Calculation of energy bands using NFE

[Judas503]

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.

 

[NateD]

I think I'm doing something wrong while calculating the Fourier co-efficients. Can someone please tell me if I'm doing it correctly?My calculations are as follows:\begin{align}U(x,y) &= -4UCos[\frac{2\pi x}{a}]Cos[\frac{2\pi y}{a}] \\U_G &= \frac{1}{a^2} \int_{-a/2}^{a/2} \int_{-a/2}^{a/2} U(x,y)e^{-iG.r}dxdy \\&= -4U \frac{1}{a^2} \int_{-a/2}^{a/2} \int_{-a/2}^{a/2} Cos[\frac{2\pi x}{a}]Cos[\frac{2\pi y}{a}]e^{-iG.r}dxdy \end{align}where G.r=Gx+Gy.Now I split the integral into two parts:\begin{align}U_G &= -4U \frac{1}{a^2} \int_{-a/2}^{a/2} \int_{-a/2}^{a/2} Cos[\frac{2\pi x}{a}]e^{-iGx}dx Cos[\frac{2\pi y}{a}]e^{-iGy}dy \\&= -4U \frac{1}{a^2} \int_{-a/2}^{a/2} Cos[\frac{2\pi x}{a}]e^{-iGx}dx \int_{-a/2}^{a/2} Cos[\frac{2\pi y}{a}]e^{-iGy}dy\end{align}The two integrals are now decoupled and can be solved separately:\begin{align}U_G &= -4U \frac{1}{a^2} \int_{-a/2}^{a/2} Cos[\frac{2\pi x}{a}]e^{-iGx}dx \