01-IntroductionDay2.ipynb 3.19 KB
 Yury Lysogorski committed Mar 08, 2021 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Day 2 - Parameterization of interatomic potentials" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this tutorial we will do simple fits for three different interatomic potentials.\n", "* Embedded Atom Method Potential\n", "* Neural Network Potential\n", "* Atomic Cluster Expansion\n", "\n", "Some details of these potentials will be summarized in the following." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Embedded Atom Method Potential" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* Atomic descriptors: pair functions\n", "\n", "$\\rho_i = \\sum_j \\phi(r_{ij})$ (density)\n", "\n", "$V_i = \\sum_j V(r_{ij})$ (pair repulsion)\n", " \n", "* Atomic energy\n", "\n", "$E_i = F( \\rho_i ) + V_i$ \n", "\n", "with non-linear embedding function $F$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Neural Network Potential" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* Atomic descriptors: pair and three-body symmetry functions\n", "\n", "$G_i = \\sum_j \\phi(r_{ij})$\n", "\n", "$G_i = \\sum_{jk} \\phi(r_{ij},r_{ik}, \\cos_{jik})$\n", " \n", "* Atomic energy\n", "\n", "$E_i = NN(G_i)$ \n", "\n", "with neural network $NN$. Various different $G_i$ are the inputs to the $NN$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Atomic Cluster Expansion" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* Atomic descriptors: pair, three-body, ... many-body basis functions\n", "\n", "$A_i = \\sum_j \\phi(\\pmb{r}_{ij})$ (many different basis functions that depend on direction and length of $r_{ij}$)\n", "\n", "$\\varphi_i = c_1 A_i + c_2 A_i A_i + c_3 A_i A_i A_i + ...$\n", " \n", "* Atomic energy\n", "\n", "$E_i = F(\\varphi_i)$ \n", "\n", "with general non-linear function $F$ and several $\\varphi_i$. In the tutorial we will use $E_i = \\sqrt{\\varphi^{(1)}_i} + \\varphi^{(2)}_i$ to make contact to the Embedded Atom Method.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Reference data" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The potentials are parameterized by fitting to reference data. Here we use DFT data for Cu that we generated with the FHI-aims code. In the following we summarize key properties of the dataset." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.6" } }, "nbformat": 4, "nbformat_minor": 5 }