'hf done'

_toc.yml

@@ -65,6 +65,7 @@ parts:
     - file: ch07/note01
     - file: ch07/note02
     - file: ch07/note03
+    - file: ch07/note04
   - file: ch08/index
     sections:
     - file: ch08/note01

ch07/note04.ipynb

@@ -0,0 +1,233 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Hartree-Fock"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## **1. Introduction**\n",
+    "\n",
+    "- **Objective**: Approximate the electronic structure of atoms and molecules by treating electrons as independent particles in an average field created by all other electrons.\n",
+    "- **Motivation**: The Schrödinger equation for multi-electron systems is too complex to solve exactly.\n",
+    "\n",
+    "## **2. Key Concepts**\n",
+    "\n",
+    "### **Wavefunction**\n",
+    "- The total wavefunction $\\Psi(1, 2, \\cdots, N)$ is approximated as a single Slater determinant to satisfy the Pauli exclusion principle.\n",
+    "  $$ \\Psi(1, 2, \\cdots, N) = \\frac{1}{\\sqrt{N!}} \\begin{vmatrix} \\psi_1(1) & \\psi_2(1) & \\cdots & \\psi_N(1) \\\\ \\psi_1(2) & \\psi_2(2) & \\cdots & \\psi_N(2) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\psi_1(N) & \\psi_2(N) & \\cdots & \\psi_N(N) \\end{vmatrix} $$\n",
+    "\n",
+    "### **Hartree-Fock Equations**\n",
+    "- Each molecular orbital (MO) $\\psi_i$ satisfies the self-consistent field (SCF) equation:\n",
+    "  $$ \\hat{F} \\psi_i = \\varepsilon_i \\psi_i $$\n",
+    "  where $\\hat{F}$ is the **Fock operator** and $\\varepsilon_i$ are the orbital energies.\n",
+    "\n",
+    "### **Fock Operator**\n",
+    "- The Fock operator $\\hat{F}$ is a sum of the one-electron Hamiltonian and a mean-field potential:\n",
+    "  $$ \\hat{F} = \\hat{h} + \\sum_{j} \\left( \\hat{J}_j - \\hat{K}_j \\right) $$\n",
+    "  - $\\hat{h}$: Kinetic energy and nuclear attraction operators.\n",
+    "  - $\\hat{J}_j$: Coulomb operator (electron-electron repulsion).\n",
+    "  - $\\hat{K}_j$: Exchange operator (arises from antisymmetry requirement of the wavefunction).\n",
+    "\n",
+    "### **Self-Consistent Field (SCF) Method**\n",
+    "1. **Guess** initial molecular orbitals $\\psi_i$.\n",
+    "2. **Construct** the Fock operator $\\hat{F}$ using the current $\\psi_i$.\n",
+    "3. **Solve** the Hartree-Fock equation $\\hat{F} \\psi_i = \\varepsilon_i \\psi_i$ to obtain new orbitals $\\psi_i$.\n",
+    "4. **Repeat** steps 2-3 until convergence (orbitals do not change significantly).\n",
+    "\n",
+    "\n",
+    "\n",
+    "\n",
+    "## **3. Total Energy of the System**\n",
+    "- Total electronic energy $E_{HF}$ in the Hartree-Fock approximation:\n",
+    "  $$ E_{HF} = \\sum_{i} \\langle \\psi_i | \\hat{h} | \\psi_i \\rangle + \\frac{1}{2} \\sum_{i,j} \\left( J_{ij} - K_{ij} \\right) $$\n",
+    "  - $J_{ij}$: Coulomb integral.\n",
+    "  - $K_{ij}$: Exchange integral.\n",
+    "\n",
+    "\n",
+    "\n",
+    "\n",
+    "## **4. Strengths and Limitations**\n",
+    "\n",
+    "### **Strengths**\n",
+    "- **Efficient**: Reduces the complexity of the multi-electron Schrödinger equation.\n",
+    "- **Exchange interactions**: Includes effects of exchange through the Slater determinant.\n",
+    "\n",
+    "### **Limitations**\n",
+    "- **No dynamic correlation**: Electron-electron correlation is only partially captured.\n",
+    "- **Post-Hartree-Fock methods**: Needed for more accurate energy calculations (e.g., MP2, CCSD).\n",
+    "\n",
+    "\n",
+    "\n",
+    "\n",
+    "## **5. Summary**\n",
+    "- Hartree-Fock approximates the multi-electron wavefunction as a Slater determinant.\n",
+    "- The Fock operator accounts for one-electron and mean-field electron-electron interactions.\n",
+    "- The SCF procedure ensures convergence to a consistent set of orbitals.\n",
+    "- While efficient, it lacks dynamic correlation, motivating the development of post-Hartree-Fock methods.\n",
+    "\n",
+    "**Important Equations**\n",
+    "1. **Slater Determinant**:\n",
+    "\n",
+    "   $$ \\Psi(1, 2, \\cdots, N) = \\frac{1}{\\sqrt{N!}} \\begin{vmatrix} \\psi_1(1) & \\psi_2(1) & \\cdots & \\psi_N(1) \\\\ \\psi_1(2) & \\psi_2(2) & \\cdots & \\psi_N(2) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\psi_1(N) & \\psi_2(N) & \\cdots & \\psi_N(N) \\end{vmatrix} $$\n",
+    "\n",
+    "2. **Hartree-Fock Equations**:\n",
+    "\n",
+    "   $$ \\hat{F} \\psi_i = \\varepsilon_i \\psi_i $$\n",
+    "\n",
+    "3. **Fock Operator**:\n",
+    "\n",
+    "   $$ \\hat{F} = \\hat{h} + \\sum_{j} \\left( \\hat{J}_j - \\hat{K}_j \\right) $$\n",
+    "\n",
+    "4. **Electronic Energy**:\n",
+    "\n",
+    "   $$ E_{HF} = \\sum_{i} \\langle \\psi_i | \\hat{h} | \\psi_i \\rangle + \\frac{1}{2} \\sum_{i,j} \\left( J_{ij} - K_{ij} \\right) $$\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "![](./images/HF1.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Practical use of HF "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "* We need to install a few things before we get started\n",
+    "  * [PySCF](https://pyscf.org/) (for the quantum chemistry)\n",
+    "  * [py3DMol](https://3dmol.csb.pitt.edu/) for visualizing the molecule\n",
+    "  * [plotly](https://plotly.com/python/) and kaleido for plotting"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "%pip install -q pyscf py3DMol plotly kaleido"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "#@title PYSCF code to calcualte energies of Alkali Metals\n",
+    "\n",
+    "from pyscf import gto, scf\n",
+    "\n",
+    "# Function to calculate orbitals and energies for a given atom\n",
+    "def calculate_orbitals_energies(atom_symbol):\n",
+    "    \"\"\"\n",
+    "    Calculate the orbitals and energies for a given atom using PySCF.\n",
+    "    \n",
+    "    Parameters:\n",
+    "    atom_symbol (str): Symbol of the atom (e.g., 'He', 'Li').\n",
+    "    \n",
+    "    Returns:\n",
+    "    dict: A dictionary containing the total energy, MO coefficients, and MO energies.\n",
+    "    \"\"\"\n",
+    "    # Define the atom and basis set\n",
+    "    mol = gto.Mole()\n",
+    "    mol.atom = [(atom_symbol, (0, 0, 0))]  # Specify the atom and its position\n",
+    "    mol.basis = 'sto-3g'  # Basis set\n",
+    "    \n",
+    "    \n",
+    "    mol.spin = 1  # 2S = 1, since Li, Na, K has 1 unpaired electron\n",
+    "    \n",
+    "    mol.build()\n",
+    "    \n",
+    "    # Perform Hartree-Fock calculation\n",
+    "    mf = scf.RHF(mol)  # Restricted Hartree-Fock\n",
+    "    total_energy = mf.kernel()  # Calculate the total electronic energy\n",
+    "    \n",
+    "    # Extract molecular orbital coefficients and energies\n",
+    "    mo_coefficients = mf.mo_coeff  # MO coefficients\n",
+    "    mo_energies = mf.mo_energy  # MO energies\n",
+    "\n",
+    "    # Determine the HOMO (highest occupied molecular orbital) energy\n",
+    "    num_electrons = mol.nelectron  # Total number of electrons in the system\n",
+    "    homo_index = num_electrons // 2 - 1  # HOMO index for closed-shell systems (0-indexed)\n",
+    "    homo_energy = mo_energies[homo_index] if homo_index >= 0 else None\n",
+    "\n",
+    "  # Generate the cube file for the HOMO\n",
+    "    if homo_index >= 0:\n",
+    "        cube_filename = f\"{atom_symbol}_HOMO.cube\"\n",
+    "        cubegen.orbital(mol, cube_filename, mf.mo_coeff[:, homo_index])\n",
+    "        print(f\"HOMO cube file saved as: {cube_filename}\")\n",
+    "        \n",
+    "        # Visualize the cube file using py3Dmol\n",
+    "        cube_view = py3Dmol.view(width=400, height=400)\n",
+    "        with open(cube_filename, 'r') as file:\n",
+    "            cube_data = file.read()\n",
+    "        cube_view.addVolumetricData(cube_data, \"cube\", {'isoval': -0.03, 'color': \"red\", 'opacity': 0.85})\n",
+    "        cube_view.addVolumetricData(cube_data, \"cube\", {'isoval': 0.03, 'color': \"blue\", 'opacity': 0.85})\n",
+    "        cube_view.addModel(mol.tostring(format=\"xyz\"), 'xyz')\n",
+    "        cube_view.setStyle({'stick': {}, \"sphere\": {\"radius\": 0.4}})\n",
+    "        cube_view.setBackgroundColor('0xeeeeee')\n",
+    "        cube_view.show()\n",
+    "    \n",
+    "    results = {\n",
+    "        'atom': atom_symbol,\n",
+    "        'total_energy': total_energy,\n",
+    "        'mo_coefficients': mo_coefficients,\n",
+    "        'mo_energies': mo_energies,\n",
+    "        'homo_energy': homo_energy\n",
+    "    }\n",
+    "    \n",
+    "    return results"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# List of atoms to calculate\n",
+    "atoms = ['Li', 'Na']\n",
+    "\n",
+    "# Store the results for each atom\n",
+    "results_dict = {}\n",
+    "\n",
+    "# Loop over each atom and calculate orbitals and energies\n",
+    "for atom in atoms:\n",
+    "    results = calculate_orbitals_energies(atom)\n",
+    "    results_dict[atom] = results\n",
+    "    \n",
+    "    print(f\"Results for {atom} atom:\")\n",
+    "    print(\"Total Energy (Hartree):\", results['total_energy'])\n",
+    "    print(\"Molecular Orbital Energies (Hartree):\", results['mo_energies'])\n",
+    "    print(\"HOMO energy (Hartree):\", results['homo_energy'])\n",
+    "    #print(\"Molecular Orbital Coefficients:\")\n",
+    "    #print(results['mo_coefficients'])\n",
+    "    #print(\"\\n\" + \"-\"*50 + \"\\n\")\n",
+    "    print(\"\\n\")"
+   ]
+  }
+ ],
+ "metadata": {
+  "language_info": {
+   "name": "python"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}