ARTICLE pubs.acs.org/JPCA
Theoretical Study of MþRG Complexes (M = Ga, In; RG = HeRn) Adrian M. Gardner, Kayla A. Gutsmiedl,† and Timothy G. Wright* School of Chemistry, University of Nottingham, University Park, Nottingham NG7 2RD, U.K.
Edmond P. F. Lee School of Chemistry, University of Southampton, Highfield, Southampton SO17 1BJ, U.K.
W. H. Breckenridge Department of Chemistry, University of Utah, Salt Lake City, Utah 84112, United States
Shristi Rajbhandari, Chivone Y. N. Chapman, and Larry A. Viehland Science Department, Chatham University, Pittsburgh Pennsylvania 15232, United States
bS Supporting Information ABSTRACT: We present potential energy curves calculated at the CCSD(T) level of theory for GaþRG and InþRG complexes (RG = HeRn). Spectroscopic parameters have been derived from these potentials and compared to previously calculated parameters for the AlþRG and TlþRG complexes. Additionally, for some cases, we compare these parameters with those obtained from electronic spectroscopic studies on excited states of the neutral species, arising from atomic-based d r p excitations. The GaþRG and InþRG potentials have also been used to calculate the transport coefficients for Mþ traveling through a bath of RG atoms.
1. INTRODUCTION In recent years, there has been significant interest in the weakly bound MþRG (RG = HeRn) complexes, as they are prototypical systems to study fundamental electrostatic interactions.1 The complexes of the alkali metal cations, Alkþ, with the RGs have been shown to follow a model based on purely electrostatic interactions,2 whereas the same approach fails dramatically when applied to the complexes of open-shell alkaline earth metal cations, AlkEþ, with rare gas atoms.3 Combined with this, anomalous trends in the equilibrium bond length, Re, were observed with increasing RG atomic number for the AlkEþRG complexes.46 This was in contrast to the expected trend of increasing Re with RG atomic number as observed in the AlkþRG complexes, in line with the increasing van der Waals radii of the RG atoms.710 The anomalous trend is a decrease in Re along the AlkEþHe, AlkEþNe, and AlkEþAr series, after which an (expected) increase in Re is observed for AlkEþKr, AlkEþXe, and AlkEþRn. A marked increase in the dissociation energy, De, between the AlkEþNe and AlkEþAr complexes has also been observed, most noticeably for the BeþRG4 and BaþRG6 complexes. The intriguing trends in Re and De observed in the AlkEþRG complexes have been rationalized as an increase in the degree of sd mixing in the heavier metal complexes r 2011 American Chemical Society
(Caþ, Srþ, Baþ, and Raþ)5,6 with increasing RG atomic number and sp mixing within the BeþRG and MgþRG complexes.4 Very recently, we have extended these series of studies to the AlþRG complexes,11 which are closed shell, unlike the corresponding AlkEþRG complexes. However, a decreasing trend in Re was also observed for the first members of the AlþRG series, RG = He, Ne, Ar, and Kr, with Re increasing slightly on moving to the AlþXe and AlþRn complexes. Despite this, De was found to increase with increasing RG atomic number approximately in line with that expected owing to the increasing polarizability of the RG. This is in contrast to the trends previously observed in the TlþRG complexes,12 in which a slight decrease in Re between the TlþHe and TlþNe complexes was observed, after which there was a monotonically increasing trend in Re. The aim of the present study is to extend these studies to the GaþRG and InþRG complexes. Both gallium and indium cations are involved in chemical vapor deposition (CVD) and/or Special Issue: J. Peter Toennies Festschrift Received: December 23, 2010 Revised: March 7, 2011 Published: March 24, 2011 6979
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The Journal of Physical Chemistry A analysis of substances formed by the CVD process; consequently, understanding the interactions of the atomic cations with the rare gases,13,14 and hence the movement of these cations, is important in precise modeling of such processes. Potential energy curves, PECs, have been calculated at the CCSD(T) level, utilizing basis sets of (approximately) augmented quintuple-ζ quality. We have used such methods in our previous work, where tests against mobility and spectroscopic data have shown that PECs obtained at these levels of theory are highly reliable.410 (This comment applies to all-electron calculations, and those where “small-core” effective core potentials have been used; often the latter have necessitated augmentation of standard basis sets with tight basis functions to describe the outer-core/valence correlation.) Spectroscopic parameters have been determined from these potentials and are compared to the only previous reports of calculations on the GaþRG and InþRG complexes: configuration interaction (CI) calculations by Dunning et al.15 for GaþAr and the results of a more recent CASSCF and MRCI study of GaþAr by Park et al.16 In addition, we shall compare the spectroscopic parameters we obtain for GaþRG to those obtained experimentally for excited states of GaRG obtained by Stangassinger et al.,17 and our previously calculated results for the TlþRG complexes will be compared to those obtained, again experimentally, for the excited states of TlRG.18 Finally, we shall discuss trends in the values of the spectroscopic quantities in group 13 and present results on the transport coefficients.
2. COMPUTATIONAL METHODOLOGY For HeAr, the standard aug-cc-pVQZ and aug-cc-pV5Z basis sets1921 were employed; for KrRn 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.22 For Ga and In, the relativistic effective core potentials ECP10MDF and ECP28MDF were utilized,23 respectively, together with the valence aV5Z-PP basis sets.24 All calculations were carried out using the CCSD(T) procedure as implemented in MOLPRO.25 For Ne only the 1s electrons were frozen, while for Ar the 1s2s2p electrons were frozen. For all calculations involving He, Ne, and Ar, all of the metal electrons that were not described by the ECP were explicitly included in the correlation treatment. To describe the correlation of the outer core electrons, additional “tight” functions were incorporated into the metal basis set. For the 3s, 3p and 3d electrons of Ga, four s functions (ζ = 31.7952, 13.248, 5.52 and 2.3); four p functions (ζ = 6.0, 3.0, 1.5 and 0.375); four d functions (ζ = 14.0, 7.0, 3.5 and 1.75); three f functions (ζ = 15.0, 5.0 and 1.66667); two g functions (ζ = 9.6 and 2.4); and two h functions (ζ = 8.0 and 2.0) were added to the aV5Z-PP basis set. For the 4s, 4p and 4d electrons of In, four s functions (ζ = 15.7437, 7.497, 3.57 and 1.7), three p functions (ζ = 8.47, 3.85 and 1.75), four d functions (ζ = 8.8, 4.4, 2.2 and 1.1), three f functions (ζ = 5.0, 2.0 and 0.8), two g functions (ζ = 4.9 and 1.4), and two h functions (ζ = 6.0 and 1.5) were added to the aV5Z-PP basis set. The outermost p orbitals of Kr, Xe, and Rn are lower in energy than each of the 3s3p and 4s4p orbitals of Gaþ and Inþ, respectively, and therefore in these cases, only the valence RG electrons (and the Ga 3d4s4p and In 4d5s5p electrons) were included in the correlation treatment. This means that the
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additional tight s and p basis functions described above for Ga and In are not required for calculations involving the heavier RGs, and thus for these calculations only the 4d3f2g2h additional sets of functions were added to the standard aV5Z-PP metal basis sets. The full counterpoise correction was applied at each of about 50 nuclear separations, R. Wide ranges of R (0.620.0, 0.920.0, and 2.120.0 Å for HeAr, respectively; and 1.620.0 Å for KrRn) were covered to include the short-, medium- and longrange regions. The rovibrational energy levels for each potential energy curve were obtained using the LEVEL26 program and the lowest three calculated rotational or vibrational energy levels were leastsquares fitted to standard expressions to obtain the reported spectroscopic quantities. The most abundant naturally occurring isotope of each element was used in all cases (69Ga, 113In, 4He, 20 Ne, 40Ar, 84Kr, 132Xe, and 222Rn). We calculated the transport cross sections for Mþ in each RG from the ab initio interaction potential energy curves as functions of the ionneutral collision energy using the classical-mechanical program PC27 that is an improved version of the earlier program QVALUES.28,29 The cross sections converged within 0.1% for He, 0.25% for NeKr, 0.8% for Xe and 0.9% for Rn. The cross sections as a function of collision energy were used in the program GC28,30,31 to determine the standard mobility, K0, 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. Calculations were performed for each of the naturally-occurring isotopes of Gaþ and Inþ, but in all cases the rare gas was assumed to be the naturally occurring mixture of isotopes. The calculated mobilities are generally precise within the precision of the cross sections at E/n0 values below 100 Td (1 Td = 1021 V m2). The results are progressively less precise as E/n0 increases to 1000 Td, but these details, as well as the mobilities and other transport properties, can be obtained from the tables placed in the database that has recently been moved from Chatham University to a Web site maintained from the University of Toulouse.32
3. RESULTS AND DISCUSSION The calculated RCCSD(T) extrapolated potential energy curves for each of the species are available as Supporting Information. In Table 1 are presented the calculated spectroscopic parameters for the GaþRG complexes, from the present work. Also shown are what appear to be the only previously reported results for these cations: the CASSCFþMRCI results for GaþAr16 and the CI results for GaþKr.15 As may be seen for GaþAr, the agreement between the present results and the CASSCFþMRCI ones is extremely poor, with the previous De value being only about 50% of the present value, and the ωe value being significantly smaller also. It could be that there is not enough dynamic correlation in the previous calculation, as the basis set employed there was quite large; even so, the discrepancy is far larger than one might have anticipated. On the other hand, the polarized CI (POL-CI) calculations of Dunning et al.15 for GaþKr are in much better agreement with the present calculations, although the absolute differences are still significant. The large basis sets employed in the present work, and our use of the CCSD(T) method, would suggest that the present results are the more reliable. 6980
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Table 1. Calculated Spectroscopic Constants for GaþRG (RG = Rare Gas)a Re (Å)
De (cm1)
D0 (cm1)
ωe (cm1)
ωexe (cm1)
Be (cm1)
R (cm1)
DMorse (cm1) e
DMorse /De e
k (N m1)
κ
GaþHe
3.189
117.3
85.5
66.7
0.443
0.0845
107
0.91
0.992
42.9
GaþNe
3.098
267.1
241.8
52.1
2.90
0.113
0.00632
234
0.88
2.48
44.8
GaþAr
3.094
1070.7
1033.5
75.2
1.44
0.0697
0.00138
981
0.92
8.43
37.9
3.37b
584b
0.906
0.0459
0.000594
1490
0.93
12.0
36.6
GaþKr
10.4
51b
3.116
1603.5
3.14c
1900c
1567.0
73.4
GaþXe
3.204
2411.9
2372.3
79.5
0.697
0.0363
0.000332
2270
0.94
16.9
36.1
GaþRn
3.224
2972.8
2932.8
80.7
0.559
0.0309
0.000232
2910
0.98
20.2
35.4
83c
a
The quantities refer to the following isotopes: 69Ga, 4He, 20Ne, 40Ar, 84Kr, 132Xe, and 222Rn. Re is the equilibrium bond length, De is the depth of the potential, D0 is the difference in energy between the zero-point and the Gaþ(1S) þ RG(1S) 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, R is the spin-rotation constant, and κ is the reduced curvature value. b CASSCFþMRCI calculations from ref 16. c CI calculations from ref 15.
Figure 1. Standard mobilities of Gaþ in each of the rare gases at 300 K.
Figure 2. Standard mobilities of Inþ in each of the rare gases at 300 K.
It is also of interest to compare the present results to the experimentally determined parameters for excited state neutral GaRG complexes using photoionization spectroscopy. Stangassinger et al.17 vaporized a gallium antimonide alloy rod, and mixed the metal vapor with an inert gas, or mixture of inert gases, before expanding into vacuum. The subsequent supersonic jet expansion led to the formation of GaRG complexes, which were internally cold. These were probed by photoionization using a two-color resonance-enhanced multiphoton ionization (REMPI) scheme, leading to the production of an electronic
absorption spectrum measured by means of the production of ions. Of interest here are the results obtained for the electronic transitions that correspond to the atomic 4d r 4p excitation. The Ga(42P1/2,3/2) þ RG asymptotes can give rise to 2Π1/2,3/2 and 2Σ1/2þ states, with the ground state being the X2Π1/2 state. The Ga(42D3/2,5/2) þ RG asymptote can give rise to 2Δ3/2,5/2, 2 Π1/2,3/2, and 2Σ1/2þ states, with the lowest energy states expected to be 2Δ3/2,5/2. In Figure 1 of ref 17 are shown schematics of the orientation of the upper state 4d orbitals with respect to the internuclear axis, together with the outer s or p 6981
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Table 2. Comparison of Spectroscopic Constants for GaRG and GaþRG (RG = Ar, Kr, and Xe) state
D0 (cm1)
ωe (cm1)
ωexe (cm1)
Table 3. Comparison of Spectroscopic Constants for TlRG and TlþRG (RG = Ar, Kr, and Xe) state
D0 (cm1)
ωe (cm1)
ωexe (cm1)
GaAr(2Π1/2)
176
38.8
1.93
TlAr(2Π1/2)
246
41.7
1.63
GaAr( Π3/2) GaAr(2Δ3/2)
199 793
44.3 71.2
2.22 1.53
TlAr(2Π3/2) TlAr(2Δ3/2)
297 725
44.4 62.2
1.54 1.28
GaAr(2Δ5/2)
869
73.2
1.48
TlAr(2Δ5/2)
746
61.5
1.22
Ga Ar( Σ )
1033.5
75.2
1.44
TlþAr(1Σþ)
919
62.5
1.20
GaKr(2Π1/2)
366
45.1
1.31
TlKr(2Π1/2)
449
38.7
0.80
GaKr(2Π3/2)
379
46.3
1.33
TlKr(2Π3/2)
456
40.1
0.84
GaKr( Δ3/2)
1294
71.0
0.95
TlKr(2Δ3/2)
1125
54.5
0.65
GaKr(2Δ5/2)
1308
71.7
0.96
TlKr(2Δ5/2)
1130
54.0
0.63
GaþKr(1Σþ) GaXe(2Π1/2)
1567.0 743a
73.4 53.7a
0.906 0.94a
TlþKr(1Σþ) TlXe(2Π1/2)
1295.3 790
53.4 41.8
0.62 0.54
GaXe(2Π3/2)
898a
57.1a
0.88a
TlXe(2Π3/2)
a
a
2
þ
1 þ
2
933
39.6
0.41
GaXe( Δ3/2)
2442
86.0
0.74
TlXe(2Δ3/2)
1927
57.0
0.42
GaXe(2Δ5/2)
2508a
85.5a
0.72a
TlXe(2Δ5/2)
1871
57.2
0.43
GaþXe(1Σþ)
2372.3 (2412)b
80.7 (79.8)b
0.697 (0.70)b
TlþXe(1Σþ)
1870.7
52.8
0.41
2
a
a
Experimental values are for 69Ga129Xe. b The parentheses contains the values for the 69Ga129Xe isotopologue, for which the experimental values were reported.
shells of the approaching RG atom. It is clear that the isotropic RG outer s or p shells can interact with the Gaþ core more easily for the 2ΔΩ states than they can for the 2ΠΩ states and, further, that the interaction for the 2Σþ state is expected to be weak. We thus expect the interaction energies, in the absence of any other state mixings, to be in the order 2Δ > 2Π > 2Σþ. Of course, the spinorbit interaction can lead to states of the same Ω interacting, and this will lead to perturbations in the potential energy curves. The state for which such mixing will be essentially absent is the 2Δ5/2 state, as there are no Ω = 5/2 states close in energy; we thus expect this state to be “pure-δ”. Given the orientation of the orbitals, we expect the dissociation energy for the 2ΔΩ states to be most like the cation, GaþRG. In Table 2 we collect the spectroscopically derived D0 values for the GaAr, GaKr, and GaXe excited state neutrals, together with the present calculated values for the cations. As may be seen, the similarity between the dissociation energies of the 2Δ5/2 state and those calculated here for the 1Σþ states of the cation is reasonably good, supporting the physical picture presented above. Additionally, it is clear that the 2ΠΩ states are much more weakly bound than the 2 ΔΩ ones (the 2Σþ ones have not been observed and have been hypothesized as being either very weakly bound or perhaps dissociative). It is also noteworthy that the 2Δ5/2 state is the most strongly bound, with the 2Δ3/2 state presumably mixing with the higher 2Π3/2 state, which could explain the slight lowering of its D0 value. As mentioned above, we have previously presented potential energy curves and derived spectroscopic parameters for the TlþRG complexes.12 In Table 3, we present a summary of this spectroscopic data, together with the experimental spectroscopic data obtained by Stangassinger et al.18 using laser-induced fluorescence (LIF) on the neutral TlRG states. In that work, the complexes were formed in a jet expansion, as for GaRG, but the thallium vapor was produced via electric discharge. As may be seen from the values in Table 3, a very similar story may be told as for GaþRG, with the 2ΠΩ states being significantly more weakly bound than the 2ΔΩ ones, as expected from the
orientation of the d orbitals. Further, the 2Δ5/2 state is the most strongly bound, as anticipated, owing both to the relatively clear view that the RG atom has of the Tlþ nuclear charge and to the fact that that it is a “pure-δ” state, since there are no other Ω = 5/2 states nearby, with which it could mix. As before, we hypothesize that there is mixing in of 2Π3/2 character into the 2Δ3/2 state, leading to its being slightly less strongly bound than the corresponding 2Δ5/2 one. It is generally true that the dissociation energies of the 2Δ5/2 neutral states are closest to (but lower than) that of the cationic states for both gallium and thallium cases. For the TlXe and GaXe states, this general statement does not hold, with the 2Δ states apparently being more strongly bound than the cation. This might be rationalized by noting that, because of the higher binding energy of Xe complexes, the interactions may be complicated by interloping states arising from higher asymptotes; such has been discussed recently by some of us in the case of the AuXe complex.33,34 However, some caution is merited here for the GaþRG complexes since the errors in the experimental dissociation energies17 appear to be ca. 80100 cm1. (For TlRG the errors were not given18 but likely are about the same order of magnitude.) This will be discussed further below in relation to similar MgXe 3Δ states. It should be noted that, formally, the electrostatic interaction in the 2Δ states is mainly quadrupole/induced-dipole, where the quadrupole moment on the M atom is extremely large due to the very diffuse 3dδ orbital. Also, simple, direct atomic/molecular spinorbit coupling should be of minor importance in the multiplet splittings in these excited valence Δ and Π states, because the atomic nd orbitals do not penetrate well to the nuclei: even for the “heavy-atom” Tl case, the J = 3/25/2 splitting is only 82 cm1; for Ga, it is only 6 cm1.35 For the AlRG cases, only for AlAr have the 3d 2Δ and 2Π states been characterized.36 The spectroscopic parameters follow the same general trends as for the GaAr and TlAr states, in that the Δ states are much more strongly bound than the Π states, and are less bound than the AlþAr ion. Similar D0 trends to those of Tables 2 and 3 have been observed for the 3d 3Δ1 and 3d 3Π0 excited valence states of 6982
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Table 4. Calculated Spectroscopic Constants for InþRG (RG = Rare Gas)a Re (Å)
De (cm1)
D0 (cm1)
ωe (cm1)
ωexe (cm1)
Be (cm1)
R (cm1)
DMorse (cm1) e
DMorse /De e
k (N m1)
κ
InþHe
3.396
99.3
71.3
58.5
9.42
0.383
0.0754
0.91
0.778
44.9
InþNe
3.288
231.3
209.2
45.5
2.57
0.0918
0.00520
202
0.87
2.08
48.7
InþAr
3.307
889.5
858.7
62.3
1.21
0.0520
0.00101
802
0.90
6.80
42.0
InþKr
3.344
1298.3
1269.9
57.2
0.691
0.0311
0.00038
1190
0.91
9.37
40.6
InþXe
3.435
1926.0
1896.3
59.8
0.501
0.0233
0.00020
1780
0.93
12.9
39.9
InþRn
3.448
2367.2
2337.8
59.0
0.394
0.0187
0.00013
2210
0.93
15.5
39.2
90.7
a
The quantities refer to the following isotopes: 113In, 4He, 20Ne, 40Ar, 84Kr, 132Xe and 222Rn. Re is the equilibrium bond length, De is the depth of the potential, D0 is the difference in energy between the zero-point and the Inþ(1S) þ RG(1S) 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, R is the spin-rotation constant, and κ is the reduced curvature value.
Table 5. Re Values (Å) for the MþRG Complexesa
Table 6. De Values (cm1) for the MþRG Complexesa
RG
Alþ
Gaþ
Inþ
Tlþ
RG
Alþ
Gaþ
Inþ
Tlþ
He
3.357
3.189
3.396
3.28
He
95.1
117.3
99.3
119.0
Ne
3.183
3.098
3.288
3.23
Ne
225.6
267.1
231.3
263.1
Ar
3.122
3.094
3.307
3.32
Ar
1040.6
1070.7
889.5
919.0
Kr Xe
3.109 3.182
3.116 3.204
3.344 3.435
3.38 3.48
Kr Xe
1561.1 2429.0
1603.5 2411.9
1298.3 1926.0
1295.3 1870.7
Rn
3.192
3.224
3.448
3.50
Rn
3027.9
2972.8
2367.2
2262.0
a
Values for AlþRG are from ref 11, values for GaþRG and InþRG are from the present work, and values for TlþRG are from ref 12.
Values for AlþRG are from ref;11 values for GaþRG and InþRG are from the present work; and values for TlþRG are from ref.12
MgAr, MgKr, and MgXe by Breckenridge and coworkers,3739 who also determined D0 accurately for MgþAr, MgþKr, and MgþXe by two-color photoionization threshold measurements40,41 in the same apparatus. The D0 values for the 3 Δ states are all much greater than those of the 3Π states, and approach (or exceed, for Xe; see below) the D0 values of the corresponding MgþRG ground state ions. In the MgXe case, the difference in D0 values for the 3Δ neutral state (larger) and the MgþXe ion was determined accurately to be 312 ( 10 cm1. A reliable value was possible because low v0 values for the Δ state were directly accessed spectroscopically down to v0 = 1 from the same MgXe (3Π; v = 0) metastable state from which the two-color photoionization threshold to produce MgþXe was measured accurately. This unusual result was rationalized tentatively by the authors39 as being due to an “extra” dispersive C6 attraction between the diffuse 3dδ orbital, beyond the mostly “Mgþ/Xe”-type core interaction in the 3Δ state. For InþRG, we present the spectroscopic parameters in Table 4. To our knowledge, there are no previous values for these species (nor for the neutral InRG states) to which to compare; consequently, we shall only refer to these in the below when examining trends in the group IIIA cation/RG spectroscopic parameters. A. Trends in the Spectroscopic Parameters with Increasing RG Atomic Number. The spectroscopic parameters determined from the PECs for each GaþRG and InþRG complex are shown in Tables 1 and 4, respectively. A summary of the Re and De values presented in Tables 5 and 6, together with those from our previous work on AlþRG11 and TlþRG.12 There is a trend of increasing De with increasing RG atomic number for both metals. The ratios of successive De values are 2.3, 4.0, 1.5, 1.5, and 1.2 for the GaþRG complexes and 2.3, 3.8, 1.5, 1.5, and 1.2 for the InþRG complexes, which are in close
Table 7. WrightBreckenridge Radii (RWB) of Mþ Ions and De Values for the MþHe Complexes (Details in Text)
a
Mþ
De(MþHe) (cm1)
RWB (Å)
Alþ
95
1.87
Gaþ
117
1.70
Inþ Tlþ
99 119
1.91 1.79
agreement with the ratios of the RG polarizabilities of 1.9, 4.1, 1.5, 1.6, and 1.3. This is in line with expectations deduced from the AlþRG series that it is the charge/induced dipole interaction that is the dominant attractive term in the potential energy function.11 The corresponding ratios for TlþRG are 2.2, 3.5, 1.4, 1.4, and 1.2, allowing a similar conclusion to be reached in that case. As observed in the AlkEþRG series,46 as well as for the AlþRG11 and TlþRG12 complexes, Re initially decreases as the RG atomic number increases, before the expected monotonically increasing trend is observed. The minimum in Re occurs at different places for the different metals, being at argon for all AlkEþRG complexes, while in the group IIIA metal cation RG complexes it occurs at AlþKr, GaþAr, InþNe, and TlþNe. The rationale for these observations is that there is a subtle balance between the increasing attractive terms as the RG becomes heavier and more polarizable, and the increasing repulsive terms owing to the increasing number of electrons of the RG. This balance can be further complicated in cases where an Rdependent increase in hybridization of the metal valence orbital is observed, such as the sd hybridization observed in the BaþRG (RG = Ar, Kr, Xe, and Rn) complexes;6 however, this effect can usually be identified by noting unusual DeMorse/De values (DeMorse = ωe2/4ωexe), whereas for all GaþRG and InþRG complexes these appear to be within the expected ranges. 6983
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The Journal of Physical Chemistry A The ωe values of the GaþHe and InþHe complexes are considerably higher than those determined for the corresponding neon complexes; one rationale for this is the lack of pπpπ repulsion in the helide complexes. Thereafter, for increasing RG atomic number, there is generally an increase in ωe, which reaches a plateau for the heaviest RGs. The latter arises as a result of the competing effects of increasing reduced mass and increasing interaction. The force constant, k, is often a more insightful parameter, in which the mass dependence of ωe has been removed; for this, the trend follows that of De, as might be expected. Also presented for each complex in Tables 1 and 4 are the reduced curvature values, κ. This parameter42 is essentially the Sutherland parameter put forward in 1938,43 and is denoted Δ in recent work by Xie and Gong;44,45 it has been termed the “reduced curvature of the potential at Re”.42 For potentials which have similar forms of interaction, the value of κ, should be similar. In general, this is true for all GaþRG and InþRG complexes, although we note that the values of κ decrease slightly with increasing RG atomic number, reflecting the increase in De and the interaction potentials becoming more “Morse like”. This is also consistent with the lower κ values determined for the more strongly bound GaþRG complexes compared to the corresponding InþRG complexes. B. Trends with the Group IIIA Metal. Tables 5 and 6 summarize the Re and De values for AlþRG, GaþRG, InþRG, and TlþRG. As we have argued earlier,46 one consistent measure of the apparent “size” of an atomic ion, RWB(Mþ), is the Re value for the MþHe complex, minus the radius of the He atom (taken to be one-half of the diatomic van der Waals molecule HeHe bond distance, Re). Shown in Table 7 are such radii for the ions of interest in this paper. These radii make good qualitative sense. There is a substantial drop in ionic radius from Alþ to Gaþ, owing to the addition of the poorly shielding 3d10 shell of electrons. The Inþ radius is much larger than the Gaþ radius, because the electronic configurations of the outer shell electrons are just the same as for Gaþ, but the quantum number of those orbitals increases by one. Finally, the Tlþ radius decreases from that of Inþ, owing to the lanthanide contraction and, perhaps more importantly, the relativisitic contraction of the outer-shell s orbital, which for thallium penetrates to a nucleus with a much larger charge. The De values for the MþHe complexes directly reflect the magnitudes of RWB, with small values for the larger Alþ and Inþ ions, and larger values for the smaller Gaþ/Tlþ ions. It is likely that in all MþHe complexes, the 1/R4 attractive, ion-induced-dipole term is dominant, and that the repulsive interaction between Mþ and He comes in quite suddenly near Re in a “hard-sphere” type interaction. The MþNe trends in Re and De are quite similar to those for the MþHe complexes discussed above, suggesting that our arguments are also qualitatively valid for MþNe complexes. The trends for the other MþRG ions (RG = ArRn) do not always follow those of the RG = He, Ne ions. With the larger, more polarizable RG atoms, two factors come increasingly into play that are subtle and more difficult to rationalize: first, C6 and C8 dispersion attractive terms become larger and are dependent on the polarizibilities of both the RG atom and the Mþ ion; and second, some distortion of the Mþ electron cloud can occur, as the attractive forces increase, to minimize repulsion; these serve to complicate the R dependence of the repulsion forces, which will also vary with Mþ and RG.
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It should be noted, however, that in all the RG cases, the Re values for the InþRG ions are substantially larger than the Re values for the GaþRG ions, with the De values being correspondingly smaller. This is consistent with a simple change of quantum number of all the outer-shell electrons on the metal ion of þ1. C. Transport Coefficients. We show the calculated mobilities for each of 69Gaþ and 113Inþ in RG at 300 K as a function of E/n0 in Figures 1 and 2; there appear to be neither experimental nor previous theoretical data to which to compare. The main observation of note here is that the weaker interaction (smaller well depth) for MþHe leads to that mobility curve having no maximum at 300 K, in contrast to the other MþRG systems. At 100 K not shown, even He shows a mobility maximum. It is hoped that these mobilities, and other data deposited in the database, will be useful to future workers interested in the gaseous transport of these ions.
4. CONCLUSIONS We have presented spectroscopic parameters for the GaþRG and InþRG complexes determined from high level ab initio PECs. As with the AlþRG and TlþRG complexes, these complexes have interaction potentials that behave similarly to the closed shell AlkþRG complexes. There are subtle differences, such as the nonmonotonically increasing trend in Re with increasing RG atomic number, which has been observed previously for the AlkEþRG complexes. We have compared our results to previous calculated results for GaþAr and GaþKr, obtaining reasonably good agreement for the latter, but significant disagreements with the former. We have also compared our calculated spectroscopic parameters to those obtained for excited electronic states of analogous neutral GaRG complexes, obtained from photoionization spectroscopy. We obtain generally good agreement with the 2ΔΩ state parameters, which are expected to be closest to those of the cations; particularly the “pure-δ” 2Δ5/2 state. We extend these comparisons to the case of thallium in a similar way, by comparing our previous calculated TlþRG parameters to those of electronic excited states of neutral TlRG, obtained from laser-induced fluorescence experiments. Again, good agreement is generally obtained. Since there are neither previous experimental nor theoretical results for the corresponding indium species, no detailed discussion is possible. We note, however, that the previously good agreement for AlþRG, together with generally good agreement with the experimental 2Δ5/2 parameters, suggests that the PECs obtained here are of a good quality. The exception to these comments arises for the cases of Ga and Tl complexed to Xe, where the dissociation energies for the 2 ΔΩ states seem to be larger than those of the corresponding cations. We suggest that the stronger binding in these species might arise from interactions with other electronic states. ’ ASSOCIATED CONTENT
bS
Supporting Information. Lists of internuclear separations and interaction energies. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Present Addresses †
Chemistry and Biochemistry, Carroll University, 100 N. East Ave. Waukesha, WI 53186, U.S.A.
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’ ACKNOWLEDGMENT T.G.W. is grateful for the provision of computing time under the auspices of the NSCCS. A.M.G. is grateful to the EPSRC-GB and the University of Nottingham for provision of a DTA studentship. K.A.G. is grateful to Carroll University and the Edgar A. and Edna M. Thronson International Travel Fund for funding her visit to the University of Nottingham. W.H.B. is grateful for travel grants from the University of Utah, allowing visits to the University of Nottingham to be made. The work of S.R., C.N.Y.C., and L.A.V. was supported by the U.S. National Science Foundation. ’ REFERENCES (1) Bellert, D.; Breckenridge, W. H. Chem. Rev. 2002, 102, 1595. (2) Breckenridge, W. H.; Ayles, V. L.; Wright, T. G. J. Phys. Chem. A 2007, 127, 294308. (3) Gardner, A. M.; Graneek, J. B.; Wright, T. G.; Breckenridge, W. H. Manuscript in preparation. (4) Gardner, A. M.; Withers, C. D.; Graneek, J. B.; Wright, T. G.; Viehland, L. A.; Breckenridge, W. H. J. Phys. Chem. A 2010, 114, 7631. (5) Gardner, A. M.; Withers, C. D.; Wright, T. G.; Kaplan, K. I.; Chapman, C. Y. N.; Viehland, L. A.; Lee, E. P. F.; Breckenridge, W. H. J. Chem. Phys. 2010, 132, 054302. (6) McGuirk, M. F.; Viehland, L. A.; Lee, E. P. F.; Breckenridge, W. H.; Withers, C. D.; Gardner, A. M; Plowright, R. J.; Wright, T. G. J. Chem. Phys. 2009, 130, 194305. (7) Hickling, H.; Viehland, L. A.; Shepherd, D. T.; Soldan, P.; Lee, E. P. F.; Wright, T. G. Phys. Chem. Chem. Phys. 2004, 6, 4233. (8) Lozeille, J.; Winata, E.; Soldan, P.; Lee, E. P. F.; Viehland, L. A.; Wright, T. G . Phys. Chem. Chem. Phys. 2002, 4, 3601. (9) Viehland, L. A.; Lozeille, J.; Soldan, P.; Lee, E. P. F.; Wright, T. G. J. Chem. Phys. 2003, 119, 3729. (10) Viehland, L. A.; Lozeille, J.; Soldan, P.; Lee, E. P. F.; Wright, T. G. J. Chem. Phys. 2004, 121, 341. (11) Gardner, A. M.; Gutsmiedl, K. A.; Wright, T. G.; Breckenridge, W. H.; Chapman, C. Y. N.; Viehland, L. A. J. Chem. Phys. 2010, 133, 164302. (12) Gray, B. R.; Lee, E. P. F.; Yousef, A.; Srestha, S.; Viehland, L. A.; Wright, T. G. Mol. Phys. 2006, 104, 3237. (13) See, for example: Hensen, E. J. M.; García-Sanchez, M.; Rane, N.; Magusin, P. C. M. M.; Liu, P.-H.; Chao, K.-J.; van Santen, R. A. Catal. Lett. 2005, 101, 79. (14) See, for example: Fischer, R.; Weiss, J.; Rogge, W. Polyhedron 1998, 17, 1203. (15) Dunning, T. H., Jr.; Valley, M.; Taylor, H. S. J. Chem. Phys. 1978, 69, 2672. (16) Park, S. J.; Kim, M. C.; Lee, Y. S.; Jeung, G.-H. J. Chem. Phys. 1997, 107, 2481. (17) Stangassinger, A.; Knight, A. M.; Duncan, M. A. J. Chem. Phys. 1998, 108, 5733. (18) Stangassinger, A.; Mane, I.; Bondybey, V. E. Chem. Phys. 1995, 201, 227. (19) Woon, D. E.; Dunning, T. H., Jr. J. Chem. Phys. 1994, 100, 2975. (20) Dunning, T. H., Jr. J. Chem. Phys. 1989, 90, 1007. (21) Woon, D. E.; Dunning, T. H., Jr. J. Chem. Phys. 1993, 98, 1358. (22) Yousaf, K. E.; Peterson, K. A. Manuscript in preparation. (23) Metz, B.; Stoll, H.; Dolg, M. J. Chem. Phys. 2000, 113, 2563. (24) Peterson, K. A. J. Chem. Phys. 2003, 119, 11099. (25) MOLPRO is a package of ab initio programs written by H.-J. Werner and P. J. Knowles, with contributions from J. Alml€of, R. D. Amos, A. Berning, M. J. O. Deegan, F. Eckert, S. T. Elbert, C. Hampel, R. Lindh, W. Meyer, A. Nicklass, K. Peterson, R. Pitzer, A. J. Stone, P. R. Taylor, M. E. Mura, P. Pulay, M. Schuetz, H. Stoll, T. Thorsteinsson, and D. L. Cooper . (26) LeRoy, R. J. Level 7.2 A computer program for solving the radial Schr€odinger equation for bound and quasibound levels and
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