complete 3d overlap equation

Author: hatemhelal

notebooks/gto_integrals.ipynb

@@ -50,7 +50,7 @@
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-   "execution_count": 2,
+   "execution_count": 1,
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     {
@@ -59,7 +59,7 @@
        "5"
       ]
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+     "execution_count": 1,
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@@ -87,7 +87,7 @@
   },
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-   "execution_count": 3,
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     {
@@ -96,7 +96,7 @@
        "15"
       ]
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-     "execution_count": 3,
+     "execution_count": 2,
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@@ -497,19 +497,20 @@
     "\\left(x_B^{l_\\nu} e^{-\\alpha_\\nu x_B^2} \\right) dx = \\\\\n",
     "\\int_{-\\infty}^\\infty \n",
     " e^{-\\alpha_\\mu \\alpha_\\nu (X_A - X_B)^2 / \\gamma} e^{-\\gamma x_C^2}\n",
-    "\\sum_{s=0}^{l_\\mu+ l_\\nu} B(l_\\mu, l_\\nu, C_A, C_B, s) x_C^s dx\n",
+    "\\sum_{s=0}^{l_\\mu+ l_\\nu} B(l_\\mu, l_\\nu, CA_x, CB_x, s) x_C^s dx\n",
     "$$\n",
     "Swapping the order of integration and the summation we see that we need to evaluate\n",
     "integrals of the form:\n",
     "$$\n",
+    "\\tag{27}\n",
     "\\int_{-\\infty}^\\infty t^s e^{-a t^2}  dt\n",
     "$$\n",
     "This has a known analytic solution that we can evaluate using SymPy:"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 51,
+   "execution_count": 10,
    "metadata": {},
    "outputs": [
     {
@@ -528,7 +529,7 @@
        "-∞             "
       ]
      },
-     "execution_count": 51,
+     "execution_count": 10,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -544,7 +545,7 @@
   },
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-   "execution_count": 52,
+   "execution_count": 11,
    "metadata": {},
    "outputs": [
     {
@@ -563,7 +564,7 @@
        "        2⋅√a               2⋅√a    "
       ]
      },
-     "execution_count": 52,
+     "execution_count": 11,
      "metadata": {},
      "output_type": "execute_result"
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@@ -581,7 +582,7 @@
   },
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    "cell_type": "code",
-   "execution_count": 57,
+   "execution_count": 12,
    "metadata": {},
    "outputs": [
     {
@@ -594,7 +595,7 @@
        "0"
       ]
      },
-     "execution_count": 57,
+     "execution_count": 12,
      "metadata": {},
      "output_type": "execute_result"
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@@ -613,7 +614,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 65,
+   "execution_count": 13,
    "metadata": {},
    "outputs": [
     {
@@ -627,7 +628,7 @@
        "a        ⋅Γ(s + 1/2)"
       ]
      },
-     "execution_count": 65,
+     "execution_count": 13,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -640,13 +641,29 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Using this result allows us to write the one-dimensional overlap integral as:\n",
+    "This result can also be written in terms of the [double factorial](https://en.wikipedia.org/wiki/Double_factorial#Additional_identities) which gives us two possible computation strategies for this integral:\n",
+    "$$\n",
+    "\\tag{28}\n",
+    "G(a, s) = \\int_{-\\infty}^{\\infty} t^{2s} e^{-a t^2} dt \\\\\n",
+    "= a^{-s - \\frac{1}{2}} \\Gamma\\left(s + \\frac{1}{2}\\right) \\\\\n",
+    "= \\frac{(2s-1)!!}{(2a)^s} \\sqrt{\\frac{\\pi}{a}}.\n",
+    "$$\n",
+    "The last form agrees with Equation (3.15) derived by [Fermann and Valeev](http://arxiv.org/abs/2007.12057).\n",
+    "\n",
+    "Using the function $G(a, s)$ allows us to write the one-dimensional overlap integral as:\n",
     "$$\n",
     "\\tilde{S}_{\\mu \\nu}^{(x)} = \n",
     "e^{-\\alpha_\\mu \\alpha_\\nu (X_A - X_B)^2 / \\gamma}\n",
-    "\\sum_{s=0}^{\\lfloor(i+j)/2 \\rfloor} B(l_\\mu, l_\\nu, C_A, C_B, 2s)\n",
-    "\\int_{-\\infty}^\\infty \n",
-    "x_C^{2s} e^{-\\gamma x_C^2} dx\n",
+    "\\sum_{s=0}^{\\lfloor(l_\\mu + l_\\nu)/2 \\rfloor} B(l_\\mu, l_\\nu, CA_x, CB_x, 2s)\\;G(\\gamma, 2s)\n",
+    "$$\n",
+    "substituting this back into Equation (11) gives us the overlap of two primitive Gaussians:\n",
+    "$$\n",
+    "\\tag{11}\n",
+    "\\tilde{S}_{\\mu \\nu} = \\iiint p_\\mu(\\br) p_\\nu(\\br) dx dy dz \\\\\n",
+    "= N_\\mu N_\\nu e^{-\\alpha_\\mu \\alpha_\\nu |\\mathbf{A}-\\mathbf{B}|^2 / \\gamma}\n",
+    "\\sum_{s=0}^{\\lfloor(l_\\mu + l_\\nu)/2 \\rfloor} B(l_\\mu, l_\\nu, CA_x, CB_x, 2s)\\;G(\\gamma, 2s) \\\\\n",
+    "\\times \\sum_{s=0}^{\\lfloor(m_\\mu + m_\\nu)/2 \\rfloor} B(m_\\mu, m_\\nu, CA_y, CB_y, 2s)\\;G(\\gamma, 2s)\n",
+    "\\sum_{s=0}^{\\lfloor(n_\\mu + n_\\nu)/2 \\rfloor} B(n_\\mu, n_\\nu, CA_z, CB_z, 2s)\\;G(\\gamma, 2s)\n",
     "$$"
    ]
   }