Hexadecapole moment, dipole and quadrupole polarizability of sulfur hexafluoride

22 October 1999

Chemical Physics Letters 312 Ž1999. 255–261 www.elsevier.nlrlocatercplett

Hexadecapole moment, dipole and quadrupole polarizability of sulfur hexafluoride George Maroulis

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Department of Chemistry, UniÕersity of Patras, GR-26500 Patras, Greece Received 5 July 1999; in final form 16 August 1999

Abstract We have obtained SCF and MP2 values for the polarizability and hyperpolarizability of sulfur hexafluoride. A large w8s6p5d2fr6s4p4d1fx basis set, consisting of 335 Gaussian-type functions, provides near-Hartree–Fock values for all properties of interest: y24.76 ea40 for the hexadecapole moment ŽF ., 26.54 e 2 a20 Ey1 for the dipole polarizability Ž a ., h 2 4 y1 gz z z z s 738, gx x z z s 317 and g s 823 e 4 a40 Ey3 for the dipole hyperpolarizability, 227.5 e a 0 E h for the mean quadrupole h 2 4 y1 Ž . Ž . polarizability C , 101.7 e a 0 E h for the dipole–octopole polarizability E and B z z , z z s y134, B x z , x z s y156 e 3 a40 Ey2 h for the dipole–dipole–quadrupole hyperpolarizability. At MP2 level the dipole polarizability varies with the S–F bond length Žsymmetric stretch. as w a Ž R . y a Ž R 0 .xre 2 a 20 E hy1 s 25.38D R q 13.72D R 2 y 0.99D R 3 where D Rra0 s Ž R SyF y R 0 . and R 0 is the experimental S–F separation. The value of F is lower than any experimental result, and E is lower than the best experimental estimate. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction The electric properties of SF6 are currently of much interest because of their relevance to an impressive range of experimental investigations w1–5x. Of particular importance is the contribution of dipole and quadrupole polarizability to the interpretation of spectroscopic observations. In their pioneering paper, Samson and Ben-Reuven w6x demonstrated how the higher polarizabilities of SF6 could be used to develop a rational approach to the analysis of the measurements of the forbidden vibrational Raman bands of gaseous and liquid SF6 reported by Holzer and Ouillon w7x. Collision- and interaction-induced spectroscopy emerges as a research field where computational quantum chemistry can be expected to make significant contributions by providing accurate values for the polarizability and hyperpolarizability of atoms and molecules w8x. This has been forcefully borne out recently in paradigmatic papers w9–13x. From a computational point of view, SF6 is a molecule of some size. The application of high-order theories, of confirmed predictive capability but computationally demanding, is severely restricted. Previous theoretical efforts are limited to coupled Hartree–Fock ŽCHF. or self-consistent field ŽSCF. studies w14–17x. The goal of

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0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 9 3 9 - 2

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this preliminary study is to obtain reference SCF results for all electric properties of interest. We also report estimates of the electron correlation effects for all these properties and information about the bond length or R-dependence of the properties for the symmetric stretch of the S–F bond. As the extension of theoretical efforts to problems involving interactions with other molecules, or even primarily the dimer ŽSF6 . 2 , is never lost sight of, we conclude this study with a short discussion on possible computational strategies for such systems. With the notable exception of bond lengths, atomic units are used for all other properties throughout this work. Conversion factors to SI units are: energy, 1 Eh s 4.3597482 = 10y1 8 J; length, 1 a0 s 0.529177249= 10y1 0 m; hexadecapole moment F , 1 ea40 s 1.256363 = 10y6 0 C m4 ; dipole polarizability a , 1 e 2 a02 Ey1 h s y6 5 4 4 y3 1.648778 = 10y4 1 C 2 m2 Jy1 ; dipole hyperpolarizability g , 1 e 4 a40 Ey3 s 6.235378 = 10 C m J ; h y62 2 4 y1 quadrupole Ž C . or dipole octopole Ž E . polarizability, 1 e 2 a 40 Ey1 s 4.617048 = 10 C m J ; and dipole– h y6 3 dipole–quadrupole hyperpolarizability B, 1 e 3 a40 Ey2 C 3 m4 Jy2 . h s 1.696733 = 10

2. Theory and computational strategy Relatively few electric multipole moment and polarizability tensors are needed to describe the distortion of a molecule of high symmetry, as SF6 , in a weak electric field. Up to the fourth rank, these are the hexadecapole moment ŽFa b gd ., dipole polarizability Ž a a b ., second dipole hyperpolarizability Žga b gd ., quadrupole- Ž Ca b,gd . and dipole–octopole Ž Ea ,b gd . polarizability, and the dipole–dipole–quadrupole hyperpolarizability Ž Ba b,gd . w18x. Adopting the conventional molecular orientation, the sulfur atom on the origin Ž0,0,0. and the fluorine centers at Ž"R,0,0., Ž0," R,0. and Ž0,0," R ., we need few Cartesian components to specify the above tensors. There is only one independent component for either the hexadecapole moment or the dipole polarizability. We write simply F ŽF ' Fx x x x s Fy y y y ' Fz z z z . and a Ž a ' a x x ' a y y ' a z z . for these components. For ga b gd we select gz z z z and gx x z z , for Ca b,gd , C z z , z z and C x z , x z and for Ba b,gd , Bz z , z z and B x z , x z . There is also only one independent component for Ea ,b gd , E ' Ez , z z z . Details about the finite-field approach to the calculation of a a b , ga b gd and Ca b,gd may be found in previous work w19,20x. Both SCF and second-order Møller–Plesset perturbation theory ŽMP2. w21x values for these properties were thus obtained from suitably perturbed molecular energies. F , E and Ba b,gd were easily obtained from the induced multipole moments. MP2 values were conveniently extracted from the electric multipole moments obtained through the MP2 density w22x.

3. Basis sets and other computational details We have designed basis sets for SF6 that should be useful in the present work but also in other studies involving computational efforts in intermolecular interactions. The largest basis set used in this Letter is Ž13s9p5d2fr11s7p4d1f.w8s6p5d2fr6s4p4d1fx, hereafter D, consisting of 433 primitive Gaussian-type functions ŽGTF. or 335 contracted GTF ŽCGTF.. It is built upon a flexible Dunning substrate of Ž11s7p.w6s4px for S w23x and Ž9s5p.w4s2px for F w24x. The construction follows a computational philosophy detailed in previous work w25x. In three steps, the initial substrate is augmented with diffuse s- and p-GTF, tight d-GTF with exponents chosen to minimize the energy of the free molecule and diffuse d-GTF with exponents chosen to maximize the dipole polarizability. A smaller basis set was built upon a Ž10s7pr7s4p.w4s3pr3s2px DZV substrate w26x. According to the method mentioned above, it was increased to w5s4p2dr4s3p2dx or 165 CGTF, hereafter A0. Larger basis sets A1 Ž170 GTF., A2 Ž200 GTF. and A3 Ž177. were obtained by the addition of d- and f-GTF to A0. A basis set even smaller than A0, T0 ' w4s3p2dr3s2p2dx, consisting of 137 CGTF, was built upon a Ž10s7pr10s7p. w3s2pr2s1px substrate w27,28x. We are particularly interested in testing the quality of this small T0 basis, as it is of a size that could allow calculations on much larger systems of interest such as ŽSF6 . 2 . In some detail, the composition of the basis sets employed in this work was as follows:

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D s w 6s4pr4s2px qS: s s 0.054342, 0.019441, p s 0.043466, 0.012545, d s 0.9966, 0.5478, 0.3011, 0.1655, 0.0910, f s 0.5478, 0.1655 qF: s s 0.109339, 0.032898, p s 0.079604, 0.023187, d s 0.6893, 0.2066, 0.1131, 0.0619, f s 0.2066 A0 s w 4s3pr3s2px qS: s s 0.05116083, p s 0.04674535, d s 0.5476, 0.0979 qF: s s 0.10374752, p s 0.08991665, d s 0.6408, 0.2017 A1 s A0 q S: d s 0.2315 A2 s A1 q F: d s 0.1132 A3 s A1 q S: f s 0.0979 T0 s w 3s2pr2s1px qS: s s 0.051171, p s 0.045837, d s 0.5507, 0.0805 qF: s s 0.087137, p s 0.063414, d s 0.6602, 0.1782 Electric fields of 0.005, 0.01 and 0.02 ey1 ay1 0 E h were used for the calculation of the dipole polarizability and hyperpolarizability. E and Ba b,gd were obtained from the induced multipole moments using fields of 0.005 ˚ w x and 0.01 ey1 ay1 0 E h . The experimental bond length is R 0 s 1.56072 A 29 . All calculations were performed with GAUSSIAN 92 and GAUSSIAN 94 w30,31x.

4. Results and discussion Our results are shown in Tables 1–3. In Table 3 we include a selection of previous theoretical efforts and experimental data. The hexadecapole moment of SF6 merits special attention. Our best SCF value for Frea04 is y24.76 and is obtained with basis D. We expect this value to be very close to the HF limit. This high electric moment displays strong basis set dependence. T0 and A0 give values larger and smaller in magnitude than the presumably most accurate D result. The sequence A0–A3 converges to a value of y24.10, close to our best result. The inclusion of electron correlation effects drastically reduces the magnitude of F . The MP2 values show a uniform decrease of
Table 1 Basis set dependence of the electric hexadecapole moment F r e a40 of SF6 Basis set

CGTF

SCF

MP2

T0 w4s3p2dr3s2p2dx A0 w5s4p2dr4s3p2dx A1 w5s4p3dr4s3p2dx A2 w5s4p3dr4s3p3dx A3 w5s4p3d1fr4s3p2dx D w8s6p5d2fr6s4p4d1fx

137 165 170 200 177 335

y31.29 y21.97 y21.07 y21.83 y24.10 y24.76

y19.33 y7.61 y6.75 y8.05 y9.39 –

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Table 2 Geometry dependence Žsymmetric stretch. of the electric properties of SF6 . Basis set A0 ' w5s4p2dr4s3p2dx. The geometry parameter is ˚ defined as Ž R Sy F y R 0 .rA Property

Method

a

SCF MP2 SCF MP2 SCF MP2 SCF MP2 SCF MP2 SCF MP2 SCF MP2 SCF SCF SCF SCF

gz z z z gx x z z ga Cz z , z z Cx z , x z Cb

F E Bz z , z z Bx z , x z a b

y0.2

y0.1

19.57 21.96 484 808 186 306 513 852 111.6 126.9 65.2 74.1 145.2 165.1 y28.63 33.5 y114 y105

22.40 25.26 568 965 216 363 600 1014 143.9 163.9 76.1 86.8 177.6 202.5 y26.78 60.0 y126 y117

0 26.16 29.52 669 1100 268 432 723 1178 189.9 214.6 88.3 101.6 219.8 250.7 y21.97 102.8 y122 y134

0.1 31.34 34.84 789 1190 362 567 908 1394 258.7 283.1 101.6 118.2 277.2 311.7 y15.01 173.4 y86 y155

0.2 38.43 41.04 855 1039 517 753 1134 1527 363.8 367.7 116.1 136.9 357.5 384.9 y6.39 286.0 12 y181

g s Ž3r5.Žgz z z z q 2gx x z z .. C s Ž3r5.ŽC zz,zz q 2C xz,xz ..

y14.3. Given the complexity of the issue w17x, the comparison of our MP2 to the CIA result points to a certain convergence. For the dipole polarizability are 2 a 02 Ey1 we report stable SCF values of 26.81 ŽT0., 26.16 ŽA0. and a h near-HF result of 26.54 ŽD.. MP2 calculations with T0 and A0 lead to second-order corrections ŽD2. of 3.48 ŽT0. and 3.36 ŽA0.. The stability of the D2 correction is encouraging. We expect the two MP2 values, 30.29 ŽT0. and 29.52 ŽA0. to be reasonably accurate theoretical estimates of the dipole polarizability of SF6 . In addition, we have obtained the R-dependence Žsymmetric stretch. of this property at the SCF the MP2 levels of theory. We find 2 3 a Ž R . y a Ž R 0 . re 2 a02 Ey1 h s 23.22D R q 19.88D R q 12.10D R Ž SCF .

a Ž R. ya Ž R0 .

2 3 re 2 a 02 Ey1 h s 25.38D R q 13.72D R y 0.99D R

Ž MP2.

Ž 1. Ž 2.

where D Rra0 s Ž R y R 0 .. Thus, the first derivative of the dipole polarizability at R 0 is estimated at Ždard R . f 25 e 2 a 0 Ey1 w x h . SCF values for the dipole polarizability have been reported by Lazzeretti et al. 14 , Spackman w15x and Fowler et al. w16,17x. Our respective SCF results are in fair agreement with those of Spackman w15x and Fowler et al. w17x. The experimental static dipole polarizability is 30.35, obtained from refractive index ŽRI. measurements w1x. There is also a semi-empirical result of slightly lower magnitude at 30.04, obtained from dipole oscillator strength distributions ŽDOSD. w34x. Our MP2 values are quite close to the RI and DOSD results. our large SCF calculations give gz z z z s 738, For the second dipole hyperpolarizability ga b gdre 4 a04 Ey3 h gx x z z s 317 and g s 823. The T0 and A0 basis sets display the same pattern as in the case of the dipole polarizability. The SCFrT0 values are larger and the SCFrA0 smaller than the SCFrD values. The D2 correction for the mean g is quite large, 541 ŽT0. and 445 ŽA0.. The contents of Table 2 show a strong R-dependence for the components of ga b gdre 4 a 40 Ey3 h . At the SCF level all components increase smoothly and

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Table 3 Experimental data and theoretical predictions for the electric properties of SF6 Property

F a gz z z z gx x z z g Cz z , z z Cx z , x z C E Bz z , z z Bx z , x z

Theory

Experiment

SCF a

SCF b

MP2 b

SCF c

MP2 c

Previous

y24.76 26.54 738 317 823 194.5 92.4 227.5 101.7 y134 y156

y24.10 d 26.16 669 268 723 189.9 88.3 219.8 102.8 y122 y134

y9.39 d 29.52 1100 432 1178 214.6 101.6 250.7 111.5 y211 y191

y31.29 26.81 769 346 876 193.3 92.1 226.5 100.3 y155 y162

y19.33 30.29 1237 562 1417 216.4 107.5 258.9 101.8 y241 y232

y14.6 e , y25.7 f 26.04 j, 26.97 e , 26.93 f

32 " 6 g , y14.3 h , 28.5 " 2.0 i 30.35 k , 30.04 l

1565 m

207.66 n , 227 e , 229 f 101e , 107 f

220 o

a

Basis set D ' w8s6p5d2fr6s4p4d1fx, 335 CGTF. Present investigation. Basis set A0 ' w5s4p2dr4s3p2dx for all properties, with the exception of the hexadecapole moment. c Basis set T0 ' w4s3p2dr3s2p2dx. d Present investigation, basis set A3 ' w5s4p3d1fr4s3p2dx. e Fowler et al. w17x, basis set w7s6p2dr5s3p2dx. f Fowler et al. w17x, basis set as above augmented with f-GTF. g
the mean varies with R as 2 3 g Ž R . yg Ž R 0 . re 4 a40 Ey3 h s 813D R q 680D R q 62D R

Ž SCF.

Ž 3.

At the MP2 level, the gz z z z component reaches a maximum and lowers again but the gx x z z one increases ˚ g wy0.2, 0.2x and steadily. This results in a nearly linear dependence of the mean for bond lengths RrA 2 g Ž R . yg Ž R 0 . re 4 a40 Ey3 Ž MP2. h s 915D R y 12D R

Ž 4.

The experimental estimate of the mean hyperpolarizability of SF6 is 1565 and has been obtained from electric field-induced second harmonic generation ŽESHG. at 694.3 nm w35x. This value includes pure vibrational effects. Our MP2 results allow no more than an inspired guess of the static limit of g at this stage, a value of 2 Ž . gre 4 a04 Ey3 h s 13 " 1 = 10 , admittedly a conservative estimate but nevertheless an unambiguously reasonable one. The quadrupole polarizabilities Ca b,gdre 2 a40 Ey1 calculated with basis sets T0 and A0 are within a few h percent of the most accurate D values. It should be noted that the small T0 basis yields remarkably good results for this property. The MP2 values for the mean C are 258.9 ŽT0. and 250.7 ŽA0., 14.3 and 14.1%, respectively, above the SCF values of 226.5 ŽT0. and 219.8 ŽA0.. At the MP2rA0 level of theory, the mean

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value varies with R as 2 3 C Ž R . yC Ž R 0 . re 2 a 04 Ey1 h s 257.8D R q 222.0D R q 161.8D R

Ž MP2.

Ž 5.

The quadrupole polarizability has attracted some theoretical attention. SCF values of 207.66 w15x and 227, 229 w17x have been reported for C, in close agreement with the present values. The dipole–octopole polarizability is of considerable importance for SF6 . We report SCF values of 103.8 ŽT0., 102.8 ŽA0. and 101.7 ŽD. for Ere 2 a04 Ey1 h . Electron correlation changes the T0 and A0 values by 1.5 and 8.5%, respectively. Our SCF values agree very well with those reported by Fowler et al. w17x. The R-dependence of this property, calculated at the SCFrA0 level, shows a strong dependence on bond length. The experimental estimates of this value are invariably higher than the theoretical results w17x. A value that has found some use in applications is the 220 obtained by El-Sheikh et al. w36x from measurements on the SF6 –Xe system. We conclude this discussion with our findings for the Ba b,gdre 3a 40 Ey2 hyperpolarizability. Our best values h are Bz z , z z s y134 and B x z , x z s y156. Electron correlation has a very strong effect on both components. The SCFrA0 R-dependence of these components ŽTable 2. shows a different behaviour, as the former decreases and the latter increases in magnitude with the S–F bond length.

5. Conclusions We have calculated electric polarizabilities and hyperpolarizabilities for SF6 . To the best of our knowledge, this study reports the first post-HF values for these properties. We have also obtained the R-dependence Žsymmetric stretch. for all properties. Very good agreement is observed between our MP2 results and the available experimental data for the dipole polarizability. We have neglected vibrational averaging, but then our calculations pertain to R 0 and not to R e , so some of the effect is recovered. We expect our discussion of the hexadecapole moment to be of help in further investigations. We anticipate, based on the precedent of methane and carbon tetrafluoride, the contribution of our quadrupole polarizability values to collision-induced absorption ŽCIA. and light scattering ŽCILS. studies involving SF6 . Last, we have designed small sized but flexible basis sets that might be of use in further computational endeavours in interactions of SF6 with other systems. The small but flexible T0 ' w4s3p2dr3s2p2dx could easily be used in calculations on small ŽSF6 . n clusters. We recommend the larger A0 ' w5s4p2dr4s3p2dx basis set for calculations on systems involving interactions of SF6 with atoms or small molecules.

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