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   "source": [
    "# Obara Saika Scheme\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f43933a5",
   "metadata": {},
   "source": [
    "## Overlap\n",
    "\n",
    "Let us frist consider the simple overlap integral\n",
    "\n",
    "$$\n",
    "S_{ab} = \\langle G_a|G_b \\rangle\n",
    "$$\n",
    "\n",
    "it can be factorized in the three Cartesian directons\n",
    "\n",
    "$$\n",
    "S_{ab} = S_{ij}S_{kl}S_{mn}\n",
    "$$\n",
    "\n",
    "The Obara-Saika recurrence relations for the Cartesian overlap integral over one direction is:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "S_{i+1,j} &= X_{PA}S_{ij} + \\frac{1}{2p}(iS_{i-1,j}+jS_{i,j-1})\\\\\n",
    "S_{i,j+1} &= X_{PB}S_{ij} + \\frac{1}{2p}(iS_{i-1,j}+jS_{i,j-1})\\\\\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "With the boundary condtion\n",
    "\n",
    "$$\n",
    "S_{0,0} = \\sqrt(\\frac{\\pi}{p}) exp(-\\mu X_{AB}^2)\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9692f9d3",
   "metadata": {},
   "source": [
    "## Kinetic\n",
    "\n",
    "Consider the integral of electrons kinetic energy\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "T_{a,b} &= \\langle G_a  \\vert -\\sum^N_{i=1}\\frac{\\hbar^2}{2m_i}\\boldsymbol{\\nabla}_i^2 \\vert G_b \\rangle\\\\\n",
    "T_{a,b} &= T_{ij}S_{kl}S_{mn} + S_{ij}T_{kl}S_{mn} + S_{ij}S_{kl}T_{mn}\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "\n",
    "The Obara-Saika recurrence relations for the Cartesian kinetic integrals over one direction is:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "T_{i+1,j} &= X_{PA}T_{ij} + \\frac{1}{2p}(iT_{i-1,j}+jT_{i,j-1}) + \\frac{b}{p}(2aS_{i+1,j}-iS_{i-1,j})\\\\\n",
    "T_{i,j+1} &= X_{PB}T_{ij} + \\frac{1}{2p}(iT_{i-1,j}+jT_{i,j-1})+ \\frac{a}{p}(2bS_{i,j+1}-jS_{i,j-1})\\\\\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "With the boundary condtion\n",
    "\n",
    "$$\n",
    "T_{0,0} = \\left[a-2a^2(X_{PA}^2 + \\frac{1}{2p})\\right] S_{00}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cb7a7840",
   "metadata": {},
   "source": [
    "## Nuclear Attraction\n",
    "\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "V_{a,b} &=\\langle G_a \\vert -\\sum^N_{i=1}\\sum^M_{\\alpha=1} \\frac{Z_\\alpha e^2}{\\textbf{r}_{i\\alpha}} \\vert G_b \\rangle \\\\\n",
    "V_{a,b} &= V_{ijklmn}^{000} = \\Theta_{ijklmn}^{0}\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "\n",
    "The Obara-Saika recurrence relations for the Cartesian nuclear attraction integrals:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\Theta_{i+1,j,k,l,m,n}^{N} &= X_{PA}\\Theta_{ijkln}^{N}\n",
    "+\\frac{1}{2p}(i\\Theta_{i-1,j,k,l,m,n}^{N} + j\\Theta_{i,j-1,k,l,m,n}^{N})\\\\\n",
    "&-X_{PC}\\Theta_{ijklmn}^{N+1}\n",
    "-\\frac{1}{2p}(i\\Theta_{i-1,j,k,l,m,n}^{N+1} + j\\Theta_{i,j-1,k,l,m,n}^{N+1})\\\\\n",
    "\\Theta_{i,j+1,k,l,m,n}^{N} &= X_{PB}\\Theta_{ijkln}^{N}\n",
    "+\\frac{1}{2p}(i\\Theta_{i-1,j,k,l,m,n}^{N} + j\\Theta_{i,j-1,k,l,m,n}^{N})\\\\\n",
    "&-X_{PC}\\Theta_{ijklmn}^{N+1}\n",
    "-\\frac{1}{2p}(i\\Theta_{i-1,j,k,l,m,n}^{N+1} + j\\Theta_{i,j-1,k,l,m,n}^{N+1})\\\\\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "With the boundary condtion\n",
    "\n",
    "$$\n",
    "\\Theta_{000000}^{N} = \\frac{2\\pi}{p}(-2p)^{-N}K_{ab}^{xyz}R_{000}^{N} \n",
    "= \\frac{2\\pi}{p} K_{ab}^{xyz}F_N(pR_{PC}^2)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "be27bf0f",
   "metadata": {},
   "source": [
    "## Electron Repulsion\n",
    "\n",
    "\n",
    "$$       \n",
    "V_{abcd} = \\langle G_a G_b\\vert \\sum^N_{i=1}\\sum^N_{j>i} \\frac{e^2}{\\textbf{r}_{ij}}\\vert G_c G_d \\rangle \n",
    "$$\n",
    "\n",
    "The source and target integrals:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\Theta_{0000;0000;0000}^{N} &= \\frac{2\\pi^{2.5}}{pq\\sqrt{p+q}}\n",
    "K_{ab}^{xyz}K_{cd}^{xyz}F_N(\\alpha R_{PQ}^2)\\\\ \n",
    "\\Theta_{i_xj_xk_xl_x;i_yj_yk_yl_y;i_zj_zk_zl_z}^{0} &= g_{i_xj_xk_xl_x;i_yj_yk_yl_y;i_zj_zk_zl_z}\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "The Obara-Saika two electron recurrence relation\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\Theta_{i+1,j,k,l}^{N} &= \n",
    "X_{PA}\\Theta_{ijkl}^{N}\n",
    "-\\frac{\\alpha}{p}X_{PQ}\\Theta_{i,j,k,l}^{N+1} \n",
    "+\\frac{i}{2p}\\left(\\Theta_{i-1,j,k,l}^{N}-\\frac{\\alpha}{p}\\Theta_{i-1,j,k,l}^{N+1}\\right)\\\\\n",
    "&+\\frac{j}{2p}\\left(\\Theta_{i,j-1,k,l}^{N}-\\frac{\\alpha}{p}\\Theta_{i,j-1,k,l}^{N+1}\\right)\n",
    "-\\frac{k}{2(p+q)}\\Theta_{i,j,k-1,l}^{N+1} \n",
    "-\\frac{l}{2(p+q)}\\Theta_{i,j,k,l-1}^{N+1} \n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "Using the horizontal recurrence relation, a similar relation may be written down for increments in j, replaceing $X_{PA}$ with $X_{PB}$. \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\Theta_{i,j,k+1,l}^{N} &= \n",
    "X_{QC}\\Theta_{ijkl}^{N}\n",
    "-\\frac{\\alpha}{q}X_{PQ}\\Theta_{i,j,k,l}^{N+1} \n",
    "+\\frac{k}{2q}\\left(\\Theta_{i,j,k-1,l}^{N}-\\frac{\\alpha}{q}\\Theta_{i,j,k-1,l}^{N+1}\\right)\\\\\n",
    "&+\\frac{l}{2q}\\left(\\Theta_{i,j,k,l-1}^{N}-\\frac{\\alpha}{q}\\Theta_{i,j,k,l-1}^{N+1}\\right)\n",
    "-\\frac{i}{2(p+q)}\\Theta_{i-1,j,k,l}^{N+1} \n",
    "-\\frac{j}{2(p+q)}\\Theta_{i,j-1,k,l}^{N+1} \n",
    "\\end{aligned}\n",
    "$$"
   ]
  }
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