Nearest-neighbour interactions crystal structure

Machlin (1974, 1977) developed a semi-empirical treatment which used a constant set of nearest-neighbour interactions and was one of the earliest semi-empirical attempts to obtain the relative enthalpies of formation between different crystal structures. This successfiilly predicted the correct ground states in a substantial number of cases, but the treatment was generally restricted to transition metal combinations and a limited number of crystal structures. 

The valence and coordination symmetry of a transition metal ion in a crystal structure govern the relative energies and energy separations of its 3d orbitals and, hence, influence the positions of absorption bands in a crystal field spectrum. The intensities of the absorption bands depend on the valences and spin states of each cation, the centrosymmetric properties of the coordination sites, the covalency of cation-anion bonds, and next-nearest-neighbour interactions with adjacent cations. These factors may produce characteristic spectra for most transition metal ions, particularly when the cation occurs alone in a simple oxide structure. Conversely, it is sometimes possible to identify the valence of a transition metal ion and the symmetry of its coordination site from the absorption spectrum of a mineral. 

Figure 8.19 Crystal structure of the methyl deuterated lithium acetate dihydrate. Left view of the unit cell with thermal ellipsoids. Right projection onto the (a,b) plane. The lines along nearest neighbour interactions are guides for the eye. For clarity, the hydrogen atoms of the water molecules are hidden. Reproduced with permission from Chem. Phys. 290, B. Nicolai, A. Cousson and F. Fillaux, The effect of methyl deuterat ion on the crystalline structure in Lithium acetate dihydrate, 101, Copyright (2003) Elsevier.

Leaving aside this dynamical problem, we can make some further remarks about the equilibrium structure of Van der Waals molecules. Some attempts have been made to predict this structure from the molecular properties, multipole moments, polarizabilities, which are reflected in the long range interactions (electrostatic, dispersion). Other authors have assumed that the equilibrium structure of Van der Waals dimers resembles the structure of nearest neighbour pairs in molecular crystals. The latter approach could possibly be justified by packing considerations (short range repulsion). An example of the first approach is the prediction of a T-shaped (0 = 90°, 0 = < ) = 0°) equilibrium structure for the Nj-dimer, mainly... 

As indicated in 12.5.1, it is difficult to compare two crystal structures. Structures with different space groups may actually be quite similar, and the comparison of spatial symmetry and even of cell parameters can be quite misleading. A much safer comparison is between the molecular coordination spheres, using 0(J)> U) nd E(j) (see 12.3.2.4). The similarity of 0 j) and R(j) may be less compelling, but the partitioning of the energy over molecule-molecule interactions usually provides a very strict test of similarity. 1 vo crystal structures with the same E(J) fo " few nearest neighbours are likely to turn out to be the same. 

Returning for a moment to the description of bonding inside the crystal, " those d-orbitals whose interactions are responsible for bonding nearest neighbours (viz. the t2g family) will form a band which is broader than that formed by the 6g family, since interactions between next-nearest neighbours are less strong. Extending this concept to surface atoms, we see on the (KX)) surface for example that the absence of atoms above the plane means that the overlap of dxz and d y orbitals has decreased and their band is narrowed, while the dyz orbitals in the surface plane are unaffected, and their band remains broader. Similar but smaller effects will occur with the Cg and 5-orbitals. The modification of electronic structure of atoms at steps and kinks is then easily rationalised, and the story will be resumed in Chapter 2, where other concepts developed in the context of small metal particles will be considered. 

In crystalline solids the various distances r, — ry of a given atom i to its nearest and remote neighbours j are fixed by the crystal structure so that the total interaction per moment or the exchange field experienced by a given moment i can be calculated by performing the summation LjF(2A p r, — r ) exp( — r, — rj / ). It has been discussed in more detail in section 5.1 that the neighbour distances r- — r in amorphous solids are described in terms of radial distribution functions which are different for the individual atoms, so that one has to take averages. This means that in amorphous... 

---