minor empirical modifications provide a reasonable way to do this
4.2 Comparison of model and experimental results
4.2 Comparison of model and experimental results
For the systems investigated in [9], model calculations were performed, the results of which are shown in Figure 4.3. The difference in solvation energy∆SEwas calculated as
∆SE(n) = VDE(NgnX−)−VDE(X−) (4.15)
from the experimental values ofVDE, the vertical detachment energy, i.e. the energy required to remove an electron from the anion. Thus, the difference in solvation en-ergies can be interpreted as the increase of the energy needed for the removal of an electron from the anion caused by the presence of the noble gas atoms in the cluster.
The theoretical counterpart is
∆SE(n) = Vneutralmin −Vanionmin +
∑
i
1 2
ωneutrali −ωanioni (4.16)
whereVminis the depth of the potential energy well at (for the anionic cluster) or clos-est to (for the neutral cluster) the global energy minimum for the anionic cluster, and the sum represents the difference in zero point energies for the anion and the neutral cluster. Thus, ∆SE(n) with the X state as a reference corresponds to EAX(NgnX) in Figure 4.1.
The comparison of theory and experiment rests on the assumption that the electron detachment takes place mostly to the vibrational ground state of the neutral cluster. In other words, it is assumed that a minimum in the potential energy function of the neu-tral cluster exists for which the geometry of the cluster does not differ greatly from the anionic global minimum, such that the Franck–Condon factor for the 0-0 vibrational transition is the dominant one. This may well be a local minimum of the potential en-ergy function of the neutral state. Additionally, the experiment was usually not able to resolveX and I states. Noticing that obvious deviations are substantially greater than what would be expected to be introduced by either of these simplifications, however, they appear acceptable.
It should be noted that some changes in the parameters used have been made, resulting in values differing somewhat from those presented in [9]. The potentials and parameters used here are listed in Appendix C. All model results exhibit shortcomings when compared to the experimental values, which shall be analyzed in this section.
Beginning with the fluorine systems, it is obvious that only for the case of ArnF
Chapter 4. Interactions in weakly bound noble gas–halide clusters
4.2. Comparison of model and experimental results Figure 4.3:(facing page) Comparison of experimental and model results for the difference in solvation energy∆SEfor various halide–noble gas systems. “Bin. model” refers to calculations without many-body effects. e = 140.54 meV andranionm = 4.05 Å refer to changes in the Xe–Br−binary potential parameters (see text).
could a qualitative agreement be achieved. Of course, xenon and krypton are well-known to form compounds with fluorine, while compounds for argon with fluorine and hydrogen have been found only fairly recently[185], a fact that underscores the strength of the interaction between fluorine and these noble gases. The very short equilibrium distances of the corresponding anionic potentials lead to an exaggeration of the largest repulsive contribution, induction nonadditivity. As already pointed out elsewhere[3], an uncertainty in equilibrium distance in the binary potentials translates into an uncertainty in energy from all non-additive potential contributions.
Additionally, an uncertainty in the shape of the more distant part of the potential certainly has a detrimental effect on the accuracy of the energy from the second sol-vation shell onward. For instance, the quadrupole contribution to the Xe–F− binary potential (MMSV or Morse-Morse-Switching-van der Waals-type) has obviously not been determined appropriately, while the quadrupole contributions in the Ar–F− and Kr–F− potentials appear exaggerated. In the case of the latter two, the outermost no-ble gas atoms (see Figure 4.5) are close to or in the range of the multipole part of the MMSV potential (i.e.rNg−X >x2·rm) and indeed∆SEfor KrnF in the binary potential (Figure 4.3) level off much too strongly upon completion of the first shell of noble gas atoms (n > 6). However, it is obvious that strong deviations from the experimental values start much earlier: Even the values for clusters with only a single noble gas atom do not agree well with the experiment for XeF, and even more so for KrF, which is clearly due to a wrong well depth of the binary MMSV potential.
Figure 4.4 depicts the ratio of induction nonadditivity to the sum of all binary interactions between the halide anion and the surrounding noble gas atoms. The effect of the former starts out to be small for every combination of halide and noble gas and rises, to then level off as the first shell of noble gas atoms around the halide closes.
With respect to the total energy, this corresponds to a decrease in relative contribution of induction nonadditivity, as the contribution of the noble gas–noble gas interactions becomes dominant. Generally, the value for a cluster with only two noble gas atoms is below 0.1, except for Kr2F and Xe2F, where it is roughly 0.15 and 0.22, respectively.
Likewise, the upper limit for this ratio is generally about 0.25 to 0.3, while it is between 0.45 and 0.5 for KrnF and XenF. The upper limit of XenBr of 0.37 appears to be shared
Chapter 4. Interactions in weakly bound noble gas–halide clusters
1 5 10 15 20 25
0 0.1 0.2 0.3 0.4 0.5
n
Vind−nonadd
∑ iV Rgi−X−
ArnF KrnF XenF KrnCl XenCl ArnBr KrnBr XenBr
KrnI
Figure 4.4:Ratio of induction nonadditivity to the sum of binary potential contributions be-tween the halide anion and all noble gas atoms as a function of cluster size.
by ArnF and seems to be still within the validity of the assumptions for induction nonadditivity, given that ∆SE(n) does not become erratic as in the case of XenF and KrnF after closure of the first solvent shell. A substantial contribution to the deviation from experimental values can thus certainly be attributed to overestimated induction nonadditivity in XenF and KrnF.
Another large deviation was found for XenBr-type clusters. Note that in the pre-sentation of the experimental data given in [9], these were in rather good agreement with the experimental values due to a change of the anionic potential well depth from 126.92 to 140.54 meV, as shown in Figure 4.3. Obviously, however, this is not a good correction as the binary potential should agree with the experimental value of XeBr (i.e. the case of a single interaction partner) and increasing the binding energy of the binary potential necessarily leads to worse agreement for all small clusters.
An increase in the anionic equilibrium distance, on the other hand, can have a similar effect in correcting the deviations from the experimental values, as also shown in Figure 4.3, by decreasing the mostly repulsive contributions of the non-additive potential contributions. Changes in this parameter leave the energy of the smallest cluster unaffected, but the slope of ∆SE for clusters with a closed first solvent shell obviously decreases too strongly. Ultimately, the given experimental data is not
suffi-4.2. Comparison of model and experimental results cient to determine which potential parameters require improvement and to what ex-tent, especially when considering that the binary potentials of the neutral species offer yet more parameters. It may, however, be assumed that parameters pertaining to the non-additive contributions to the potential energy, e.g. dipole and quadrupole polar-izabilities, are largely correct as this large deviation is not observable in other clusters containing either xenon or bromine. Similar problems appear to exist for the ArnF species, although to a much lesser degree.
For the remaining clusters, the greatest difficulty for the model appears to be to describe the closing of the first shell of noble gas atoms correctly. In KrnI, ArnBr, KrnBr, and KrnCl, the ∆SE value of the cluster n =14, 12, 13, and 13, respectively, exhibit an irregularly small increase in ∆SE. From the halide–noble gas distances shown in Figure 4.5, it can be inferred that these are due to some rearrangement within the not yet filled first solvation shell. For KrnBr and KrnCl, distances from the halide core have a larger spread in these instances, whereas in KrnI and ArnBr the spread in noble gas–halide distance is smaller in these cases than in the next larger and smaller clusters.
The most straightforward explanation seems to be that the equilibrium distances of the corresponding halide–noble gas potentials are insufficiently accurate.
Additionally, in many cases (especially KrnI, ArnBr, KrnBr, KrnCl) the slope of∆SE does not decrease after the completion of the first solvation shell as strongly as is the case for the experimental data. This indicates deficiencies in the long range part of the potential: most likely an overly attractive anionic potential, because it is the greatest contribution to impact on an increase of∆SE. This is true, although the binary noble gas interactions are the largest component of the total energy of the anion. For the computation of ∆SE these are indeed practically completely eliminated as they also exist, with almost identical magnitude, in the neutral clusters (see Figure 4.6). In other words: their significance extends mostly to the determination of the cluster geometry.
Geometries are of interest in particular with regard to the size of the first solvation shell. From the plot of distances in the various clusters (Figure 4.5), it is apparent that for KrnCl, KrnBr, XenCl, and XenBr, a tendency to form icosahedral structures is prevalent. In two systems, namely XenCl and XenBr, the icosahedron constitutes the inner shell of noble gas atoms as further atoms are added to the second shell. For XenBr even the start of a third shell can be seen. After it is energetically favorable for the first few noble gas atoms of the second shell to be located over the triangular faces of the icosahedron of the first shell, eventually forming a pentagon, it is then apparently more favorable to add the next xenon atom centered over that pentagon. It is thus also
Chapter 4. Interactions in weakly bound noble gas–halide clusters
1 5 10
3 5 7 9
ArnF
n rNgX/Å
1 5 10
KrnF
n 1 5
XenF
n
1 5 10 15
3 5 7 9
KrnCl
n rNgX/Å
1 5 10 15
XenCl
n
1 5 10 15
3 5 7 9
ArnBr
n rNgX/Å
1 5 10 15
KrnBr
n 1 5 10 15 20 25
XenBr
n
1 5 10 15
3 5 7 9
KrnI
n rNgX/Å
Figure 4.5:Halide–noble gas distances obtained from the model calculations for the respective anionic clusters.
4.2. Comparison of model and experimental results
1 5 10 15 20 25
−2.5
−2
−1.5
−1
−0.5 0 0.5
n
e/eV
Xe–Xe (anion) Xe–Br−(anion) induction nonadditivity (anion) exchange (anion)
dispersion multipole (anion) triple dipole (anion) Xe–Xe (neutral) Xe–Br (neutral) triple dipole (neutral)
Figure 4.6:Contributions to the total cluster solvation energy (SE) for the XenBr system (data for the neutral system is for theXstate; the unmodified potential was used for the anion).
on top of one of the tips of the icosahedron. This is a manifestation of the growing importance of xenon-xenon interactions versus xenon-bromide interactions as a result of increasing shielding of the attraction of the anion.
For KrnCl and KrnBr, the preference of an icosahedral geometry is noticeable from a substantially smaller spread in halide–noble gas distances, as compared to both smaller as well as larger clusters. In both cases, the icosahedron is subsequently dis-torted to enable the first solvation shell to accommodate two more noble gas atoms and eventually less symmetrical polyhedrons form the first solvation shell. Contrary to what was assessed earlier[9]about structures of clusters with an incomplete first sol-vent shell in these systems, they do not resemble a capped square antiprism forn =10, but ratherarachno[186]-type icosahedrons (i.e. icosahedrons with two apex atoms miss-ing).
In the cases of ArnBr and KrnI, shell closure occurs only with the 15th added noble gas atom, as the smaller ratio of van-der-Waals radii of noble gas and halide would
Chapter 4. Interactions in weakly bound noble gas–halide clusters lead to expect.
Marking the opposite end of the scale, fluorine clusters prefer much smaller sol-vation shells, with Ar8F being a square antiprism, of which subsequently (n = 9 and 10), caps are added to the squares and then the to the triangular faces (n > 11). Kr6F and Xe6F are both octahedral, but in both cases the addition of further noble gas atoms leads to distortion of the octahedron. While for KrnF (n > 6) subsequently added atoms are substantially farther from the halide, Xe8F is of square antiprismatic shape, although this is again distorted greatly in Xe9F.
This entire analysis of geometries, it should be noted, depends on the somewhat questionable (vide supra) accuracy of the equilibrium distances of the interaction po-tentials.
4.3 Conclusion
Improvements over the original work of Yourshawet al.[3]have been achieved by find-ing a concise analytic solution of the Lawrence and Apkarian-Hamiltonian[164] and the adoption of a hypertensor[177]-like treatment of induction nonadditivity. The lat-ter proved helpful for finding analytic derivatives of this energy contribution and thus avoiding the more costly numerical derivation.
While interesting insights into cluster geometries could be gleaned from model cal-culations, weaknesses of the underlying binary potentials were discovered by compar-ing the model results to experimental data. It seems that particularly the equilibrium distances of these potentials require greater accuracy, since many-body potential con-tributions are very sensitive to them. At the same time, the model may have general difficulties in describing larger clusters with a closed first solvent shell.
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