As stated before, a periodic function like *u*_{k}
can be expanded as a sum of reciprocal lattice vector waves like:

*N* exp(*i***G·r**) = *f*_{i}

where **G** is a reciprocal lattice vector and
*N*=1/(*a*^{2} sin 60°)^{½} is
a normalizing constant that forces the integral over a real lattice cell to be
1. (*a*^{2} sin 60° is the area of a lattice cell;
the square-magnitude of the exponential part is 1.)
The result is a Fourier series:

Now *u*_{k} must satisfy the Schrödinger's
equation:

which we write simply as:

*H* *u*_{k} = *E*(**k**) *u*_{k}

We now do the usual Fourier trick to isolate a particular expansion
coefficient *a*_{i}. We multiply both sides of the
Schrödinger's equation by *f*_{i}^{*} and
integrate over a lattice cell...orthonormality of the
*f*_{i} reduces the sum on the rhs to a single term; on the lhs we have a sum over all
possible reciprocal lattice vectors labeled by *j*.

< *i* | *H* | *j* > *a*_{j} =
*E* *a*_{i} or

[*H*]·[*a*] = *E* [*a*]

where:

Notice that we diagonalize the kinetic energy part of the Hamiltonian
at the expense of spreading *V*--which was diagonal in the grid approximation--all
over the matrix. *Mathematica* (or knowing a bit about special functions)
allows us to express the *V* integral in terms of Bessel functions: *J*.

So far all is exact: instead of solving Schrödinger's differential equation
we are now requested to solve an infinite-sized matrix eigenproblem. Our approximation
consists of truncating the infinite matrix, thus including only a finite number
of reciprocal lattice vectors (a.k.a., Fourier components). We justify this by noting
that in the limit of "small" *V*_{0}, the solutions are single waves and
that the *V* matrix elements grow small for larger **G** differences...we
say *V* causes only a small coupling of distant **G**. On the other hand,
as *V*_{0} grows large, large **G** will be required to localize the ground state
to a small region in space. Usually the only way to see if the truncation is a good
approximation is to expand the number of included Fourier components and check to
see that little is affected.

Below is a plot of the *V* matrix elements as a function of the
magnitude of difference
in **G** vector. The dots show the **G** vector differences in the actual
reciprocal lattice. Notice that the diagonal *V* matrix elements (which are
much larger: about -0.1)
are not plotted since they produce only a uniform energy shift

We now compare the results of band structure calculations using the 37 reciprocal
lattice vectors within 3× the nearest neighbor distance (i.e., a 37×37 matrix eigenproblem);
to the *h*=1/16 grid (which produces a 256×256 matrix eigenproblem).
In the below plots the red lines are the new plane-wave approximation method
and the black lines are the grid approximation. They agree closely, but the
37×37 matrix eigenproblem is solved about ten times faster.