From c89cae8ef991ab02acbe9f34ce84264fcd76c8c5 Mon Sep 17 00:00:00 2001 From: Davit Potoyan Date: Wed, 11 Dec 2024 11:51:09 -0600 Subject: [PATCH] 'hf done' --- _toc.yml | 1 + ch07/images/HF1.png | Bin 0 -> 182975 bytes ch07/images/HF2.png | Bin 0 -> 186860 bytes ch07/images/cat5.gif | Bin 0 -> 29802 bytes ch07/note04.ipynb | 233 +++++++++++++++++++++++++++++++++++++++++++ 5 files changed, 234 insertions(+) create mode 100644 ch07/images/HF1.png create mode 100644 ch07/images/HF2.png create mode 100644 ch07/images/cat5.gif create mode 100644 ch07/note04.ipynb diff --git a/_toc.yml b/_toc.yml index 43df07e9..85269fe2 100644 --- a/_toc.yml +++ b/_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 diff --git a/ch07/images/HF1.png b/ch07/images/HF1.png new file mode 100644 index 0000000000000000000000000000000000000000..5fd2186cbc558a995472b103cd43befd77cbec55 GIT binary patch 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**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 +}