Theoretical Study of M+− RG and M2+− RG Complexes and Transport

J. Phys. Chem. A 2010, 114, 7631–7641

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Theoretical Study of M+-RG and M2+-RG Complexes and Transport of M+ through RG (M ) Be and Mg, RG ) He-Rn) Adrian M. Gardner, Carolyn D. Withers, Jack B. Graneek, and Timothy G. Wright* School of Chemistry, UniVersity of Nottingham, UniVersity Park, Nottingham NG7 2RD, U.K.

Larry A. Viehland Science Department, Chatham UniVersity, Pittsburgh, PennsylVania 15232, USA

W. H. Breckenridge Department of Chemistry, UniVersity of Utah, Salt Lake City, Utah 84112, USA ReceiVed: April 28, 2010; ReVised Manuscript ReceiVed: June 9, 2010

We present high level ab initio potential energy curves for the Mn+-RG complexes, where n ) 1, 2, RG ) rare gas, and M ) Be and Mg. Spectroscopic constants have been derived from these potentials, and they generally show very good agreement with the available experimental data. The potentials have also been employed in calculating transport coefficients for M+ moving through a bath of RG atoms, and the isotopic scaling relationship is examined for Mg+ in Ne. Trends in binding energies, De, and bond lengths, Re, are discussed and compared to similar ab initio results involving the corresponding complexes of the heavier alkaline earth metal ions. We identify some very unusual behavior, particularly for Be+-Ne, and offer possible explanations. 1. Introduction We have previously published high-level ab initio potential energy curves for Ba+ and Ba2+ interacting with rare gas (RG) atoms,1 and the corresponding species involving calcium, strontium, and radium atomic ions.2 In the present work we address the lightest species, completing the above series: Ben+-RG and Mgn+-RG (n ) 1, 2). To our knowledge, there have been no experimental studies on the Be+-He nor Be+-Ne species. A preliminary study reported the observation of an unidentified emission in a microwave discharge of BeCl2 in Ar.3 Later work identified the spectrum as being due to Be+-Ar,4 with the mechanism for production of such species having been discussed in that work, and also by Goble et al.5 Follow-on studies also identified the spectra of Be+-Kr4 and Be+-Xe.6 Analysis of high-resolution spectra for Be+-Ar7 and Be+-Kr8 allowed the determination of precise spectroscopic constants for the ground and first excited states. Although there are no experimental studies on the lightest two species, there have been some theoretical studies. As part of a wider study of the interactions between singly and doubly charged first row cations with He and Ne, Frenking et al.9,10 have reported the results of MP4(SDTQ) calculations employing 6-311G(2df, 2pd) basis sets. Leung and Breckenridge11 have reported spectroscopic constants for Be+-He and Be+-Ne at the QCISD(T) level using quite large basis sets, both for the ground and some excited states. In addition, Be+-He has been studied by Bu et al.12,13 at the MP2/6-311+G(3df,3pd) level as part of a study investigating Be+-Hen clusters. We also note that the spectroscopic data from the emission spectroscopy studies4,7,8,6 have been employed to investigate the potential and * To whom correspondence [email protected].

should

be

addressed.

E-mail:

interactions in the Be+-RG species.14–17 Some of the above work is contained in a wide-ranging review on the interactions between M+ and RG atoms.18 No work appears to have been reported on Be+-Rn. There has been very little work on the beryllium dications, with apparently the only recent work being that on Be2+-He and Be2+-Ne.9,11 The study by Frenking et al.9 reported an optimized bond length and vibrational frequency for Be2+-Ne at the MP2 level, as well as an MP4//MP2 dissociation energy. The only other study appears to be that of Leung and Breckenridge,11 where Be2+-He and Be2+-Ne potential energy curves were calculated at the QCISD(T) level and bond lengths, vibrational frequencies, and dissociation energies were reported. Work on Mg+-RG has also been reviewed in ref 18. The experimental data comes from two main sources: the photodissociation work of Duncan and co-workers19–21 and the later photoionization threshold experiments of Breckenridge and co-workers.22–24 Discussion of extrapolations of the photodissociation data19–21 to obtain dissociation energies on these species has also been provided by Le Roy,25 prior to the appearance of the photoionization work. In addition, there have been a number of theoretical studies, with one including Mg+ interacting with He, Ne. and Ar being reported by Partridge et al. using the modified coupled-pair functional (MCPF) method.26 Further calculations by two of the same workers at the CISD level were reported for Mg+-Ar and Mg+-Kr.27 Various levels of theory, up to QCISD(T), were employed by Leung et al.28,29 and yielded spectroscopic constants for Mg+-He and Mg+-Ne within a wider study into the ground and excited states. In investigations looking at solvation of Li+, Na+. and Mg+ by helium atoms, Sapse et al.30 reported Re and De values for Mg+-He at the MP2 and QCISD(T) levels, with ωe values at the MP2 level only. More recently, Bu and Zhong have reported calculations on Mg+-He as part of a wider study looking at Mg+ interacting

10.1021/jp103836t  2010 American Chemical Society Published on Web 06/25/2010

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with small clusters of helium.31 Some of the above work is contained in a wide-ranging review on the interactions between M+ and RG atoms.18 No work appears to have been reported on Mg+-Rn. Work on the dications, Mg2+-RG, appears to have been limited to QCISD(T) calculations on Mg2+-He and Mg2+-Ne by Leung et al.,28,29 and by Petrie32 as part of a series of calculations aimed at the establishing of a magnesium dication affinity scale. The above studies provide a wealth of data allowing trends between the species to be examined,18 but in some cases the uncertainties on the derived quantities are rather large, and additionally some have been obtained by long spectroscopic extrapolations. It would be useful, therefore, to study the complete set of Be+-RG and Mg+-RG species at a high, and consistent, level of theory to test the previous values, and also to allow direct comparison between the species and thus identify trends more precisely. In the present work we shall consider the set of Group 2 M+-RG and M2+-RG complexes, M ) Be and Mg, and compare these results to those we have reported previously for Ca+-RG to Ra+-RG.1,2 In many cases the present results represent the only such data available. 2. Computational Details A. Potential Energy Curves. For He-Ar, the standard augcc-pVQZ and aug-cc-pV5Z basis sets33–35 were employed; for Kr-Rn the basis sets employed were the small-core relativistic effective core potentials (ECPs), ECP10MDF, ECP28MDF, and ECP60MDF for Kr, Xe, and Rn, respectively; these were used in conjunction with the aug-cc-pV5Z-PP valence basis sets.36 Additionally, we included tight functions to describe the correlation of the inner-valence d orbitals (see ref 37 for functions and exponents). For beryllium and magnesium, augcc-pVQZ and aug-cc-pV5Z basis sets were employed, but each was augmented with tight functions of each angular momentum type, to allow description of core-valence correlation.38,39 Each of the basis sets was augmented by an additional set of diffuse basis functions of each angular momentum type, obtained in an even-tempered way from the two lowest exponents in each case, so that each basis set was overall doubly augmented with diffuse functions. All calculations were carried out using the RCCSD(T) procedure40 as implemented in MOLPRO.41 For Kr, Xe, and Rn, we explicitly correlated the outermost occupied d orbitals as well as the valence s and p orbitals. For Be, all electrons were correlated, and for Mg and Ne, only the 1s orbital was frozen; in the case of Ar, the 1s, 2s, and 2p orbitals were frozen. The full counterpoise correction was applied at each nuclear separation, R, and a wide range of R (generally 40-50 points in the range 1-10 Å) was covered to include short-, medium-, and long-range. For RG ) He-Ar, we extrapolated the interaction energy at each R to the basis set limit employing the two-point formula of Halkier et al.42,43 B. Spectroscopic Parameters. The rovibrational energy levels for each potential energy curve were obtained using the LEVEL44 program, and the few lowest calculated energy levels were least-squares fitted to standard expressions to obtain the reported spectroscopic quantities. The most abundant naturally occurring isotope of each element was used in all cases (9Be, 24 Mg, 4He, 20Ne, 40Ar, 84Kr, 132Xe, and 222Rn). C. Ion Transport Properties. We calculated the transport cross sections for M+ in each RG from the ab initio interaction potential energy curves as functions of the ion-neutral collision energy using the classical-mechanical program QVALUES.45,46

Gardner et al. The cross sections as a function of collision energy were used in the program GC45,47,48 to determine the standard mobility and the other gaseous ion transport coefficients as functions of E/n0 (the ratio of the electric field to the gas number density) at gas temperatures of 100, 200, 300, 400, and 500 K. The calculated mobilities are generally precise within 0.1%, which means that the numerical procedures within programs QVALUES and GC have converged within 0.1% for the ion-neutral interaction potential. Calculations were performed for every isotope of the ion which is naturally occurring, but in all cases the rare gas was assumed to be the naturally occurring mixture of isotopes. 3. Results and Discussion The calculated RCCSD(T) extrapolated potential energy curves for each of the 24 species are available as Supporting Information. A. Spectroscopic Constants. Be+-RG. The derived spectroscopic constants for Be+-RG are given in Table 1, where the force constant, k, has also been calculated, using the simple harmonic relationship. (For the interested reader, we also include the corresponding quantities for Be+-He, Be+-Ne, and Be+-Ar, as calculated at the RCCSD(T)/daVQZ and RCCSD(T)/ daV5Z levels of theory, as Supporting Information.) For Be+-He, the only previous results are theoretical, and we see that there is generally good agreement between the calculated Re values, with the present value being the shortest; the latter is consistent with the fact that the present binding energy is also the highest. All three ωe values are in excellent agreement. For Be+-Ne, there is generally good agreement with the previous QCISD(T) results from ref 11 for Re, De, and ωe; however, there is stark disagreement with the previously reported values from Frenking et al.,10 who report a significantly shorter bond length than that obtained here, at both the MP2 and MP4 levels of theory. In addition, the reported ωe value is almost four times as high as the present value, whereas there is good agreement between our value and the QCISD(T) value from ref 11. The larger ωe value in ref 10 has led to a large difference between De and D0 reported in that work. We also note that the reported BSSE in ref 10 is over 1200 cm-1 and suggests that the binding energy of 314 cm-1 is unreliable, and the anomalously high ωe value also suggests that the values in ref 10 are in error. The fact that the QCISD(T) values from ref 11 and the present results are in good agreement for Be+-He and Be+-Ne and that the present values were obtained at the higher level of theorysfrom extrapolated quadruple- and quintuple-ζ resultsssuggests that these are the most reliable. For Be+-Ar there are high-resolution spectroscopic results available from emission spectra, reported by Subbaram et al.7 as a follow-on from their initial, lower-resolution experiments on Be+-Ar.3,4 These high-resolution results allowed a very precise bond length and very precise vibrational constants to be obtained. They did not, however, allow a precise dissociation energy to be determined. As may be seen from Table 1, the agreement between the present calculated Re, ωe, De, Be, and R values and those obtained from the high-resolution spectroscopic experiments is really quite remarkably good. With regard to the dissociation energy, this could only be estimated in ref 7 from approximate potentials, and a “best” average value of De ) 4112 ( 200 cm-1 (D0 ) 3933 ( 200 cm-1) was reported therein; in a similar way (and employing the same data), but considering other model potential forms, Goble et al.14 suggested that a value of De ) 4157 cm-1 should provide a close lower bound to the true dissociation energy. A consideration of the form of near-dissociation expansions by Le Roy and Lam, again

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TABLE 1: Spectroscopic Constants for Be+-RG Calculated at the RCCSD(T) Level, See Text for Basis Setsa De (cm-1)

Re (Å) Be+-He

Be+-Ar

2.924 3.132b 2.96d 3.104e 2.454 (2.462) 1.856f 2.054g 2.59h 2.084

Be+-Kr

2.0855 ( 0.0006 2.221

Be+-Xe

2.2201k 2.407

Be+-Ne

D0 (cm-1)

133.1

97.8 70c

76.3 68b 73d 65e 65.2 (65.6) 259f

124d 60e 375.3 (371.7) 175 175g

407.2 (403.7) 314g 359h 4427.7 4500 ( 700i

68h 364.1

4247.7

j

8239.8 10000 ( 2000i

2.486

9490.5

DMorse e (cm-1)

R (cm-1)

DMorse e (De)

k (Nm-1)

11.7

0.723

0.142

125

0.94

0.950

1.47 (1.63)

0.456 (0.453)

0.00607 (0.00592)

723

1.78

1.555

3710

0.84

57.4

8.94

0.528

0.0147

362.7 ( 0.1 365.5

8.92 ( 0.05 5.82

0.5271 ( 0.0003 0.420

0.0145 ( 0.0003 0.00816

5740

0.95

64.0

8054.3

367.14k 372.8

6.21k 4.04

0.42030k 0.345

0.00821k 0.00486

8600

1.04

69.0

9305.7

∼367 371.0

∼3.7 3.32

0.315

0.00381

10400

1.10

70.1

j

l

Be+-Rn

Be (cm-1)

3933 ( 200 5871.8

j

6053.2 6500 ( 1000i

ωexe (cm-1)

ωe (cm-1)

a

9

4

20

j

j

j

l

40

84

132

222

The quantities refer to the following isotopes: Be He, Ne, Ar, Kr, Xe and Rn. Re is the equilibrium bond length, De is the depth of the potential, D0 is the energy between the zero-point and the asymptote, ωe is the harmonic vibrational frequency, ωexe is the anharmonicity constant, Be is the equilibrium rotational constant at the minimum, k is the harmonic force constant, and R is the spin-rotation constant, Quantities in bold are results from the present study, with the values in parentheses for Be+-Ne being the calculations where the 1s electron of Ne was also correlated; see text for details. b MP2/6-31G** from ref 9. c MP4/6-311G(2df/2pd)//MP2/6-31G** from ref 9. d QCISD(T) using a 6-311++G(3df,3dp) basis set for Be, and a aug-cc-pVQZ basis set for He, from ref 11. e MP2/6-311+G(3df,3pd) from ref 13. f MP2/6-31G** from ref 10. g MP4/6-311G(2df,2pd)//MP2/6-31G** from ref 10. h MP2/6-311+G(3df,3pd) from ref 12. i Recommended value18 from review of data up until 2001; see text for discussion of derived Be+-Xe value. j From high-resolution emission experiments, ref 7. k From high-resolution emission experiments, ref 8. l From emission experiment, ref 6.

TABLE 2: Spectroscopic Constants for Be2+-RG Calculated at the RCCSD(T) Level, See Text for Basis Setsa Re (Å) Be2+-He Be2+-Ne Be2+-Ar Be2+-Kr Be2+-Xe Be2+-Rn

1.428 1.339b 1.43d 1.577 1.58d 1.867 2.009 2.197 2.288

De (cm-1)

D0 (cm-1)

ωe (cm-1)

ωexe (cm-1)

Be (cm-1)

7574.5

7131.3 6610c

902.0 829b 877d 675.0 664d 691.9 634.9 594.3 563.5

27.4

2.981

10.5

7389d 10983 10601d 24874 29875 36974 40496

10648 24530 29559 36677 40216

5.15 3.91 3.37 3.11

R (cm-1)

DMorse (cm-1) e

DMorse (De) e

k (Nm-1)

0.121

7423

0.98

133

1.091

0.0235

10850

0.99

167

0.658 0.513 0.414 0.373

0.00738 0.00480 0.00342 0.00292

23240 25750 26200 25530

0.93 0.86 0.71 0.63

207 193 175 162

a The symbols are defined in the footnotes to Table 1. Quantities in bold are results from the present study. b MP2/6-31G** from ref 9. MP4/6-311G(2df/2pd)//MP2/6-31G** from ref 9. d QCISD(T) using a 6-311++G(3df,3dp) basis set for Be, and aug-cc-pVQZ basis sets for He and Ne, from ref 11.

c

using the results of the high-resolution experiments of ref 7, led to the reported value15 of De ) 4500 ( 50 cm-1. Taking into account the approximate nature of the previous determinations, Bellert and Breckenridge18 recommended a value of 4500 ( 700 cm-1 in their 2002 review. The excellent agreement of the present calculated results and the other precise quantities that were determined from the high-resolution results, suggests that the present calculated value of De ) 4427.7 cm-1 is reliable. This value is in very good agreement with the value reported by Le Roy and Lam,15 and is comfortably within the error range given by Bellert and Breckenridge.18 It is also consistent with the lower bound value of 4157 cm-1 given by Goble et al.14 Moving on to Be+-Kr, again analyses of high-resolution emission spectra by Coxon et al.8 have yielded very precise experimental values, and again excellent agreement is seen with the present calculated results for the bond length and vibrational and rotational constants. No dissociation energy was reported in that study, although the earlier lower-resolution study4 reported a value of D0 ∼ 6000 cm-1, with Bellert and Breckenridge18 estimating De ) 6500 ( 1000 cm-1. The former value is in very good agreement with the present De value of 6053.2 cm-1, and the latter is consistent within the substantial uncertainties.

There has been no high-resolution emission study of Be+-Xe, although a low-resolution one was reported.6 That study yielded vibrational constants that are in very good agreement with the values obtained herein. An estimated value6 for the dissociation energy, De of ∼9000 cm-1, is in fair agreement with the value of 8239.8 cm-1 obtained herein. An estimate of 11 000 ( 2000 cm-1 is given in the 2002 review18 and seems to be too high based on the present results. In fact, both the rough estimates of De for Be+-Kr and Be+-Xe in ref 18 were made assuming they would have the same ratios to De of Be+-Ar as the analogous Mg+-RG complexes. Upon recalculation, it seems that the estimate of De for Be+-Xe of 11 000 cm-1 in ref 18 was a typographical error, with the true estimate being 10 000 cm-1. There are no data on Be+-Rn, to our knowledge, and so the values in Table 1 represent the first such values for this species. Be2+-RG. The derived spectroscopic constants for Be2+-RG are given in Table 2. There have been no experimental studies reported involving any of these dications, but there have been two theoretical studies on Be2+-He and one on Be2+-Ne. Frenking et al.9 employed the MP2 method to obtain equilibrium internuclear separations and the MP4 method to yield a Be2+-He dissociation energy; Leung and Breckenridge used

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Gardner et al.

TABLE 3: Spectroscopic Constants for Mg+-RG Calculated at the RCCSD(T) Level, See Text for Basis Setsa Re (Å) +

Mg -He 3.482 3.565b 3.56c 3.611d 3.685e Mg+-Ne 3.145 3.15f 3.298b 3.25g 3.17 ( 0.05h,i Mg+-Ar 2.822

De (cm-1) 73.2 70b 65c 30d 76.6e 203.1 216 ( 100f 170b 179g 1298.6

2.81 ( 0.03f 1290 ( 60f 2.894b 1140b 2.854l 2.825 ( 0.007h,m Mg+-Kr 2.884 ∼2.80f

1978.1 1949 ( 100f

2.886l Mg+-Xe 3.018 ∼2.90f Mg+-Rn 3.064

2972.7 2910 ( 100f 3639.0

D0 (cm-1) 51.7

ωe (cm-1) ωexe (cm-1) Be (cm-1) R (cm-1) DMorse (cm-1) DMorse (De) k (Nm-1) e e 7.68

0.412

0.0900

2.44

0.157

0.00987

2.53

0.141

2.08

134.8

124f 4182m 120j,k 3299 ( 1654n 3568.8 141.1

182.3

96 ( 50i 1246.7 1237 ( 40k 1041l 1281m 1210 ( 165n 1919.4 1891 ( 80k 1923m 1863l 1812 ( 591n 2905.8 2848 ( 150k

45.8 21b 44c 44d 42e 43.1 46f 33b 39g 41i,j 104.8

68.3

0.95

0.424

191

0.94

1.19

0.00407

1090

0.84

9.71

0.109

0.00213

1690

0.85

15.4

1.66

0.0912

0.00123

2740

0.92

21.7

1.41

0.0830

0.000919

3530

0.97

25.4

100 ( 3f 92b 90j,l 96j,m 118.4 116f 112j,m 115j,l

a The quantities refer to the following isotopes: 24 Mg, 4He, 20Ne, 40Ar, 84Kr, 132Xe and 222Rn. The symbols are defined in the footnotes to Table 1. Quantities in bold are results from the present study. b MCPF calculations from ref 26. c QCISD(T) calculations from ref 28. d MP2 calculations from ref 31. e MP2/6-311+G(3df, 3pd) calculations from ref 30. f Recommended value18 from review of data up until 2001. g QCISD(T) calculations from ref 29. h R0 value. i Obtained from vibrationally and rotationally resolved photodissociation spectroscopy, ref 20. j ∆G1/2 value. k Derived from photoionization measurements.23,50,51 l CISD values from ref 27. m Obtained from vibrationally21 and rotationally resolved19 photodissociation experiments. n Re-evaluated dissociation energies based upon data from ref 21 and consideration of neardissociation behavior.25

the QCISD(T) method to obtain similar quantities for Be2+-He and Be2+-Ne.11 As may be seen, these methods yield values that are in good agreement with each other and with the present results. Mg+-RG. All of the present calculated spectroscopic quantities are presented in Table 3, together with those from previous studies. (For the interested reader, we also include the corresponding quantities for Mg+-He, Mg+-Ne and Mg+-Ar, as calculated at the RCCSD(T)/daVQZ and RCCSD(T)/daV5Z levels of theory, as Supporting Information.) As in the case of Be+-RG, we note that Bellert and Breckenridge18 assessed all of the data available up until 2001, and recommended values for Re, De, and ωe, which are also indicated in Table 3. For Mg+-He, there are only theoretical values available, and these are presented in Table 3 together with those from the present work. The MCPF calculations of Partridge et al.26 yielded values in good agreement with the present ones for Re and De; however, the value of ωe is almost half of the present value. In contrast, the QCISD(T) calculations from Leung et al.28 yield a value of ωe ) 44 cm-1, very close to the present value, and confirmed by the recent MP2 values from Sapse et al.30 and Bu and Zhong;31 together, these indicate that the low value in ref 26 is unreliable. The ultraviolet photodissociation experiments of Duncan provided the first experimental values of a range of spectroscopic quantities. The first paper21 reported results on Mg+-Ar, Mg+-Kr, and Mg+-Xe, and yielded ground state dissociation energies (calculated from derived upper state values), D0, of 1281, 1923, and 4182 cm-1, respectively. The first two may be

seen to be in good agreement with the present calculated values in Table 3, whereas the experimental Mg+-Xe value is significantly higher (see below). Vibrational hot bands allowed ground state fundamental vibrational spacings to be measured as ∆G1/2 ) 96, 112, and 120 cm-1. Breckenridge49 has reanalyzed the original data yielding revised values of 98, 114, and 122 cm-1. We can compare these directly with the calculated ∆G1/2 values, which are 99.8, 114.3, and 131.4 cm-1; clearly the agreement is excellent for the two lighter species, but is poorer for Mg+-Xe. Later, high-resolution spectra19 for Mg+-Ar allowed rotational structure to be obtained, and the ground state rotational constant, B0, was obtained as 0.1409 ( 0.0007 cm-1, allowing the bond length, R0, to be derived as 2.825 ( 0.007 Å. The values compare well with our theoretical values of B0 ) 0.13919 cm-1 and R0 ) 2.843 Å. A few years later, corresponding spectra of the Mg+-Ne complex were obtained,20 allowing a ground state dissociation energy of D0 ) 96 ( 50 cm-1 (De ) 117 ( 50 cm-1) to be derived from the Birge-Sponer extrapolated upper state value; this value is much smaller than the present values, even taking the rather large experimental uncertainties into consideration. Hot band structure yielded ∆G1/2 ) 41 cm-1, which is in good agreement with the present ωe value, and can be compared directly with the calculated ∆G1/2 value of 38.2 cm-1. The R0 value was determined to be 3.17 ( 0.05 Å, which is in very good agreement with the present value of 3.192 Å. The only other theoretical values appear to be those of Leung et al.29 who reported QCISD(T) valuesssee Table 1; the agreement with the present values for Re, De, and ωe is generally good, although

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TABLE 4: Spectroscopic Constants for Mg2+-RG Calculated at the RCCSD(T) Level, See Text for Basis Setsa Re (Å)

De (cm-1)

D0 (cm-1)

ωe (cm-1)

ωexe (cm-1)

Be (cm-1)

2526.8

462.3 441b 434c

21.1

1.38

Mg -He

1.885 1.91b 1.910c 2.054d 1.909f

2752.7 2550b 2144c

Mg2+-Ne

2.035 2.08g 2.058d 2.066f

4224.1 3806g

Mg2+-Ar

2.318 2.396 2.345

10894

Mg2+-Kr Mg2+-Xe Mg2+-Rn

2.453 2.632 2.711

13668 17849 20113

2+

2170f 4072.2

R (cm-1)

DMorse (cm-1) e

DMorse (De) e

k (Nm-1)

0.0747

2532

0.92

43.2

306.6 285g

6.05

0.373

0.00859

3884

0.92

60.4

3600f 10730.5

328.2

2.55

0.209

0.00209

10560

0.97

95.2

9881f 13520 17707 19978

296.8 284.3 271.0

1.63 1.20 1.02

0.150 0.120 0.106

0.00111 0.000699 0.000556

13510 16840 18000

0.99 0.94 0.89

96.9 96.7 93.7

a The symbols are defined in the footnotes to Table 1. Quantities in bold are results from the present study. b QCISD(T) calculations from ref 28. c MP2 calculations from ref 31. d MP2 calculations from ref 32. e B3LYP calculations from ref 32. f Obtained using a nonstandard dG2thaw(QCI)//B3LYP/dB4G method; see ref 32. g QCISD(T) calculations from ref 29.

the values in that work indicated slightly weaker bonding than the present potential indicates. In a similar way to that discussed above for Be+-Ar, consideration of near-dissociation behavior of vibrational energy spacings has led Le Roy25 to derive improved estimates of D0 values, including error ranges, for Mg+-Ar, Mg+-Kr, and Mg+-Xe, using the excited state vibrational information from Pilgrim et al.21 Ground state values could then be derived from these values as: 1210 ( 165 cm-1 for Mg+-Ar; 1812 ( 591 cm-1 for Mg+-Kr; and 3299 ( 1654 cm-1 for Mg+-Xe. These values are consistent with the present calculated values, and with those of ref 21 within the large error ranges derived in ref 25. In a different set of experiments, Breckenridge and co-workers employed photoionization spectroscopy to find ionization thresholds22,50 and used this data to estimate ground state dissociation energies. These likely result in the most accurate experimental D0 values since the photoionization thresholds can be determined very precisely (within a few cm-1), so the error is essentially in the estimate of the D0 values of the weakly bound neutral Mg-RG (3Π0) states from which the photoionization occurs. The errors are thus small overall, although estimated conservatively. For Mg+-Ar,22 they obtained a value of D0 ) 1237 ( 40 cm-1, which is (just about) in agreement with the above photodissociation results, within experimental error, and only slightly below the present calculated value. Similar experiments on Mg+-Kr and Mg+-Xe23,51 allowed dissociation energies, D0, to be derived as 1891 ( 80 and 2848 ( 150 cm-1. The Mg+-Kr value is consistent with the photodissociation value but is slightly above the present calculated value. On the other hand, for Mg+-Xe the photoionization value23 is significantly lower than the photodissociation21 value, but it is the former that is consistent with the present calculated value. No previous studies on Mg+-Rn appear to have been reported. Mg2+-RG. Table 4 contains the calculated spectroscopic parameters for the complexes involving the magnesium dication. As may be seen, the dissociation energies are quite substantial, even for the Mg2+-He species; these quantities are, however, smaller than those of the Be2+-RG species (see Table 2), likely owing to the extremely small size of the Be2+ ion. Although there appear to be no experimental values, there have been some

calculations reported for each of the three lightest Mg2+-RG species. QCISD(T) calculations on Mg2+-He from Leung et al.28 yield values in very good agreement with those obtained herein; this is also true of the MP2 values by Bu and Zhong.31 Petrie32 has recently investigated a magnesium dication affinity scale, and within those calculations reports Re values for Mg2+-He in good agreement with the other values. A nonstandard approach to calculating dissociation energies yields a D0 value that is somewhat lower than the value obtained herein. For Mg2+-Ne, similar comments to the above hold for the QCISD(T) results29 and for those from Petrie.32 For Mg2+-Ar, Petrie’s values32 seem to be the only ones available, and again the Re value is in reasonable agreement with that obtained herein, whereas the D0 value is somewhat low. B. Trends. In previous work on Ca+-Ra+ complexed with RG atoms,1,2 we have noted that there are unusual trends in the spectroscopic parameters, which were attributed to sd mixing. Such mixing causes the electron density on M+ to move offaxis, away from the incoming RG atom, reducing electron repulsion and hence allowing a closer approach, and so a synergistic increase in the attractive terms: especially the dispersion terms that appear to be very important in these species. This effect was particularly noteworthy in Ba+ owing to the low-lying 5d orbitals. In the cases of Be+ and Mg+, the lowest-lying d orbitals lie above the lowest-lying p orbitals and so, as the plots in Figure 1 show, the main effect here is the movement of electron density away from the incoming RG atom, but on-axis, owing to sp mixing. These plots also indicate that there are small amounts of charge transfer as the atomic number of the RG increases, although the multireference character is still very small in these species, as indicated by the T1 diagnostic52,53 values, which were 1, with an excepe tionally high value for Be+-Ne. Looking at the Birge-Sponer plots for the two heavier species, we see that there is a “convex” portion of the curve in the mid-V region, before the curve reverts back to a shallowing of the slope at high-V. The plot for Be+-Ne is highly unusual, having an extended “S” shape, similar to an ogive, with a shallow slope at low V (which gives rise to the low value of ωexessee Table 1, and Vide supra), followed by a steepening of the curve, before reverting to a shallow slope at high V in line with the heavier Be+-RG species. The ωe and ωexe values determined from the lowest few levels yield a high /De value, since the DMorse value is obtained assuming a DMorse e e linear Birge-Sponer plot following the behavior of the lowest few levels: this is clearly far from the case, and we shall return to this below. /De ratios For the dication species, the majority of the DMorse e in Tables 2 and 4 are close to unity, consistent with the linearity

of the Birge-Sponer plots over a wide range of V, with deviation to positive curvature at high V. For some of the heavier Be2+-RG species, the numbers are rather low, and this seems to be linked to curve crossings and charge transfer at long R. This is the subject of ongoing work involving multireference calculations, as noted above. D. Reduced Potential Plots and the K Parameter. The use of reduced parameter plots has been employed5,16 in comparing species with very different interaction potential well depthsssuch plots are argued to reveal relatiVe differences between species more obviously. Reduced potential plots have been examined by many workers58,59 with a view to examining whether a universal diatomic function exists.60,61 Reduced potentials are obtained by plotting E/De vs R/Re for each species, and this is done for Be+-RG in Figure 4 and for Mg+-RG in Figure 5. As may be seen, for Mg+-RG, the curves are nicely clumped together, confirming that interactions in these species are similar in nature. For Be+-RG, again for RG ) Ar-Rn, the curves are clumped closely together, but for Be+-Ne the reduced potential is clearly very different from the others; the other exception is Be+-He, which, although being close to the RG ) Ar-Rn curves, is slightly steeper throughout. In fact, if one looks at the Mg+-RG curves more closely, one sees similar behavior for the Mg+-He and Mg+-Ne curves, but it is nowhere near as marked. Another way of getting a handle on the difference in interactions is via the κ parameter16 (which is essentially the Sutherland parameter put forward in 1938,62 and has been denoted ∆ in recent work by Xie and Hsu61). This has been

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Gardner et al.

Figure 6. Plot of κ for the titular complexesssee text for details.

Figure 5. Reduced potentials for Mg+-RGssee text for details. The solid bold line is the curve for Mg+-He; the dashed bold line is the one for Mg+-Ne. The other curves are all very close to each other.

termed the “reduced curvature of the potential at Re”,16 and is defined as:

( )

Re2 d2V k) De dR2

(1)

Re

which may be expressed as

κ ) ωe2/2BeDe

(2)

where ωe, Be, and De are defined in the caption to Table 1. As discussed in ref 5, smaller values of κ indicate a “softer” potential near the minimum. Winn16 presents κ values for a range of species, and it is found that for strongly bound chemical species, κ values are between 0-20, whereas for Be+-Ar and Be+-Kr, the values were determined to be κ ) 27.5 and 29.1, respectively. It was initially expected62 that such values should be constant for particular groups of complexes; however, that assertion was based on a rather simple form of interaction potential. One might, however, expect there to be similar values of κ for similar species, and that there may be trends in such values for similar species. In Figure 6, we plot the κ values for all of the titular complexes, using the spectroscopic values reported herein. First, note that κ is not constant, but the values do lie close to each other for the same series of M+-RG or M2+-RG. The M2+-RG complexes have very similar trends, with a decreasing κ value from He to Rn, with the exception of the small increase from M2+-He to M2+-Ne. Interestingly, very similar trends are seen for the complexes between an alkali metal cation (Alk+)

and a RG atom, where we have selected Li+-RG and Na+-RG, as they are isoelectronic with the dication species. To calculate the κ values for these species, we have used (or derived) the relevant spectroscopic quantities from the data presented in refs 63–65. Turning now to Mg+-RG, again the values of κ are similar to each other, but this time they are increasing with the atomic number of the RG atom, in contrast to the values for the Be+-RG complexes. The most striking feature about the κ values, however, is the exceptionally low value for Be+-Ne compared to the other Be+-RG complexes; except for that one, the rest of the Be+-RG values seem to behave similarly to the Alk+-RG or M2+-RG complexes. If one now considers again the Mg+-RG species, the increasing κ value with the atomic number of RG would be consistent with the potential becoming “harder” as the atomic number of the RG increases. In contrast, the slope of the Be+-RG κ line is decreasing with increasing atomic number. To gain some insight into this, we examine the HOMO plots in Figure 1 (for Be+-RG), where it can be seen that from RG ) Ar onward, these species have significantly reduced electron density along the internuclear axis; that is, the RG atom is able to interact more strongly with the dicationic core. Thus, it is perhaps not so surprising that these complexes have κ trends more similar to the M2+-RG complexes, than to the other M+-RG complexes. That the HOMO plots show that this drop in on-axis electron density happens very sharply between Ne and Ar is completely consistent with the large rise in κ at this point. We have already noted that for M+-RG the lowest unoccupied atomic orbitals on Be+ and Mg+ are p orbitals, and that the unpaired electron density moves into an spσ mixed orbital as the RG atom approaches. For Be+, however, the charge density is very high, and therefore it is not surprising that in this case the interaction of the RG atom with the dicationic core is strong enough that it allows even more distortion of the 2s orbital to allow this interaction to occur better, with the sharp increase in κ resulting (as discussed above) because the polarizabilities of Ar-Rn are significantly higher than those of He and Ne. That the potentials then generally become softer with increasing atomic number for the dications is likely attributable to the increasingly diffuse nature of the RG outershell orbitals. Although Be+-He “has about the right value of κ”, when compared to the other complexes, the value for Be+-Ne is very low. We believe the following is a reasonable rationale for this observation. We know that the 2s electron density on Be+ moves away from the pσ orbital of the approaching RG atom, and does this by the mixing in of 2pσ character. This movement of Be+ electron density away from the approaching RG atom “softens”

Beryllium and Magnesium Cationic Complexes the potential, leading to a lower κ value. The suggestion is, therefore, that this softening is significantly more efficient, in a relative sense, in the case of Be+-Ne than it is for the other M+-RG species considered here. We believe this is a juxtaposition of two effects: (i) the 2pσ orbital of Ne has a higher electron density than the corresponding orbitals of the heavier RG atoms, allowing for a stronger relative directional interaction with the Be+ electron density; (ii) in the case of Be+-Ne, subtle changes in the repulsion terms as a function of R are more obvious because of the small total interaction energy; whereas for the heavier species the much larger polarizability means the attractive terms, and the resulting sp mixing on Be+ to reduce repulsion (Vide supra), are much larger, and hence such subtle changes in the repulsive potential are not so obvious. Point (i), combined with the small size of Ne and in the light of the HOMO plots in Figure 1, suggests that the 2pσ orbital of Ne may be able to penetrate partially into the nodal region of the 1s orbital of Be+. Indeed, the unusual form of the Be+-Ne potential likely arises from the fact that the Ne(2pσ) orbitals begin to sample the nodal region of the Be+ 2s orbital at small R, resulting in less relative repulsion near Re. One might wonder why Be+-He has a higher κ value than Be+-Ne: we believe this is due to the fact that the 2s electron density in He is spherical whereas that in the 2pσ orbital of Ne is much more directional, and so there is a higher directed charge density for Ne than for He. Wording this in reverse, the “harder” potential of Be+-He relative to Be+-Ne complexes (see κ value, and reduced potential plot in Figure 2) is likely due to the fact that the softening effect on the potential from Be+ s/p mixing in Be+-Ne is not present in Be+-He because of the isotropic nature of the He 1s orbital. The difference in behavior between Be+-RG and Mg+-RG (aside from the markedly different Be+-Ne case) is that there is a general increase in κ for the latter case, but a decrease for the former. We have already noted in the above that the HOMO plots in Figure 1 demonstrate that a large proportion of the Be+ 2s electron density moves away from the approaching RG atom, and the κ variation very much resembles that of the dications. For the larger Mg+, however, there is still significant electron density between the Mg2+ core and the approaching RG atom. As the atomic number of RG increases, so do the electron repulsion terms, and in this case they appear to rise relatively faster than the attraction terms, giving an increasingly harder repulsive potential. That the κ values of Mg+ are higher than those of Be+, can be attributed to the fact that Mg+ has p electrons, and so there is additional pπ-pπ repulsion for RG ) Ne-Rn compared to Be+; and similarly for the dications. Indeed, this effect is likely contributing to the fact that the HOMOs in Mg+-RG at Re are less distorted than those for Be+-RG. The lack of the anomalous behavior for Mg+-Ne, as seen for Be+-Ne, also likely lies in the pπ-pπ repulsion in the former case, which may prevent the Ne from penetrating far enough into the 3s orbital to experience the first nodal region. E. Mobility Results. The numerous results have been placed in the database maintained at Chatham University.66 We show the calculated mobilities at 300 K as a function of E/n0 in Figures 7 and 8 for Be+ in RG and Mg+ in RG. There appear to be no experimental nor previous theoretical data to which to compare, and so these present mobilities, and other data deposited in the Chatham database, will hopefully be useful to future workers interested in the gaseous transport of these ions. The first approximation of the two-temperature theory of gaseous ion transport67 predicts that the mobility is inversely

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Figure 7. Calculated standard mobilities for Be+ in a bath of RG (RG ) He-Rn).

Figure 8. Calculated standard mobilities for Mg+ in a bath of RG (RG ) He-Rn).

proportional to the square root of the ion-neutral reduced mass. This means that the biggest difference between mobilities of different isotopes of the same ion should occur when the RG

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Figure 9. Graph of Krel 0 , the ratio of the calculated standard mobility for the indicated isotope of Mg+ to the calculated standard mobility of 26 Mg+, vs E/n0. Note that the broad wobbles are not significant and arise as the order of approximation changes as a result of the changing demands of meeting convergence criteria at each E/n0.

has approximately the same molar mass; in the case of Mg+ ions, the biggest mobility difference should occur in Ne. Higher approximations of the two-temperature theory and even the first approximations of more elaborate theories, such as the Gram-Charlier theory68 upon which program GC is based, indicate that the differences between the ion mobilities of different isotopes should vary with E/n0. To test the significance of this, we show in Figure 9 the ratios of the standard mobilities of the various Mg+ isotopes to that of 26Mg+. The differences at very low E/n0 are essentially those predicted from the reduced masses: 24Mg+ and 25Mg+ are 1.80 and 0.87% above 26Mg+, respectively. However, these differences increase slightly as E/n0 increases and then drop to values approximately half the size predicted from the reduced masses alone; the slight variations seen from smooth behavior are reflections of the limited precision (0.1%) to which each set of standard mobility values was determined. Figure 9 shows that the mobilities of the three isotopes of Mg+ in Ne differ by ∼1%, where the expected trend of the heavier isotope having the lower mobility is seen. It is likely that deviations from the usual reduced mass scaling will only be significant in experiments that are of a higher precision than those generally undertaken. 5. Conclusions We have calculated accurate ab initio PECs for the Ben+-RG and Mgn+-RG complexes, for n ) 1 and 2. These lead to excellent agreement with the very precise spectroscopic constants obtained from high resolution emission spectra for

Gardner et al. Be+-Ar and Be+-Kr, and there is good agreement also with the results of the lower resolution emission spectra for Be+-Xe. The most surprising results are those for Be+-Ne, which indicate that it has a potential with a very different relative curvature to the other species, something which is clearly seen in both the reduced potential plots, and also in the κ parameter. There are no experimental results to which to compare the calculated spectroscopic parameters of Be+-He, Be+-Ne, and Be+-Rn, although good agreement is seen with the results of previous calculations for the lighter two species. The reduced potential plots are very similar for the heavier Be+-RG complexes, but Be+-Ne appears to be “softer”, and Be+-He appears to be “harder” than these. We have interpreted the former in terms of the 2pσ orbital of Ne having a high electron density, and so being able to interact strongly and effectively with the Be+ 2s electron density, and leading to enhanced relative sp mixing, moving electron density away from the incoming Ne; additionally, we have suggested that the unusual form of the reduced potential and the Birge-Sponer plots, which are markedly different in shape to the other species, could be caused by the experiencing of the nodal region of the Be+ 2s orbital by the approaching Ne 2pσ orbital. For the heavier RG atoms, the larger polarizabilities lead to the subtleties in the repulsive terms being overwhelmed by the attractive ones; for He, the effect is reduced, owing to the lack of p orbitals, together with the lower charge density in the spherical He 1s orbital relative to the directional Ne 2pσ orbital. The results for Mg+-RG are much more straightforward: where reliable experimental values are available, good agreement is seen. We note that the dissociation energies derived from photoionization experiments appear to be in better agreement with the present values than are those from photodissociation experiments. The reduced potential and κ values indicate that the Mg+-RG potentials are all very similar to each other, particularly when R is close to Re. The potentials of the magnesium complexes are harder than those of the corresponding beryllium ones, owing to the presence of pπ-pπ repulsion in the former; this also explains the nonanomalous behavior of Mg+-Ne relative to Be+-Ne. Finally, we have presented mobilities for Be+ and Mg+ in each of the rare gases for the first time and examined the dependence of the ratio of the standard mobilities of different isotopes of Mg+ in Ne; this dependence is found to be small. Acknowledgment. The work of L. A. V. was supported by the National Science Foundation under Grant CHE-0718024. T. G. W. is grateful for the provision of computing time under the auspices of the NSCCS. The EPSRC is also thanked for funding, and A. M. G. and C. D. W. are grateful to the EPSRC and the University of Nottingham for the provision of studentships. Professors Michael Morse and Jack Simons (Utah) are both thanked for useful discussions. Supporting Information Available: The supplementary data consists of 24 files containing the potentials and one file containing additional Tables. The potentials are given in Ångstrom and Hartree, and the files are obviously named, and cover all of the titular complexes. The additional tables have the RCCSD(T)/d-aug-cc-pCVQZ and RCCSD(T)/d-aug-ccpCV5Z results for the spectroscopic constants, and in the case of Be+-NE/Ne, there are also the CASSCF+MRCI+Q values. This information is available free of charge via the Internet at http://pubs.acs.org.

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