Cluster densities

Fernandez, E.M., Soler, J.M. and Baibas, L.C. (2006) Planar and cagelike structures of gold clusters Density-functional pseudopotential calculations. Physical Review B - Condensed Matter, 73, 235433-1-235433-8. 

Hill, J.-R., Freeman, C. M., Delley, B., 1999, Bridging Hydroxyl Groups in Faujasite Periodic vs Cluster Density Functional Calculations , J. Phys. Chem. A, 103, 3772. 

Figure 10.19 A histogram of Au cluster density after the indicated treatment normalized to the cluster density after nucleation at room temperature. The Au coverage in each of the experiments was 0.4 ML Au. The first three columns compare the cluster density on Ti-free Si02, TiOJt(8%)-Si02, and Ti0x(17%)-Si02thin films, respectively, after a 850 K anneal. The fourth column shows the normalized Au cluster density of a Ti0x(17%)-Si02thin film after a CO oxidation reaction (CO/02 = 2 1,60 torr, 370 K, and 120 min). (Reprinted from Min, B.K. et al., J. Phys. Chem. B, 108, 14609-14615, 2004. Copyright 2004. With permission from American Chemical Society.)...

A cluster density of states of Niia, literally (not just formally) broadened by 0.2 eV, is presented in Figure 2a for the icosahedral geometry near the equilibrium structure. The general features are roughly similar to that derived from a spin-polarized band structure calculation of bulk nickel. Near the Fermi level a very high density of states of the minority spin is found, the d band of the... 

Figure 2. Cluster density of states (in arbitrary units) generated by Gaussian broadening of the one-electron energies and cluster Fermi energy Cf a) icosahedral Niia (spin-polarized calculation, majority spin above the axis a = 0.2 eV). b) NiirNa (solid line a = 0.3 eV). Also shown are the sum of the contnbutions from the s and p populations of the nickel atoms (dashed line) and the population of the sodium atom (dotted line).

Fig. 4 shows the ionisation potentials as a function of a value which is proportional to K, where R is the radius of an assumed spherical cluster. If F = 4itR /3 is the volume of a cluster of radius R one has, neglecting geometric and packing effects, V = nv, where v is the volume of an atom. Hence R cc n"For the data points at n = 90 and 95 the experimental results had to be averaged over 2 cluster sizes in order to obtain, accurate threshold data for n = 100 an average over 5 cluster sizes was necessary. The cluster density in the beam, the sensitivity of the detector and the electron current available at threshold all decrease for n > 100, making a threshold determination impossible. 

G. E. Scuseria, Int. ]. Quantum Chem., 55, 165 (1995). On the Connections Between Brueckner-Coupled-Cluster, Density-Dependent Hartree-Fock, and Density Functional Theory. 

More important is the evaluation of the integral of the CO2 pulse. The mass spectrometer was calibrated by measuring the reflected CO from the MgO substrate of a known pulse from the piezo-electric valve. The error of this measurement is estimated to be about 10%. As the cluster density on the surface (see above) is also known, turn-over frequencies (TOF) for the oxidation of CO can be extracted by simply calculating the ratio of the total number of CO2 formed and the number of clusters on the surface times the pulse duration. 

Fig. 17.11. The formation of CO2 at various reaction temperatures and at a constant O2 pressure of 5 x 10 mbar. The TOFs shown in the inset are obtained by integrating the CO2 peaks and normalizing with the cluster density.

The total charge density of the system can be represented as the sum of cluster density and environmental host density. 

The cluster densities are determined from molecular orbital (MO) wave functions, j/j j), that are solutions of Schrodinger equations of the Hamiltonian in Eq. (1). With the Fermi-Dirac occupation numbers n, the cluster density is written as... 

In the case of clusters representing a solid, to the cluster density (and model density) are added the densities of several shells of neighboring external atoms at the crystal sites ... 

In summary, to better asses the quality of V. africana seeds, we propose additional standards, such as percent of clusters, density, ashes and total alkaloids (Table III). In Ghana, the seeds are purchased according to their weight. Thus, the seed clusters and seed density ate important parameters to assess the quality of voacanga seeds. 

To quantitatively characterize the percolation cluster as a whole, the notion of infinite cluster density is introduced, Pooip) It is the ratio of the number of the bonds belonging to the infinite cluster, nk to all black bonds on the percolation lattice N ... 

The increase of infinite cluster density (p) near the percolation threshold p, (Ap/p c lj is characterized by a critical index (1... 

Figure 16. Dependence of infinite cluster density on concentration.

Here, the percolation cluster (infinite cluster) density PKl (p) can be represented as... 

The density of clusters of finite dimensions, ns(p) is connected to the percolation cluster density Pooip) by the equation... 

---