# Calculation of energy bands using NFE

## Homework Statement

Consider a square lattice in two-dimensions with crystal potential
$U = -4UCos[\frac{2\pi x}{a}]Cos[\frac{2\pi y}{a}]$
Apply the central field equation to find approximately the energy gap at the corner point $(\frac{\pi}{a},\frac{\pi}{a})$ of the Brillouin zone. It will suffice to solve a 2 x 2 determinantal equation.

## Homework Equations

The central field equation is
$(\frac{\hbar ^{2} k^{2}}{2m}-E)C(k)+\sum U_{G}C(K-G)$

## The Attempt at a Solution

I know that to solve this problem, we need to know the Fourier co-efficient of U(x,y) and the energy gap is 2|U_G|.
However, my calculated Fourier co-efficients are coming out to be really complicated and I'm not able to simplify it.