diff --git a/Lesson4_DeepPot.ipynb b/Lesson4_DeepPot.ipynb
index a38b0ea..8f366fd 100644
--- a/Lesson4_DeepPot.ipynb
+++ b/Lesson4_DeepPot.ipynb
@@ -78,7 +78,7 @@
       "metadata": {
         "id": "jvwwiTzS_yxR"
       },
-      "execution_count": 1,
+      "execution_count": null,
       "outputs": []
     },
     {
@@ -155,7 +155,7 @@
       "metadata": {
         "id": "v9RdVoWhpVBi"
       },
-      "execution_count": 2,
+      "execution_count": null,
       "outputs": []
     },
     {
@@ -188,12 +188,12 @@
         "s(R_{1i}) & s(R_{1i}) \\frac{x_{1i}}{R_{1i}} & s(R_{1i}) \\frac{y_{1i}}{R_{1i}} & s(R_{1i}) \\frac{z_{1i}}{R_{1i}} \\\\ \\\\\n",
         "s(R_{2i}) & s(R_{2i}) \\frac{x_{2i}}{R_{2i}} & s(R_{2i}) \\frac{y_{i1}}{R_{2i}} & s(R_{2i}) \\frac{z_{2i}}{R_{2i}}  \\\\ \\\\\n",
         "\\cdots & \\cdots & \\cdots & \\cdots  \\\\ \\\\\n",
-        "s(R_{ni}) & s(R_{ni}) \\frac{x_{ni}}{R_{ni}} & s(R_{ni}) \\frac{y_{ni}}{R_{ni}} & s(R_{ni}) \\frac{z_{ni}}{R_{ni}}\n",
+        "s(R_{n_ii}) & s(R_{n_ii}) \\frac{x_{n_ii}}{R_{n_ii}} & s(R_{n_ii}) \\frac{y_{n_ii}}{R_{n_ii}} & s(R_{n_ii}) \\frac{z_{n_ii}}{R_{n_ii}}\n",
         "\\end{pmatrix}\n",
         "\\end{eqnarray}$$\n",
         "<br/>\n",
         "\n",
-        "where $n$ is the number of neighbors for the atom. $s$ is the weighting function.\n",
+        "where $n_i$ is the number of neighbors for atom $i$ and $s$ is the weighting function.\n",
         "\n",
         "$$\\begin{eqnarray}\n",
         "s(R_{ji}) =\n",
@@ -233,7 +233,7 @@
       "metadata": {
         "id": "6CksVhkaqA9e"
       },
-      "execution_count": 3,
+      "execution_count": null,
       "outputs": []
     },
     {
@@ -262,7 +262,7 @@
         "Now we use embedding neural networks to transform each of the $s$ values into $M_1$ numbers. This gives us the local embedding matrix $\\mathcal{G}$. Note the embedding neural network parameters depend on the chemical species of atom $i$ and atom $j$.  \n",
         "\n",
         "$$\\begin{eqnarray}\n",
-        "{\\mathcal{G}}_i\n",
+        "g_i\n",
         "= \\begin{pmatrix}\n",
         "\\left( G[s(R_{1i})] \\right)_1  &\n",
         "\\left( G[s(R_{1i})] \\right)_2  &  \\cdots  &\n",
@@ -270,17 +270,17 @@
         "\\left( G[s(R_{2i})] \\right)_1  &\n",
         "\\left( G[s(R_{2i})] \\right)_2  &  \\cdots  & \\left( G[s(R_{2i})] \\right)_{m_1}  \\\\\n",
         "\\cdots & \\cdots & \\cdots & \\cdots \\\\\n",
-        "\\left( G[s(R_{ni})] \\right)_1  &\n",
-        "\\left( G[s(R_{ni})] \\right)_2  &  \\cdots  & \\left( G [s(R_{ni})] \\right)_{m_1}\n",
+        "\\left( G[s(R_{n_ii})] \\right)_1  &\n",
+        "\\left( G[s(R_{n_ii})] \\right)_2  &  \\cdots  & \\left( G [s(R_{n_ii})] \\right)_{m_1}\n",
         "\\end{pmatrix}\n",
         "\\end{eqnarray}$$\n",
         "\n",
-        "Two local embedding matrices are used: $\\mathcal{G}_{i1}$ is $n\\times M_1$ dimensions, while $\\mathcal{G}_{i2}$ is $n\\times M_2$ dimensions. The dimensions $M_1$ and $M_2$ represent the number of neural network parameters, where $M_1$ is larger than $M_2$.\n",
+        "Two local embedding matrices are used: $g_{i}^{1}$ is $n\\times M_1$ dimensions, while $g_{i}^{2}$ is $n\\times M_2$ dimensions. The dimensions $M_1$ and $M_2$ represent the number of neural network parameters, where $M_1$ is larger than $M_2$.\n",
         "\n",
-        "By multiplying our local embedding matrices and environment matrices, we can preserve the translational, rotational, and permutational symmetry in the form of the encoded feature matrix $\\mathcal{D_i}$  \n",
+        "By multiplying our local embedding matrices and environment matrices, we can preserve the translational, rotational, and permutational symmetry in the form of the encoded feature matrix $D_i$  \n",
         "\n",
         "$$\\begin{eqnarray}\n",
-        "{\\mathcal{D}}_i = \\left( {\\mathcal{G}}_i^1   \\right)^T {\\mathcal{R}}_i  {\\mathcal{R}}_i^T {\\mathcal{G}}_i^2\n",
+        "D_i = \\left( g_i^1   \\right)^T {\\mathcal{R}}_i  {\\mathcal{R}}_i^T g_i^2\n",
         "\\end{eqnarray}$$\n",
         "\n",
         "The local feature matrix is then mapped to the atomic energy using the fitting neural network. Finally, the atomic energies are summed to yield the total energy of the molecule."
@@ -303,7 +303,7 @@
       "source": [
         "### The Feature Class\n",
         "\n",
-        "Here we define the feature class, which uses the local environment matrix ($\\mathcal{R}_i$) and local embedding matrices ($\\mathcal{G}^1_i$ and $\\mathcal{G}^2_i$) to construct the encoded feature matrix ($\\mathcal{D}_i$)."
+        "Here we define the feature class, which uses the local environment matrix ($\\mathcal{R}_i$) and local embedding matrices ($g^1_i$ and $g^2_i$) to construct the encoded feature matrix ($D_i$)."
       ],
       "metadata": {
         "id": "w7XZmhe7d1gW"
@@ -348,7 +348,7 @@
       "metadata": {
         "id": "WY8A0MmNqnvM"
       },
-      "execution_count": 4,
+      "execution_count": null,
       "outputs": []
     },
     {
@@ -381,7 +381,7 @@
       "metadata": {
         "id": "KWF2_lClqvpN"
       },
-      "execution_count": 5,
+      "execution_count": null,
       "outputs": []
     },
     {
@@ -431,7 +431,7 @@
       "metadata": {
         "id": "kfEi4Ffrq196"
       },
-      "execution_count": 6,
+      "execution_count": null,
       "outputs": []
     },
     {
@@ -489,7 +489,7 @@
         },
         "outputId": "82951cc4-9756-46bc-dc5e-a6443b3462df"
       },
-      "execution_count": 7,
+      "execution_count": null,
       "outputs": [
         {
           "output_type": "stream",
@@ -562,7 +562,7 @@
         },
         "outputId": "5be04806-9a15-46ab-99e0-cc92f5effe23"
       },
-      "execution_count": 8,
+      "execution_count": null,
       "outputs": [
         {
           "output_type": "display_data",
@@ -606,7 +606,7 @@
         },
         "outputId": "61f07938-3b0e-4b03-9197-3fdf5d9896c5"
       },
-      "execution_count": 9,
+      "execution_count": null,
       "outputs": [
         {
           "output_type": "display_data",