Property clusters

The most fiindamental of cluster properties are the bond strengths and entropy changes for the process [125]... 

Much of the work done on metal clusters has been focused on the transition from cluster properties to bulk properties as the clusters become larger, e.g. the transition from quantum chemistry to band theory [127]. 

The cluster properties of the reactants in the MM model at criticality have been studied by Ziff and Fichthorn [89]. Evidence is given that the cluster size distribution is a hyperbolic function which decays with exponent r = 2.05 0.02 and that the fractal dimension (Z)p) of the clusters is Dp = 1.90 0.03. This figure is similar to that of random percolation clusters in two dimensions [37], However, clusters of the reactants appear to be more solid and with fewer holes (at least on the small-scale length of the simulations, L = 1024 sites). 

Vanadium atom depositions were further studied in alkane matrices 109) in an effort to observe the influence of other low-temperature, matrix environments on the optical spectra and clustering properties of metal atoms. Thus, vanadium atoms were deposited with a series of normal, branched, and cyclic alkanes over a wide range of temperature. The atomic spectra were somewhat broadened compared to those in argon, but the matrix-induced, frequency shifts from gas-phase values were smaller. As shown in Fig. 3, these shifts decrease with in-... 

On the Extensivity Problem in Coupled-Cluster Property Evaluation... 

Bonadc-Koutecky, V., Mitric, R., Burgel, C. and Schafer-Bung, B. (2006) Cluster properties in the regime in which each atom counts. Computational Material Science, 35, 151-157. 

To sum up, theory plays an important role in the understanding of cluster properties. Certainly, theory provides the conceptual basis from which experiments can be planned and analyzed. The calculations discussed here all... 

Eurther decrease in redox potential, -200 mV, only affects the [3Fe-4S] cluster properties (one electron reduced in the [3Ee-4S] state). The EPR spectrum now shows a g = 12 feature due to a S = 2 spin system. Mossbauer data clearly illustrates this point, ruling out the possibility of a [3Ee-4S] to [4Ee-4S] cluster interconversion. The two [4Ee-4S] clusters remain in the diamagnetic state, indicating that they have a lower redox potential. 

It cannot be stated, however, that a systematic cluster chemistry exists yet. In most cases each metal has its own structural and chemical cluster properties, and only a few interrelations exist. Predictability is low for cluster compounds and reactions. This is underlined by the fact that there is still no systematic way of preparing clusters, And the few generalizations that can be drawn from the large number of observations justify the conclusion that this field of chemistry is still in its beginning stages. In order to keep cluster research free from the odor of being a I art pour I art science, considerable efforts toward a better understanding wiU have to be made. 

The letter m represents the position coordinate Rm of atom m. Here, /i(l,l ) is the dipole moment induced in the dissimilar pair 1,1 in total isolation from other atoms, etc. The U functions have the cluster property of being zero unless all the molecules appearing in the argument are interacting at close enough range so that their spectral contributions are significant. The function U2 represents the dipole moment induced in the pair of... 

In many situations one expects the dots to be statistically dependent only over short time intervals. A formal expression of this cluster property is... 

The following problem is in a certain sense the inverse of the one treated in the two preceding sections. Consider a photoconductor in which the electrons are excited into the conduction band by a beam of incoming photons. The arrival times of the incident photons constitute a set of random events, described by distribution functions/ or correlation functions gm. If they are independent (Poisson process or shot noise) they merely give rise to a constant probability per unit time for an electron to be excited, and (VI.9.1) applies. For any other stochastic distribution of the arrival events, however, successive excitations are no longer independent and therefore the number of excited electrons is not a Markov process and does not obey an M-equation. The problem is then to find how the statistics of the number of charge carriers is affected by the statistics of the incident photon beam. Their statistical properties are supposed to be known and furthermore it is supposed that they have the cluster property, i.e., their correlation functions gm obey (II.5.8). The problem was solved by Ubbink ) in the form of a... 

The integration extends over an m-dimensional cube of volume m but owing to the cluster property the integrand vanishes unless all variables lie within a distance of order tc from each other. The region where that is so is the neighborhood of the diagonal of the cube and its volume is t 1. Hence (4.8) is of order... 

Since the d orbitals are not allowed to relax in a one electron ECP it may appear that a third prerequisite is that the frozen d approximation should be valid, i.e. the relaxation of the d orbitals should not influence the bonding appreciably. In reality the effect of d-shell relaxation on various metal cluster properties is appreciable e.g. the d-shell relaxation effect on chemisorption energy of oxygen on a Nis cluster is about 40 kcal/mol[23]. However, a small d orbital relaxation is not a necessary prerequisite for the development of a one electron ECP provided that the relaxation is not dominated by covalency effects. The covalent contribution to the bonding of an oxygen atom to a Cus cluster (where the d-shell relaxation contribution to the binding energy is 17 kcal/mol) is only a few kcal/mol[24]. 

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