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J. Phys. Chem. B 2006, 110, 18272-18277
Structural Isomers and Reactivity for Rh6 and Rh6+ D. Harding,† S. R. Mackenzie,† and T. R. Walsh*,†,‡ Department of Chemistry and Centre for Scientific Computing, UniVersity of Warwick, CoVentry, CV4 7AL, United Kingdom ReceiVed: April 28, 2006; In Final Form: July 21, 2006
The structure, energetics, and interconversion of isomers of Rh6 and Rh6+ are studied by using density functional theory with Gaussian basis sets, using guess structures derived from basin-hopping simulations, and obtained by using the Sutton-Chen potential. A large range of spin multiplicities is considered for each isomer. Our calculations suggest two low-lying structures as possible structural isomers: a square bipyramid and a trigonal prism. The reactivity of these two candidate structural isomers with respect to adsorption of nitric oxide is studied via location of reaction transition states and calculation of reaction barriers. Similarities and differences with surface reaction studies are highlighted. These data provide powerful evidence that structural isomerism, and not different spin states, is responsible for the observed biexponential reaction kinetics.
1. Introduction Transition metal (TM) nanoclusters are the subject of considerable research activity in the chemistry community as a result of both their catalytic and size-dependent properties. The discovery of the complex correlation between the size of TM clusters and reactivity toward small molecules has led scientists to study the interplay between geometrical structure and electronic structure that must govern size-reactivity relationships.1,2 Direct experimental determination of cluster structures remains, however, extremely challenging although very recent advances in photodissociation action spectroscopy3 and highresolution photionization techniques hold great promise.4,5 Rhodium clusters have been extensively studied experimentally, most recently with the Fourier transform ion cyclotron resonance (FTICR) kinetics studies of Ford et al.,6 who reported biexponential kinetics for the first step (of many) in the reaction of Rh6+ with nitric oxide (Rh6+ + NO f [Rh6 NO]+). Biexponential reaction kinetics are frequently observed in the reaction of small molecules with TM clusters and have been interpreted in terms of structural isomerism, where different structures are hypothesized to react at different rates.6-12 Confirmation of structural isomerism in TM clusters poses a considerable challenge for both theory and experiment. Indirect experimental evidence of different structural isomers for a given cluster size is available: along with reactivity studies, ion mobility studies can distinguish markedly different structural forms13 and action spectroscopy techniques3 can also be used to identify different structures. As experimental methods improve, theory can play a complementary role in determining the structures of reactive and unreactive isomers, so that the geometric and electronic motifs that control heterogeneous catalysis may be identified. The modeling of the structure, dynamics, and properties of transition metal clusters remains a significant challenge. Application of sophisticated all-electron ab initio methodologies is possible only for the smallest of clusters.14 For clusters of the size we wish to study, density functional theory (DFT) is † ‡
Department of Chemistry. Centre for Scientific Computing.
more practical. Although multiconfigurational methods are shown to compare well with experiment for very small, highsymmetry TM clusters, it is not yet clear that larger clusters with little or no symmetry possess substantial multiconfigurational character. We note that the development and validation of exchange-correlation functionals for use on TM clusters remains an ongoing and important effort.15 Numerous recent studies have reported structural optimization of Rh clusters (mostly the neutral state) for a range of sizes using DFT.16-20,23 The focus for many of these studies has been the magnetic properties of these clusters. Many of these previous studies have concluded that the global minimum of Rh6 is octahedral or nearoctahedral, and have typically reported one or two other structures and energies for comparison. There is also some disagreement in the literature regarding the lowest energy spin state of this structure. Of the studies reported, only Bae et al.21 identified the trigonal prism as a candidate low-energy structure (these authors identified this as the global minimum). Despite these numerous studies, further work is required to probe the relative energies of candidate minima as a function of a wide range of spin multiplicities. Furthermore, barriers to reaction and interconversion for these isomers have not yet been reported. Previous theoretical studies of molecules interacting with transition metal clusters have mostly addressed physisorption: see for example the work of Gro¨nbeck and Rose´n.24 Few have used electronic structure theory to find possible reaction pathways, a recent example being the study by Pedersen et al.25 An emphasis on relaxation dynamics is important since the most favored isomer in energetic terms is not necessarily the isomer identified in experiment, especially if other isomers are kinetically favored. Since the results of cluster reactivity experiments have been interpreted in terms of structural isomerism, we seek here to determine the likely structural isomers of Rh6 and Rh6+ using DFT calculations to identify minima and transition states for isomer interconversion for a range of spin multiplicities. We have then used two candidate structural isomers for the cation case in reaction pathway calculations for the reaction with nitric oxide. Similarities and differences with experiments26,27 and calculations28-30 of the rhodium surface interaction with NO are also highlighted. We have chosen the
10.1021/jp062603o CCC: $33.50 © 2006 American Chemical Society Published on Web 08/25/2006
Structural Isomers and Reactivity for Rh6 and Rh6+ Rh6/Rh6+ system as Ford et al.6 observed biexponential kinetics in the reaction with nitric oxide, and our calculations will allow comparison with this experimental work. In particular, we seek to explore whether it is variations in geometrical structure or spin state (either of which could conceivably account for the experimental findings) that are responsible for the biexponential reaction kinetics. The small cluster size ensures the calculations, especially for the reaction pathways, remain computationally tractable. 2. Methods Many past studies16-23 suggest that the putative global minimum is the octahedral structure. To check this further, a quick search was performed using basin-hopping simulations31,32 with the Sutton-Chen potential33 to find possible minimum structures. Comprehensive tests performed by others31,34 suggest that basin-hopping should be able to find the global minimum for a cluster of six atoms. For anything but the most pathological of potential energy surfaces, any reasonably robust global optimization method should work well for a six-atom system: e.g., approaches such as the genetic algorithm35 would equally be applicable here in place of basin-hopping. However, we recognize that the application of basin-hopping should be applied with great care for considerably more complex systems such as proteins. All of the minimum structures located in this procedure were reoptimized by using density-functional theory (DFT). In the basin-hopping runs the raw-structures variant36 was used, with a temperature of 500 K (reflecting the energy scale of the potential). The procedure followed was very similar to that reported by Lai et al.37 To prevent fragmentation, a sizedependent spherical confining potential was used. As an additional check, seeding and freezing runs were also implemented, using the structures corresponding to the lowest three energies (from corresponding runs of Rh5 and Rh7 respectively) as input. The results of these basin-hopping runs were not the only data we used to inform our choice of viable minima. The reason for this is the lack of reliability of empirical potentials as applied to Rh clusters. For example, one of our key structures, the trigonal prism, was not located with our basin-hopping runss rather we have taken this structure from the previous study by Bae et al.21 This failure was not due to the global optimization scheme, but rather the potential; we found that the trigonal prism structure collapsed to a capped-bipyramidal structure upon geometry optimization using the Sutton-Chen potential. We tried other potentials for Rh (e.g., the many-body potential of Chien et al.38) and similarly could not find the prism minimum. This is not surprising given that many potentials for Rh are fitted to bulk properties, and are not expected to perform comprehensively for clusters. This also highlights the need for new potentials that are specifically designed for application to TM cluster systems. Instead of relying solely on the basin-hopping results, we have examined each of the low-energy structures found using basinhopping, plus other structures not included in this set including those geometries reported by others.18,21,23 We used DFT (see below for details) to optimize each of these structures, and then considered plausible connectivities between these structures including diamond-square-diamond and cap-migration mechanisms, as well as other new mechanisms (see the Connectivities section). We found that the set of geometries comprising the low-energy basin-hopping structures plus the trigonal prism structure (our four key structures as listed in the Results section)
J. Phys. Chem. B, Vol. 110, No. 37, 2006 18273 can interconvert via our considered mechanisms. Unlike noble metal clusters (e.g., gold), Rh6 does not support low-energy twodimensional clusters that are energetically competitive with more compact three-dimensional (3D) geometries. Furthermore, bridgeand atop-cappings of Rh5 geometries are either not supported or lie at much higher energies than the structures included heres see the work of Wang and Ge18 for examples of high-energy flat and linear-chain structures. Further, we suspect that some of the optimized structures reported by others using full symmetry may not be genuine minima when allowed to relax without symmetry restrictions. For example, in our symmetryfree optimizations the C2V “bow-tie” (comprising two edge-fused tetrahedra) reported by Wang and Ge18 collapsed to the cappedbipyramidal structure. In general, considering 3D geometries from previous calculations18,23 led either to minima with binding energies significantly higher (around 1 eV or greater) than our four key structures or collapsed to one of the four key structures. This suggests that the barriers to convert from one of our four structures to these high-energy minima will be at the Very least 1 eV. These results and the results of others strongly suggest, but of course do not prove, that the four compact structures reported in this work represent the structures of lowest energy. The fact that plausible mechanisms exist that support mutual interconversion among this group provides further compelling evidence to justify the methods used here to obtain our four key structures. The low-energy structures (found via basin-hopping plus consideration of other published structures) were used as input for DFT geometry optimizations, using the Gaussian03 package.39 The PBE exchange-correlation functional40 was used throughout in this work. The SDD41 effective core potential (ECP) and valence Gaussian basis functions were used, with each rhodium atom represented with 17 valence electrons. In accord with some previous theoretical work on rhodium clusters16,17,21,23 a range of spin multiplicities was considered for both the neutral and cationic states. No symmetry restrictions were placed on the structures during optimization, since it was found that the low-symmetry versions of high-symmetry structures were often found to be lower in energy, in agreement with our previous work,42 and the work of others.43 Analytic second derivatives of the potential energy were used during optimization. Confirmation of genuine minima and transition states was obtained by calculation of the vibrational frequencies. The integration grid over which the exchange-correlation contribution was evaluated corresponded to a pruned grid with 99 radial shells and 590 angular points per shell. Cluster reactivities were studied by location of true transition states for reactions of Rh6+ with NO. In each case the reaction pathway was followed both parallel and antiparallel to the transition vector by using Eigenvector Following,44-49 to locate reactant (molecularly adsorbed minima) and dissociated product states, providing the potential energy barriers for each reaction. The basis sets used in these calculations were D95V50 on N and O, and the SDD ECP41 on Rh. 3. Results 3.1. DFT Minima Structures. The four structures lowest in energy, as determined by our DFT calculations at the PBE/SDD level, were the following: a square bipyramid structure (1), a trigonal prism (2), a capped square-pyramid (3) and a capped trigonal-bipyramidal structure (4), as shown in Figure 1. In all cases, the structures have C1 symmetry, though all are only slightly distorted from their more symmetric counterparts. For Rh6, spin multiplicities of 1 through to 11 were treated. Similarly
18274 J. Phys. Chem. B, Vol. 110, No. 37, 2006
Harding et al. TABLE 2: Binding Energies (eV) of Isomers for Various Spin States of Rh6+ Calculated at the PBE/SDD Level of Theory isomer
multiplicity
〈S2〉
-Eb
1
2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12
0.78 4.00 8.75 15.80 24.82 35.81 1.37 3.97 8.75 15.75 24.83 35.81 2.66 4.59 8.96 15.82 24.82 35.81 3.27 5.19 8.91 15.81 24.83 35.80
19.62 19.65 19.86 20.12 20.12 19.62 19.59 19.71 19.84 19.80 19.79 19.10 19.02 19.17 19.16 19.28 19.33 19.11 18.91 18.95 19.15 19.33 19.56 19.23
2
3 Figure 1. DFT optimized structures labeled with numbering taken from Table 1.
TABLE 1: Binding Energies (eV) of Isomers for Various Spin States of Rh6 Calculated at the PBE/SDD Level of Theorya isomer
multiplicity
〈S2〉
-Eb
1
1 3 5 7 9 11 1 3 5 7 9 11 1 3 5 7 9 11 1 3 5 7 9 11
0.00 2.17 6.15 12.03 20.09 30.07 0.00 2.81 6.53 12.07 20.06 30.07 0.00 3.39 6.26 12.21 20.07 30.06 0.00 3.96 -a 12.12 -a 30.06
17.89 17.83 17.95 18.19 17.96 17.97 17.38 17.73 17.82 17.92 17.91 17.87 16.98 17.50 17.49 17.48 17.62 17.71 16.66 17.25 -a 17.48 -a 17.86
2
3
4
a
Collapsed structures.
for Rh6+, multiplicities of 2 through to 12 were considered here. The binding energies and calculated 〈S2〉 values are given for each structure in each spin multiplicity in Tables 1 and 2 for Rh6 and Rh6+, respectively. In the neutral case, the binding energy is reported with respect to the quartet state of the atom. In the cation case, the binding energy is reported with respect to five neutral atoms and one rhodium cation atom. The main difference between the neutral and cationic cases is the lack of converged minima for structure 4, some of which collapsed upon optimization for the neutral spin states. A similar behavior has been noted previously in a study of Nb10 isomers.42 In general, the low-spin cases (doublet and triplet) were difficult to converge, and in some cases the expectation value of 〈S2〉 has strayed outside the acceptable range of (10% of s(s + 1). In both the neutral and cationic cases, the square bipyramid and trigonal prism structures are favored over the other two structures. The binding energies show spin-dependence, in the cation case exhibiting a maximum in binding (as a function of spin multiplicity) for all structures in the octet and dectet cases. The neutral case is less clear-cut, although the binding energy
4
is a maximum for the septet of structure 1. For the octet and dectet cases, structures 1 and 2 are clearly favored over the other two structures. The same can be said for the neutral septet state. On the basis of these data, we suggest that two possible structural isomers for this system are the square bipyramid and the trigonal prism. In light of recent far-infrared vibrational resonance enhanced multiple photon dissociation (FIR-MPD) experiments,43,51,52 we give examples of infrared spectra (simulated with a halfwidth of 10 cm-1) for structures 1 and 2, shown in Figure 2. These particular spectra were calculated for the dectet state, but other spin states yielded similar spectra. 3.2. Connectivities. A natural choice for a likely rearrangement mechanism is the diamond-square-diamond53 (DSD) mechanism, as has been reported in a previous study of Nb10.42 On the basis of geometries, we inferred that possible DSD connectivities may exist among structures 1, 3, and 4, but no such rearrangement connects 1 and 2, even in an indirect sense. Instead, we have identified a new rearrangement mechanism for interconversion between 1 and 2s an anti-geared twisting mechanism of the triangular faces of the trigonal prism. The transition state for this rearrangement is shown in Figure 3. The energetic barriers for this interconversion as a function of spin multiplicity are given in Table 3. To compare with a typical DSD rearrangement for structure 1, an example barrier for converting from 1 and 4 was calculated to be 0.63 eV, with a reverse barrier of 0.07 eV (the transition state is structurally very similar to structure 4) for the dectet state (the state of lowest energy for both isomers). Although the numbers differ, a similar picture is presented for the other electronic states. This suggests that this DSD interconversion to other structures will be shortlived, relative to the new twist mechanism suggested here. The overall picture our data indicate is a low-energy double-welled basin that contains structures 1 and 2, where the barriers to enter this basin are much smaller than either the barrier to escape the well or the intrabasin barrier. This gives further support to the identification of the square bipyramid and trigonal prism as our candidate structural isomers. It is also noted that the prism is a relatively “floppy” structure with three soft modes between 84 and 91 cm-1 that suggest interconversion pathways between 2-1, 2-3, and 2-4, respectively.
Structural Isomers and Reactivity for Rh6 and Rh6+
J. Phys. Chem. B, Vol. 110, No. 37, 2006 18275 TABLE 4: Energies of Reaction Barriers (eV) for Selected Spontaneous Reaction Pathways, Calculated for a Range of Spin Multiplicitiesa multiplicity
Figure 2. Infrared spectra calculated for structures 1 and 2 in the dectet state.
forward
reverse
7 9 11
Path 1-2 0.93 1.27 1.32
1.38 1.33 1.62
7 9 11
Path 2-1 1.11 1.33 1.12
1.41 1.37 1.48
7 9 11
Path 2-3 0.23 0.36 0.33
1.07 0.88 0.90
a The labeling follows that given in Figure 4. ZPE corrections are not included, since the correction was found to be very small for all cases.
Figure 3. Structure of the transition state for the rearrangement from square bipyramid to prism. Arrows show the transition vector.
TABLE 3: Barriers (eV) of the 1 - 2 Transition State for Rh6 and Rh6+ Calculated at the PBE/SDD Level of Theory for Various Spin States multiplicity
forward barrier
reverse barrier
1 3 5 7 2 4 6 8
0.72 0.52 0.61 0.75 0.50 0.41 0.61 0.68
0.21 0.42 0.49 0.49 0.48 0.47 0.59 0.37
3.3. Reactivites. Existing experimental work6 on the reaction kinetics of Rh6+ with nitric oxide molecules suggests that structural isomers with markedly different reactivities may be present in significant fractional abundance. We have taken structures 1 and 2 and searched for transition states corresponding to reaction of NO on the cluster surface. We have calculated the reaction pathways using Eigenvector Following,44-49 and deduced reaction barriers from these pathways. We have modeled the complex for spin multiplicities ranging from the septet to the 11-tet. Two of the reaction pathways, 1-1 and 2-2, yield products that are higher in energy than reactants (even when zero-point energy (ZPE) effects are accounted for), and so are not predicted to occur spontaneously. Of the three remaining pathways that are predicted to occur spontaneously, the reaction barriers, as a function of multiplicity, for both the forward and backward reactions are summarized in Table 4. Despite intensive searching, we did not locate a transition state for reaction of NO along the edge of the square bipyramid structure. Similarly we also searched for reactions where the NO bond is oriented across an edge bond in the cluster (such that the adsorbate atoms could adsorb on separate cluster faces), without success. The energies reported here are based on differences of the potential energy: inclusion of ZPE corrections (to all three structures on the pathway) made very little difference (around 0.03 eV) to the barriers calculated here and they are not shown. Of the three reactions summarized in Table 4 that are predicted to occur spontaneously, the 2-3
pathway has the lowest barrier, ranging from 0.23 to 0.36 eV (depending on spin state). The other pathways present substantially larger barriers with 0.93 to 1.32 eV for 1-2 and 1.11 to 1.33 eV for 2-2. Across the range of spin multiplicities, while the barrier heights show some variation, the trends in barrier heights hold. All of the stationary points for all of the pathways reported here are bound with respect to the separated bare cluster and NO molecule. The magnitude of this binding depends on the reference spin state of the separated bare cluster. The interaction energies defined in this way range from -1.41 to -3.23 eV. 4. Discussion On the basis of the structures considered here, the square bipyramid structure and the trigonal prism structure are identified as our candidate structural isomers to account for the experimental findings of Ford et al.6 Many other structures, taken from the Sutton-Chen simulations considered here were all substantially higher in energy than structures 3 and 4, and therefore were not investigated further. As already noted in the Results section, the structure of the potential energy landscape suggests that structures 1 and 2 appear both kinetically and thermodynamically favored. The intensities of the infrared spectra shown in Figure 2 are, as expected, very low for structures 1 and 2. If the infrared spectra of these species could be resolved, it should be possible to distinguish these geometries on this basis. As the data in Table 4 suggest, there is little difference in the relative reaction barriers for the forward reaction across the pathways, e.g., the forward barrier for pathway 2-3 remains the smallest barrier across all spin multiplicities. The relative reaction barriers are also reasonably insensitive to both the exchange-correlation functional and the ECP used. We repeated some of our calculations by changing both, and found that the trends established here held. Overall, these data strongly support an interpretation of current experimental data attributed to structural isomers, and not due to species with different spin states. While bearing in mind that an extended transition-metal surface and a transition-metal cluster are very different entities, we have looked for similarities and differences with Rh-surfaceNO data from the literature. The rectangular sides of the trigonal prism structure bear some resemblance to the arrangement of atoms on the Rh(100) surface. For the molecularly adsorbed minima, Loffreda and co-workers28 identified a bridge geometry where the nitrogen atom bonds to the surface, with an adsorption
18276 J. Phys. Chem. B, Vol. 110, No. 37, 2006
Figure 4. Pathways calculated for the reaction of nitric oxide on the square bipyramid and prism structures of Rh6+. The reactant minimum is shown on the left, the transition state (with transition vector) is shown in the middle, and the dissociated product is shown on the right. Pathway labels are taken from text and Table 4.
energy (where a negative sign implies binding) of -2.68 eV, and a “lying down” geometry with an adsorption energy of -2.61 eV. These geometries bear strong resemblance to our reactant minima for the 2-1 and 2-3 paths, with adsorption energies of -2.48 and -2.27 eV respectively for the septet state of the complex evaluated with respect to the separated bare octet cluster and doublet molecule (adsorption energies for other spin states of the complex show a similar relationship). Although our magnitudes of the interaction are different from those of Loffreda and co-workers, the trend is reflected in our cluster calculations. In terms of transition states and reaction barriers, Loffreda and co-workers28 have identified a single mechanism on the Rh(100) surface that appears very similar to the transition state found for the 2-3 pathway. Since the reactant minimum for the Rh(100) is also very similar to the reactant minimum for our 2-3 pathway, it is legitimate to compare barriers here; Loffreda et al. estimate the barrier at 0.47-0.5 eV (coverage dependent), which compares very well with the experimental estimate from Villarrubia and Ho26 of 0.46 ( 0.03 eV, while our corresponding barrier is significantly lower at 0.23-0.36 eV (depending on spin state). Furthermore, Loffreda et al.30 report a dissociation transition state on the Rh(111) surface that has similarities with our 2-1 transition-state structure. We cannot expect that the Rh(111) barrier is directly comparable with our data since the Rh(111) reactant minimum is a 3-fold hcp geometry, while our reactant minimum is a bridging geometry. However, we note that their Rh(111) barrier (calculated to be 1.61 eV) is significantly higher than their Rh(100) barrier. Our 2-1 barrier is estimated at 1.11 eVssimilarly larger than the 2-3 barrier. The experimental estimate27 for the Rh(111) barrier is 0.67 ( 0.06 eV. The lack of 3-fold adsorption geometries for our cluster structures is a distinct difference from surface studies, since both theory and experiment agree that the favored adsorption site for oxygen on the Rh(111) surface is the fcc hollow site.29 As shown in Figure 4, all of our dissociated product structures feature only bridge and atop binding geometries for the dissociated oxygen. While the study of binding geometries of
Harding et al. various oxides of Rh6+ is currently underway in our labs, our preliminary results indicate that the 3-fold adsorption geometry is not supported for the square bipyramid Rh6O+ structure at all, and the corresponding 3-fold Rh6O+ geometry for the trigonal prism is a high-energy structure. Rather, our preliminary results suggest that bridging sites for oxygen are preferred, in agreement with the calculations of Mainardi and Balbuena.20 Many previous computational studies have established that small metal clusters are often inappropriate when used as models of the single-crystal metal surfaces (rather than as clusters for their own sake), since the cluster may not sufficiently recover the electronic structure of the surface (see Yudanov et al.54 for a recent review). It is therefore not surprising that the Rh6 cluster does not support all possible Rh-surface binding geometries. At present, we rationalize the absence of 3-fold adsorption sites on the square bipyramid structure by the fact that this geometry has near D4h symmetry, with the equatorial bonds all being around 0.13 Å shorter than the other bonds, as opposed to the perfect Rh(111) surface where all bonds are the same length. The highest occupied molecular orbital for this near-D4h structure is situated in the plane of the equatorial bonds (with no contribution from the apexes), supporting the idea that edge bonding is preferred. If transition-state energies remain of comparable energy, the lack of strongly bound 3-fold adsorption sites on the cluster should have the effect of reducing barrier heights in the forward direction compared with some surface reactions, by virtue of raising the energy of the reactant minimum. The reactivity studies of this system indicate at least three reaction pathways that are predicted to occur spontaneously. Two of these reactions take place on the same structure, namely the prism, where one has a relatively low barrier and is predicted to proceed faster than the other (that has a higher barrier). The final reaction pathway of the three is associated with the square bipyramid structure, and presents a barrier height similar to the slow-reacting prism pathway. These data therefore indicate that the differing reactivities observed by Ford et al.6 could be attributed to different structural isomers. 5. Conclusions Our calculations of the energetics of four candidate minimumenergy structures of Rh6 and Rh6+ identified two geometries, roughly isoenergetic, as likely structural isomers: the square bipyramid structure and the trigonal prism structure. Using these two structures, we found five pathways for the reaction of the cation cluster with nitric oxide, three of which are predicted to proceed spontaneously. Of these three pathways, one path for the prism system presents a relatively low barrier of 0.23 to 0.36 eV, while the other two barriers have values of 0.93 to 1.32 eV and 1.11 to 1.33 eV for the square bipyramid and prism systems, respectively. The lowest barrier reaction mechanism bears close resemblance to the calculated dissociation mechanism of nitric oxide on the Rh(100) surface.28 Our data therefore give strong support to an interpretation of observed biexponential reaction kinetics based on structural isomerism, and not different spin states. Acknowledgment. The authors gratefully acknowledge the computing facilities of the Centre for Scientific Computing, University of Warwick. D.H. is funded through a DTA award from the EPSRC. S.R.M. thanks the EPSRC for an Advanced Research Fellowship award. References and Notes (1) Knickelbein, M. B. Annu. ReV. Phys. Chem. 1999, 50, 79. (2) O’Hair, R. A. J.; Khairallah, G. N. J. Cluster Sci. 2004, 15, 331.
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