Model Core Potential and All-Electron Studies of Molecules

Jun 4, 2010 - Department of Chemistry, University of Alberta, Edmonton, AB T6G 2G2, ... Molecules HRgF (Rg = Ar, Kr, Xe, Rn) were studied at levels of...
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J. Phys. Chem. A 2010, 114, 8786–8792

Model Core Potential and All-Electron Studies of Molecules Containing Rare Gas Atoms† Amelia Fitzsimmons,‡ Hirotoshi Mori,§ Eisaku Miyoshi,| and Mariusz Klobukowski*,‡ Department of Chemistry, UniVersity of Alberta, Edmonton, AB T6G 2G2, Canada, DiVision of AdVanced Sciences, Ocha-dai Academic Production, Ochanomizu UniVersity, 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan, and Graduate School of Engineering Sciences, Kyushu UniVersity, 6-1 Kasuga Park, Fukuoka 816-8580, Japan ReceiVed: February 26, 2010; ReVised Manuscript ReceiVed: April 26, 2010

Molecules HRgF (Rg ) Ar, Kr, Xe, Rn) were studied at levels of theory that included electron correlation, taking relativistic effects into account either by using the recently developed parametrization of the extended model core potentials and basis sets or by using the Douglas-Kroll method with all-electron basis sets. Charge distributions were calculated according to Mulliken, Lo¨wdin, and natural bond orbital methods of population analysis and the results of these methods were compared, confirming that bonding in these molecules corresponds to interaction between the fluoride anion and the RgH+ moiety. In contrast to previously reported results, the present calculations show that the radon compound, HRnF, is more stable than compounds of the lighter congeners. Trends in the first ionization energies, bond lengths, energies of formation and decomposition, and harmonic vibrational frequencies were discussed and found to be consistent with the periodic trends of the atomic properties of the rare gas atoms. Introduction Interest in rare gas chemistry has been greatly increasing, as clearly shown in the recent overview of the field.1 Among the rare-gas atoms, xenon has been the most extensively studied and has been found to form many compounds with halogen atoms and organic moieties. Other rare gas compounds have been formed with argon and krypton.2 The chemistry of the lighter rare gases was reviewed by Frenking and Cremer,3 with focus on both experimental findings and theoretical predictions. Radon’s history in the chemical literature begins as early as that of xenon, but it is significantly less voluminous. Divalent compounds of Rn were studied experimentally4 soon after the experimental discovery of xenon compounds was reported by Bartlett in 1962.5 In 1975, Liebman6 commented on the apparent nonexistence of RnF4. In the same year Pitzer7 discussed bonding and stability of RnF2 from the perspective of relativistic quantum mechanics and elucidated the strongly ionic character of the Rn-F bond in radon difluoride. The 1979 review of structural chemistry of fluorides (and oxide fluorides) of nonmetals by Seppelt8 included discussion of radon fluorides, with comment that RnF8 should not be expected. After a few years, as the interest in the field of radon chemistry grew, Avrorin et al. summarized the experimental results,9 suggesting that radon’s chemistry “should be at least as varied as the chemistry of the lighter inert gases (in particular, xenon)”. The authors also discussed the possibility of radiochemical synthesis of radon compounds. In the mideighties, Selig and Holloway discussed the presence of Rn2+, RnF+, and Rn2F3+ in solution in their review of ionic complexes of rare gas atoms.10,11 The chemical properties of radon were again reviewed by Stein in 198712 who reported that experiments carried out at Argonne National Laboratory failed to confirmed earlier reports9 of †

Part of the “Klaus Ruedenberg Festschrift”. * To whom correspondence should be addressed E-mail: mariusz. [email protected]. ‡ University of Alberta. § Ochanomizu University. | Kyushu University.

existence of radon in higher oxidation states, especially RnO3. Radon fluorides were mentioned in the review by Jørgensen and Frenking13 and compared with krypton and xenon compounds. Dolg et al. studied four fluorides of radon, RnF2, RnF4, RnF6, and RnF8, using the MP2 method with both non- and scalar-relativistic pseudopotentials.14 In 1996, Kaupp et al. carried out MP2 calculations using effective core potentials to elucidate the structure and stability of RnF6 in the context of studies on compounds isoelectronic with XeF6.15 Two years later, Runeberg and Pyykko¨ carried out very detailed MP2 and CCSD(T) calculations on the van der Waals compounds with radon, XeRn and Rn2,16 again employing pseudopotentials and keeping only eight electrons in the valence space of each atom. In the same year, Liao and Zhang17 used DFT methodology to study RnF2 both in the gas phase and in the solid state. In 1999, compounds of the type RnFn (n ) 2,4) were studied computationally by Han and Lee18 using two-component spin-orbit method. In the same year, Ra¨sa¨nen and co-workers reviewed experimental and computational aspects of novel neutral compounds containing rare gases of the HXY type (where X ) Kr or Xe and Y ) electronegative atom), synthesized in solid state matrices,19 and a year later Lundell et al. carried out detailed computational studies of the related systems HRgF, where R ) He, Ne, Ar, Kr, Xe, or Rn.20 Christe, in his 2001 review of the rare gas chemistry, indicated the existence of “many known Xe, Kr, and Rn compounds”;21 commenting on the stability of rare gas hydrogen fluoride molecules HRgF, he quotes Frenking22 stating that such molecules “are only kinetically stable; their stability depends on the energy barriers toward decomposition which can be quite low, particularly in the condensed neat phase.” In 2002, Schrobilgen reviewed experimental techniques for synthesis of radon fluoride and fluorocations of radon,23 while Malli carried out Dirac-Fock calculations on radon carbonyl and predicted that the molecule RnCO should be stable.24 A year later, in all-electron relativistic calculations at the MP2 level of theory, Filatov and Cremer25 found that radon hexafluoride is a bound species. Their calculations shed light on some earlier results for RnF6 obtained by Malli.26 In the same

10.1021/jp101765m  2010 American Chemical Society Published on Web 06/04/2010

Molecules Containing Rare Gas Atoms year, Peterson et al. reported results of calculations for RnH+ using new small-core pseudopotentials.27 In a tribute to Bartlett, Liebman and Deakyne28 discussed the structure of condensed phase binary radon fluoride. In 2004, Gerber29 reviewed novel rare gas molecules formed in low-temperature matrices, commenting on radon compounds. The next year, the possibility of existence of the ionic species O3RntN- was discussed by Pyykko¨ et al.30 In a recent review,31 Khriachtchev et al. mentioned interesting prospects for the studies of unexpected radon compounds. Radon is an unstable element; among the isotopes of radon the most stable is 222Rn, whose half-life of is 3.82 days.32 At present, experimental studies involving radon have been limited by the radioactivity and short half-life of radon, thus computational studies should provide insight into the reactions and properties of radon-containing compounds. In the present work, we report results of the application of the model core potential (MCP) method33 to the study of the simplest polyatomic heavier rare gas compounds of the type HRgF (Rg ) Ar, Kr, Xe, and Rn), compare the MCP results with those from all-electron calculations as well as with results previously reported,20 and examine the bonding in HRnF in comparison to bonding of the other compounds in this family. Computational Methodology Geometry optimization of the rare gas compounds was carried out at three levels of theory: MP2, CCSD, and CCSD(T); the CCSD methods were implemented in GAMESS-US34 by Piecuch et al.35 Single-point calculations of reaction energies were done at the CCSD(T) and MP2 optimized geometries using the CR-CC(2,3) method.36,37 To reduce the computational effort and to (implicitly) introduce scalar relativistic effects, we replaced the core electrons using the model core potential methodology.33 We used the basis sets and parameters from the latest MCP compilation,38 designed for accurate studies. This recent parametrization of the MCP method was calibrated in the studies of geometries of 90 molecules containing main-group elements.39 While the calibration results showed that the MCPqzp basis set is superior, the MCP-tzp basis set gave satisfactory results. In view of planned calculations on large molecules containing radon, we decided to use the more economical MCPtzp basis set. To increase the flexibility of the basis sets in the valence region and ensure a proper description of bonding in these highly ionic systems, we augmented the MCP-tzp basis sets with diffuse Gaussian functions, whose exponents were taken from an online compilation.40 The valence subshells that were not replaced by core potentials were: nsnp for F and Ar, (n - 1)dnsnp for Kr, Xe, and Rn (18 valence electrons). The corresponding valence basis sets were rather extensive: for fluorine, (5s5p4d3f) were contracted to [4s4p3d2f] ) (2111/ 2111/211/21); for argon, (5s5p5d3f) were contracted to [4s4p3d2f] ) (2111/2111/311/21); for krypton, (8s7p9d3f) were contracted to [5s5p4d2f] ) (41111/31111/5211/21); for xenon, (9s8p9d3f) were contracted to [5s5p4d2f] ) (51111/41111/6111/21); for radon, (10s9p11d4f) were contracted to [5s5p4d2f] ) (61111/ 51111/7211/31). These basis sets were designated aug-MCPtzp (mcpT in the abbreviated form), and the resulting sizes of variational space (in terms of spherical Gaussian functions) were 113 for HArF and 122 for HKrF, HXeF, and HRnF. The segmented Gaussian basis set for hydrogen was used in the contraction (411/21/2) based on Matsuoka’s uncontracted basis set,41 with correlating functions taken from the work of Noro et al.42 and augmented with diffuse functions; the hydrogen basis set is available online.40

J. Phys. Chem. A, Vol. 114, No. 33, 2010 8787 TABLE 1: First Ionization Energies of Rare Gas Atoms (in eV)a,b atom

MP2

CCSD

CR-CC(2,3)

expc

Ar Kr Xe Rn

15.691 14.102 (14.033) 12.424 (12.327) 11.655 (11.534)

15.494 13.954 (13.911) 12.284 (12.226) 11.521 (11.457)

15.541 13.996 12.325 11.562

15.814 14.218 12.562 12.023

b

a Values in parentheses correspond to frozen nd subshell. aug-MCP-tzp basis set. c Reference 32.

Additional calculations using all-electron basis sets were carried out to check the reliability of the new MCP parametrization and basis sets in the application to studies of rare gas compounds. The aug-cc-pVTZ basis sets43-45 for H, F, Ar, and Kr were used to study HArF and HKrF; for brevity, these basis sets were denoted by accT. For systems containing atoms H, F, Ar, Ke, and Xe (HArF, HKrF, and HXeF) we also used the nonrelativistic aug-TK/NOSeC-TZP basis sets41,42,46-52 in nonrelativistic all-electron MP2 calculations; these aug-TK/NOSeCTZP basis sets were named nasT. Finally, to have a more direct comparison with the results obtained using the MCP basis sets and parameters (which by their design implicitly include scalar relativistic effects for heavy atoms), we used the basis sets augDK3-Gen-TK/NOSeC-TZP (designated later as rasT) that were optimized using the Douglas-Kroll Hamiltonian53 in atomic calculations.54 These basis sets were used in all-electron calculations that were carried out using the third-order Douglas-Kroll Hamiltonian. The nasT and rasT basis sets were downloaded from the Segmented Gaussian Basis Set collection in Sapporo.40 Furthermore, to elucidate basis set effects, we carried out the MP2 calculations using the basis sets of Lundell et al.20 in the GAMESS-USprogram.34 Inthiscase,thestandard6-311++G(2d,2p) basis set was used for H, F, and Ar, while for the heavier rare gas atoms (Kr, Xe, and Rn) the averaged relativistic core potentials of Christiansen and co-workers55-57 were employed; this basis set was also defined in terms of spherical Gaussian functions and is referred to as LCG. Spin-restricted open-shell MP2 calculations for the fluorine atom were carried out using the restricted Møller-Plesset theory for open-shell molecules.58,59 In the post-Hartree-Fock calculations the nd electrons present in Kr, Xe, and Rn were correlated together with the 16 valence electrons from the H, F, and Rg atoms. All calculations were carried out using the GAMESS-US program34 on Linux clusters and Apple Macintosh computers. Results and Discussion The proposed model of binding in the HRgF compounds is that of the RgH+ cation interacting with the fluoride ion.20,60,61 It is thus important that the method and basis set used in our work are capable of reasonably approximating the experimental first ionization energy of the rare gas atoms. We evaluated these ionization energies using the basis set aug-MCP-tzp at the MP2, CCSD, and CR-CC(2,3) levels of theory using the ROHF Hamiltonian62-64 the calculations for the positive ions were carried out in C2V symmetry. The results are compared in Table 1 with experimental values32 corrected for the spin-orbit splitting using the experimental values of spin-orbit coupling;65-67 the comparison shows that the computed ionization energies are in reasonable agreement with the experimentally determined values. They decrease with increasing atomic number implying that, as expected, radon

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TABLE 2: Bond Lengths (in Å) in HArF and Energies (in kJ/mol) of Reactions Involving H, Ar, and F method/basis

re(Ar-H)

re(Ar-F)

∆E1a

∆E2b

TAEec

MP2/mcpT MP2/accT MP2/nasT MP2/rasT MP2/LCGd MP2/LCGe CCSD/mcpT CCSD(T)/mcpTf CCSD(T)/LCGd CR-CC(2,3)/mcpTg CR-CC(2,3)/mcpTh

1.304 1.313 1.314 1.314 1.326 1.326 1.296 1.319 1.355

1.954 1.968 1.976 1.975 1.996 1.998 1.970 1.964 2.005

-66.8 (-43.2) -49.2 (-40.2) -48.2 (-35.4) -46.7 (-36.3) -6.8 -4.0 (7.8) -21.2 (-0.4) N/A -49.2 -41.6 (-23.5) -41.5 (-22.6)

-546.7 -551.1 -552.8 -553.4 -584.3 -585.2 -546.6 -545.8 -579.6 -551.0 -551.1

613.5 600.3 601.0 600.1 591.1 589.2 567.8 N/A 628.8 592.6 592.6

a Energy change in the formation reaction H + F + Rg f HRgF (values in parentheses are corrected for BSSE). b Energy change in the decomposition reaction HRgF f Rg + HF. c Molecular total atomization energy, eq 3. d Values from ref 20. e This work. f Present implementation of CCSD(T) in GAMESS-US is restricted to closed-shell RHF reference wave function (entries N/A in the table). g At CCSD(T)/mcpT optimized geometry. h At MP2/mcpT optimized geometry.

should be the most reactive of the rare gas atoms studied, with its p electrons least strongly bound. It is interesting to observe that contributions from correlating the ten electrons from the nd subshell have only a very small effect on the values of ionization energies. The calculated bond lengths in the rare gas hydrogen fluoride molecules are collected in Tables 2-5, together with the energy changes in the two reactions that determine thermodynamic stability of these compounds (∆E1 and ∆E2) and the atomization energy of hydrogen fluoride, TAEe:

H(g) + F(g) + Rg(g) f HRgF(g) HRgF(g) f HF(g) + Rg(g) H(g) + F(g) f HF(g)

∆E1 ∆E2

∆E1 + ∆E2 ) -TAEe

(1) (2) (3)

Energy difference ∆E1 in the reaction of formation of HRgF from the constituent atoms H, Rg, and F, eq 1, was corrected for the basis set superposition error (BSSE)68,69 using full counterpoise (CP) correction.70 There is no unique way of defining the corresponding basis set extension error for the energy of the second reaction, ∆E2, and these values are reported below without any correction. The value of TAEe was calculated using ∆E1 and ∆E2 uncorrected for the basis set effects. HArF. The MP2 results in Table 2 show that there is a reasonable agreement between the MCP and all-electron bond lengths, with the MCP values smaller by about 0.01 and 0.02 Å for Ar-H and Ar-F bonds, respectively. The energy of formation of HArF is overestimated in the MCP approach by about 20 kJ/mol (MP2/mcpT vs MP2/rasT); the difference decreases to about 7 kJ/mol when the CP corrections are taken into account. It appears that the formation energy of HArF depends strongly on the basis set used, as indicated by the large discrepancy between the results of Lundell et al.20 (MP2/LCG) and the present all-electron results, indicating that reaction 1 is more exothermic by about 40 kJ/mol. The difference could be attributed to different basis sets used: it should be noted that the 6-311G basis set for fluorine is not of triple-ζ valence quality in the s space, where it should be described as 63-11, with the 6- and 3-member contracted functions describing only the 1s orbital.71 Another difference between the present basis sets and those used by Lundell et al. is that only two d- and p-type polarization functions are used with the 6-311++G(2d,2p) basis set, whereas both the mcpT and all-electron basis sets have an

extensive set of contracted polarization and correlating functions. Interestingly, the CR-CC(2,3)/mcpT values of ∆E1 are quite close to the CCSD(T) value of Lundell et al.20 The CP corrections for the basis set effects reduce the value of ∆E1 by 10-20 kJ/mol. The energy is further reduced by considering the zero-point energy (ZPE) correction: Using the value of ZPE for HArF from MP2/mcpT calculations, where it was found to be 26.3 kJ/mol, the ZPE-corrected values of the best ∆E1 from CR-CC(2,3)/ mcpT//MP 2/mcpT calculation become essentially zero. The ZPE correction will increase the value of ∆E2 only marginally by about 1 kJ/mol, as the ZPE for hydrogen fluoride equals 24.9 kJ/mol (MP2/mcpT). Computed atomization energy of HArF, which is equal to -∆E1, was reported previously. In 2001 Runeberg et al.72 found the CCSD(T)/accT value of -∆E1 equal to 23.2 kJ/mol, which was reduced to 13.5 kJ/mol when basis set extension corrections were added. One year later Panek et al.,73 using the MP2 method as well as several density functionals with the accT basis set, found the range for the energy of HArF relative to the neutral atom dissociation limit (∆E1) to be from -30.3 to -11.4 kJ/ mol; their MP2/accT value was -40.5 kJ/mol, fairly close to our value of -49.2 kJ/mol. In 2004 Xie and co-workers reported74 that the combined energy of H + Ar + F relative to the minimum energy of HArF (-∆E1) equals 17.6 kJ/mol when evaluated at the MR-AQCC/accT level of theory. Three years later Hu and co-workers75 calculated the atomization energy of HArF to be 40.6 kJ/mol at the MP2/accT level of theory. HKrF. Data in Table 3 show that the MP2/mcpT structural parameters agree within few mÅ with the relativistic all-electron values (MP2/rasT), and the discrepancy in the reaction energies evaluated at the MP2 level is less than 10 kJ/mol. Relativistic effects together with appropriate basis sets lead to the shortening of bonds. The MP2/LCG bond length of Kr-F is about 0.1 Å longer than the best all-electron value (MP2/rasT) and the HKrF formation energy, ∆E1, is underestimated at the MP2/LCG level, possibly due to the absence of polarization functions in the LCG basis set for krypton and poorer representation of the correlating functions for fluorine. The values of ∆E1 are reduced by the CP correction, which is the largest for the LCG basis set. The ZPE of HKrF equals 24.5 kJ/mol (MP2/mcpT value), and it will reduce the energy of formation of HKrF from the elements even further. There will be no significant change in the energy of dissociation to Kr and HF, because the ZPEs for HKrF and HF are essentially equal. HXeF. Results for HXeF in Table 4 reveal similar agreement between the MCP and all-electron MP2 results as found earlier for HKrF. The formation energy of HXeF is again underesti-

Molecules Containing Rare Gas Atoms

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TABLE 3: Bond Lengths (in Å) in HKrF and Energies (in kJ/mol) of Reactions Involving H, Kr, and F method/basis

re(Kr-H)

re(Kr-F)

∆E1a

∆E2b

TAEec

MP2/mcpT MP2/accT MP2/nasT MP2/rasT MP2/LCGd MP2/LCGe CCSD/mcpT CCSD(T)/mcpTf CCSD(T)/LCGd CR-CC(2,3)/mcpTg CR-CC(2,3)/mcpTh

1.456 1.461 1.459 1.454 1.423 1.423 1.455 1.474 1.439

2.034 2.035 2.036 2.031 2.134 2.134 2.039 2.040 2.138

-137.6 (-105.4) -128.5 (-110.1) -133.3 (-109.6) -133.7 (-110.4) -91.5 -90.2 (-33.7) -93.2 (-65.0) N/A -131.8 -114.5 (-88.0) -114.4 (-87.1)

-475.8 -471.8 -467.7 -466.3 -499.6 -499.1 -492.5 -474.3 -497.0 -478.1 -478.3

613.4 600.3 601.0 600.0 591.1 589.3 585.7 N/A 628.8 592.6 592.7

a Energy change in the formation reaction H + F + Rg f HRgF (values in parentheses are CP corrected for BSSE). b Energy change in the decomposition reaction HRgF f Rg + HF. c Molecular total atomization energy, eq 3. d Values from ref 20. e This work. f Present implementation of CCSD(T) in GAMESS-US is restricted to closed-shell RHF reference wave function (entries N/A in the table). g At CCSD(T)/mcpT optimized geometry. h At MP2/mcpT optimized geometry.

TABLE 4: Bond Lengths (in Å) in HXeF and Energies (in kJ/mol) of Reactions Involving H, Xe, and F method/basis

re(Xe-H)

re(Xe-F)

MP2/mcpT MP2/nasT MP2/rasT MP2/LCGd MP2/LCGe CCSD/mcpT CCSD(T)/mcpTf CCSD(T)/LCGd CR-CC(2,3)/mcpTg CR-CC(2,3)/mcpTh

1.633 1.644 1.638 1.665 1.666 1.639 1.651 1.681

2.100 2.100 2.102 2.146 2.147 2.096 2.102 2.150

∆E1a -240.3 -240.2 -234.7 -188.7 -187.4 -199.7 N/A -231.1 -220.0 -219.9

(-190.6) (-204.4) (-199.1) (-157.1) (-155.7) (-177.4) (-176.3)

∆E2b

TAEec

-373.1 -360.8 -365.3 -402.2 -401.8 -386.1 -369.7 -397.7 -372.6 -372.8

613.4 601.0 600.0 590.9 589.2 585.8 N/A 628. 8 592.6 592.7

a Energy change in the formation reaction H + F + Rg f HRgF (values in parentheses are CP corrected for BSSE). b Energy change in the decomposition reaction HRgF f Rg + HF. c Molecular total atomization energy, eq 3. d Values from ref 20. e This work. f Present implementation of CCSD(T) in GAMESS-US is restricted to closed-shell RHF reference wave function (entries N/A in the table). g At CCSD(T)/mcpT optimized geometry. h At MP2/mcpT optimized geometry.

TABLE 5: Bond Lengths (in Å) in HRnF and Energies (in kJ/mol) of Reactions Involving H, Rn, and F method/basis

re(Rn-H)

re(Rn-F)

∆E1a

∆E2b

TAEec

MP2/mcpT MP2/rasT MP2/LCGd MP2/LCGe CCSD/mcpT CCSD(T)/mcpTf CCSD(T)/LCGd CR-CC(2,3)g CR-CC(2,3)h

1.736 1.716 1.868 1.870 1.743 1.755 1.942

2.175 2.163 2.809 2.808 2.173 2.177 2.849

-287.0 (-223.2) -277.7 (-224.8) +39.7 +40.7 (53.3) -245.8 (-190.2) N/A -2.5 -266.3 (-211.9) -266.0 (-210.7)

-326.4 -322.3 -630.7 -629.9 -340.0 -323.7 -626.2 -326.4 -326.6

613.4 600.0 591.0 589.2 585.8 N/A 628.7 592.7 592.6

a Energy change in the formation reaction H + F + Rg f HRgF (values in parentheses are CP corrected for BSSE). b Energy change in the decomposition reaction HRgF f Rg + HF. c Molecular total atomization energy, eq 3. d Values from ref 20. e This work. f Present implementation of CCSD(T) in GAMESS-US is restricted to closed-shell RHF reference wave function (entries N/A in the table). g At CCSD(T)/mcpT optimized geometry. h At MP2/mcpT optimized geometry.

mated at the MP2/LCG level and, given the energy balance of eq 3, it leads to an increased value of the energy in the dissociation reaction 2. The CP corrections reduce the energy of formation of HXeF by 30-50 kJ/mol, while ZPE of 23.6 kJ/mol for HXeF (MP2/mcpT) will reduce it even further. The ZPE of HF is only slightly larger than that of HRgF, leading to an insignificant reduction of ∆E2 by about 1 kJ/mol. HRnF. The agreement between the all-electron values obtained at the DK3 level of theory with a suitable basis set and the MCP values shown in Table 5 is very good, with the deviation of 0.02 Å for the H-Rn bond length. There is a large difference between the present results and those obtained by Lundell et al.,20 with the present bond lengths significantly shorter than the ones reported previously. To check whether there is a second local minimum on the potential energy surface for HRnF, we carried out geometry optimization at the MP2/

mcpT level of theory, starting from the initial geometry as reported by Lundell et al.;20 however, optimization converged to the geometry reported in Table 5. As we were able to reproduce Lundell’s results using the GAMESS program, the difference may be attributed to the different pseudopotentials used by Lundell et al. and in the present work. The CP corrections lead to the reduction of ∆E1, with the vibrational correction due to the ZPE of 21.8 kJ/mol for HRnF (MP2/mcpT) reducing it further. ZPE corrections for ∆E2 will decrease its value by about 3 kJ/mol. Summary for All Systems. The bond lengths obtained using the model core potentials at MP2 and CCSD(T) levels of theory agree reasonably well, with the largest difference for the Rg-H bond smaller than 0.02 Å, and the largest difference for the Rg-F bond smaller than 0.01 Å. The Rg-H bond length increases systematically by about 0.1 Å as the size of the Rg

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Fitzsimmons et al. TABLE 6: Harmonic Vibrational Frequencies (in cm-1)a,b molecule

ν1(Σ+)

ν2(Π)

ν3(Σ+)

HArF HKrF HXeF HRnF

483.8 (481.2) 455.0 (396.1) 463.2 (438.9) 461.8 (312.6)

772.4 (743.4) 708.0 (695.2) 660.2 (657.8) 572.2 (601.2)

2371.4 (2148.9) 2230.5 (2315.9) 2160.1 (2068.9) 2046.3 (1575.0)

a Values in parentheses are from ref 20. b Modes ν1(Σ+), ν2(Π), and ν3(Σ+) correspond to Rg-F stretch, degenerate bend, and Rg-H stretch, respectively.

Figure 1. MP2/mcpT Rg-X bond lengths (in Å).

Figure 2. Energy change in the reactions involving Rg, H, and F (in kJ/mol); data from CR-CC(2,3)/ mcpT//MP 2/mcpT calculations. (CP) refers to CP-corrected energies.

increases; similarly, the bond length Rg-F increases by about 0.07 Å. The trends in the calculated geometry of the molecules, illustrated in Figure 1, display regular behavior that correlates well with the ionization energies of the rare gas atoms. It is encouraging to observe that for the argon, krypton, and xenon systems the present structural parameters agree rather well with those obtained by Lundell et al.20 However, there is a significant difference for HRgF, where the bond lengths (in particular Rg-F) were found to be much longer in the previous work.20 The difference may be attributed to different basis sets and pseudopotentials used by Lundell et al. and in the present work: Lundell et al. used the standard 6-311++G(2d,2p) basis sets for H and F, while for the heavier rare gas atoms (Kr, Xe, and Rn) they used the averaged relativistic core potentials of Christiansen and co-workers,55-57 with very limited expansion of the valence orbitals (only three s- and p -type and four d -type primitive Gaussian functions) and no polarization/correlating functions on the heavy atom. It is the deficiency in the basis sets for the rare gas atom that leads to an underestimated energy of formation of HRgF, with concomitant increase in the energy of dissociation in reaction 2. According to the data shown in Tables 2-5 and visualized in Figure 2, as the rare gas atom increases in size, the reaction of formation of HRgF, eq 1, becomes more exothermic, while the dissociation of HRgF into hydrogen fluoride and the rare gas atom, eq 2, becomes less exothermic. The HRgF compounds are unstable thermodynamically; however, as pointed out by Frenking,22 they may be stable kinetically if the energy barrier for reaction 2 is high. The best energy differences ∆E1 and ∆E2 are the ones computed in the CR-CC(2,3)/mcpT//MP2/mcpT calculations and, for the Ar, Kr, and Xe compounds, they agree reasonably well with the data reported previously.20 The large difference for HRnF may be attributed to differences in the

optimized geometry of the molecule. It is satisfying to observe that the energy differences from the CR-CC(2,3)/mcpT//MP2/ mcpT and from CR-CC(2,3)/mcpT//CCSD(T)/mcpT calculations agree very well, suggesting that for larger molecules we may rely on the less expensive MP2 optimization of geometry. The atomization energy of hydrogen fluoride, TAEe, estimated on the basis of the energy changes in the two reactions, (1) and (2), for all four systems studied depends on the basis set and method used. It equals about 613 kJ/mol at the MP2/mcpT level of theory and is reduced to about 600 kJ/mol when the allelectron basis sets are employed (accT, nasT, and rasT). The CCSD(T)/LCG value of TAEe is about 590 kJ/mol. At the CRCC(2,3) level of theory the value equals 593 kJ/mol and is reasonably close to the experimental molecular total atomization energy of 591 kJ/mol reported by Martin.76 Harmonic vibrational frequencies, calculated at the MP2/ mcpT level of theory, are compared in Table 6 with the values obtained by Lundell et al.20 The Rg-H stretch has the largest intensity, which is decreasing from the lightest (17.0 D2/(amu Å2)) to the heaviest (8.9 D2/(amu Å2)) congener. The intensity of the Rg-F stretch changes less, from 6.2 D2/(amu Å2) for HArF to 4.4 D2/(amu Å2) for HRnF. The agreement between the previous20 and present values is satisfactory, except for HRnF. Smaller vibrational frequencies corresponding to the bond stretches correlate with the longer bond lengths Rn-H and Rn-F obtained in the previous work.20 Atomic charges, calculated using three methods of population analysis: Mulliken (MPA),77 Lo¨wdin (LPA),78,79 and natural population analysis (NPA)80-82 and utilizing two electron densities, MP2 and CCSD, are shown in Table 7. With the exception of HArF, the Mulliken charges agree quite well with the NPA charge distribution. Electron densities from both MP2 and CCSD methods yield similar charge distributions for all atoms. The Lo¨wdin charges attribute less partial negative charge to fluorine than do NPA and Mulliken, but in all three cases the model of bonding is of a fluorine with a partial negative charge of about -0.8 e (NPA) interacting with the Rg-H complex, which has a partial positive charge that is predominantly localized on the rare gas atom, especially for the heavier atoms. As the size of the rare gas atom increases, the magnitude of the dipole moment decreases, with the dipole moments of Rn and Xe being the most closely spaced of all the dipole moments. In charge distribution and dipole moment, radon appears to resemble xenon more closely than it does any other rare gas atom. The charge distribution clearly supports the accepted model of bonding in the HRgF systems, HRg+F-. It is interesting to mention in this context that the Rg-H bond lengths in HRgF are only 3-4% longer than the ones found of the ions HRg+: 1.271, 1.406, 1.578, and 1.662 for Rg ) Ar, Kr, Xe, and Rn, respectively (MP2/mcpT results). Natural electron configurations80,81 are shown in Table 8. They reveal that the amount of electron transfer from the valence np subshell of the rare gas atom increases with the increase of the size of the rare gas atom. In all four molecules, the natural bond

Molecules Containing Rare Gas Atoms

J. Phys. Chem. A, Vol. 114, No. 33, 2010 8791

TABLE 7: Atomic Charges Computed According to Various Population Analysesa population analysis H molecule

method

MPA

LPA

HArF HArF HKrF HKrF HXeF HXeF HRnF HRnF

MP2 CCSD MP2 CCSD MP2 CCSD MP2 CCSD

0.67 0.61 0.13 0.11 0.11 0.09 -0.11 -0.12

-0.46 -0.44 -0.46 -0.45 -0.51 -0.50 -0.54 -0.53

a

Rg MPA

LPA

NPA

MPA

LPA

NPA

µ/Db

0.21

0.13 0.20 0.67 0.70 0.66 0.69 0.89 0.91

1.10 1.09 1.12 1.12 1.23 1.23 1.31 1.30

0.55

-0.80 -0.81 -0.80 -0.81 -0.77 -0.79 -0.78 -0.80

-0.64 -0.65 -0.66 -0.67 -0.72 -0.73 -0.76 -0.77

-0.76

6.686 6.797 6.118 6.179 5.125 5.126 4.971 4.980

0.11 -0.02 -0.07

0.65 0.79 0.86

-0.76 -0.77 -0.79

aug-MCP-tzp (mcpT) basis set used. b Dipole moment in Debye units.

TABLE 8: Natural Valence Electron Configurationsa

a

F

NPA

molecule

H

Rg

F

HArF HKrF HXeF HRnF

1s0.77 1s0.86 1s0.99 1s1.04

3s1.953p5.31 4s1.964p5.23 5s1.965p5.07 6s1.976p5.02

2s1.982p5.67 2s1.972p5.67 2s1.972p5.69 2s1.972p5.70

Based on MP2/aug-MCP-tzp electron density.

orbital analysis identifies only one bond (between the Rg and H atoms) and seven lone pairs, three on the Rg and four on the fluoride. Bond order analysis83,84 performed on the MP2 electron density reveals that the bond order for the H-Rg bond increases with increasing size of the rare gas atom, with the values 0.31, 0.81, 0.86, and 0.89 for HArF, HKrF, HXeF, and XRnF, respectively, while the bond order for the Rg-F bond diminishes to zero: 0.20, 0.17, 0.11, and 0.00, in the same series. Conclusion In the present work we reported the geometry, charge distribution, dipole moments, and reaction energetics of molecules of the type HRgF calculated at four levels of theory, MP2, CCSD, CCSD(T), and CR-CC(2,3), using an augmented model core potentials basis set as well as several all-electron basis sets. The bonding behavior of radon is consistent with the periodic trends for the rare gases, with radon binding more strongly than the lighter rare gas atoms. Examination of calculated dissociation energies shows that the HRgF compounds are thermodynamically more stable than the free atoms but less stable than HF + Rg. Importantly, the structural parameters obtained in the less time-consuming MP2 calculations are very close to those obtained in the CCSD(T) calculations, allowing for the determination of reaction energetics in single-point CR-CC(2,3) calculations at the MP2 geometry and increasing prospects of accurate computational chemistry of radon compounds. The charge distributions for all the rare gas systems studied are similar, with large partial negative charge being localized on the fluorine atom and a positive charge distributed across the RgH+ group. The dipole moment of these molecules decreases with increasing rare gas atom size. In comparison to previous studies of the HRnF molecule,20 the present determination of the Rn-H and Rn-F bond lengths found them to be much smaller than previously reported; corresponding deviations were found in the values for the harmonic vibrational frequencies of HRnF. The difference may be assigned to less flexible basis sets for the rare gas atom used in the previous work.20 The present results confirm the general model of bonding for the HRgX systems and predict that HRnF would bind in

much the same way as the HRgF compounds that contain lighter rare gas atoms. Acknowledgment. M.K. thanks Natural Sciences and Engineering Research Council (NSERC) for the support of the present project under Grant G121210414. M.K. thanks Professor Grochala for providing an unpublished manuscript of a review of noble gas chemistry. We are grateful to Professor M. S. Gordon and Dr. M. W. Schmidt for the continuing support of the GAMESS program. We greatly appreciate the insightful comments of the anonymous Reviewers which resulted in an improved version of the present report. All calculations were carried out using the computers obtained by M.K. through the NSERC Equipment Grant as well as using the Linux cluster at the Department of Chemistry, University of Alberta (maintained by Dr. S. L. Delinger), and using the computers at the Department of Academic Information and Communication Technologies, University of Alberta. References and Notes (1) Grochala, W. Chem. Soc. ReV. 2007, 36, 1632–1655. (2) Zhdankin, V. V. Russ. Chem. Bull. 1993, 42, 1849–1857. (3) Frenking, G.; Cremer, D. Struct. Bonding (Berlin) 1990, 73, 17– 95. (4) Fields, P. R.; Stein, L.; Zirin, M. H. J. Am. Chem. Soc. 1962, 84, 4164–4165. (5) Bartlett, N. Proc. Chem. Soc. 1962, 218. (6) Liebman, J. F. Inorg. Nucl. Chem. Lett. 1975, 11, 683–685. (7) Pitzer, K. S. J. Chem. Soc., Chem. Commun. 1975, 760–761. (8) Seppelt, K. Angew. Chem., Int. Ed. 1979, 18, 186–202. (9) Avrorin, V. V.; Krasikova, R. N.; Nefedov, V. D.; Toropova, M. A. Russ. Chem. ReV. 1982, 51, 12–20. (10) Selig, H.; Holloway, J. H. Top. Curr. Chem. 1984, 124, 33–90. (11) Holloway, J. H. J. Fluor. Chem. 1986, 33, 149–158. (12) Stein, L. Chemical Properties of Radon. In Radon and Its Decay Products; Hopke, P. K., Ed.; American Chemical Society: Washington, DC, 1987; Vol. 331,Chapter 18, pp 240-251. (13) Jørgensen, Ch. K.; Frenking, G. Struct. Bonding (Berlin) 1990, 73, 1–15. (14) Dolg, M.; Ku¨chle, W.; Stoll, H.; Preuss, H.; Schwerdtfeger, P. Mol. Phys. 1991, 74, 1265–1285. (15) Kaupp, M.; van Wu¨llen, Ch.; Franke, R.; Schmitz, F.; Kutzelnigg, W. J. Am. Chem. Soc. 1996, 118, 11939–11950. (16) Runeberg, N.; Pyykko¨, P. Int. J. Quantum Chem. 1998, 66, 131– 140. (17) Liao, M.-S.; Zhang, Q.-E. J. Phys. Chem. A 1998, 102, 10647– 10654. (18) Han, Y. K.; Lee, Y. S. J. Phys. Chem. A 1999, 103, 1104–1108. (19) Pettersson, M.; Lundell, J.; Ra¨sa¨nen, M. Eur. J. Inorg. Chem. 1999, 729–737. (20) Lundell, J.; Chaban, G. M.; Gerber, R. B. Chem. Phys. Lett. 2000, 331, 308–316. (21) Christe, K. O. Angew. Chem., Int. Ed. 2001, 40, 1419–1421. (22) Frenking, G. Nature 2000, 406, 836–837. (23) Schrobilgen, G. J. Noble-Gas Chemistry. In Encyclopedia of Physical Science and Technology, 3rd ed.; Hawthorne, M., Ed.; Academic Press: New York, 2002; Vol. 10, pp 449-461. (24) Malli, G. L. Int. J. Quantum Chem. 2002, 90, 611–615.

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