2D problem of nearly free electron model

[unscientific]

Homework Statement

(a) Find energies of states at $(\frac{\pi}{a},0)$.
(b) Find secular equation

Homework Equations

The Attempt at a Solution

Part(a)[/B]
In 1D, the secular equation for energy is:
$$E = \epsilon_0 \pm \left| V(x,y) \right|$$

When represented in complex notation, the potential becomes
$$V(x,y) = V_{10} \left[ e^{i\frac{2\pi x}{a}} + e^{-i\frac{2\pi x}{a}} + e^{i\frac{2\pi y}{a}} + e^{-i\frac{2\pi y}{a}} \right] + V_{11} \left[ e^{i\frac{2\pi x}{a}} + e^{-i\frac{2\pi x}{a}} \right] \left[ e^{i\frac{2\pi y}{a}} + e^{-i\frac{2\pi y}{a}} \right]$$

$$E = \epsilon_0 \pm \sqrt{V_{10}^2 + V_{11}^2 }$$

Part(b)
I know the central equation is given by
$$\left(\epsilon_0 - E \right) C_{(k)} + \sum\limits_{G} U_G ~ C_{(k-G)} = 0$$

How do I find the 4x4 matrix?

 

[unscientific]

bumpp

 

[unscientific]

bumpp