I need to create a 2-D electron energy density plot in Mathematica to compare with my STM experimental results in my lab class. This would be done by plotting the sum of the symmetric and anti-symmetric wave functions,

$$\Psi\_s(\textbf{r}) = \sum\_{i=1}^{3}a\_i\cos(\textbf{k}\_i\cdot\textbf{r})$$
$$\Psi\_a(\textbf{r}) = \sum\_{i=1}^{3}b\_i\sin(\textbf{k}\_i\cdot\textbf{r})$$
and squaring it to get this,
$$\Psi\_s^2(\textbf{r})+\Psi\_a^2(\textbf{r})=\Big[\sum\_{i=1}^{3}a\_i\cos(\textbf{k}\_i\cdot\textbf{r})\Big]^2+\Big[\sum\_{i=1}^{3}b\_i\sin(\textbf{k}\_i\cdot\textbf{r})\Big]^2$$
Where the k's are equal to,
$$\textbf{k}\_1=\frac{4\pi}{3d\_b}\hat{y}$$
$$\textbf{k}\_2=\frac{4\pi}{3d\_b}(-\frac{1}{2}\sqrt{3}\hat{x}-\frac{1}{2}\hat{y})$$
$$\textbf{k}\_3=\frac{4\pi}{3d\_b}(\frac{1}{2}\sqrt{3}\hat{x}-\frac{1}{2}\hat{y})$$
Here is what I have so far

```
Needs["VectorAnalysis`"]
r = {1, 1, 0};
k1 = 4*Pi/3 (2.46) {0, 1, 0};
k2 = 4*Pi/3 (2.46) {-(Sqrt[3]/2), -(1/2), 0};
k3 = 4*Pi/3 (2.46) {(Sqrt[3]/2), -(1/2), 0};

Subscript[\[Psi], s] = 
Cos[DotProduct[k1, r]]*x + Cos[DotProduct[k2, r]]*x + 
Cos[DotProduct[k3, r]]*x;

Subscript[\[Psi], a] = 
Sin[DotProduct[k1, r]]*y + Sin[DotProduct[k2, r]]*y + 
Sin[DotProduct[k3, r]]*y;
DensityPlot[(Subscript[\[Psi], s] x)^2 + (Subscript[\[Psi], a]
y)^2, {x, 0, 100}, {y, 0, 100}]

```

Resulting in this

[![enter image description here](https://i.stack.imgur.com/KA5cP.jpg)](https://i.stack.imgur.com/KA5cP.jpg)

However, it should be uniform distribution of "bright spots" to model the electron density of carbon atoms, something along the lines of this

[![enter image description here](https://i.stack.imgur.com/cEFFD.jpg)](https://i.stack.imgur.com/cEFFD.jpg)

Could someone help me out with this? It would be much appreciated.