diff --git a/Final/Final - Tight-binding propagation method.ipynb b/Final/Final - Tight-binding propagation method.ipynb
index 7432e96..f476ce7 100644
--- a/Final/Final - Tight-binding propagation method.ipynb	
+++ b/Final/Final - Tight-binding propagation method.ipynb	
@@ -606,7 +606,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 114,
+   "execution_count": 118,
    "metadata": {
     "deletable": false,
     "nbgrader": {
@@ -632,22 +632,19 @@
     "    positions = atomic_positions(n)\n",
     "    \n",
     "    infty = 1e3\n",
-    "    epsilon = 1e-5\n",
+    "    epsilon = 1e-3\n",
     "    x = np.arange(-infty, infty, epsilon)\n",
     "    \n",
-    "    # V is calculated by creating a matrix len(x) by len(positions) over which\n",
-    "    # we sum each row to create a matching value V(x) for each element of x.\n",
-    "    #V = lambda x: np.sum(-1/(np.abs(x - positions)) + 0.001, axis=1)\n",
-    "    #V = lambda x: np.sum( -1/( x - np.tile(positions, (len(x), 1)).T) + 0.001 ), axis=0 )\n",
     "    def V(x):\n",
-    "         value = np.zeros(x.shape)\n",
+    "         ret = np.zeros(x.shape)\n",
     "         for x_i in positions:\n",
-    "             value += -1./(np.abs(x - x_i) + 0.001)\n",
-    "         return value\n",
-    "    #V = lambda x: np.sum( -1/( np.abs(np.tile(x, (len(positions), 1)) - np.tile(positions, (len(x), 1)).T) + 0.001 ), axis=0 )\n",
-    "    #V = lambda x: np.sum( -1/( np.abs(x - np.tile(positions, (len(x), 1)).T) + 0.001 ), axis=0 )\n",
-    "    # OMG IT EXISTS\n",
+    "             ret += -1./(np.abs(x - x_i) + 0.001)\n",
+    "         return ret\n",
+    "    # Instead of using a loop, one could vectorize the problem by calculating all sum\n",
+    "    # elements in a len(x) by len(positions) matrix and then summing along the rows.\n",
+    "    # In testing I found that this was slower than using the loop, so I commented it out.\n",
     "    #V = lambda x: np.sum( -1/( np.abs(np.subtract.outer(x, positions)) + 0.001 ), axis=1 )\n",
+    "    \n",
     "    integrand = lambda x: atomic_basis(x, positions[i], sigma)*V(x)*atomic_basis(x, positions[j], sigma)\n",
     "    return integrate(integrand, x)\n",
     "\n",
@@ -663,7 +660,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 115,
+   "execution_count": 119,
    "metadata": {
     "deletable": false,
     "nbgrader": {
@@ -683,13 +680,12 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "21.9 s ± 35.9 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)\n"
+      "-0.13881260449985544 -0.13881260449985544\n"
      ]
     }
    ],
    "source": [
-    "#print(hopping(1, 0, 10), hopping(0, 1, 10))\n",
-    "%timeit hopping(1, 0, 10)\n",
+    "print(hopping(1, 0, 10), hopping(0, 1, 10))\n",
     "\n",
     "# YOUR CODE HERE\n",
     "#raise NotImplementedError()"