Band model clusters

Along with the band models, cluster methods have been widely used to describe the electronic properties of refractory crystal surfaces. Specific features of the the surface states of the (100) and (111) faces in TiC and... 

The effect of alloying is studied in the model clusters (see Fig. 17) by letting the metal orbitals simulate a d-band in a true transition metal cluster. 

The calculated photoabsorption bands for a model cluster simulating NBO in Ge02 (Figure 4.9) are located in three energy ranges, 1.7-1.9eV (22A ),... 

Figure 8.6 Calculated energy diagrams for the Si110H114, B6Si108Hn4, and BB6Sin08H1i4 model clusters with optimization. Unoccupied levels are denoted by broken lines and occupied levels are denoted by solid lines. Open and filled circles are unoccupied and occupied states, respectively. The B6Si108H114 has a double acceptor level located immediately above the top of the valence band. B< B6Si108Hn4 has many mid-gap levels localized to the B B6 cluster and may act as recombination centers.

Figure 8.8 Calculated energy diagrams for (a) Bi2-icoSii60Hi00, (b) V< , (c) Cr<5>, (d) Mn< , (e) Fe< , and (f) Co B12-icoSii60Hi0o model clusters with optimization. The symbols and notations are the same as in Figure 6. B12-ico has a double acceptor level lying above the top of the valence band. Diagrams (b-f) have many mid-gap levels that localize to X Bi2-ico clusters.

The group of Zhao is studying a broad spectrum of clusters with a fixed set of methods. They use EA approaches on TB and empirical potentials, sometimes followed by GGA-DFT refinements electronic and magnetic properties are studied with an spd-band model Hamiltonian in the unrestricted Hartree-Fock (UHF) approximation. Among other systems, they have studied pure clusters of Ag [86],Rh [87],V [88] andCr [89], and mixed clusters of similar atom types, for example, Co/Cu [90] or V/Rh [91], for cluster sizes up to n=13-18. 

Fig. 4.30. Molecular-orbital/band models to illustrate the electronic structures of hematite and ilmenite and based on MS-SCF-Za calculations on FeO , FeOs", and TiO/ clusters. The double arrows labeled (a), (b), (c) refer to electronic transitions giving rise to optical properties (after Vaughan and Tossell, 1978).

Fig. 6.1. The electronic structure of ZnS (a) simplistic qualitative one-electron MO energy-level diagram for a ZnS4 (tetrahedral) cluster (b) simplistic one-electron band model for ZnS.

Fig. 6.6. Energy-level diagram to illustrate the electronic structure of iron-bearing sphalerite and based on MS-SCF-A a cluster calculations on the FeS/ tetrahedron (after Vaughan et al., 1974). Discrete MO energy levels (spin up,, and spin down, i ) are shown on the right on the left is a simplistic band model based on this, with filled (or partly filled) bands shown shaded (lines crystal-field-type band dots sulfur nonbonding band dashes metal-sulfur bonding band).

Next we study the effects of the cluster size on DOS. When the cluster size is increased, the interactions between neighboring atoms with long distances are taken into account, then the electronic state approaches that of btilk. Figure 12 compares DOS of the clusters Nig, Nij3, Nijj and Ni j, as well as bulk crystaP by a band structure calculation. Usually the band structure of the bulk crystal can be rather well reproduced if we take several ten atoms in the model cluster for transition elements, though the small cluster model provides somewhat narrower d band. In the case of the element with a d band which is almost completely occupied, for example the case of silver, the size effect is not very large, but a small cluster already well represents the band structure of bulk as shown in Fig. 13. 

Further investigation has been made for dependence of the valence electronic structure upon the geometrical structure. As an example, we take two model clusters, one of which is for fee lattice and the other bcc. These clusters are shown in Fig.l4 (a) is fee cluster model Mjg and (b) bcc cluster M jg. In order to compare the electronic states and to clarify the difference between fee and bcc lattices, we take Fe metal as an example, and DOS curves are displayed in Fig.15. In this figure, (a) is the case of crystal and (b) of cluster. The solid line denotes DOS for fee crystal and dotted line for bcc. These results for the crystals havebeen obtained by band structure calculations by other authors. In the case of bcc, roughly speaking the d band is split into three bands, and the difference between two structures are clearly seen. If we use the cluster model to investigate the difference of the d band between the two lattice structures, we obtain the DOS for fee and bcc clusters as shown in Fig.l5(b). As is mentioned above, the width of the d band of these small clusters are somewhat narrower compared with that of bulk, but the essential characteristics of the band structure of the fee and bcc... 

Following a short review of early studies of the valence band of tetrahedral oxyanions of types X04" some recent results of this field are discussed. High resolution XPS measurements of the valence band of phosphorus and sulphur oxyanions made possible a rigorous test of various theoretical models. For DV-Xa cluster MO calculations, experimentally determined crystal structure information were used to set up realistic model clusters. In the case of the S04 cluster, the results of several model calculations ab initio, DV-Xa, hybrid models) are presented. From the comparison of these results a better understanding of the role of the contributions from different effects to the MO one-electron energies can be obtained. 

Theoretical cluster model MO calculations were used for the interpretation of these spectra. The DV-Xa method combined with realistic cluster approach (in which the model clusters were set up using available crystal structure information from diffraction experiments) proved to be a powerful tool to interpret the structural changes in the valence band spectrum due to the changes in the superstructure of the consisting PO4 tetrahedra. We found noticeable differences in the calculated spectra when the cluster geometry was changed, especially in the case of the hydrous and anhydrous pyrophosphate and similarly for the ring and spiral forms of the tetra-metaphosphate. 

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