Dimers and larger clusters

In this expression, designates the distance of closest approach which is of the order of the root a of the interaction potential. The duration of optical interactions is then simply given by 

In Chapter 4 we will see that two types of induced dipole functions are of a special importance the overlap-induced dipole, Eq. 4.2, an exponential with a range Rq O.lcr, and the multipole-induced dipole, Eq. 4.3, which falls off as R N (N = 4, 5.). For these, the optical range becomes 

Intermolecular forces are repulsive at close range (R Kmm) and attractive at distant range (R Rmj Rm n is the position of the minimum of the pair interaction potential). At high enough densities most of the common gases form dimers, also called double molecules or (binary) van der Waals molecules. Higher than binary complexes are expected, too, especially at high densities and temperatures comparable to, or lower than the well 

The key to a treatment of molecular clusters in situations of thermal equilibrium are the N-particle partition functions. Specifically, the classical two-particle partition function, Z2(T), is given by [183, 184, 377] 

F(J ) is the pair interaction potential, T the temperature and k the Boltzmann factor. The integral to the right is over the volume v of the container it is often referred to as the configuration integral, Q2(T). For large enough volumes, the integral may be replaced by the volume. With the help of Hill s effective potentials, Eq. 2.28, this expression may be written as a sum of free and bound pair state sums, respectively [183, 184], 

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Ferritin, 36 449-451, 43 363, 399-400 amino acid sequences, 36 465-467 bacterial, see Bacfer biochemistry, 36 450-451 dimer and larger cluster formation, 36 479-481... 

Not surprisingly, many complexes containing both Cp and CO ligands are known. These include half sandwich compounds such as (Tf 5-C5H5)Mn(CO)3 and dimeric and larger cluster molecules. Selected examples are illustrated in Figure 5-14. As... 

In 1987 Howe et al. [32] found that the exchange was facilitated when a critical point was approached, in parallel with a clustering of the droplets. The formation of dimers and larger clusters in droplet microemulsions has long been claimed to occur. An account of the prehistory is given in a more recent paper by Koper et al. [54], who also present a simple quantitative model of cluster formation. 

The O H stretching spectra of ethanol trimers and larger clusters cannot be conformationally resolved in a slit jet expansion [65, 77, 157], VUV-IR spectra [184] are even broader, sometimes by an order of magnitude, and band maxima deviate systematically by up to +50 cm 1 from the direct absorption spectra. We note that ethanol dimers and clusters have also been postulated in dilute aqueous solution and discussed in the context of the density anomaly of water ethanol mixtures [227], Recently, we have succeeded in assigning Raman OH stretching band transitions in ethanol-water, ethanol water, and ethanol water2 near 3550, 3410, and 3430cm, respectively [228],... 

The alkali halides, have long been known to form clusters In the gas phase. With the rapid growth of the Interest In molecular dimers and larger aggregates In general, several groups have again turned their attention to the chemistry of alkali halide vapors (12-17). 

It should be noted that broad emission spectra are also observed in some cases for molecular dimers and larger aggregates, e.g., in the case of naphthalene [11]. The excited state in these homogeneous clusters is characterized as an excimer, and the interaction between the two components is mainly due to exchange interactions, although the contribution from charge-transfer states such as A A " is also important [12, 13]. 

The leading nonadditive term in the many-body expansion of a potential is the three-body interaction. Similarly like dimers, trimers (and larger clusters) can be selectively studied by molecular beam spectroscopy. A number of such trimers have been the subjects of investigations. Among them are the Rg2-diatom trimers mentioned above, with the most extensive data available for Ar2-HF [64]. Both empirical [29,30] and ab initio [33] nonadditive potentials have been obtained for this system. A large number of spectral data are available also for the water trimer [65,66]. An accurate three-body potential for water has recently been developed [34]. 

With increasing solution concentration there is a tendency for dissolved species to interact with each other to form dimers, trimers, and larger clusters that are in dynamic equilibrium with the monomers. The concentration of each cluster size can be expressed by an equilibrium constant for each reaction step. 

The intennolecular forces between water molecules are strongly non-additive. It is not realistic to expect any pair potential to reproduce the properties of both the water dimer and the larger clusters, let alone liquid water. There has therefore been a great deal of work on developing potential models with explicit pairwise-additive and nonadditive parts [44, 50, 51]. It appears that, when this is done, the energy of the larger clusters and ice has a nonadditive contribution of about 30%. 

Besides water dimer, larger clusters of water molecules were extensively investigated by means of the DFT calculations87 111 114 127 128. Laasonen et al.113 studied the structure, the energies, and the vibrational frequencies of small water clusters (up to eight molecules)... 

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