Matrix elements between Bloch sums

The state may be written as a linear combination of the three Bloch sums 1xj(I )> from Eq. (6-7) for a = 1, 2, and 3. The three bond types are illustrated below. Construct matrix elements between Bloch sums to obtain the three-by-three Hamiltonian matrix. 

When k is in a special direction such as along the z-axis, cancellation among terms in the sum in fiq. (20-2) will cause many of the H i, values to vanish. Which ones vanish can be learned readily by symmetry, using group theory, or by direct evaluation. It is found in fact that there arc matrix elements between Bloch sums based upon the s state and upon the d state of symmetry 3z — but all other for P a. vanish. For these others, Eq. (20-3) becomes a set of independent equations E. = // , and we evaluate them directly. 

There arc two atoms per primitive cell, so two Bloch sums of the form ofEq. (6-7) are needed (with a bond orbital replaced by p orbitals oriented perpendicular to the graphite plane). The expectation value of the Hamiltonian with respect to cither of the Bloch sums taken individually is simply Cp. The matrix element between the two is of a form analogous to Eq. (6-9), but with only a single type of nearest-neighbor matrix element and with three rather than two terms. They may be written for arbitrary k in terms of the three nearc.st-neighbor vectors dj, da, and da. ... 

For an intraband transition between the states s) and s ) of the electron-hole pair one may simple average t(r) c(r) and ipl(r)ipv(r) over the unit cell using the Bloch functions at the band extremes, since the principal term does not already vanish. As a result, the corresponding matrix element of the charge density is given by the sum of the electron and hole contributions ... 

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