Extension of the LCAO Treatment to Crystalline Solids

The most general expression for a periodic function is the plane wave, e , in which d is a parameter equal to the vector dot product k R. Hence, the wave function, ip(r), of Eq. 5.24 is of the form  

Equations 5.25 and 5.26 are equivalent expressions. If the physical location in real space is shifted by R, only the phase of the wave function will change. Because ip(r) over any 

For each t)q)e of atomic orbital in the basis set, which is the chemical point group, or lattice point, one defines a Bloch sum (also known as Bloch orbital or Bloch function). A Bloch sum is simply a linear combination of aU the atomic orbitals of that type, under the action of the infinite translation group. These Bloch sums are of the exact 

In this equation, N is equal to the number of unit cells in the crystal. Note how the function in Eq. 5.27 is the same as that of Eq. 5.19 for cyclic tt molecules, if a new index is defined ask = liij/Na. Bloch sums are simply symmetry-adapted linear combinations of atomic orbitals. However, whereas the exponential term in Eq. 5.19 is the character of the yth irreducible representation of the cychc group to which the molecule belongs, in Eq. 5.27 the exponential term is related to the character of the Mi irreducible representation of the cychc group of infinite order (Albright, 1985). This, in turn, may be replaced with the infinite linear translation group because of the periodic boundary conditions. It turns out that SALCs for any system with translational symmetry are con-stmcted in this same manner. Thus, as with cychc tt systems, there should never be a need to use the projection operators referred to earher to generate a Bloch sum. 

how Bloch sums combine to form COs must be considered, for example, hke the cr- or 7r-combinations between the Bloch sums of metal d orbitals and oxygen p orbitals in a transition metal oxide. Bloch sums are used as the basis for such COs  

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