diff --git a/Notebooks/F12_matrix.ipynb b/Notebooks/F12_matrix.ipynb
index 8b1b0c6..7ba560b 100644
--- a/Notebooks/F12_matrix.ipynb
+++ b/Notebooks/F12_matrix.ipynb
@@ -864,6 +864,8 @@
    "source": [
     "$i,j$ : orbital indices of MOs occupied in $|I\\rangle$.\n",
     "\n",
+    "$\\sum_i \\sum_a \\hat{T}_{i}^{a} \\hat{T}_{a}^{i}$\n",
+    "\n",
     "\\begin{align}\n",
     "\\sum_{a} \\sum_{i}\n",
     "        \\left( h_{ia} +\n",
@@ -943,6 +945,9 @@
     "\n",
     "$k$   : orbital indices of MOs unoccupied in $|I\\rangle$.\n",
     "\n",
+    "$\\sum_i \\sum_a \\sum_j \\sum_k \\hat{T}_{ij}^{ak} \\hat{T}_{ak}^{ij}$\n",
+    "\n",
+    "\n",
     "\\begin{align}\n",
     "\\sum_{a} \\sum_{k} \\sum_{i} \\sum_{j<i}\n",
     "            \\langle i j || a k \\rangle \n",
@@ -1047,6 +1052,8 @@
    "source": [
     "$i,j$ : orbital indices of MOs   occupied in $|I\\rangle$.\n",
     "\n",
+    "$\\sum_i \\sum_a \\sum_j \\sum_b \\hat{T}_{ij}^{ab} \\hat{T}_{ab}^{ij}$\n",
+    "\n",
     "\\begin{align}\n",
     "\\sum_{a} \\sum_{b>a} \\sum_{i} \\sum_{j<i}\n",
     "            \\langle i j || a b \\rangle \n",
@@ -1099,7 +1106,7 @@
     "     ) 0. [ (mos_a ki,Spin.Alfa) ; (mos_b ki,Spin.Beta)]\n",
     "     \n",
     "let _ =\n",
-    "  check 100 integral_value 0 2 2 2\n",
+    "  check 0 integral_value 0 2 2 2\n",
     "    "
    ]
   },
@@ -1127,12 +1134,14 @@
     "\n",
     "$j$ : orbital indices of MOs occupied both in $|I\\rangle$  and $|J\\rangle$.\n",
     "\n",
+    "$\\sum_a \\hat{T}_{i}^{a} \\hat{T}_{a}^{k}$\n",
+    "\n",
     "\\begin{align}\n",
     "    \\sum_{a} \n",
     "    \\left( h_{ia} +\\sum_{j} \\langle i j || a j \\rangle +\n",
     "    \\sum_{\\bar{j}} \\langle i \\bar{j} | a \\bar{j} \\rangle  \\right)\n",
     "    \\left( f_{ak} +\\sum_{j}  \\left[ a j || k j \\right] + \n",
-    "    \\sum_{\\bar{j}}  \\left[ a \\bar{j} | k \\bar{j} \\right]\\right) \n",
+    "    \\sum_{\\bar{j}}  \\left[ a \\bar{j} | k \\bar{j} \\right]\\right) \\\\\n",
     "\\end{align}"
    ]
   },
@@ -1177,33 +1186,14 @@
     "    | Phase.Neg -> -. result\n",
     "     \n",
     "\n",
-    "let _ = check 50 integral_value 1 1 1 1"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "let ki = det_I.(0)\n",
-    "let kj = det_I.(0)\n",
-    "\n",
-    "\n",
-    "let _ =\n",
-    "    let alphas =\n",
-    "        generate_alphas ki kj 0 1 2 2\n",
-    "    in\n",
-    "    compute_HaaF ki alphas kj\n",
-    "\n",
-    "let _ = integral_value ki kj"
+    "let _ = check 0 integral_value 1 1 1 1"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# 5. *1 1 12*"
+    "# 5. 1 1 12"
    ]
   },
   {
@@ -1219,7 +1209,22 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# 6. *1 1 21*"
+    "$|J\\rangle = \\hat{T}_{i}^{k}|I\\rangle$\n",
+    "\n",
+    "$\\sum_{j} \\sum_a \\hat{T}_{j}^{a} \\hat{T}_{ai}^{jk}$\n",
+    "\n",
+    "$j$ : orbital indices of MOs occupied both in $|I\\rangle$  and $|J\\rangle$.\n",
+    "\n",
+    "\\begin{align}\n",
+    "    \\sum_{a} \\sum_{j} \n",
+    "    \\left( h_{ja} +\\sum_{m} \\langle j m || a m \\rangle +\n",
+    "    \\sum_{\\bar{m}} \\langle j \\bar{m} | a \\bar{m} \\rangle  \\right)\n",
+    "    \\left[ a i || j k \\right] + \\\\\n",
+    "    \\sum_{\\bar{a}} \\sum_{\\bar{j}} \n",
+    "    \\left( h_{\\bar{j}\\bar{a}} +\\sum_{m} \\langle \\bar{j} m | \\bar{a} m \\rangle +\n",
+    "    \\sum_{\\bar{m}} \\langle \\bar{j} \\bar{m} || \\bar{a} \\bar{m} \\rangle  \\right)\n",
+    "    \\left[ \\bar{a} i | \\bar{j} k \\right] \n",
+    "\\end{align}"
    ]
   },
   {
@@ -1227,10 +1232,184 @@
    "execution_count": null,
    "metadata": {},
    "outputs": [],
+   "source": [
+    "let integral_value ki kj = \n",
+    "    let i, k, s, phase =\n",
+    "          match Excitation.of_det ki kj with\n",
+    "          | Excitation.(Single (phase, { hole ; particle ; spin })) ->\n",
+    "            hole, particle, spin, phase\n",
+    "          | _ -> assert false\n",
+    "    in\n",
+    "\n",
+    "    let mos_j, mos_j' =\n",
+    "      let alfa = \n",
+    "        let i = Spindeterminant.bitstring @@ Determinant.alfa ki in\n",
+    "        let j = Spindeterminant.bitstring @@ Determinant.alfa kj in\n",
+    "        Bitstring.to_list (Bitstring.logand i j)\n",
+    "      in\n",
+    "      let beta  = \n",
+    "        let i = Spindeterminant.bitstring @@ Determinant.beta ki in\n",
+    "        let j = Spindeterminant.bitstring @@ Determinant.beta kj in\n",
+    "        Bitstring.to_list (Bitstring.logand i j)\n",
+    "      in\n",
+    "      match s with\n",
+    "      | Spin.Alfa -> alfa, beta\n",
+    "      | Spin.Beta -> beta, alfa\n",
+    "    in\n",
+    "    \n",
+    "    let mos_i, mos_i' =\n",
+    "      match s with\n",
+    "      | Spin.Alfa -> mos_a ki, mos_b ki\n",
+    "      | Spin.Beta -> mos_b ki, mos_a ki\n",
+    "    in\n",
+    "\n",
+    "    let result = \n",
+    "       let s' = Spin.other s in\n",
+    "       List.fold_left (fun accu a -> accu +.\n",
+    "           List.fold_left (fun accu j -> accu +.\n",
+    "               (h_one j a s +. \n",
+    "                  (List.fold_left (fun accu m -> accu +. h_two j m a m s s ) 0. mos_i ) +.\n",
+    "                  (List.fold_left (fun accu m -> accu +. h_two j m a m s s') 0. mos_i') ) *.\n",
+    "               (f_two a i j k s s )\n",
+    "             ) 0. mos_j\n",
+    "           +.\n",
+    "           List.fold_left (fun accu j -> accu +.\n",
+    "               (h_one j a s +. \n",
+    "                  (List.fold_left (fun accu m -> accu +. h_two j m a m s' s ) 0. mos_i ) +.\n",
+    "                  (List.fold_left (fun accu m -> accu +. h_two j m a m s' s') 0. mos_i') ) *.\n",
+    "               (f_two a i j k s' s )\n",
+    "             ) 0. mos_j'\n",
+    "         ) 0. mos_cabs\n",
+    "    in\n",
+    "    match phase with\n",
+    "    | Phase.Pos -> result\n",
+    "    | Phase.Neg -> -. result\n",
+    "     \n",
+    "\n",
+    "let _ = check 0 integral_value 1 1 1 2"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# 6. *1 1 21*"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "scrolled": false
+   },
+   "outputs": [],
    "source": [
     "png_image \"1_1_21.png\""
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$|J\\rangle = \\hat{T}_{i}^{k}|I\\rangle$\n",
+    "\n",
+    "$\\sum_{j} \\sum_a \\hat{T}_{ij}^{ka} \\hat{T}_{a}^{j}$\n",
+    "\n",
+    "$j$ : orbital indices of MOs occupied both in $|I\\rangle$  and $|J\\rangle$.\n",
+    "\n",
+    "\\begin{align}\n",
+    "    \\sum_{a} \\sum_{j} \n",
+    "    \\langle i j || k a \\rangle \n",
+    "    \\left( f_{aj} +\\sum_{m} \\left[ m a || m j \\right] +\n",
+    "    \\sum_{\\bar{m}} \\left[ \\bar{m} a | \\bar{m} j \\right] \\right) + \\\\\n",
+    "    \\sum_{\\bar{a}} \\sum_{\\bar{j}} \n",
+    "    \\langle i \\bar{j} | k \\bar{a} \\rangle\n",
+    "    \\left( f_{\\bar{a}\\bar{j}} +\\sum_{m} \\left[ m \\bar{a} | m \\bar{j} \\right] +\n",
+    "    \\sum_{\\bar{m}} \\left[ \\bar{m} \\bar{a} || \\bar{m} \\bar{j} \\right]  \\right)\n",
+    "\\end{align}"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "let integral_value ki kj = \n",
+    "    let i, k, s, phase =\n",
+    "          match Excitation.of_det ki kj with\n",
+    "          | Excitation.(Single (phase, { hole ; particle ; spin })) ->\n",
+    "            hole, particle, spin, phase\n",
+    "          | _ -> assert false\n",
+    "    in\n",
+    "\n",
+    "    let mos_j, mos_j' =\n",
+    "      let alfa = \n",
+    "        let i = Spindeterminant.bitstring @@ Determinant.alfa ki in\n",
+    "        let j = Spindeterminant.bitstring @@ Determinant.alfa kj in\n",
+    "        Bitstring.to_list (Bitstring.logand i j)\n",
+    "      in\n",
+    "      let beta  = \n",
+    "        let i = Spindeterminant.bitstring @@ Determinant.beta ki in\n",
+    "        let j = Spindeterminant.bitstring @@ Determinant.beta kj in\n",
+    "        Bitstring.to_list (Bitstring.logand i j)\n",
+    "      in\n",
+    "      match s with\n",
+    "      | Spin.Alfa -> alfa, beta\n",
+    "      | Spin.Beta -> beta, alfa\n",
+    "    in\n",
+    "    \n",
+    "    let mos_i, mos_i' =\n",
+    "      match s with\n",
+    "      | Spin.Alfa -> mos_a kj, mos_b kj\n",
+    "      | Spin.Beta -> mos_b kj, mos_a kj\n",
+    "    in\n",
+    "\n",
+    "    let result = \n",
+    "       let s' = Spin.other s in\n",
+    "       List.fold_left (fun accu a -> accu +.\n",
+    "           List.fold_left (fun accu j -> accu +.\n",
+    "               (h_two i j k a s s ) *.\n",
+    "               (f_one a j s  +. \n",
+    "                  (List.fold_left (fun accu m -> accu +. h_two m a m j s  s) 0. mos_i ) +.\n",
+    "                  (List.fold_left (fun accu m -> accu +. h_two m a m j s' s) 0. mos_i') ) \n",
+    "             ) 0. mos_i\n",
+    "           +.\n",
+    "           List.fold_left (fun accu j -> accu +.\n",
+    "               (h_two i j k a s s') *.\n",
+    "               (f_one a j s' +. \n",
+    "                  (List.fold_left (fun accu m -> accu +. h_two m a m j s  s') 0. mos_i ) +.\n",
+    "                  (List.fold_left (fun accu m -> accu +. h_two m a m j s' s') 0. mos_i') ) \n",
+    "             ) 0. mos_i'\n",
+    "         ) 0. mos_cabs\n",
+    "    in\n",
+    "    match phase with\n",
+    "    | Phase.Pos -> result\n",
+    "    | Phase.Neg -> -. result\n",
+    "     \n",
+    "\n",
+    "let _ = check 11 integral_value 1 1 2 1"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "let ki = det_I.(0)\n",
+    "let kj = det_I.(10)\n",
+    "\n",
+    "\n",
+    "let _ =\n",
+    "    let alphas =\n",
+    "        generate_alphas ki kj 1 1 2 1\n",
+    "    in\n",
+    "    compute_HaaF ki alphas kj\n",
+    "\n",
+    "let _ = integral_value ki kj"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -1525,7 +1704,7 @@
  "metadata": {
   "celltoolbar": "Raw Cell Format",
   "kernelspec": {
-   "display_name": "OCaml default",
+   "display_name": "OCaml 4.07.1",
    "language": "OCaml",
    "name": "ocaml-jupyter"
   },