M+ = Li, Na, Be, Mg; RG = HeâRn - ACS Publications - American

Article pubs.acs.org/JPCA

Theoretical Study of M+−RG2 (M+ = Li, Na, Be, Mg; RG = He−Rn) Anna Andrejeva,† Adrian M. Gardner,§ Jack B. Graneek,∥ Richard J. Plowright, W. H. Breckenridge,‡ and Timothy G. Wright*,† †

School of Chemistry, University of Nottingham, University Park, Nottingham NG7 2RD, U.K. Department of Chemistry, University of Utah, Salt Lake City, Utah 84112, United States

‡

ABSTRACT: Ab initio calculations were employed to determine the geometry (MP2 level), and dissociation energies [MP2 and RCCSD(T) levels], of the MIIa+−RG2 species, where MIIa is a group 2 metal, Be or Mg, and RG is a rare gas (He−Rn). We compare the results with similar calculations on MIa+−RG2, where MIa is a group 1 metal, Li or Na. It is found that the complexes involving the group 1 metals are linear (or quasilinear), whereas those involving the group 2 metals are bent. We discuss these results in terms of hybridization and the various interactions in these species. Trends in binding energies, De, bond lengths, and bond angles are discussed. We compare the energy required for the removal of a single RG atom from M+−RG2 (De2) with that of the dissociation energy of M+−RG (De1); some complexes have De2 > De1, some have De2 < De1, and some have values that are about the same. We also present relaxed angular cuts through a selection of potential energy surfaces. The trends observed in the geometries and binding energies of these complexes are discussed. Mulliken, natural population, and atoms-in-molecules (AIM) population analyses are performed, and it is concluded that the AIM method is the most reliable, giving results that are in line with molecular orbital diagrams and contour plots; unphysical amounts of charge transfer are suggested by the Mulliken and natural population approaches. linear, while in 2003, Sapse et al.13 reported both Li+−He2 and Na+−He2 to be bent, with bond angles of 113° (130)° and 123° (167°), respectively, employing the MP2/6-311+G(3df,3pd) method with the full (frozen core) approximations. Giju et al.14 studied Na+−Arn complexes, finding Na+−Ar2 to be linear, with Na+−Ar internuclear separations of 2.781 Å at the MP2/6311+G(3df) level of theory. Nagata et al.15 also studied Na+−Arn complexes, but with geometries only optimized at the HF/6311++G(3df) level; a linear geometry with Na+−Ar internuclear separations of 2.797 Å was reported. Finally, Ben El Hadj Rhouma et al.16 also reported a linear geometry from atomistic pairwise potentials, which had been fitted to high-level ab initio data. All three of these latter papers discussed the fact that the potential is flat and that a C2v minimum is very close in energy. Bu et al.17,18 used MP2/6-311+G(3df,3pd) calculations to study a range of Be+−Hen complexes. They concluded that Be+−He2 was of C2v symmetry, with a HeBeHe bond angle of 61.3° and Be+−He internuclear separations of 3.085 Å. Subsequently, work on Be+−He2 was published by Page et al.,19 reporting a potential energy surface, which was calculated using both UCCSD(T) and MRCI approaches. Very similar results to the earlier studies were obtained, with the minimum energy geometry being concluded to be of C2v symmetry with an equilibrium bond angle of 60.2° and Be+−He internuclear separations of 2.920 Å. The energy for the removal of a single

1. INTRODUCTION In the following, we shall use M to denote a general metal, and hence M+−RG2 will be a general triatomic metal cation/di-rare gas complex. We shall refer to M+−RG2 complexes involving the group 1 (alkali metals) as MIa+−RG2, to avoid ambiguity with stoichiometries; similarly, we shall refer to M+−RG2 complexs involving the group 2 (alkaline earth metals) as MIIa+−RG2. If we wish to talk about a family of specific complexes, we shall specify the common element. Interactions between metal cations, M+, and rare gas atoms, RG, in M+−RG complexes are prototypes for the study of solvation. The whole series of MIa+−RG complexes (where MIa is a group 1 metal atom) have been studied by our group.1−5 These complexes have been shown6 to interact via almost entirely physical processes based upon the model potential of Bellert and Breckenridge,7 even when Tang−Toennies8 damping functions are included. We have also studied the MIIa+−RG complexes (where MIIa is a group 2 metal atom),9−11 which show significant interaction energies, beyond that expected on purely physical grounds. The origin of this is significant hybridization between the outermost occupied diffuse s orbital on the MIIa+ metal atom and its lowest unoccupied orbitals, which occurs as the RG atom approaches; this leads to sp hybridization in the cases of Be+−RG and Mg+−RG. In the present work, we extend these studies to the triatomic complexes M+−RG2 to investigate their behavior. Regarding the equilibrium geometries of the two lightest MIa+−RG2 complexes, these do not appear to have been established unequivocally. For example, Bauschlicher et al.12 considered Li+−Ar2 and Na+−Ar2 and deduced these to be © XXXX American Chemical Society

Special Issue: Terry A. Miller Festschrift Received: July 30, 2013 Revised: September 11, 2013

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dx.doi.org/10.1021/jp4075652 | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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He atom was 151 cm−1 and harmonic vibrational frequencies were reported. No work on the heavier Be+−RG2 complexes appears to have been published. In 2005, Bu et al.17,18,20 studied the Mg+−Hen complexes employing the MP2/6-311+G(3df,3pd) method and concluded Mg+−He2 to be of C2v symmetry, with a HeMgHe bond angle of 52° and Mg+−He internuclear separations of 3.600 Å; harmonic vibrational frequencies were also reported as well as binding energies. Sapse et al.13 reported results at the MP2/ 6-311+G(3df,3pd) level of theory for Mg+−He2, which was found to be bent with C2v symmetry, with an equilibrium bond angle of 52.5° [50.8°] and Mg+−He internuclear separations of 3.602 Å [3.678 Å] using the MP2 (full) [and MP2 (frozen core)] methods. In 2002 Sapse21 studied the Mg+−Ne2 complex using levels of theory including MP2/6-311+G(3df), obtaining an equilibrium C2v structure with a bond angle of 58.4° and Mg+−Ne internuclear separations of 3.020 Å; harmonic vibrational frequencies and the binding energy were also reported. We note that in 1990, Bauschlicher et al.12 looked at Mg+−Ar2, obtaining a C2v structure. The rationale for this was in terms of sp hybridization, and competition between steric effects and electron repulsion. The modified coupled pair functional (MCPF) approach was employed with a basis set of slightly better than triple-ζ quality, and Mg+−Ar2 was deduced to have a bond angle of 82.5° and Mg+−Ar internuclear separations of 2.942 Å; the binding energy was also reported. No work on the heavier Mg+−RG2 complexes appears to have been reported.

The quintuple-ζ basis sets employed were the standard augcc-pV5Z basis sets for He, Ne, and Ar.25−27 For Kr, Xe, and Rn, the small-core, relativistic effective core potentials, ECP10MDF, ECP28MDF, and ECP60MDF were again employed,29 with ECP10MDF_aV5Z, ECP28MDF_aV5Z, and ECP60MDF_aV5Z valence basis sets. The same electrons were correlated, as noted above for the triple-ζ calculations. For Be+ and Mg+, standard augcc-pCV5Z basis sets30 were employed. To obtain reliable binding energies for the MIIa+−RG2 species, single-point (R)CCSD(T) calculations using the largest basis set in each case were performed at the MP2optimized geometries obtained with the largest basis set. Unfortunately, for the heavier species, there was overlap between the outermost occupied rare gas d orbitals and the 1s and 2s orbitals of Be+ and Mg+, respectively. Hence, to perform reliable calculations, all of these orbitals would need to be either frozen or correlated. We chose to correlate these, and this meant that quintuple-ζ calculations were not possible, and so for the calculations involving the heavier rare gases, quadruple-ζ versions of the above basis sets were employed. For Be+ and Mg+, these were the standard aug-cc-pCVQZ basis sets,30 for the rare gases, standard aug-cc-pVQZ basis sets were used for He−Ar, and the aug-cc-pwCVQZ25 basis sets for Kr−Rn were employed, together with the same small-core, relativistic ECPs. All of the above calculations employed Molpro.28,29

3. RESULTS A. Geometries. In Table 1 we show the results of optimizing the geometry of the Li+−RG2 and Na+−RG2 complexes at the MP2/aug-cc-pVTZ level of theory. Initially, in each case, the geometry was fixed to be linear, with equal M+−RG bond lengths. (Below we shall show that the results obtained for the Be+−RG2 and Mg+−RG2 complexes are very similar when the MP2/aug-cc-pVTZ or MP2/aug-cc-pV5Z level of theory is employeed.) We calculated harmonic vibrational frequencies to ascertain whether these were minima or saddle points, although the values of the harmonic frequencies were only tens of cm−1 in some cases, and so the actual values are likely not reliable. In most cases, there were four real frequencies, showing the linear geometry to be a minimum. In several cases, however, there was a small, degenerate imaginary frequency along the bending coordinate, even when tighter geometry convergence criteria were employed. In these cases, the geometry optimization was restarted from a bent geometry, leading to convergence to a C2v structure, and three real harmonic vibrational frequencies. As will be discussed further below, the depths of these nonlinear minima were very shallow, and so these structures should be viewed as quasilinear. As expected, the internuclear separations, MIa+···RG, increase monotonically with the atomic number of RG, in line with the observed trends for the diatomic MIa+−RG complexes2−6 (Figure 1). In contrast, the optimized geometries of the MIIa+−RG2 complexes, shown in Table 2, are very bent, with angles as low as 52°. For Be+−RG2 and Mg+−RG2, the results are very similar when either the MP2/aug-cc-pVTZ or MP2/aug-cc-pV5Z method is employed. The trend is for the bond angle to increase monotonically with the atomic number of the RG atom for a particular MIIa. The trend in bond length is not monotonically increasing, however, and we observe an initial decrease in the internuclear separation with atomic number, with a minimum at Ar, then an increase thereafter; this trend is similar to that observed for the diatomic MIIa+−RG complexes9−11 (Figure 1). We explained this in terms of a delicate

2. COMPUTATIONAL DETAILS In the following, we shall report the results of geometry optimizations using the MP2 method and employing triple-ζ, and either quadruple- or quintuple-ζ quality basis sets; we shall also present the results of single-point (R)CCSD(T) binding energy calculations. All of the titular species were geometry optimized using triple-ζ basis sets, whereas the MIIa+−RG2 complexes were also optimized using quadruple- or quintuple-ζ basis sets. We also performed some relaxed angular scans using the triple-ζ basis sets, where at each angle, the bond lengths were optimized to yield a minimum energy path along that angular coordinate. For the triple-ζ calculations, standard aug-cc-pVTZ basis sets were employed for He, Ne, and Ar22−24 whereas aug-ccpwCVTZ-PP valence basis sets25 were used for Kr, Xe, and Rn together with small-core, relativistic effective core potentials, ECP10MDF, ECP28MDF, and ECP60MDF, respectively.26 Because the sole valence electron is lost from the neutral atom for Li+ and Na+, the valence electrons of these correspond to the inner-valence electrons of the neutral species; hence, augcc-pwCVTZ basis sets were employed for M = Li, Na,27 allowing the core−valence correlation to be better described. For consistency, the same type of basis sets were employed for Be and Mg,30 which would also account for any involvement of the inner-core electrons of the metal atoms in Be+−RG2 and Mg+−RG2, which each have a single electron in their outermost s orbital. For the optimizations, we employ the MP2 or RMP2 procedures, as implemented in MOLPRO. No electrons were frozen for Li+, Be+, or He; for Na, Mg, and Ne, the 1s orbitals were frozen; for Ar, the 1s2s2p orbitals were frozen; and for Kr, Xe and Rn, only the innermost s and p orbitals, not included in the ECP, were frozen. B



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Table 1. Structural Parameters for MIa+−RG2 Speciesa RG

Re2/Å

Re1/Å

Re2/Re1

θ/deg

De2/cm−1

De1/cm−1

Deav

De2/De1

(630) (1054) (2289) (2710) (3389) (3677)

(628) (1039) (2367) (2899) (3688) (4072)

(629) (1047) (2328) (2802) (3532) (3878)

(1.00) (1.014) (0.97) (0.93) (0.92) (0.90)

(331) (594) (1349) (1657) (2147) (2370)

(323) (573) (1350) (1698) (2251) (2527)

(327) (584) (1350) (1676) (2194) (2450)

(1.03) (1.04) (1.00) (0.97) (0.95) (0.94)

+

He Ne Ar Kr Xe Rn

(1.902) (2.050) (2.379) (2.520) (2.710) (2.789)

(1.901) (2.050) (2.380) (2.510) (2.697) (2.776)

1.00 1.00 1.00 1.00 1.00 1.00

He Ne Ar Kr Xe Rn

(2.346) (2.489) (2.801) (2.930) (3.106) (3.179)

(2.339) (2.490) (2.801) (2.930) (3.100) (3.179)

1.00 1.00 1.00 1.00 1.00 1.00

Li (180.0) (180.0) (164.0b) (149.9b) (180.0) (179.0b) Na+ (180.0) (180.0) (180.0) (180.0) (180.0) (180.0)

Re is the equilibrium bond length, θ is the RG−M+−RG bond angle, De1 is the dissociation energy of M+−RG, De2 is the dissociation energy of M+−RG···RG, and Deav is the average of De1 and De2. bAlthough the optimized geometry has a bond angle of 0.9. Be+−RG2 and Mg+−RG2. Moving on to the bent Be+−RG2 and Mg+−RG2 complexes, we have calculated dissociation energies using both MP2/aug-cc-pVTZ and RCCSD(T)/augcc-pVXZ methods (X = 5 for RG = He, Ne and Ar; and X = Q for RG = Kr, Xe and Rn; see above). As may be seen, the results are generally in very good agreement. We see that for the De1 values, the higher level of theory gives the higher binding energy, which is generally expected; however, for De2 this trend is broken for the MIIa+−Xe2 and MIIa+−Rn2 complexes. A rationalization of this is that there is an increased need for higher levels of theory/basis sets to describe both attractive and repulsive interactions. Notably, both the De2 and De1 (and so Deav) values are increasing monotonically with the atomic number of the RG atoms, suggesting that the attractive terms are dominating the overall trend. We also present the De2/De1 ratios in Table 2. C. Atomic Charges. In Tables 4−7, we present the results of three population analyses: Mulliken,30 natural population analysis (NPA),31 and atoms-in-molecules (AIM).32 The Mulliken and NPA analyses were performed with routines integrated into Gaussian,33 whereas the AIM analysis employed the AIMAll program34 using the WFX file produced by Gaussian, which allowed ECP-based basis sets to be employed (see ref 35 for details). In all cases the MP2 method was used with the triple-ζ basis sets, at the corresponding optimized geometry. As will be seen from these tables, although there is reasonable agreement between the methods for the lighter RG-containing complexes, there are significant deviations for those containing the heavier RG atoms; this is particularly stark for the Be+−RG and Be+−RG2 complexes, for which the NPA and Mulliken charges suggest a large amount of charge transfer. If this were the case, it would show up in molecular orbital plots, and so we have also produced these, as described in the next subsection. D. Molecular Orbital Diagrams. In Figures 3 and 4, we show a selection of MIa+−RG2 and MIIa+−RG2 molecular orbital diagrams. These were each calculated at the MP2 optimized geometry using the triple-ζ basis set. The molecular orbital diagrams show the Hartree−Fock energies of both the uncomplexed M+ orbitals, and the (paired) set of RG orbitals. We give some preliminary information regarding these plots in this subsection but expound further on these in the Discussion section below. We have included all occupied valence atomic orbitals in the plot, together with the lowest unoccupied M+ orbitals. In each diagram, we have also indicated the unoccupied and occupied molecular orbitals, with the highest occupied molecular orbital being singly occupied for the MIIa+−RG2 complexes. First we note that, in all cases, the RG orbitals are expected to decrease in energy as they come close to the M+ center, owing to Coulombic attraction. For the MIa+−RG2 complexes, we see that the occupied orbitals are essentially the RG orbitals, and that the energies and contours of these are such that very little difference to the isolated atomic orbitals is seen, excepting the Coulombic energy lowering: the contour plots are largely

Re2 is the M+−RG equilibrium bond length in M+−RG2, and Re1 is the M+−RG equilibrium bond length in M+−RG. θ is the RG−M+−RG bond angle, De1 is the dissociation energy of M+−RG, De2 is the dissociation energy of M+−RG···RG, and Deav is the average of De1 and De2. RRG−RG is the RG−RG internuclear distance in the M+−RG2 complex; Re(RG2) is the optimized RG2 Re value. Values in parentheses were obtained at the RMP2 level of theory using triple-ζ quality basis sets (see text). For the geometries, the nonparenthesized values were obtained at the RMP2 level of theory using quintuple-ζ quality basis sets (see text). For the interaction energies, the nonparenthesized values were obtained at the RCCSD(T) level of theory theory using quintuple-ζ quality basis sets (see text) for RG = He−Ar, and at the RCCSD(T) level of theory theory using quadruple-ζ quality basis sets (see text) for RG = Kr−Rn.

(3.062) (3.132) (3.701) (3.946) (4.303) (4.460) 3.134 3.198 3.681 3.861 4.187 4.298 (50.6) (58.6) (78.6) (83.3) (87.9) (90.4) 51.8 59.9 78.8 82.3 86.1 87.5 (1.00) (1.00) (1.02) (1.02) (1.03) (1.03) 1.03 1.02 1.03 1.02 1.02 1.01 (3.565) (3.200) (2.863) (2.896) (3.019) (3.064) 3.482 3.145 2.822 2.884 3.018 3.064 (3.582) (3.200) (2.922) (2.969) (3.100) (3.143) 3.587 3.203 2.900 2.934 3.067 3.108 He Ne Ar Kr Xe Rn

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a

(1.09) (1.12) (0.94) (0.87) (0.82) (0.79) 1.07 1.15 0.91 0.84 0.78 0.75 76 (68) 218 (213) 1237 (1130) 1823 (1730) 2644 (2605) 3176 (3146) 73 (65) 203 (201) 1299 (1167) 1978 (1850) 2972 (2866) 3639 (3513) 78 (71) 233 (225) 1175 (1093) 1668 (1610) 2315 (2343) 2713 (2779) (0.99) (0.99) (0.98) (0.98) (0.99) (1.01) 1.01 1.00 0.98 0.97 0.98 0.98 (3.088) (3.166) (3.760) (4.006) (4.315) (4.402)

(1.06) (1.05) (0.55) (0.55) (0.54) (0.53) 1.03 1.01 0.54 0.53 0.51 0.50 135 (111) 409 (365) 3417 (3143) 4625 (4430) 6216 (6148) 7112 (7082) 133 (108) 407 (356) 4428 (4050) 6053 (5731) 8240 (8008) 9491 (9241) 137 (114) 410 (374) 2407 (2235) 3197 (3128) 4192 (4287) 4733 (4922) (0.99) (0.98) (0.88) (0.89) (0.92) (0.94) 0.98 0.96 0.87 0.89 0.91 0.93 (3.088) (3.166) (3.760) (4.006) (4.315) (4.402) 3.009 2.595 2.197 2.302 2.475 2.544

(3.021) (2.625) (2.205) (2.327) (2.500) (2.577)

2.924 2.454 2.084 2.221 2.407 2.486

(3.039) (2.621) (2.100) (2.223) (2.397) (2.474)

1.03 1.06 1.05 1.04 1.03 1.02

(0.99) (1.00) (1.05) (1.05) (1.04) (1.04)

60.1 72.9 96.6 100.5 104.3 105.8

(60.9) (72.4) (96.8) (100.5) (104.7) (106.3)

3.014 3.083 3.281 3.540 3.909 4.058

(3.062) (3.101) (3.298) (3.578) (3.959) (4.124)

Be+ 3.090 3.204 3.760 3.980 4.287 4.369 Mg+ 3.090 3.204 3.760 3.980 4.287 4.369 He Ne Ar Kr Xe Rn

RRG−RG /Å θ/deg Re2/Re1 Re1/Å Re2/Å RG

Table 2. Structural Parameters for MIIa+−RG2 (MIIa = Be, Mg) Speciesa

Re(RG2)

RRG−RG/Re(RG2)

De2/cm−1

De1/cm−1

Deav

De2/De1


D



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Figure 2. Trends in the De1 and De2 dissociation energies for the MIa+−RGn and MIIa+−RGn complexes (n = 1 and 2).

can be considered to be (quasi)linear. Very similar results are seen for the Na+−RG2 complexes, although all complexes have a linear geometry. It is interesting to note that for Li+−Xe2, Na+−Xe2, Be+−Ar2, Be+−Xe2, and Mg+−Xe2 there is a local asymmetric M+−RG−RG minimum at 0°. That the xenoncontaining species all have a local minimum suggests that the induced dipole on the Xe atom, caused by the interaction with the M+, is sufficiently large to allow a second Xe atom to bind, with a significant binding energy, as seen from Figures 5 and 6. That Be+−Ar2 also demonstrates such a local minimum can be attributed to the small size of Be+ and the polarizability of Ar.

atomic-like and the sets of s and p orbitals are largely degenerate. Differences in energy are observed, particularly for the complexes containing Xe, but these are small. The virtual orbitals more clearly show signs of possible mixing and distortion, but these are essentially unoccupied. For the two MIIa+−He2 complexes, there is a slight distortion of the 2s orbital from the incoming He atoms, and a very small involvement of the He 1s orbitals in the HOMO, with the effect greater for Be+−He2, likely caused by the size of the Be+ atom; however, even here the distortion is quite small. For the heavier Be+−RG2 complexes, the distortions are more significant, with large distortions of the 2s orbital of Be+ by the incoming RG atoms. This distortion is evident as a bulge on the far side of the Be+ center, which is evidence of the sp hybridization. There is also significant involvement of the outermost occupied p orbital from the incoming RG atoms in the HOMO. For the Be+−RG2 complexes, there is a displacement of the 2s electron density away from the incoming RG atoms, and showing clear signs of sp hybridization; the Mg+−Ar2 plot closely resembles that published by Bauschlicher et al.;12 further discussion will be presented below. E. Relaxed Angular Plots. We noted, when discussing the equilibrium geometries above, that the MIa+−RG2 complexes were linear or quasilinear. In Figure 5 we show the cut through the MP2/aug-cc-pVTZ potential energy surface for selected angles of the MIa+−RG2 complexes, where at each angle, the internuclear separations have been independently optimized. It may be seen that these potentials are very flat, but with steeply rising regions when the two RG atoms are in close proximity. There are subtle differences in the shapes of the potential in this flat region: Li+−He2 has a shallow minimum at 180°; Li+−Ar2 has a minimum at 164°, but the potential is very flat; Li+−Xe2 has a single shallow minimum at 180°. It is possible that the nature of these shallow regions will vary with basis set; in any case, these minima are so shallow that all of the complexes will show essentially linear geometries at the zero-point level and so

4. DISCUSSION A. Geometry. Group 1. As was noted previously, in 1990 Bauschlicher et al.12 studied a range of M+−RG2 complexes, in particular Li+−Ar2 and Na+−Ar2, and concluded that these were linear, using the MCPF approach and basis sets of around triple-ζ quality. From Table 1, it can be seen that we concur with the conclusions for Na+−Ar2, but that we obtain a bent geometry for Li+−Ar2, albeit with a very shallow potential. We noted in the above that other workers14−16 have commented on the flatness of the potential energy surface in the angular direction. Interestingly, for the corresponding helium-containing complexes, Sapse et al.13 obtained bent geometries at the MP2/ 6-311+G(3df,3pd) level. The bond angle for Li+−He2 was very sensitive to the method employed and changed from 113° to 130° when the MP2(full) method was changed to MP2(FC); in the present work we obtain a linear geometry. For Na+−He2 Sapse et al.13 obtained bond angles of 123° and 167°, using the same MP2 methods, whereas, again, we obtain linear geometries. However, as we have shown above, the minimum-energy angular plots are extremely flat for the MIa+−RG2 complexes, and even with only the zero-point vibrational energy, the RG atoms will be sampling a wide range of angular space. It is clear that for a definitive answer on the E


Figure 3. Molecular orbital diagrams, including contour plots, for selected MIa+−RG2 complexes. The occasional orbital label is omitted for space reasons.

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Figure 4. Molecular orbital diagrams, including contour plots, for selected MIIa+−RG2 complexes. The occasional orbital label is omitted for space reasons.

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Figure 6. Selected minimum energy angular paths on selected Be+−RG2 and Mg+−RG2 potential energy surfaces; the internuclear separations have been independently optimized in each case. LM marks the position of a local minimum, and GM is the global minimum.

Figure 5. Selected minimum energy angular paths on selected Li+−RG2 and Na+−RG2 potential energy surfaces; the internuclear separations have been independently optimized in each case. LM marks the position of a local minimum, and GM is the global minimum.

Only the lightest member of the Be+−RG2 series, Be+−He2, has been studied previously, by Bu et al.17 and Page et al.19 The Bu et al. study used the MP2/6-311+G(3df,3pd) method, obtaining a bond angle of 61.3°, whereas the Page et al. study used a range of methods including MP2, UCCSD(T), and MRCI+Q methods. Interestingly, the bond angle was rather constant, lying in the range 59.9−61.3°, with the UCCSD(T) and MRCI+Q methods and aug-cc-p(C)VQZ basis sets, yielding 60.1° and 60.2°, respectively, which are in excellent agreement with the value of 60.1° obtained herein. The best UCCSD(T) De2 value was 136.3 cm−1, whereas the MRCI+Q method yielded 150.5 cm−1. The former value is in near perfect agreement with the RCCSD(T) result in the present work of 137 cm−1. We note that in ref 19 BSSE was corrected for, and amounted to ∼7 cm−1 in this species; it seems the method employed here benefits from a cancellation of errors at the RCCSD(T) level. The MRCI+Q results of ref 19 suggest that our De2 results could be ∼10% too low, although we note that this conclusion is not definitive, as larger basis sets led to lower De2 values (Table 2). In the work reported in the papers of Sapse et al.13 and Bu and Zhong,17,20 the MP2/6-311+G(3df,3pd) method was employed, and slightly different bond angles and bond angles for the Mg+−He2 complex are reported, but these can be

shape of the angular potential, higher levels of theory and larger basis sets would be required; however, whether the effort would be worthwhile is debatable, given the comments in the preceding sentence. Group 2. As noted above, the difference between the observed geometries for the MIa+−RG2 and MIIa+−RG2 complexes appears to have been discussed first by Bauschlicher et al.12 with regard to Mg+−Ar2. They rationalized its bent geometry in terms of sp hybridization, which allows the unpaired electron in the isotropic diffuse 3s orbital to move to the opposite side of Mg+ into an sp hybrid orbital, reducing the electron repulsion with the two incoming Ar atoms. This hybridization would not be beneficial if the Ar atoms approached the Mg+ linearly (from opposite directions), because this would simply position electron density in the way of the other Ar atom. In addition, we note that this hybridization process also allows the two Ar atoms approaching from the same side to see a larger effective nuclear charge, and so there is both an increase in attractive and a reduction in repulsive terms; hence the cost of hybridization is “paid back” by these two effects. For the MIa+−RG2 complexes, this hybridization cannot occur, because the lowest (allowed) excitation energies36 of the closed-shell Li+ and Na+ atomic cations (Table 3) are too high compared to those of the open-shell Be+ and Mg+ (Table 3) cations. H



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Table 3. Atomic Propertiesa lowest allowed transition

species Li+ Na+ Be+ Mg+ He Ne Ar Kr Xe Rn

P1 (1s2p) ← 1S0 (1s2) P1 (2p53s) ← 1S0 (2p6) 2 P1/2 (2s2p) ← 1S0 (2s2) 2 P1/2 (3s3p) ← 1S0 (3s2) 1

1

Table 4. Atomic Charges for the Li+−RGn (n = 1 and 2) Complexesa

static ionization wavenumber of lowest allowed polarizability/ energy/ −1 3 Å cm−1 transition/cm 501816 268767 31929 35669

0.029 0.148 3.7 5.5 0.205 0.396 1.642 2.519 4.044 5.103

Li+−RG2

198311 173930 127110 112914 97834 86693

Li charge

charge on each rare gas atom

Li charge

rare gas charge

He

0.99 [0.97] (1.05) 0.98 [0.97] (0.93) 0.96 [0.89] (1.00) 0.95 [0.85] (0.86) 0.94 [0.77] (0.84) 0.94 [0.74] (0.80)

0.01 [0.01] (−0.03) 0.01 [0.02] (0.03) 0.02 [0.05] (0.00) 0.02 [0.08] (0.07) 0.03 [0.11] (0.08) 0.03 [0.13] (0.10)

0.99 [0.99] (1.03) 0.99 [0.99] (0.97) 0.98 [0.97] (0.93) 0.98 [0.96] (0.87) 0.97 [0.93] (0.78) 0.97 [0.92] (0.71)

0.01 [0.01] (−0.03) 0.01 [0.01] (0.03) 0.02 [0.03] (0.07) 0.03 [0.05] (0.13) 0.03 [0.07] (0.22) 0.03 [0.08] (0.30)

Ar

Wavenumbers of lowest allowed transitions and ionization energies taken from ref 36. Static dipole polarizabilities taken from ref 7.

Kr

explained by different starting geometries and convergence criteria. Both conclude the Mg+−He internuclear separations are ∼3.6 Å and the HeMgHe bond angle is around 51−52°. These results are in good agreement with the bond angle of 51.8° obtained in the present work. Bu and Zhong20 report a very low value of 20 cm−1 for De2, whereas Sapse et al.13 report a significantly higher value of 85 cm−1 (although both appear to be using the same method); it is Sapse et al.’s value that is in better agreement with the value of 78 cm−1 obtained in the present work. Sapse has reported results on Mg+−Ne2 using the same method and same quality basis set, obtaining a bent geometry of 57−58°, which is in very good agreement with the value of 59.9° obtained herein. In that work a value for De1+De2 of 535 cm−1 was reported, which is significantly greater than the value of 436 cm−1 obtained in the present work, at more reliable levels of theory. In agreement with the conclusion of Bauschlicher et al.,12 we obtain a bent geometry for Mg+−Ar2 and their angle of 82.5° at the MCPF level is in reasonable agreement with our value of 78.8°. These authors only reported a value for De1+De2 (the energy to remove both Ar atoms), obtaining a value of 2170 cm−1, which compares reasonably well with the 2474 cm−1 value obtained here, at higher levels of theory. There do not appear to be any previous results on the heavier Mg+−RG2 complexes, and so it is not possible to make further comparisons. B. Interactions. Group 1. Looking first at the dissociation energies in Table 1, we see that for the two lightest MIa+−RG2 complexes, the De1 and De2 values are very similar, and so the De2/De1 ratios are very close to unity. However, for the heavier ones, the ratio gradually drops, even though the Re2/Re1 ratios are essentially unity. If we consider the atomic charges for the Li+−RG2 complexes (Table 4), we see that there are significant differences between the three different approaches. Although it is well-known that Mulliken populations are unreliable, particularly for extended basis sets, it is less well-known that the NPA method has also been deduced to be unreliable in metal−ligand complexes,37 giving unrealistic charges on the metal center, and the results here seem to concur with these conclusions. First, we note that the trends in the atomic charges seem reasonable: as the atomic number of the RG atom increases, the ionization energy drops and the polarization increases. It is thus unsurprising that the charge on Li+ monotonically drops with the atomic number of RG. However, the drop is rather significant for both MIa+−RG and MIa+−RG2 complexes, according to the Mulliken and NPA analyses, which was somewhat surprising, given our previous conclusions6 that

Li+−RG +

rare gas

Ne

a

+

Xe

Rn

a

AIM values are in bold, NPA values are given in square brackets, and Mulliken values are given in parentheses.

the interactions in the MIa+−RG diatomic complexes could be described well by a physical model in which the atomic charge was unity. On the other hand, the AIM charges are very much in line with our earlier deduction. Another piece of evidence for concluding that the NPA analyses are less reliable than the AIM method is that the NPA charge on the RG atom for the heavier Li+−RG2 complexes is higher than that for the corresponding Li+−RG ones: this does not make physical sense, because any charge transfer from one RG atom is expected to reduce the amount of charge transfer that a second would make (there would be some electron density in the isotropic 2s orbital). These expectations are met by the AIM charges, which calculate that the charge on the metal is close to unity in all cases, and that the charges on the RG atoms are very small, and either equal to or slightly smaller in the triatomic complexes than in the diatomic ones. We also note that the contour plots and molecular orbital diagrams for the MIa+−RG2 complexes in Figure 3 show that the molecular orbitals are very similar to the atomic ones for the whole series. Even for the Li+−Xe2 complex, the two 5s orbitals are essentially degenerate, as are the six 5p orbitals. We note that there are contributions from both Li+ and Xe orbitals to the lowest virtual orbitals, but these are essentially unoccupied. A very similar picture is borne out by the Na+−RG2 complexes, with very similar calculated atomic charges (Table 5). Interestingly, the anomaly of the NPA charges on the RG atoms in the triatomic species being larger than that in the corresponding diatomic ones is not present, although we note that these are also smaller. Given the large internuclear separations for the Na+−RG2 complexes, it seems the NPA method is behaving better: the NPA charges are closer to the AIM ones than was the case in the Li+−RG2 complexes, but there is still a significant deviation. Again, the AIM values are more in line with the expected physical nature of the interaction6 and also with the appearance of the molecular orbital diagrams and contour plots (Figure 3). The overall picture, I



Article

Table 6. Atomic Charges Analyses for the Be+−RGn (n = 1 and 2) Complexesa

Table 5. Atomic Charges Analysis (and Mulliken) Charges for the Na+−RGn (n = 1 and 2) Complexesa Na+−RG2 +

Na+−RG +

Be+−RG2 +

Be+−RG +

rare gas

Na charge


Na charge

rare gas charge

rare gas

Be charge


Be charge

rare gas charge

He

0.99 [0.99] (1.06) 0.99 [0.99] (0.96) 0.97 [0.95] (0.96) 0.96 [0.93] (0.89) 0.95 [0.89] (0.78) 0.94 [0.87] (0.71)

0.00 [0.01] (−0.03) 0.01 [0.01] (0.02) 0.02 [0.03] (0.02) 0.02 [0.04] (0.05) 0.03 [0.06] (0.11) 0.03 [0.07] (0.15)

1.00 [1.00] (1.02) 1.00 [1.00] (0.98) 0.98 [0.98] (0.98) 0.98 [0.98] (0.93) 0.97 [0.97] (0.87) 0.97 [0.96] (0.82)

0.00 [0.00] (−0.02) 0.01 [0.01] (0.02) 0.02 [0.02] (0.02) 0.02 [0.02] (0.07) 0.03 [0.04] (0.13) 0.03 [0.04] (0.18)

He

1.01 [1.00] (1.01) 1.02 [0.98] (0.95) 1.03 [0.69] (0.69) 1.01 [0.59] (0.65) 0.98 [0.45] (0.30) 0.96 [0.40] (0.15)

0.00 [0.00] (0.00) −0.01 [0.01] (0.02) -0.01 [0.16] (0.15) -0.01 [0.20] (0.18) 0.01 [0.28] (0.35) 0.02 [0.30] (0.42)

1.00 [1.00] (1.00) 1.01 [0.99] (0.98) 1.02 [0.85] (0.82) 1.00 [0.80] (0.77) 0.98 [0.73] (0.53) 0.97 [0.70] (0.43)

0.00 [0.00] (0.00) −0.01 [0.01] (0.03) −0.02 [0.15] (0.18) 0.00 [0.20] (0.23) 0.02 [0.27] (0.47) 0.03 [0.31] (0.57)

Ne

Ar

Kr

Xe

Rn

Ne

Ar

Kr

Xe

Rn

a

a



Table 7. Atomic Charges for the Mg+−RGn (n = 1 and 2) Complexesa

therefore, is that the MIa+−RG2 complexes are interacting in a physical sense, with very little chemical contribution, consistent with the picture for the MIa+−RG complexes.6 Group 2. Moving onto the group 2 systems, we have noted that the explanation for their nonlinearity is sp hybridization of the MIIa+ center, allowing the unpaired outermost s electron density to move away from the incoming RG atom’s electron density. The contour plots shown in Figure 4 show this very clearly, with the highest occupied orbital (ϕ9 for all but the MIIa+−He2 complexes, where this is ϕ3) being made predominantly of the singly occupied s orbital, but clearly distorted away from the incoming RG atoms, with the distortion being more pronounced for the species containing the heavier RG atoms, as expected. Additionally, it can be seen that there is very little contribution to the ϕ3 orbital from the He 1s orbitals in the MIIa+−He2 complexes, but that as the RG atomic number increases, the ϕ9 orbital contains a small amount of the highest occupied p orbitals from the RG atoms. This is in line with the small deviations of the AIM charges from unity (Tables 6 and 7), but not in agreement with the very large charge transfers suggested by the Mulliken and NPA analyses, particularly for the complexes with the heavier RG atoms. (It should be noted that there are significant contours in the central region of the ϕ9 (ϕ3 for MIIa+−He2 orbitals) from the inner parts of the metal’s singly occupied s orbital.) The similarity of the AIM charges (Table 4−7) for all of the M+−RG2 complexes is a little surprising, and the contour plots suggest that a greater range might have been expected. However, although the distortion of the MIIa+ singly occupied s orbitals is significant, the charge is still largely located on the MIIa+ center. It may be argued that the contribution from the p orbitals from the heavier RG atoms are suggestive of an increased charge transfer/chemical interaction, and we shall comment on this further below. (Note that the occasional negative charge calculated on the RG atom is likely an artifact,

Mg+−RG2

Mg+−RG

rare gas

Mg+ charge


Mg+ charge

rare gas charge

He

1.01 [1.00] (1.00) 1.02 [0.99] (0.97) 1.03 [0.93] (0.94) 1.02 [0.88] (0.87) 1.00 [0.81] (0.73) 0.99 [0.76] (0.62)

0.00 [0.00] (0.00) -0.01 [0.00] (0.01) −0.02 [0.04] (0.03) −0.01 [0.06] (0.06) 0.00 [0.10] (0.13) 0.01 [0.12] (0.19)

1.00 [1.00] (1.00) 1.01 [1.00] (0.99) 1.02 [0.97] (0.97) 1.01 [0.95] (0.92) 0.99 [0.91] (0.83) 0.98 [0.89] (0.76)

0.00 [0.00] (0.00) -0.01 [0.00] (0.01) −0.02 [0.03] (0.03) −0.01 [0.05] (0.08) 0.01 [0.09] (0.17) 0.02 [0.11] (0.24)

Ne

Ar

Kr

Xe

Rn

a


with its magnitude giving a lower bound to the error of the method.) C. Dissociation Energies. We now compare the dissociation energies in the M+−RG and M+−RG2 complexes and both De1 and De2 are shown in Tables 1 and 2, together with the De2/De1 ratio and the average value, Deav. We commence our discussion with the MIa+−RG2 complexes. We note from Table 1 that these have very similar De1 and De2 values, and hence the value of Deav is also close to each J



Article

of these, and the De2/De1 value is very close to unity. Both De1 and De2 show a monotonically increasing trend with atomic number of RG; this is in line with the increasing polarizability of the RG atom (Table 3). In addition, there is a trend for the De2/De1 ratio to fall slowly with the increasing atomic number of the RG atom. To explain this, we note that the ionization energy of the RG atom (Table 3) is falling with increasing atomic number, and so there is an increased propensity for charge transfer. Indeed, there is a gradual increase in the charge calculated on the RG atom as the atomic number increases, in line with this (Tables 4 and 5). The charge transfer for two RG atoms will be slightly less than twice that of a single the metal atom, because there will be more electron repulsion from the transferred charge, ameliorating the effect; this expectation is borne out by the AIM charges. This will lead to the observed small decrease in De2/De1 ratio with increasing atomic number. We have shown9 that for the diatomic MIIa+−RG complexes, as the RG atom approaches the MIIa+ cation, the unpaired outermost diffuse s electron is able to undergo sp hybridization, as noted above. This clearly costs energy but is compensated by the reduced electron repulsion between this (isotropic) s electron and the electrons on the incoming RG atom, and the increased attractive terms arising from the increased effective nuclear charge. Together, these lead to much higher binding energies than might be expected. Essentially the same effect occurs in the case of the triatomic complexes, MIIa+−RG2; however, when two RG atoms approach, there would need to be an increased amount of reduction in s electron density on the side of the approaching RG atoms, to allow for the interaction with the two rare gas atoms in the same manner. Clearly there is an increased energy demand for this to occur, and there are also considerations relating to the relocalization of the electron density. Because Be+ has smaller orbitals than Mg+, arguably the electron repulsion terms could be significant when the electron density shifts to the far side of the MIIa+ atom, and so preventing double the electron density dislocation in MIIa+−RG2. The sudden drop in De2/De1 value between the neon- and argon-containing complexes is striking. We note that this ties in with a loss of degeneracy seen in the molecular orbital plots for the outermost occupied RG p orbitals; additionally, the corresponding contour plots show that there is some overlap between these orbitals on different centers. These facts suggest that the greater polarizabilities of the RG atoms with higher atomic number,38 coupled with the lowering of electron density on the side on which the RG atom is approaching, is allowing them to get closer than they would otherwise be (in RG2) and so introducing a repulsive term, lowering De2 compared to De1. This idea is supported by the RRG−RG/Re(RG2) values, which jump down from 0.96 to 0.87 between Be+−Ne2 and Be+−Ar2 showing that the Ar atoms cannot get as close as they would do in Ar2; this does not occur for Be+−He2 or Be+−Ne2 because these are much smaller and harder. Thereafter, as the atomic number of the RG increases, there is an increased competition between the attractive and repulsive effects caused by the interaction between Be+ and the two RG atoms, whose polarizability is increasing with atomic number. In addition, the gradual fall in De2/De1 can be partly attributed to the decreasing ionization energy of the RG atom, and so increased charge transfer and hence larger δ+ charge expected on the RG atoms. The AIM atomic charges in Tables 6 and 7 partially support this idea, although the effect is perhaps smaller than may be expected: whether this is due to an underestimation of the effect by the AIM analysis or not is

unclear. Certainly, as noted above, the Mulliken and NPA charges suggest amounts of charge transfer that seem unphysical. The above ideas are reflected in the RRG−RG/Re(RG2) ratios, which gradually decrease as the atomic number of the RG atoms increases, suggesting these are resisting being moved together. It is interesting to note that overall the Re2/Re1 ratios do not appear to be sensitive to these competitive processes. A similar set of observations is made for Mg+−RG2, where again for the lightest two complexes, De2/De1 values of around unity are observed, but then we see a relatively sharp drop between Mg+−Ne2 and Mg+−Ar2, followed by much smaller drops thereafter. The effects are significantly less than they are for Be+ because Mg+ is larger; thus, the benefits of hybridization, and the concomitant processes outlined for Be+ are correspondingly reduced, even though Mg+ is more polarizable (Table 3) and so it should be easier to distort the 3s electron density. Additionally, the steric repulsion between the RG atoms will be greater in Be+−RG2, explaining the smaller fractional increase in the De2 values in the corresponding Be+−RG2 cases compared to Mg+−RG2.

5. CONCLUSIONS We have shown that for the Li+−RG2 and Na+−RG2 complexes the equilibrium geometries are linear or quasilinear, each with a very flat potential around the (quasi)linear geometry. In contrast, the MIIa+−RG2 complexes are all highly bent, and these bent geometries can be explained by sp hybridization, with it being energetically more favorable for the two RG atoms to approach from the same side, leading to electron density being able to move away from both RG atoms; something that would not be possible if both RG atoms approached from opposite sides in a linear orientation. These observations were confirmed by contour plots of the HOMOs. For the other Be+−RG2 and Mg+−RG2 complexes, there is significant distortion of the outermost singly occupied s orbital, showing sp hybridization. The hybridization process not only reduces electron repulsion but also increases the effective charge seen on the metal center by the incoming RG atoms. The equilibrium geometry is a balance of these effects, coupled with the effects of steric and induced charge repulsion between the two RG atoms. We have examined the trends in the equilibrium internuclear separations and the energy required to remove a single RG atom from MIIa+−RG2, De2, comparing these to the diatomics, MIIa+−RG, De1. We have noted very striking differences in the De2/De1 ratios for the different complexes and we have discussed these. Finally, charge transfer processes clearly are expected to affect dissociation energies, and some of the trends in the dissociation energies were attributed to this. However, it was not so straightforward to obtain reliable atomic charges, because there were obvious weaknesses seen in both the Mulliken and NPA (molecular-orbital-based) analyses. The results from the AIM (electron-density-based) analyses were much more in line with expectations, particularly for the MIa+−RG and MIa+−RG2 systems; for the group 2 species, it seems the AIM analysis may be slightly underestimating the amount of charge transfer, but the charges from this method were much more reasonable than those obtained from either Mulliken or NPA analyses; small, artifactual negative charges on the RG atoms was occasionally observed. It is striking how complicated these simple systems are and shows the value of being able to look at prototypical systems in detail. It would be interesting to undertake spectroscopic K



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investigations of these species to confirm the trends observed. In future work we shall extend these theoretical investigations to the heavier MIIa+−RG2 complexes, where the involvement of low-lying, formally unoccupied d orbitals is expected to be important, based on our work on the heavier MIIa+−RG complexes.10,11

■

AUTHOR INFORMATION

Corresponding Author

*T. G. Wright: e-mail, [email protected]. Present Addresses §

Chemistry Building, Department of Chemistry, Emory University, Atlanta, GA 30322, U.S.A. ∥ Max-Planck Research Group “Structure and Dynamics of Cold and Controlled Molecules”, Center for Free Electron Laser Science, Building 99, Notkestrasse 85 D-22607 Hamburg Germany. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors are grateful for the provision of computing time by the NSCCS. The EPSRC is thanked for funding, and A.A., A.M.G,. and R.J.P. are grateful to the EPSRC and The University of Nottingham for the provision of studentships. W.H.B. is grateful to the Department of Chemistry at the University of Utah for travel funding, allowing visits to the University of Nottingham.

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M+ = Li, Na, Be, Mg; RG = HeâRn - ACS Publications - American

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M+ = Li, Na, Be, Mg; RG = HeâRn - ACS Publications - American