Bloch Sum Basis Set - The LCAO Method

For solids with more localized electrons, the LCAO approach is perhaps more suitable. Here, the starting point is the isolated atoms (for which it is assumed that the electron-wave functions are already known). In this respect, the approach is the extreme opposite of the free-electron picture. A periodic solid is constructed by bringing together a large number of isolated atoms in a maimer entirely analogous to the way one builds molecules with the LCAO approximation to MO (LCAO-MO) theory. The basic assumption is that overlap between atomic orbitals is small enough that the extra potential experienced by an electron in a solid can be treated as a perturbation to the potential in an atom. The extended- (Bloch) wave function is treated as a superposition of localized orbitals, centered at each atom  

One defines a Bloch sum (or BO) for each atomic orbital in the chemical point group (or lattice point), and COs are then formed by taking linear combinations of the Bloch sums. 

take the simplest possible case, a monatomic solid with a primitive Bravais lattice containing one atomic orbital per lattice point. The COs are then equivalent to simple Bloch sums. The wavelength. A, of a Bloch sum is given by the following relation  

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