Clusters harmonic models

Because of their importance to nucleation kinetics, there have been a number of attempts to calculate free energies of formation of clusters theoretically. The most important approaches for the current discussion are harmonic models, " Monte Carlo studies, and molecular dynamics calcula-tions. In the harmonic model the cluster is assumed to be composed of constituent atoms with harmonic intermolecular forces. The most recent calculations, which use the harmonic model, have taken the geometries of the clusters to be those determined by the minimum in the two-body additive Lennard-Jones potential surface. The oscillator frequencies have been obtained by diagonalizing the Lennard-Jones force constant matrix. In the harmonic model the translational and rotational modes of the clusters are treated classically, and the vibrational modes are treated quantum mechanically. The harmonic models work best at low temjjeratures where anharmonic-ity effects are least important and the system is dominated by a single structure. 

Fanourgakis, G. S., Farantos, S. C., Parneix, R, 8c Brechignac, P. (1997). An effective transition state for a complex cluster isomerization process Comparison between anharmonic and harmonic models for mg ar - Journal of Chemical Physics, 106, 4954. 

Similar MC calculations were used by Trout s group to study the carbon dioxide-liquid water interface at 220 K and 4 MPa near the phase boundary of a carbon dioxide hydrate (273 K and 4MPa). Nucleation was achieved by seeding the system with a cluster of carbon dioxide hydrate. It was found that a small cluster with diameter <9.6 A dissolved into the solution readily. A hydrate crystal started to grow, however, when a hydrate cluster twice that size (19.3 A) was implanted into the system. The crystal eventually spanned the whole system (Figure 22). Thus the critical nucleus size for hydrate nucleation is estimated to be about 19 A consisting of approximately 200 water molecules. This is a considerably smaller number than that estimated from the local harmonic model of around 600 molecules. The theoretical results refuted the labile cluster hypothesis.This hypothesis speculates the agglomeration... 

If we compare results obtained with the same basis sets with the three coupled cluster models CCS, CC2 and CCSD, we find similar trends as observed in Refs. [22,45] The CCS model underestimates strongly the static hyperpolarizabilities and their dispersion. The results are usually of similar quality as those obtained with SCF. For methane, the CCS static hyperpolarizabilities are intermediate between the SCF and the CCSD values obtained in the same basis set. In Ref. [45] the CCS percentage dispersion contribution to the third harmonic generation (THG) hyperpolarizability of methane was found to be slightly smaller than for SCF, both underestimating significantly the dispersion obtained with the correlated coupled cluster models CC2 and CCSD. Accordingly the CCS dispersion coeflBcients listed in Table 3 are substantially smaller than the respective CCSD results obtained in the same basis sets. 

The prerequisites for high accuracy are coupled-cluster calculations with the inclusion of connected triples [e.g., CCSD(T)], either in conjunction with R12 theory or with correlation-consistent basis sets of at least quadruple-zeta quality followed by extrapolation. In addition, harmonic vibrational corrections must always be included. For small molecules, such as those contained in Table 1.11, such calculations have errors of the order of a few kJ/mol. To reduce the error below 1 kJ/mol, connected quadruples must be taken into account, together with anhar-monic vibrational and first-order relativistic corrections. In practice, the approximate treatment of connected triples in the CCSD(T) model introduces an error (relative to CCSDT) that often tends to cancel the... 

The bonding in gold cluster molecules has been interpreted using free electron models based on Stone s tensor surface harmonic theory [48, 49]. High similarity has... 

Diatomic harmonic oscillator levels (hv 110 cm-1) to v = 3 for the H cluster are found to contribute to the observed rates with up to 1500 cm-1 of vibrational energy in the n = 3 cluster. Table 5-7 presents the results of these calculations agreement between the calculated transfer rates (tj 1) based on the above model and the experimental results is excellent. 

Our first example is the widely discussed problem of CO chemisorption on transition metal surfaces. Ray and Anderson/106/ investigated CO chemisorption on a Pt (111) surface using the ASED-MO theory. In their work, Pt4 and Pt10 clusters were used to model the Pt (111) surface. The bulk Pt-Pt bond distance was used and other theoretical parameters are reproduced in Table II. The calculated Pt-Pt and C-O bond distances and harmonic force constants are within 8% of experimental values for isolated Pt2 and CO. The results for dissociation energies are less satisfactory 94.08 and 116 kcal/mole in comparison with experimental values of 84.5 and 256 kcal/mole, respectively. 

The remainder of this article is largely concerned with describing how some of the above observations can be rationalized using Stone s Tensor Surface Harmonic theory, and with the further imphcations of this model for dynamical processes such as cluster rearrangements. The number of example systems and electron count rationalizations will be kept relatively small in favor of explaining the theoretical foundations that underhe the method. Tables of examples and more detailed analyses of the various cases may be found elsewhere. ... 

For interpreting the frequencies of the OH stretching modes, recently more sophisticated potential models including internal and external forces as well as harmonic, cubic, and higher terms were used Furthermore, by means of ab initio calculations of clusters, such as M-OH2 or HOH - X, the frequency shifts of the water bands due to interaction with adjacent ions (see Sects. 4.2.3 and 4.2.4) could be verified . ... 

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