Atomic cell primitive basis orbital

The single-site matrix t corresponds to a scattering operator t defined by t yi = iupL for any primitive basis orbital (pi in a given atomic cell. Here Xl = (Pl — fr GoV(pL = Jl Cl<,l From this definition it follows that tJi = v The t-matrix is... 

Mattheiss used a basis of fourteen atomic orbitals per primitive cell for his LCAO analysis these were the five transition-metal d states and the nine oxygen s and p states for the three oxygens. At first, one might be inclined to include the oxygen s stales as core states and therefore drop them from the analysis, as we did for Si02, but Mattheiss s APW calculation of the bands indicated that the. s- bands had appreciable width (0.66 eV in SrTiOj) and important effects upon the other bands. It is important, therefore, to include them, and we shall even find that it simplifies the calculation of bonding properties. The APW calculation allowed this insight into the electronic structure, which would not have been available earlier. 

Figure 5.7 Application of the general spreadsheet to the calculation of the energy of the helium atom using the Pople, Hehre and Stewart sto-6g) basis set of Table 1.6 and the best Slater exponent, also reported elsewhere (8,9) from variation of the entry in cell oneel D l. For the Slater-rules exponent, 1.7, the helium energy is found to be —2.8461945 hartree with the Is orbital energy equal to —0.8918763 H. Note, the detail shown for the Vijkl term. On this spreadsheet all 1296 [6 integrals] are calculated, with the degeneracies over the primitives, colour-coded in the second diagram in the figure.

The notation Pr,r Rn) for the one-electron DM in the coordinate representation imphes that the indices r and r of the matrix vary continuously only within the primitive cell. Therefore, there is an analogy between the properties of the DM in the coordinate representation and the properties of the DM represented in terms of a set of basis functions, for example, in terms of Bloch sums of atomic orbitals (AOs) or plane waves. 

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