Crystal symmetries energy calculations

The rhombohedral structure, obtained via the optimization of crystal energy [equation (1)] without the JT term inclusion or setting IVj = 0, is stable with respect to deformations of the crystal. The experimental and calculated structure parameters are listed in Table 2. As the temperature decreases down to about 400 K [5,8] the crystal symmetry changes to monoclinic phase. The major difference between the monoclinic and rhombohedral phases is the presence of JT distortions and a doubled primitive cell. The local values of JT distortions on different Mn3+ ions could be obtained via the projection of oxygen ions coordinates onto the normal local Eg modes of each [Mn06] octahedra (the numbering of Mn3+ is carried out according to Fig. 1). 

The polarizabilities are calculated from the electrical field dependence of either the total molecular energy or the dipole moment the y2 ]ki in crystals obey the laws of crystal symmetry and are measured using powerful laser sources. 

In this cluster, which has C2, symmetry, the calculated molecular-orbital energy levels should be compcu-ed with those of the FeO cluster in Fig. 4.26(a). As for the latter cluster, the molecular orbitals can be grouped into sets in relation to their atomic-orbital character, with the dominantly Fe 3d (crystal-field-type) orbitals being at the top of the valence band. Now, however, as shown in Fig. 4.29, the orbitals below these in energy fall into four groups O 2p nonbonding orbitals, Fe-(9... 

Its rapid formation from the hexafluoride vapour and molecular oxygen makes it unlikely to be Pt(OF)2F4, 0F+[PtF50] , or OPtFgO. The crystal symmetry, which in the rhombohedral form is isomorphous with potassium hexafluoroplatinate(v), and its insolubility in carbon tetrachloride, indicate an ionic lattice. Formulation as either 02 [PtFg] or PtFg+02 is consistent with the crystal structure. Lattice-energy calculations, with Kapustinskii s equations, for the processes ... 

The role of crystal symmetry properties in determining the shape of the bands has been emphasized, but the few examples reported have also shown that the existence of a gap and the energy range of bands depend on the mutual interactions of all particles, electrons, and nuclei, in the lattice. Therefore, the correctness of a calculation is largely dependent on the kind of approximation used in the evaluation of such interactions. In fact, different approximations of the Hamiltonian can produce a variety of results and, in particular, band structures that are not only quantitatively but also qualitatively different in some cases. In Figure 18, the HF band structure of silicon is compared with that obtained with DFT methods, both in the LDA, in the form of Slater-Vosko-Wilk-Nusair ° functional, and with the Becke 3 (B3) parameter-Lee-Yang-Parr (LYP) approximation, which incorporates a part of the exact exchange... 

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