Theoretical Characterization of the Minimum-Energy Structure of (SF6) 2

Article pubs.acs.org/JPCB

Theoretical Characterization of the Minimum-Energy Structure of (SF6)2 Tijo Vazhappilly,† Aude Marjolin,‡ and Kenneth D. Jordan* Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, United States S Supporting Information *

ABSTRACT: MP2 and symmetry-adapted perturbation theory calculations are used in conjunction with the aug-cc-pVQZ basis set to characterize the SF6 dimer. Both theoretical methods predict the global minimum structure to be of C2 symmetry, lying 0.07−0.16 kJ/mol below a C2h saddle point structure, which, in turn, is predicted to lie energetically 0.4−0.5 kJ/mol below the lowest-energy D2d structure. This is in contrast with IR spectroscopic studies that infer an equilibrium D2d structure. It is proposed that the inclusion of vibrational zeropoint motion gives an averaged structure of D2d symmetry.

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INTRODUCTION The SF6 dimer has been the subject of numerous experimental studies.1−13 Several of these studies have focused on the vibrational bands derived from the 948 cm−1 triply degenerate ν3 vibration of the SF6 monomer. It has been found experimentally that the ν3 band of the monomer splits into two IR-allowed transitions in the dimer, with the higher-energy transition being about twice as intense as the lower-energy transition. The most accurate values of the two transitions are 934.0 and 956.1 cm−1 from rotationally resolved measurements.5 This splitting pattern is well-explained by a simple model, allowing for the coupling of the transition dipoles on the two monomers.14,15 The dipole−dipole model predicts that the lower band has parallel polarization and is stabilized (relative to the monomer) by a factor of 2Δ, and that the upper band has perpendicular polarization, is destabilized by Δ, and is twice as intense, which is in excellent agreement with experiment. The dipole−dipole coupling model does not provide information on the relative orientation of the two SF6 monomers in the dimer. On the basis of the rotational structure in the bands, it has been concluded that the dimer has a two-fold symmetry axis.5 Model potential calculations using atom−atom repulsion, dispersion, electrostatic intersections (via atom-centered point charges), and induced dipole interactions give a D2d structure as the global minimum of the dimer,14,15 and the recent experimental studies of (SF6)2 have also been interpreted in terms of a D2d structure.5 In the present study, we revisit the problem of the structure of the SF6 dimer and the nature of the interaction between the SF6 monomers in the dimer. This is accomplished by use of Möller−Plesset second-order perturbation theory (MP2)16 and symmetry-adapted perturbation theory (SAPT)17 calculations on the dimer using basis sets as large as aug-cc-pVQZ.18,19

identified a total of nine stationary points, two each of C2v, D2h, and Cs symmetry and one each of D2d, C2h, and C2 symmetry, the structures of which are shown in Figure S1 of the Supporting Information. Only three of these structures, one of D2h symmetry, one of C2v symmetry, and that of C2 symmetry, were found to be local minima at this level of theory as determined by the calculation of the harmonic frequencies. MP2/aug-cc-pVTZ calculations at the MP2/aug-cc-pVDZ optimized geometries place the D2h and C2v local minima energetically 2.3−2.7 kJ/mol above the global minimum structure, and as a result, these structures were not considered further. The low-energy D2d, C2h, and C2 structures, depicted in Figure 1a−c, respectively, were singled out for further study. For these species, the geometry was reoptimized at the MP2 level using the aug-cc-pVTZ18,19 and aug-cc-pVQZ basis sets. The resulting energies are reported in Table S1, and the coordinates of the MP2/aug-cc-pVQZ structures are reported in Table S2. The MP2 energies were corrected for the basis set superposition error (BSSE) using the counterpoise procedure.20 The MP2/aug-cc-pVTZ optimized structures were used to carry out harmonic frequency calculations at the same level of theory. All MP2 calculations were carried out using density fitting using the MP2fit sets of functions.21,22 DFT-SAPT calculations were carried out using the aug-cc-pVQZ basis set and the MP2/aug-cc-pVQZ optimized geometries. The latter calculations were carried out using the DF-DFT-SAPT approach, which employs orbitals from density functional theory together with density fitting to reduce the computational cost.23 “JK” fitting sets were employed for the Hartree−Fock and DFT calculations.24 The asymptotically corrected25 PBE0 functional26,27 was employed, with the experimental ionization potential28 of SF6 being used to determine the asymptotic

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Special Issue: Bruce C. Garrett Festschrift

COMPUTATIONAL DETAILS A wide range of possible structures for (SF6)2 were optimized at the MP2/aug-cc-pVDZ18,19 level of theory. These calculations © XXXX American Chemical Society

Received: September 27, 2015 Revised: November 29, 2015

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DOI: 10.1021/acs.jpcb.5b09419 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B

Table 1. Interaction Energies (kJ/mol) of the D2d, C2h, and C2 Forms of (SF6)2a structure

MP2

DFT-SAPTb

D2d C2h C2

−3.85 −4.28 −4.44

−3.75 −4.23 −4.29

a

Results obtained using the aug-cc-pVQZ basis set. bThe DFT-SAPT results do not include δHF corrections, which are very small (less than 0.17 kJ/mol in magnitude and impact the relative energies by less than 0.02 kJ/mol).

C2h and C2 structures. The vibrational frequency calculations at the MP2/aug-cc-pVTZ level of theory predict that of the three structures, only the C2 structure is a local minimum; the D2d and C2h structures are predicted to be second and first order saddle points, respectively. The individual contributions from the DFT-SAPT calculations are summarized in Table 2, where E(1)elst, E(1)ex, E(2)ind, E(2)ex−ind, E(2)disp, and E(2)ex−disp denote the electrostatics, exchange−repulsion, induction, exchange−induction, dispersion, and exchange−dispersion contributions, respectively. Additional details on the DFT-SAPT calculations are given in Table S3. As seen from Table 2, the dispersion contribution is the major factor responsible for the greater stability of the C2h structure over the D2d structure. The exchange−repulsion contribution is 0.76 kJ/mol more repulsive for the C2 than that for the C2h structure, while the electrostatics and dispersion interactions (including exchange−dispersion in the latter) favor the C2 structure by 0.35 and 0.53 kJ/mol, respectively, with the net result being that the C2 structure is only slightly more stable than the C2h structure. The induction plus exchange−induction contributions nearly cancel for each of the three structures. Additional insight into the role of electrostatic interactions in establishing the relative stability of the D2d, C2h, and C2 structures can be seen from the examination of Table 3, which reports the electrostatic repulsion contributions from the model of Bladel and van Avoird15 as well as from the atomcentered moments through quadrupoles determined from a Gaussian distributed moment analysis (GDMA)32 of the B3LYP33,34/aug-cc-pVTZ charge distribution. The electrostatic interactions from the model of Bladel and van Avoird are significantly larger in magnitude than those from our GDMA calculations. In particular, we note that the D2d species is favored by the electrostatic interactions over the C2 species by 2.4 kJ/mol in the model of Bladel and van Avoird but by only 0.5 kJ/mol with the GDMA electrostatics. This is due, in part, to the fact that the atomic charges are much larger in magnitude in the former model. If we retain only the atomic charges from the GDMA analysis, our electrostatic contributions are two times smaller in magnitude than those of the model of Bladel and van Avoird. As seen from Table 2, the DFT-SAPT calculations give electrostatic contributions of −2.17, −2.11, and −2.46 kJ/mol for the D2d, C2h, and C2 isomers, respectively. The differences between the electrostatic contributions from the GDMA model and those of the SAPT calculations are largely the result of charge penetration effects that are included in the SAPT calculations but not treated explicitly in the models. We have fit the difference of the SAPT2/aug-cc-pVTZ and GDMA electrostatic contributions for several geometries of the dimer using the expressions for charge penetration introduced by Freitag et al.35 (The detailed

Figure 1. MP2/aug-cc-pVQZ structures of (SF6)2 optimized under the constraint of C2h, D2d, and C2 symmetry.

correction. The MOLPRO package29 was used to optimize the geometries at the DF-MP2 level and to carry out the DF-DFTSAPT calculations. The vibrational frequency calculations were carried out using the Turbomole code30 following reoptimization of the geometries. In analyzing the electrostatics contribution to the interaction energies, we also made use of the SAPT2 procedure as implemented in the PSI4 package.31

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RESULTS AND DISCUSSION There is a significant reduction in the BSSE in going from the aug-cc-pVTZ to the aug-cc-pVQZ basis set (Table S2). For this reason, unless mentioned otherwise, we focus on results obtained with the aug-cc-pVQZ basis set. (The exceptions are the vibrational frequencies that were calculated using the augcc-pVTZ basis set.) The MP2/aug-cc-pVQZ optimizations of (SF6)2 give SS distances of 4.836, 4.730, and 4.726 Å for the D2d, C2h, and C2 structures, respectively. The calculated distances for the latter two structures are in good agreement with the experimental value of 4.754 Å reported in ref 5. The MP2 and DFT-SAPT interaction energies are summarized in Table 1. The MP2 results reported in Table 1 include the counterpoise correction for BSSE, while the SAPT procedure, by design, is BSSE free. The excellent agreement between the MP2 and DFT-SAPT results indicates that high-order correlation effects do not play an important role in describing the interaction between the SF6 monomers in the dimer. The MP2 and SAPT calculations predict the C2h structure to be more stable than the D2d structure by 0.43 and 0.48 kJ/mol, respectively, and the C2 species to be 0.06−0.16 kJ/mol more stable than the C2h structure. This suggests that there is largeamplitude motion along the coordinate that interconverts the B

DOI: 10.1021/acs.jpcb.5b09419 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B Table 2. Contributions (kJ/mol) to the SF6 Dimer Interaction Energy from DF-DFT-SAPT Calculationsa D2d C2h C2 a

E(1)elst

E(1)ex

E(2)ind

E(2)ex−ind

E(2)disp

E(2)ex−disp

Etot

−2.17 −2.11 −2.46

7.60 7.86 8.62

−2.22 −2.28 −2.54

2.27 2.33 2.60

−10.00 −10.86 −11.39

0.76 0.80 0.88

−3.75 −4.23 −4.29

Results obtained with the aug-cc-pVQZ basis set.

derived from the ν3 mode of the monomer, which are forbidden in the D2d structure, acquire intensity and thus should have been observable experimentally. Calculated frequencies and intensities of all vibrations of the three structures are reported in Table S5. Table 4 also reports the calculated intensities for the strongly IR active vibrational modes derived from the ν3 vibrations of the monomer. The calculated ratio of the intensities of the high-frequency to low-frequency vibrations is about 1.57 for all three structures. (Here, we have summed the intensities of the degenerate and nearly degenerate pairs of higher-frequency vibrations.) This should be compared to the 2.0 value of the ratio predicted by the dipole−dipole coupling model. It is relevant to note that experimental studies have given values of the ratio of the intensities ranging from 1.70 to 1.80.1−3,6 Thus, it is clear that the ab initio calculations are also not fully satisfactory in this regard. This may reflect the need to include anharmonic effects as well as nonlinear terms in the expansion of the dipole moment operator. The vibrational zero-point energies (ZPEs) calculated using the harmonic frequencies are 112.1, 112.1, and 112.4 kJ/mol for the D2d, C2h, and C2 structures, respectively. With the inclusion of the ZPE corrections, the C2h structure becomes the most stable structure with the C2 and D2d structures being less stable by ∼0.1 and 0.4 kJ/mol, respectively. These energy differences could potentially be further reduced by the inclusion of vibrational anharmonicity in the ZPE corrections. As a result, we believe that it is reasonable to expect that (SF6)2 could have a vibrationally averaged D2d structure. The ZPE is calculated to be 3.9 kJ/mol greater for two SF6 molecules than that for the C2 form of the dimer. Combining this result with our SAPT value of the electronic interaction energy of the C2 form of the dimer, we get a value of −8.2 kJ/mol for the ZPE-corrected binding energy of the dimer.

Table 3. Electrostatic Interactions (kJ/mol) in the D2d, C2h, and C2 Structures of (SF6)2a GDMAb structure

c, d, q

c

GDMA+ charge penetration

SAPT2

model of ref 15

D2d C2h C2

−0.9 0.2 −0.4

−1.4 0.3 −0.2

−2.5 −2.3 −2.7

−2.51 −2.37 −2.73

−2.8 0.6 −0.4

a

Results reported for MP2/aug-cc-pVQZ optimized geometries. bThe “c” indicates results obtained using only the atomic charges from the GDMA analysis, while “c, d, q” indicates that atomic moments through the quadrupole are employed.

results of the SAPT2 calculations are reported in Table S4.) When these results are combined with the electrostatic contributions from using the GDMA moments, the resulting net electrostatic contributions are in excellent agreement with the results of the DFT-SAPT calculations (see Table 3). This is encouraging as the charge penetration model of Freitag et al. could be incorporated in an improved force field for SF6. Interestingly, the model of ref 14 employed atomic charges that are even larger in magnitude than those used in ref 15, with the enhanced charges being chosen to reproduce a more recent measurement of the transition moment. However, as is clear from the analysis presented above, atom-centered point charge models are not adequate for describing the electrostatic interactions in (SF6)2. This has been found to be true in general for molecular dimers at their equilibrium geometry.36 Table 4 summarizes the calculated and measured frequencies of the strongly IR active vibrations of (SF6)2 derived from the Table 4. MP2/aug-cc-pVTZ Harmonic Frequencies and Intensities of the Strongly IR-Active Vibrations of (SF6)2 Derived from the ν3 Mode of SF6 structure D2d C2h C2 expt.b

freq (cm−1) 938.8, 939.4, 938.9, 934.0,

963.0, 963.0 962.4, 965.7 963.8, 964.3 956.1

splitting (cm−1)a

intensity (km/mol)

24.2 24.6 25.2 22.1

943.0, 746.4, 746.4 938.8, 747.5, 738.4 942.5, 742.6. 741.5

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CONCLUSIONS On the basis of high-level electronic structure calculations, we conclude that the potential energy minimum of (SF6)2 is not of D2d symmetry, in contrast to the conclusion of earlier studies. Both DFT-SAPT and DF-MP2 calculations predict that the D2d species lies energetically approximately 0.6 kJ/mol above the C2 global minimum. Harmonic frequency calculations at the DFMP2/aug-cc-pVTZ level indicate that the D2d structure is a second-order saddle point. Our calculations also locate a C2h structure that lies energetically only slightly (∼0.07−0.16 kJ/ mol) above the C2 structure in the absence of vibrational ZPE corrections. Harmonic frequency calculations show that the C2h structure is a first-order saddle point for interconversion between C2 structures. However, with ZPE corrections, the C2h structure becomes favored by 0.1 kJ/mol. The calculated SS distances of the C2h and C2 structures are closer to the experimentally determined SS distance than that calculated for the D2d structure. The SAPT calculations reveal that the dispersion contribution is primarily responsible for the C2h structure being

a

The average of the two high-frequency vibrations was used in calculating the splitting. bReference 5.

ν3 vibration of the SF6 monomer. The calculated frequencies and splittings for all three structures are in excellent agreement with experiment. In particular, the calculated splittings are 24− 25 cm−1 compared to the experimentally observed splitting of 22 cm−1. Thus, the calculated splittings of the vibrational levels, by themselves, do not provide an unambiguous structure assignment for the SF6 dimer. We note that the calculations predict the upper transition of the C2h and C2 structures to be split by 3.3 and 0.5 cm−1, respectively. Such splittings should have been resolvable in the high-resolution spectroscopic studies. Moreover, in the C2 structure, the other three bands C

DOI: 10.1021/acs.jpcb.5b09419 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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ACKNOWLEDGMENTS This research was supported by Grant CHE1362334 from the National Science Foundation. The calculations were performed on computers in the University of Pittsburgh’s Center for Simulation and Modeling.

energetically favored over the D2d structure. The stability of the C2 over the C2h structure is more subtle; the SAPT calculations show that electrostatic, dispersion, and exchange−repulsion interactions are all greater in magnitude in the C2 structure than those in the C2h structure, with the net result being that the C2 and C2h structures are nearly isoenergetic. The trends in the electrostatic contributions to the interaction energy are very different in the SAPT calculations than predicted in the earlier model potential studies. Our analysis suggests that the earlier model potential studies of (SF6)2 were biased toward the D2d structure due to the use of atomic charges that were too large in magnitude and due to the neglect of higher-order atomic multipoles and charge penetration. For all three structures, the calculated splittings of the intense bands derived from the IR-active ν3 mode of the monomer are in excellent agreement with experiment. In addition, the symmetry lowering from D2d removes the degeneracy of the higher-energy intense band and, in the case of the C2 structure, imparts weak intensity to the other vibrations derived from the ν3 mode of the monomer. Although the electronic structure calculations favor a C2 structure for (SF6)2, the experimental vibrational spectra are more consistent with D2d structure. The most likely resolution of this apparent inconsistency between theory and experiment is that the dimer undergoes largeamplitude motion in its vibrational zero-point level and has a vibrationally averaged D2d structure.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.5b09419. Nine stationary points of the SF6 dimer optimized at the MP2/aug-cc-pVDZ level with Molpro; the symmetry groups, nature of the stationary points, and MP2/aug-ccpVTZ BSSE-corrected interaction energies at the MP2/ aug-cc-pVDZ optimized geometries (Figure S1). DFMP2/aug-cc-pVQZ optimized geometries of the C2, C2h, and D2d structures of (SF6)2. Coordinates (Table S1). MP2, BSSE-corrected MP2 (MP2-BSSE), and DFTSAPT interaction energies (Eint) of the optimized C2, C2h, and D2d dimers calculated using the aug-cc-pVTZ and aug-cc-pVQZ basis sets (Table S2). DFT-SAPT components of the optimized C2, C2h, and D2d structures of (SF6)2 calculated using the aug-cc-pVQZ basis set (Table S3). SAPT2 components of the optimized C2, C2h, and D2d structures of (SF6)2 calculated using the augcc-pVTZ basis set (Table S4). Harmonic frequencies (cm−1) and IR intensities (km/mol) of the low-energy C2, C2h, and D2d structures of (SF6)2 calculated at the MP2/aug-cc-pVTZ level of theory (Table S5) (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone number: +1-(412) 624-8690. Present Addresses †

T.V.: Theoretical Chemistry Section, Bhabha Atomic Research Center, Mumbai, India. ‡ A.M.: Pittsburgh Quantum Institute, University of Pittsburgh, Pittsburgh, PA, 15260. Notes

The authors declare no competing financial interest. D

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DOI: 10.1021/acs.jpcb.5b09419 J. Phys. Chem. B XXXX, XXX, XXX−XXX