An Aside on Periodic Solids -space Methods

Because of the presence of the regularity associated with a crystal with periodicity, we may invoke Bloch s theorem which asserts that the wave function in one cell of the crystal differs from that in another by a phase factor. In particular, using the notation from earlier in this section, the total wave function for a periodic solid with one atom per unit cell is 

The first sum is over cells, the of which is specified by the Bravais lattice vector Rj, while the second sum is over the set of orbitals a which are assumed to be centered on each site and participating in the formation of the solid. If there were more than one atom per unit cell, we would be required to introduce an additional index in eqn (4.68) and an attendant sum over the atoms within the unit cell. We leave these elaborations to the reader. Because of the translational periodicity, we know much more about the solution than we did in the case represented by eqn (4.58). This knowledge reveals itself when we imitate the procedure described earlier in that instead of having nN algebraic equations in the nN unknown 

Imitating the treatment given in eqn (4.60), we can once again elucidate our variational strategy, but this time with the added knowledge that the wave function takes the special form given in eqn (4.68). In particular, the function we aim to minimize is 

The energy eigenstates in the periodic case are labeled by the wave vector k. If we now seek those a s that minimize eqn (4.70), and introduce the notation 

The structure of our result is identical to that obtained in eqn (4.64) in the absence of any consideration of crystal symmetry. On the other hand, it should be remarked that the dimensionality of the matrix in this case is equal to the number of orbitals per atom, n, rather than the product nN (our statements are based on the assumption that there is only one atom per cell). As a result of the derivation given above, in order to compute the eigenvalues for a periodic solid, we must first construct the Hamiltonian matrix H(k) and then repeatedly find its eigenvalues for each -point. Thus, rather than carrying out one matrix diagonalization on a matrix of dimension nN x nN, in principle, we carry out N diagonalizations of an n x n matrix. 

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