J . Phys. Chem. 1994,98, 3513-3517
3513
Microscopic Study of Fluoride-Water Clusters Jaime E. Combariza' and Neil R. Kestner' Department of Chemistry, Louisiana State University, Baton Rouge, Louisiana 70803 Received: August 31, 1993; In Final Form: November 1 1 , 1993@
W e have carried out a b initio calculations on the structure and energetics of F-(H20),, n = 1-6, clusters at the self-consistent mean-field (SCF) and second-order Moller-Plesset (MP2) levels of theory. The rather strong F--H20 interactions determine the strength of the hydrogen bonds between the water molecules and thus the structure of the clusters. Our data show that a transition from a predominant surface state to an interior state is observed at n = 4 and 5 . By comparison with our previous calculations of other halide-water clusters, the strength of the halide-water interactions appears to be a n important determining factor in the size at which a surface to interior transition is observed. The inclusion of zero-point energies and entropy corrections seems to reverse the energetic stability, making the interior states preferred over the surface states.
I. Introduction The microscopic study of solvation processes, in particular of ions solvated by polar molecules, is of extreme importance in understanding the chemical physics of ion solvation, excess electrons in fluids, and ion-molecule reactions, to name but a few important processes in the condensed phase. However, many ions have very different properties depending on whether or not they are solvated. For example, halide ions have very high-energy excited states until they are dissolved in polar media. Then a totally new low-energy spectral feature, the so-called charge transfer to solvent (CTTS) state, suddenly emerges. Sodium chloride is a stable molecule in the gas phase but dissociates into ions in very polar media. These properties must depend on the number of solvated molecules surrounding the ion. Thus, we are naturally drawn to a study of clusters in which we can watch the evolution of properties as a function of cluster size. The size of the clusters determines specific properties such as conductivity and insulation,' the energetics and thermodynamics,24dynamics,6J and chemical behavior.618-10 For instance, it is of great interest to obtain an idea about the minimum size required to induce a specific chemical process.s.10 A detailed study of the size effects on specific properties of clusters likewise provides a quantitative description of the gradual transition from small finite clusters to larger clusters and eventually to the infinite condensedmatter system, as extensively discussed in numerous papers by Jortner." From the experimental point of view, spectroscopic techniques have been widely used to study electron attachment:-l0 as well as the kinetics,12 and thermodynamic properties1s15 of ions in water solutions. Bowen et al.8.9 carried out studies on the photodetachment of electronsfrom water clusters, following initial mass spectroscopic detections of water anions by Haberland.10 Theoretical calculations16 suggest the transition occurs via two stages: a surface indented state of intermediate sizes and the true interior state at clusters sizes beyond 32 water molecules. The hydration of alkali-metal cations and halide ions in clusters has been studied in some detail by Kebarle,13 Castleman,14 and others.Is Recently, Cheshnovsky et al." reported photoelectron spectroscopy (PES) of negatively charged ions solvated in water clusters. In these PES experiments, vertical transition energies can be obtained which systematically study properties of halide anions solvated in water clusters of various sizes. Many computational studies have been performed on trapped and solvated electron states in clusters16.18-2' and on ions solvated 0
Abstract published in Aduancs ACS Absrracrs. March 1, 1994.
0022-3654/94/2098-35 13$04.50/0
in water cluster^.^^-^^ Recent calculations have made use of molecular dynamics simulations in an attempt to reproduce thermodynamic data. These simulations suggest that the type of potentials used to describe the water-ion and water-water interactions can affect the results. For instance, Berkowitz and PereraZ4obtained two general types of structures from their calculations. A surface structure was obtained when the SPCE/ POL25 potentials were used and a solvated structure when the TIP4P26modelwas used. This latter model representsthe waterion interactionsas an effectivepair potential. The main difference between these two types of potentials is that the SPCE/POL model includes polarization of water molecules and ions. Other simulations on halide anions water clusters also suggest that the surface structure is preferred over the solvated structure. Previously, we reported a theoretical study of the structure and energetics of halide anion solvation in water clusters.27 Calculations for the energetics and equilibrium configurations for the X-(H20),, n = 1-6 and X = CI, Br, and I, clusters were compared with the experimental data from photoelectron spectroscopy by Che~hnovsky.~~ Specifically, those calculations explored the effects of size (number of solvent molecules) on ion solvation in polar clusters. For small clusters (n< 5 ) only surface states were found. In this paper we address the interaction of very small anions with water molecules and, more specifically, clusterswith strong anion-water interactions. Our choiceof anion is fluoride. Several studies have been camed out on the interaction of a fluoride anion with a water m o l e c ~ l e ? ~but f ~to our knowledge no theoretical study of the sequentialaddition of water molecules to form the ionic cluster has been carried out. The size of the fluoride anion, 1.33 & is ca. 0.5 A sqaller than the size of the C1-, B r , and I- anions (1.81, 1.96, and 2.20 A) which where studied previously.27 In addition, fluorineis more electronegative than any other element, and thus the competition between the F-HzO and H20-H20 interactions is more important. During the revision process for this paper, it was called to our attention a concurrent molecular dynamics simulations being carried out by Perera and B e r k o ~ i t z . They ~ ~ find minimum-energy configurations (0 K) for F-(H20),, n = 6-8, in which the anion is located on the surface. At higher temperature (250 K) the fluoride anion is located inside the water molecules, and thus they concluded that the solvation of the fluorideanion is due to entropy effects. In section 2 we give some details of our calculations as well as a description of the model used, ending with a description of the two general types of structures considered in this study. The results are presented in section 3 followed by their interpretation and comparison with experimental data. 0 1994 American Chemical Society
3514 The Journal of Physical Chemistry, Vol. 98, No. 13, 1994
d
2
b
Combariza and Kestner
a
d
e
b
e
C
f
161A
0 I61 A
C
f
Figwel. Optimidgeometries(SCF)for theF-(H20),,n= 1-4,clusters, F-H, 0-0,and H-O distances are given for each structure.
2. Method and Model
Standard ab initio molecular orbital calculations were carried out using extended basis sets. Optimization at the self-consistentfield (SCF) level were performed without any constraints in the geometry, and full geometry optimizations at the Moller-Plesset (MP2) level were carried out, but only for the clusters F-(HzO),, n = 1-3, due to size limitations. Consequently, single-point calculations at the MP2 level were carried out for all clusters using the SCF-optimized geometries. Vertical ionization potentials were obtained from the MP2 energy difference for the anion and neutral species. In addition, frequency calculations were carried out to differentiate local minima from transition states and obtain zero-point energies. These calculations were done using the standard quantum chemistry packages Gaussian 903’ and Basis Sets. In ab initio calculations it is important to select good basis sets which properly describe the electronic properties of the system being studied. A common practice is to use experimental information to assess the reliability of these basis sets. We are primarily interested in obtaining electronicenergies, ionization potentials (IP), and thermodynamic properties that are in good agreement with experimental data.I3 The water molecules were represented by the standard 6-3 1+G*33 basis sets. Full double-l basis sets (M~Lean-Chandler3~) were used as the parent functions to represent the fluoride anion. A set of diffuse sp functions was added along with three d functions whose exponents were optimized using the trudge procedure as implemented in the Gamess program. The exponents obtained were 0.006 415 for the diffuse sp functions and 7.098 035, 1.871 244, and 0.472 510 for the three d functions. Using these basis sets, we obtained excellent agreement between calculated and experimental properties for the F- anion and the F-(H20) cluster. For instance, we obtained an ionization potential value of 3.5 eV at the MP2 level (3.3 eV at the QCISD(T) level) for the F- anion
Figure 2. Optimized surface and interior structures (SCF) for the F-(HzO),, n = 5 and 6, clusters. Hydrogenbonds between water molecules
are represented by dashed lines.
vs an experimental value of 3.4 eV.35 For the F-(H20) cluster we obtained an IP of 4.7 eV at the CI level vs 4.74 eV obtained by K a l d ~ using r ~ ~ the coupled-cluster method. The enthalpy of formation for F-(H20) obtained in this study was 24.5 kcal mol-’ vs an experimental value of 23.3 kcal mol-’, and a calculated value of 23.85 kcal mol-’ obtained using Monte Carlo simulations.28 Model. Hydrogen bonding between the water molecules has been shown to determine thestructure of smallpolar clusters.~4~*7~30 The interaction of an anion with a water molecule is very complex. For instance, consider the interactionof a F-with a water molecule. The negative charge is located on the halide, and thus it will interact most strongly with one of the hydrogen atoms on the water molecule,thus producing an asymmetricstructureas shown in Figure 1. The second hydrogen atom can interact with other water molecules even in the same solvation shell and induce an associationof water molecules, held together by networksof waterwater hydrogen bonding. Thus, these short-range interactions forces play an important role in the energetics, electronic properties, and structure of these clusters as demonstrated in our previous studies.27 The sequential addition of water molecules to the cluster generates two types of structures. The water molecules can interact directly with the halide anion or form hydrogen-bonded structures by interacting most strongly with the other water molecules. The addition of water molecules to the cluster generates several isomers with energies determined according to the dominant type of interactions. Large number of isomers are more evident for n > 3 as shown in Figures 1 and 2. While the above may rigorously apply to halide-water clusters with halides such as C1-, B r , or I-, the picture may drastically change for the fluoride anion. F- interacts more strongly with
The Journal of Physical Chemistry, Vol. 98, No. 13, 1994 3515
Microscopic Study of Fluoride-Water Clusters
TABLE 1: Energy Differences and Zero-Point Energies (kcal mol-') for the Clusters F-(H20), n = 1-6. ~ n structure MscF AESCF 4- ZPE "2 A E M4-~ZPEb 2 2 4 4 5 5 5 6 6 6
H-B
0.414 0.0 -0.908 3-1 0.0 prism 1.052 4-1 0.0 0.076 3-2 V-shaped 2.336 dist octahedral 0.0 -0.031 octahedral
linear pyramidal
0.090 0.0 -1.310 0.0 -0.924 0.0 -0.05 0.430 0.0 1.015
0.588 0.0 -0.413 0.0 3.608 0.0 -0.430 5.866 0.0 -0.485
0.264 0.0 -0.831 0.0 1.630 0.0 -0.556 3.960 0.0 0.560
Positive difference indicates more stability. * Zero-point energies calculated at the SCF level. a water molecule than any of the other halides, and in fact, this interaction is a lot stronger than the hydrogen bond between two water molecules. This strong interaction of the F- anion with the water molecules increases the H20-H20 repulsion, and thus the hydrogen bonds between the water molecules become weakened. In addition, the F- anion has a much lower polarizability that the other anions. Figure 1 shows the interaction of 2-4 water molecules with a F- anion as determined in this study. Surface and Interior States. The existence of interior and surface states was thoroughly discussed in our previous papers.2' An interesting effect yet to be understood is the size of the cluster at which a transition from the dominant surface state to an interior state occurs. We found some evidence for I-(H20), that this transition may occur at n = 6. However, a general theory on the (S) (I) transition is still not clear. Due to the small size of the fluoride anion in addition to the strong electrostatic interactions, this transition should be easily observed in F-(H20),, n = 1-6, clusters, and thus this study may provide some valuable hints to develop a general theory on this effect.
-
3. Results and Discussion Energetics and Ground State. The first electronic property we consider is the total energy. In Table 1 we present the energy differences for the F-(H2O), isomers (n = 2,4,5,6) at the S C F level and at the MP2 level, using the optimized S C F geometries. Energy differences corrected for zero-point energies are also given in Table 1. The energy differences at the SCF are very small for all cases considered while the energy differences a t the MP2 level for n = 5 , 6 are more pronounced. The largest difference a t the SCF level is 2.336 kcal mol-' and at the MP2 level is 5.866 kcal mol-' for the surface cluster with six water molecules with respect to the interior (distorted octahedral) structure. However, it is worth noting that zero-point energy corrections reduce these energydifferencesat both the SCFand MP2 levels. Optimizations at the MP2 level for the clusters with fewer than three water molecules did not substantially change the geometry or energetic differences. From the energetic data we observed the following features. (a) For n = 2, the energy difference both a t the SCF and MP2 level is very small. Zero-point energy corrections are very small for these two clusters, and thus it is not expected to affect the energetic stability. In Figure lb,c we present the optimized geometries for the two isomers considered. We observe clearly that the hydrogen bond in these two structures is very weak, and thus the dominant force is the F--HzO interactions. (b) The clusters with four water molecules (Figure le,f) show some interesting features induced by the strong F--H20 interactions. The hydrogen bonds in the surface state are very weak. The H20-Hz0 repulsion, induced by the proximity of the Fanion to the water molecules, is high, and thus thewater molecules are further apart than in the Cl-(H20)4 clusters27 (0-0= 3.31 A in F- clusters vs 2.99 8,in C1- clusters). The addition of the fourth water molecule to form an interior state (3-1) reduces the H2O-H20 repulsion as compared with the cluster for n = 3,
F-(H20)3, Thus, from the final geometries for the clusters with
n = 3 and 4 water molecules we concluded that as the F--HzO interactions are weakened, the H20-H20 interactions or hydrogen bonds between the water molecules are enhanced. The small energy difference between the interior and surface states, at the SCF and MP2 levels, suggests that a compensatory effect exists between these two interactions. Inclusion of zero-point energies enhances the stability of the interior state (n = 4), with respect to the surface state;however, due to the small energeticdifference, a definite assignment of the preferred structure cannot be made solely based on the energetics. (c) For n = 5 we considered three isomers. The first structure started as a surface state with a pentagonal pyramid geometry. Interestingly, full optimization of this structureresulted in a geometry correspondingto an interior state with two groups of water molecules (3-2) above and below the fluoride anion (Figure 2a). The second structure considered (Figure 2b) was an interior state (4-1). As in the case for n = 4, the addition of the fifth molecule on top of the cluster produces an enhancement of the HzO-HzO interactions (hydrogen bonds). The third structure considered was a highly hydrogen-bonded structure (Figure 2c). The F- anion and H20 molecules are arranged in a triangular prismatic shape. Three of the water molecules interact strongly with the halide anion while the other two interact with the other water molecules and are located farther away from the halide. Energetically, this cluster is more stable by about 1 kcal mol-' a t the SCF level and 3.6 kcal mol-' a t the MP2 level. However, correction for zero-point energies reverses the energetic stability of the surface state at the S C F level and reduces the energy difference at the MP2 level by ca. 2 kcal mol-' (see Table 1). One can always expect that the inclusion of zeropoint energies at the MP2 level will further reduce this energetic difference and even reverse the energetic stability a t the MP2 level, making the interior state also more stable. The energetic stability of the surface structure can be explained by the combination of strong F--H20 interactions and strong hydrogen bonds between the water molecules. A possible rearrangement of the two outer water molecules, which do not interact directly with the anion, may be sterically hindered by the three water molecules which interact directly with the halide. (d) For n = 6 we also considered three structures. The first one started as a surface state with the water molecules forming an hexagon and the F- anion located on top of the water molecules. Full optimization resulted in an interior structure with the water molecules surrounding the halide anion (octahedral shape) and very weak hydrogen bonds between them. The F--HzO interactions on the contrary are very strong (Figure 2d). The second structurewas a distorted octahedron (Figure 2e). In this structure the two subgroups of water molecules are strongly bound by hydrogen bonds, and thus the F--H20 interactions are weaker. As in previous cases, the small energy difference indicates a compensating effect between these two types of interactions. The final geometry considered was a V-shaped structure according to the results from Perara et al.24 This structure is highly hydrogen bonded, and thus it is energetically more stable than the previous two structures by about 2.3 kcal mol-' at the S C F level and 5.9 kcal mol-' a t the MP2 level. Similar to the surface state for n = 5, this structure presents strong F--H20 interactions as well as strong hydrogen bonds between the water molecules. As noted for the clusters with n = 4,5, the addition of zero-point energies reduces the energetic difference between the surface and interior states. At theSCFleve1 the twostructuresarealmost isoenergetic, and a t the MP2 level this difference is reduced by ca. 2.0 kcal mol-'. The reduction of the energetic difference due to the zeropoint energies leads us to expect that entropy effects may play a substantial role in determining the energetic stability of the interior and surface states. These effects will be discussed later. Ionization Potentials. The use of ionization potentials and specifically ionization potential differences(shifts) has been proven
Combariza and Kestner
3516 The Journal of Physical Chemistry, Vol. 98, No. 13, 1994
TABLE 2 Vertical Ionization Potentials and Integral and Differential Shifts for the F-(HzO), D = 1-6. n structure IPv(n)b dIv(n)c DIV(n)d 1 2 2 3 4 4 5 5 5 6 6 6
1.549 2.966 3.072 3.791 3.992 4.039 3.970 4.576 4.841 4.291 4.873 4.689
5.049 6.466 6.572 7.291 7.491 7.539 7.470 8.076 8.341 7.797 8.373 8.189
H-B
linear pyramidal 3-1
prism 4-1 3-2
1.459 1.417 1.523 0.825 0.200 0.248 -0.021 0.537 0.802 0.327 0.297 0.113
V-shaped dist octahedral octahedral 0 All units are eV. Vertical ionization potential as given in eq 1. Integral shift. Equation 2. Differential shift. Equation 3.
TABLE 3: Total Bimling and Sequential Enthalpies of Solvation ( k d mol-') and Vibrational Entropy for the Clusters F-(H20), D = 1-6
1 2 2 3 4 4 5 5 5 6 6 6
H-B
linear pyramidal 3-1
prism 4-1 3-2
24.535 44.577 44.163 61.999 75.955 76.864 90.298 89.246 89.322 103.744 101.408 101.378
24.535 20.042 19.628 17.42 13.957 14.865 13.342 13.29 13.366 (12.458) 13.446 (14.422) 11.1 10 (12.160) 11.080 (12.132)
23.30 19.2
15.3 13.9 12.2
1.88 19.47 17.33 28.64 42.80 46.94 46.09 59.26 58.69 54.69 66.32 72.85
V-shaped 10.9 dist octahedral octahedral Total binding energy according to eq 4. Sequential enthalpy of solvation according to eq 5. For n = 5, 6 the solvation enthalpics are given with respect to the surface states and in parentheses with respect to the interior states. Experimental values taken from ref 13. Vibrational entropies in cal mol-' K-I. @
6.0 5.0
1
1
. t
p 3.0 x e 2.0
." ,
1
,.oJ 0.0
1
2
-o,51 x;:::: 1 0.0
3
4
5
6
n, Number of Water Molecules
-1.0
1
L------
0
1
2
3
4
5
6
n, Number of Water Molecules
Figure3. Comparisonof experimentaland calculatedionizationpotential shifts: (left)integral shift vs number of water molecules; (right)differential shift vs number of water molecules. useful in elucidating the preferred surface or interior structure in cases where the energetic difference is very ~ma11.3~We will present the IP results as differences between vertical ionization potentials obtained using the expression given in eq 1. The ionization potentials are given in Table 2.
IPv = E M P ~ [ F ( H ~ O-)E*MP~[F(H~O),I ~I
(1)
where Ehipf[F-(H~O)~l and E*MPZ[F(HZO)~] are the energies at the MP2 level for the anion and its respective neutral counterpart at the fixed geometry of the anion. The first shift considered is the integral shift. We subtract the ionization potential of the bare anion from the ionization potential of the cluster with n number of water molecules. dIv(n) = IPv[F(H2O)nl- IPv[Fl (2) A second shift was also defined in order to better analyze the results. The differential shift is defined as
(3) DIv(n) = IPv[F(H2O),1 - IPv[F(H20),11 The values for the integral and differential shifts are given in Table 2 and plotted against the number of water molecules in Figure 3. We observe the following trends: (a) the integral shift increases monotonically up to n = 3. For n = 4 the dIv(4) for both the interior and surface states flattens out. It is interesting to observe that there is very little difference between the dI(n) and DI(n) for the interior and surface states (n = 1-4). These findings once again suggest that the F-H20 and H20-H20 interactions have similar strengths, in particular for n = 4. (b) For n = 5, the dIv(5) for the surface state is lower than the dIv(4). As a consequence, the DIv(5) is negative! On the other hand, the dIv(5) for the interior state is higher than dIv(4), keeping the positiveslopeoftheplot. TheDIv(5)for theinterior statechanges
the slope of the curve. This feature has been observed experimentally17 for the I-(H20) clusters. (c) The curve obtained by using the interior state values for n = 5, 6 resembles closely the experimental and calculated trend observed for the I-(H20), clusters.17J7 The small energetic difference between surface and interior states for n = 4-6, plus the effects of correcting for zero-point energies and the behavior of the IP, for the clusters with n = 5 and 6 water molecules, indicatesthat a transition froma dominant surface state (n = 1-3) to an interior state (n = 4-6) occurs at n = 4,s. Our findingsagree with concurrent molecular dynamics calculations on F-(H20), clusters by Berkowitz and Perera.3O They find that entropy effects place the anion inside the cluster of water molecules. ThermodynamicData. We have also calculated the sequential enthalpies of hydration and total enthalpies of formation for the clusters F-(H20),, n = 1-6. Sequential enthalpies of hydration (LW,dn))
w&)= W-(H,O)I
-~[F(H,O),,I-
and total enthalpies of formation
(A&ind
E[H,OI (4)
(n))
The results for our calculations are given in Table 3 along with the corresponding experimental enthalpies. We found good agreement between experimental and calculated results for the sequential enthalpies of solvation. For n = 5 and 6,the enthalpy of solvation for the interior states is always higher than the enthalpies for the interior states. Due to the small differences obtained, these enthalpies cannot be used to definitely identify dominant surface or interior states. We have also evaluated the vibrational entropy contribution for the surface and interior states. These are also given in Table 3. Similar to our previous findings27 on C1-, B r , and I--water clusters, we find that entropy favors the interior states by about 11-18 cal mol-' K-1 at 298 K. This rather crude evaluation of entropy effects does not take into consideration the number of isomers or their energy distribution. However, these results do indicate that the distribution of surface and interior states is temperature dependent, in good agreement with Perara's simula tions.30 4. Summary
We have performed ab initio molecular orbital calculations on fluoride anion-water clusters using extended basis sets which reproducethe experimentalionization potential for the bare anion.
Microscopic Study of Fluoride-Water Clusters Our results were compared with available experimental thermodynamic data, and we reach the following conclusions. The strong interactions between the fluoride anion and the water molecules determines the structure of the clusters as well as the strength of the hydrogen bonds between the water molecules which interact directly with the fluoride anion. For instance, for n = 2 there is no hydrogen-bonded structure since the strong interaction of both water molecules with the fluoride anion increasesthe water-water repulsion. For n = 5 and 6we obtained highly hydrogen-bonded surface structures, which are energetically favored against the interior states. However, the inclusion of zero-point energies and entropy effects reverses the energetic ordering making the interior states energeticallyfavored or almost isoenergetic with the surface states. The combination of both strong F--H2O and hydrogen bonds accounts for the stability. However, we noticed that the strong hydrogen bonds are located in those water molecules which do not interact directly with the halide anion. It is important to note that similar effects of temperature (or entropy) were found independently in the classical molecular dynamics studies of Perera and B e r k o ~ i t z .Both ~~ widely different methods lead to the same conclusions. Analysis of the IP, data reveals the transition from surface to interior states at n = 4, 5 since the IP,(5) value for the surface state produces a negative DI,(5). For the clusters with n = 1-4, the dI(n) and DI(n) values are remarkably close, in contrast with thevalues for the C1-, B r , and I--water clusters where the halide anion-water interactionsare weaker. The small energy difference between surface states characterized by strong F--H20 and weaker hydrogen bonds and interiorstatescharacterized by weaker halidewater interactions and stronger hydrogen bonds indicates the presence of a counterbalancing effect between these two types of interactions. The size of the anion is of importance in determining the structure of these anion-water clusters, since the strength of the interactions, X--H2O, X = halide and H20-Hz0, depends on the size of the anion. The bigger size of the other halide anions (Cl-, B r , I-, >O.S A than F-) produces weaker halide-water interactions,and thus the dominant force appears to be the strength and number of hydrogen bonds between the water molecules. We expect other anions such as OH- will exhibit this transition at around n = 4-6. Experimental work is currently underway in this area.37
Acknowledgment. We express our thanks for continued collaborations with professor Joshua Jortner, Max Berkowitz, and Ori Cheshnovsky. We acknowledge receiving relevant preliminary data from U. Kaldor, K. Bowen, and W. Castleman and a preprint of ref 30. References and Notes (1) Berry, R. S. Structure and Dynamics of Clusters, In The Chemical Physics of Atomic and Molecular Clusters; Scoles, G., Ed.; Elsevier: Amsterdam, 1990 and references therein. (2) The Physics and Chemistry of Small Clusters; NATO AS1 Series; Jena, P.; Rao, B. K., Khanna, S. N., Eds.; Plenum: New York 1986. (3) Microclusters; Sugano, S., Nishima, Y., Onishi, S., Eds.; Springer: Heidelberg, 1987. (4) Lmge Finite Systems; Jortner, J. Pullman, A,, Pullman, B., Eds.; D. Reidel: Utrecht, 1987. ( 5 ) Elemental and Molecular Clusters; Benedek, G., Martin, T. P., Springer: Heidelberg, 1988. Pacchioni, G., Us.; (6) Jortner, J.; Scharf, D.; Ben-Horin, N.; Even, U.; Landman, U. Size Effects in Clusters, In The Chemical Physics of Atomic and Molecular Clusters; Scoles, G., Ed.; Elsevier: Amsterdam, 1990; p 43 and references therein. (7) Jortner, J.; Levine, R. D.; Rice, S.A. Adv. Chem. Phys. 1988, 70, 1.
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