCSD(T) values and experiment is a rather satisfactory 0.5 atomic units for the remaining 11 halomethanes. More experimental work is needed for the polarizabilities of CHCl2 F, CCl3 F, and CH2 Cl2 . Additivity approximations work moderately well for α = S(−2) and progressively less well for S(−2k − 2) as k increases.

1

Introduction

Methane and its chlorinated and fluorinated isomers, CHm Cln F4−m−n , constitute a relatively simple family of 15 related molecules which can and have been studied extensively in the gas phase. Moreover, an important subset of these molecules is the freons, CCln F4−n , which are of particular interest for their role in the photodynamics of atmospheric ozone [1]. 1

The focus of this work is on the static electronic dipole polarizability α(0) and the Cauchy expansion of the frequency-dependent electronic dipole polarizability [2] α(ω) for these molecules. The Cauchy expansion is valid for frequencies ω below the lowest excitation frequency ω1 . It is given by α(ω) = S(−2) + ω 2 S(−4) + ω 4 S(−6) + ω 6 S(−8) + · · ·

(1)

in which S(−2) = α(0) is the static electronic dipole polarizability and the S(−2k − 2) with k = 1, 2, . . . are called Cauchy moments. Electronic static polarizabilities obtained from gas-phase experimental measurements are available for 14 of these 15 halomethanes [3]. Moreover, the Cauchy moments for the freons have been extracted from experimental oscillator strength distributions [4]. All of these 15 molecules are part of the TABS database [5] and density functional theory (DFT) calculations of their polarizabilities and other properties using the B3LYP functional have been reported [6]. Further, a recent assessment of DFT methods for the calculation of polarizabilities identified the best functionals for this purpose [7]. Since then, both 2nd-order Møller-Plesset (MP2) perturbation theory [8] and these recommended functionals [7] have been used [9] to calculate polarizabilities for 145 molecules including the 14 halomethanes for which experimental data is available. Not surprisingly, many high-level ab initio calculations have been reported for methane; see, for example, Refs. [10–12]. This brief Letter reports coupled-cluster computations of the static electronic dipole polarizabilities and the Cauchy moments for all 15 halomethanes. The computations are described in Section 2. The results are compared with experiment and previous calculations in Section 3.1. Additivity models are discussed and assessed in Section 3.2. A few concluding remarks are made in Section 4. Atomic units are used throughout. The 2010 CODATA recommendations for the fundamental physical constants [13] give the SI value of one atomic unit of the dipole polarizability α as 1.648 777 × 10−41 C2 m2 J−1 . Multiplication of a value of α in atomic units by 0.148 184 7 yields the polarizability volume in ˚ A3 .

2

Computations

The structures of all the molecules were determined, using Gaussian [14], by energy minimization using the CCSD method, the coupled-cluster (CC) method [15, 16] with single (S) and double (D) substitutions (CCSD) [17].

2

The basis set used was the augmented correlation-consistent polarized valence triple-zeta (aug-cc-pVTZ) set [18–20]. Next, we used Dalton [21] to calculate the Cauchy moments S(−2k), 1 ≤ k ≤ 4, of the dipole oscillator strength distribution at the CCSD level using both aug-cc-pVTZ and its quadruple-zeta counterpart (aug-cc-pVQZ). The results are listed in Table 1. We did not use the doubly augmented counterparts of the aug-cc-pVℓZ basis sets, even though convergence with respect to cardinal number ℓ might be superior, because numerical experiments led us to conclude that the computer time demands would be too great for our resources. The CCSD Cauchy moments were extrapolated to the basis set limit using the formula [15, 22] Pℓ = P∞ + aℓ−3

(2)

in which Pℓ is a property calculated with a basis set of cardinal number ℓ. The cardinal numbers for the aug-cc-pVTZ and aug-cc-pVQZ basis sets are 3 and 4, respectively. The results are listed in Table 2. Recall that S(−2) = α(0). Finally, we used Molpro [23] to calculate the static dipole polarizability with the CCSD(T) method [24] which adds a non-iterative correction for triple substitutions to CCSD. The aug-cc-pVTZ basis set was used in these calculations. The best estimate αext for the static polarizability α = S(−2) was obtained by combining α∞ , the CCSD value extrapolated to the basis set limit, with a triples correction obtained at the aug-cc-pVTZ level. Thus, we use αext = α∞ + α(CCSD(T)/aug-cc-pVTZ) − α(CCSD/aug-cc-pVTZ). (3) Obviously, αext can equally well be thought of as the CCSD(T)/aug-ccpVTZ value corrected for basis set effects at the CCSD level. The values of the CCSD(T)/aug-cc-pVTZ and αext polarizabilities are listed in Table 3.

3 3.1

Discussion Comparison with previous work

For comparison purposes, Table 3 includes experimental values recommended by Hohm [3] after a careful analysis of published gas-phase measurements. Comparison with our static electronic polarizabilities is appropriate because Hohm’s recommendations are for zero-frequency electronic polarizabilities. He made extrapolations to zero frequency when required and corrections for the vibrational and orientational contributions when required. Nine of 3

Table 1: Cauchy moments (au) for halomethanes CHm Cln F4−m−n at the CCSD/aug-cc-pVnZ level. The upper and lower entries are for n = 3 and n = 4, respectively. Molecule S(−2) S(−4) S(−6) S(−8) CH4 16.47 54.69 231.5 1141. 16.43 54.55 231.8 1143. CH3 F 16.74 45.57 177.2 860.8 16.73 45.37 175.1 842.8 CH2 F2 17.61 40.41 139.4 611.1 17.47 39.53 134.4 580.8 CHF3 18.34 34.70 100.8 364.0 18.34 34.65 99.01 351.0 CF4 18.99 28.68 64.84 180.8 19.01 28.83 64.29 176.0 CH3 Cl 29.32 104.7 526.2 3494. 29.41 106.8 537.0 3481. CH2 ClF 30.11 99.20 475.6 2907. 30.17 100.6 481.5 2888. CHClF2 30.87 91.96 408.9 2295. 30.90 93.02 413.1 2286. CClF3 31.25 80.82 313.8 1531. 31.28 81.81 318.1 1536. CH2 Cl2 42.91 162.5 876.8 6117. 43.03 165.1 885.3 6027. CHCl2 F 43.66 154.2 783.8 5031. 43.72 156.2 790.3 4981. CCl2 F2 44.06 142.5 677.1 4135. 44.10 144.2 683.6 4116. CHCl3 56.47 218.8 1210. 8600. 56.59 221.5 1215. 8454. CCl3 F 56.94 207.1 1090. 7445. 57.02 209.4 1096. 7358. CCl4 69.67 271.8 1534. 11300. 69.81 274.6 1535. 11090.

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Table 2: Cauchy moments (au) for halomethanes CHm Cln F4−m−n extrapolated to the CCSD complete basis set limit. Molecule S(−2) S(−4) S(−6) S(−8) CH4 16.39 54.45 231.9 1144. CH3 F 16.72 45.22 173.6 829.6 CH2 F2 17.37 38.90 130.8 558.7 CHF3 18.35 34.61 97.70 341.4 CF4 19.03 28.93 63.90 172.4 CH3 Cl 29.48 108.4 544.9 3472. CH2 ClF 30.21 101.6 485.7 2874. CHClF2 30.92 93.79 416.1 2280. CClF3 31.30 82.53 321.2 1539. CH2 Cl2 43.11 167.0 891.6 5962. CHCl2 F 43.77 157.7 795.1 4945. CCl2 F2 44.12 145.5 688.4 4101. CHCl3 56.68 223.6 1219. 8347. CCl3 F 57.08 211.2 1100. 7296. CCl4 69.91 276.7 1536. 10950. his recommended values are based on refractive index measurements, four are from oscillator strength distributions obtained by analysis of mainly photoabsorption cross-section measurements, and one is from dielectric permittivity measurements. Table 3 also includes previously reported theoretical values obtained [9] using second-order Møller-Plesset (MP2) perturbation theory [8] as well as computations [9] based on density functional theory (DFT) using four different exchange-correlation functionals. The latter functionals are those recommended for polarizabilities in a recent assessment [7]. The selected methods are the Minnesota functionals M11 [25] and its predecessor M06-2X [26], the ωB97 functional [27], and LC-τ HCTH [7, 28, 29]. Three of these functionals (M11, ωB97, and LC-τ HCTH) are range-separated hybrids which contain an increasing amount of Hartree-Fock exchange as the interelectronic distance increases. On the other hand, M06-2X is a global hybrid containing a fixed 54% of Hartree-Fock exchange. The gradient approximation component of the ωB97 exchange-correlation functional depends only upon the electron density and its gradient whereas the other three functionals (M11, M06-2X, and LC-τ HCTH) also include terms that depend on the kinetic energy density.

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Table 3: Comparison of four density functional theory (M11, M06-2X, ωB97, LC-τ HCTH), 2nd-order Møller-Plesset perturbation theory (MP2), coupledcluster CCSD(T), and experimental static dipole polarizabilities (au) for halomethanes CHm Cln F4−m−n . DFT CCSD(T) Molecule M11a M06-2Xa ωB97a LCb MP2c aTZd exte Expt.f CH4 17.14 16.69 16.65 16.77 16.50 16.45 16.37 17.240±0.026(R) CH3 F 17.10 16.81 17.08 16.97 16.79 16.69 16.66 17.46±0.30(R) CH2 F2 17.52 17.37 17.95 17.73 17.53 17.38 17.14 17.89±0.18(R) CHF3 18.21 18.10 18.92 18.67 18.41 18.22 18.23 18.63±0.25(R) CF4 18.82 18.67 19.64 19.37 19.08 18.85 18.88 19.10±0.20(O) CH3 Cl 29.73 29.52 29.22 29.66 29.29 29.16 29.32 29.98±0.32(R) CH2 ClF — — — — — 29.93 30.03 — CHClF2 30.79 30.71 31.03 31.10 30.80 30.67 30.72 30.6±1.0(R) CClF3 31.13 30.98 31.54 31.47 31.20 31.03 31.07 32.04±0.60(O) CH2 Cl2 43.10 43.04 42.40 43.10 42.74 42.64 42.84 44.89±0.50(R) CHCl2 F 43.64 43.59 43.29 43.72 43.45 43.36 43.47 38.5±1.4(D) CCl2 F2 43.95 43.86 43.84 44.04 43.85 43.73 43.80 42.85±0.50(O) CHCl3 56.49 56.53 55.49 56.30 56.14 56.06 56.27 56.22±0.05(R) CCl3 F 56.90 56.85 56.13 56.61 56.60 56.49 56.63 53.3±1.0(O) CCl4 69.59 69.59 68.21 69.01 69.23 69.09 69.33 69.23±0.09(R) a DFT polarizability calculations in a def2-QZVPD basis set at a CAMB3LYP/def2-TZVP geometry, Ref. [9]. b LC-τ HCTH polarizability calculations in a def2-QZVPD basis set at a CAM-B3LYP/def2-TZVP geometry, Ref. [9]. c MP2 polarizability calculations in a def2-QZVPD basis set at a CAMB3LYP/def2-TZVP geometry, Ref. [9]. d CCSD(T)/aug-cc-pVTZ calculations, this work. e CCSD(T) results extrapolated to the basis set limit using Eq. (2), this work. f Recommended experimental values, Ref. [3]. The letter in parentheses denotes the type of experiment: O-dipole oscillator strength distribution, R-refractive index, D-dielectric permittivity.

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In the previous paper reporting MP2 and DFT calculations [9], the recommended experimental value of α for CHCl2 F was considered to be clearly in error. Moreover, three other halomethanes, CCl3 F, CH2 Cl2 , and CH3 F, were among the 32 molecules identified as requiring further experimental and theoretical work. One way of spotting some of these outliers is to organize the molecules systematically. Table 3 orders the molecules by increasing number of chlorine atoms, and secondarily by increasing numbers of fluorine atoms. This ordering is expected to be the same as ordering by the size of α(0) because chlorine atoms are the most polarizable of those occurring in the halomethanes; see, for example, the CCSD(T) polarizabilities of the free atoms given elsewhere [30, 31]. Table 3 shows that this molecule ordering indeed orders them by α(0) for all calculation methods except for a single exception with M11. On the other hand, one immediately spots two anomalies in the recommended experimental values for CHCl2 F and CCl3 F which are almost certainly in error. Moreover, the experimental value for CH2 Cl2 seems too large compared with the experimental value for CCl2 F2 . Table 4: The mean absolute deviations (MAD) of calculated static electronic polarizabilities with respect to the recommended experimental values from Ref. [3]. The MAD are in atomic units and the bias parameter B is unitless. The error measures are listed for all 14 molecules and for the subset of 11 molecules which excludes the three molecules for which experimental values are in error. Method MAD(14) B(14) MAD(11) B(11) LC-τ HCTH 1.04 0.00 0.39 −0.09 M11 1.08 −0.14 0.42 −0.27 MP2 1.09 −0.43 0.44 −0.64 CCSD(T)/aTZ 1.15 −0.43 0.53 −0.64 CCSD(T)/ext 1.16 −0.14 0.54 −0.27 M06-2X 1.18 −0.14 0.54 −0.27 ωB97 1.17 0.00 0.57 −0.09 The mean absolute differences (MAD) between the 14 experimental polarizabilities and their various calculated counterparts are shown in Table 4. For all methods, there is a drop in the MAD from about 1.0 to about 0.5 atomic units when the three molecules whose recommended values are clearly in error (CHCl2 F, CCl3 F, and CH2 Cl2 ) are excluded. Remarkably, the LC-τ HCTH, M11, and MP2 methods have slightly lower MADs than the CCSD(T) results. However, in view of the error bars on the experimental values, these very small differences are unlikely to be significant. We list a 7

bias parameter which provides information about the signs of the deviations. It is defined by B = f+ − f− in which f+ and f− are the fractions of errors that are positive and negative respectively. The bias ranges from B = −1 when all the deviations are negative to B = +1 when they are all positive. A small negative bias is reasonable because the calculated values do not include the effects of zero-point vibrational averaging which would tend to increase α. This is the reason for the larger deviations for the lightest molecules such as CH4 and CH3 F. It is satisfying to note that CCSD(T)/ext has a lower |B| than CCSD(T)/aTZ showing that extrapolation reduces the bias. Overall, we think the agreement with the 11 experimental values is rather good keeping in mind that some of the experimental error bars are somewhat over-optimistic. More experimental work is needed for the polarizabilities of the other three molecules: CHCl2 F, CCl3 F, and CH2 Cl2 . A comparison of our extrapolated CCSD Cauchy moments from Table 2 with those obtained by Bulanin and Kislyakov [4] from dipole oscillator strength distributions (DOSDs) for the freons CCln F4−n is made in Table 5. The comparison of the α(0) = S(−2) values has already been made because Hohm recommended these values for the freons. The mean absolute percent deviations (MAPDs) for the CCSD/ext moments relative to their DOSD counterparts are 4.1%, 9.5%, and 19% for S(−4), S(−6), and S(−8), respectively. We turn next to a consideration of additivity of the polarizability and the higher Cauchy moments.

Table 5: Cauchy moments (au) for freons CCln F4−n . Upper entries are CCSD/ext values from Table 2 and lower entries are DOSD values from Ref. [4]. Molecule S(−4) S(−6) S(−8) CF4 28.93 63.90 172.4 30.85 73.38 213.7 CClF3 82.53 321.2 1539. 88.35 376.8 2174. CCl2 F2 145.5 688.4 4101. 145.7 733.7 4794. CCl3 F 211.2 1100. 7296. 204.4 1149. 8552.

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3.2

Additivity

An additive model expresses a molecular property as a weighted sum of transferable contributions from its constituent parts. Additive models can be placed in a hierarchical classification [32] that brings order to the models in the literature. A brief recapitulation of the first three levels of the hierarchy follows. 1. The base of the hierarchy is a Level 1 (L1) model [32], also called a free-atom additive model (FAM), which can be expressed as: X (4) P ≈ P L1 = P FAM = ni Pi0 i

in which P is a molecular property, ni is the number of atoms of element i in the molecule, and Pi0 is the value of the pertinent property of a free atom of element i. If this model were accurate there would be no molecular chemistry! Nevertheless, such a primitive model has a conceptual role. For example, discussions of the minimum polarizability principle can be couched in terms of an L1 model [33]. 2. A Level 2 (L2) model [32], sometimes referred to as a dressed-atom additive model (DAM), can be written as: X P ≈ P L2 = P DAM = ni Pid (5) i

in which ni is the number of atoms of element i in the molecule, and Pid is the dressed or effective property of an atom of element i in a molecule. A dressed atomic property incorporates the influence of an average molecular environment. 3. The next step up is a Level 3 (L3) model [32] which can be written as X P ≈ P L3 = ni Pit (6) i

in which the sum is over atom types, ni is the number of atoms of type i in the molecule, and Pit is the effective property of an atom of type i in a molecule. An L3 model reduces to an L2 model if the atomic number is the sole criterion used to distinguish atom types. One way of creating additional atom types is to subdivide atoms of the elements by their coordination number [34]. For example, carbon 9

atoms are almost always tetra-, tri- or bi-coordinate; the corresponding atom types can be denoted C4 , C3 , and C2 , respectively. A nearly identical classification into atom types is obtained if one subdivides by hybridization instead. The H, C, Cl, and F atoms have the same coordination number in all the halomethanes. Thus, for just this class of molecules, there is no difference between the L2 and L3 models. Moreover, the degrees of freedom available for an additive model of halomethane properties are even more restricted by the stoichiometric condition nH + nCl + nF = 4.

(7)

This stoichiometric condition can be used to reduce the L2 (and equivalent L3) model for the halomethanes to one involving only three, rather than four, parameters. The choice of parameters is not unique and we choose to write P = p0 + p1 nCl + p2 nF (8) in which the constants pj are related to the dressed atom properties by p0 = PC + 4PH ,

(9)

p1 = PCl − PH ,

(10)

p2 = P F − P H .

(11)

and The striking additivity of the static electronic dipole polarizability is illustrated in Fig. 1. If attention is restricted to the chloromethanes (nF = 0) or to the fluoromethanes (nCl = 0), then a linear relationship is expected and this is seen clearly in Fig. 1. Various other linear cuts corresponding to fixed values of nF are shown as diagonal lines. Similarly, the nearly horizontal lines in Fig. 1 correspond to fixed values of nCl . A least squares fit to the CCSD(T)/ext values of α(0) from Table 3 leads to a quite good additive model with a MAPD of 0.59%: α(0) = 16.18 + 13.29 nCl + 0.61 nF .

(12)

Analogous additive models for the Cauchy moments are not quite as good. For example, for the extrapolated CCSD values of S(−4) in Table 2, we find S(−4) = 55.27 + 54.37 nCl − 7.55 nF (13) 10

70

60

CCl4

CCl3F

CHCl3

α (au)

50 CCl2F2

CH2Cl2 CHCl2F

40 CClF3 30

CH3Cl CHClF2

CH2ClF

20 CF4

CHF3

CH2F2

1

2 nH

CH3F

CH4

10 0

3

4

Figure 1: Additivity of CCSD(T)/ext static electronic dipole polarizabilities of the halomethanes, CHm Cln F4−m−n (+) from Table 3. Top diagonal line: nF = 0, chloromethanes; succeeding diagonal lines: nF = 1, nF = 2, nF = 3; horizontal lines from top to bottom correspond to nCl = 3, 2, 1, and 0, respectively, with the bottom line (nCl = 0) joining the fluoromethanes. where the MAPD is a larger 3.0%. The MAPDs of the additive models for S(−6) and S(−8) are progressively worse at 13% and 48%, respectively. As k increases, the Cauchy moments S(−2k−2) become increasingly dominated by the lowest frequency excitation and hence increasingly non-additive.

4

Concluding remarks

Coupled-cluster calculations of the static electronic dipole polarizabilities and Cauchy moments are reported for all 15 halomethanes CHm Cln F4−m−n . Comparison was made with experimental static polarizabilities and previous calculations using the MP2 method and the best available density functional methods. Excluding three experimental values which seem to be in error, the mean absolute deviation between the CCSD(T) values and experiment is 0.5 atomic units. Overall, we think this agreement with the 11 experimental values is rather good keeping in mind that some of the experimental 11

error bars are somewhat over-optimistic. More experimental work is needed for the polarizabilities of CHCl2 F, CCl3 F, and CH2 Cl2 . Simple additivity approximations work moderately well with a MAPD of 0.6% for α = S(−2) but progressively less well for S(−2k − 2) as k increases.

Acknowledgments Computations were carried out using the resources of the SKIF-Cyberia supercomputer, Tomsk State University.

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