{
 "cells": [
  {
   "cell_type": "markdown",
   "source": [
    "# Periodic problems and plane-wave discretisations"
   ],
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "source": [
    "In this example we want to show how DFTK can be used to solve simple one-dimensional\n",
    "periodic problems. Along the lines this notebook serves as a concise introduction into\n",
    "the underlying theory and jargon for solving periodic problems using plane-wave\n",
    "discretizations. This example provides a high-level overview focused on plane-wave\n",
    "discretisations. For a more hands-on introduction walking you through some exercises\n",
    "on these topics see Comparing discretization techniques and\n",
    "Modelling atomic chains."
   ],
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "source": [
    "## Periodicity and lattices\n",
    "A periodic problem is characterized by being invariant to certain translations.\n",
    "For example the $\\sin$ function is periodic with periodicity $2π$, i.e.\n",
    "$$\n",
    "   \\sin(x) = \\sin(x + 2πm) \\quad ∀ m ∈ \\mathbb{Z},\n",
    "$$\n",
    "This is nothing else than saying that any translation by an integer multiple of $2π$\n",
    "keeps the $\\sin$ function invariant. More formally, one can use the\n",
    "translation operator $T_{2πm}$ to write this as:\n",
    "$$\n",
    "   T_{2πm} \\, \\sin(x) = \\sin(x - 2πm) = \\sin(x).\n",
    "$$\n",
    "\n",
    "Whenever such periodicity exists one can exploit it to save computational work.\n",
    "Consider a problem in which we want to find a function $f : \\mathbb{R} → \\mathbb{R}$,\n",
    "but *a priori* the solution is known to be periodic with periodicity $a$. As a consequence\n",
    "of said periodicity it is sufficient to determine the values of $f$ for all $x$ from the\n",
    "interval $[-a/2, a/2)$ to uniquely define $f$ on the full real axis. Naturally exploiting\n",
    "periodicity in a computational procedure thus greatly reduces the required amount of work.\n",
    "\n",
    "Let us introduce some jargon: The periodicity of our problem implies that we may define\n",
    "a **lattice**\n",
    "```\n",
    "        -3a/2      -a/2      +a/2     +3a/2\n",
    "       ... |---------|---------|---------| ...\n",
    "                a         a         a\n",
    "```\n",
    "with lattice constant $a$. Each cell of the lattice is an identical periodic image of\n",
    "any of its neighbors. For finding $f$ it is thus sufficient to consider only the\n",
    "problem inside a **unit cell** $[-a/2, a/2)$ (this is the convention used by DFTK, but this is arbitrary, and for instance $[0,a)$ would have worked just as well).\n",
    "\n",
    "## Periodic operators and the Bloch transform\n",
    "Not only functions, but also operators can feature periodicity.\n",
    "Consider for example the **free-electron Hamiltonian**\n",
    "$$\n",
    "    H = -\\frac12 Δ.\n",
    "$$\n",
    "In free-electron model (which gives rise to this Hamiltonian) electron motion is only\n",
    "by their own kinetic energy. As this model features no potential which could make one point\n",
    "in space more preferred than another, we would expect this model to be periodic.\n",
    "If an operator is periodic with respect to a lattice such as the one defined above,\n",
    "then it commutes with all lattice translations. For the free-electron case $H$\n",
    "one can easily show exactly that, i.e.\n",
    "$$\n",
    "   T_{ma} H = H T_{ma} \\quad  ∀ m ∈ \\mathbb{Z}.\n",
    "$$\n",
    "We note in passing that the free-electron model is actually very special in the sense that\n",
    "the choice of $a$ is completely arbitrary here. In other words $H$ is periodic\n",
    "with respect to any translation. In general, however, periodicity is only\n",
    "attained with respect to a finite number of translations $a$ and we will take this\n",
    "viewpoint here.\n",
    "\n",
    "**Bloch's theorem** now tells us that for periodic operators,\n",
    "the solutions to the eigenproblem\n",
    "$$\n",
    "    H ψ_{kn} = ε_{kn} ψ_{kn}\n",
    "$$\n",
    "satisfy a factorization\n",
    "$$\n",
    "    ψ_{kn}(x) = e^{i k⋅x} u_{kn}(x)\n",
    "$$\n",
    "into a plane wave $e^{i k⋅x}$ and a lattice-periodic function\n",
    "$$\n",
    "   T_{ma} u_{kn}(x) = u_{kn}(x - ma) = u_{kn}(x) \\quad ∀ m ∈ \\mathbb{Z}.\n",
    "$$\n",
    "In this $n$ is a labeling integer index and $k$ is a real number,\n",
    "whose details will be clarified in the next section.\n",
    "The index $n$ is sometimes also called the **band index** and\n",
    "functions $ψ_{kn}$ satisfying this factorization are also known as\n",
    "**Bloch functions** or **Bloch states**.\n",
    "\n",
    "Consider the application of $2H = -Δ = - \\frac{d^2}{d x^2}$\n",
    "to such a Bloch wave. First we notice for any function $f$\n",
    "$$\n",
    "   -i∇ \\left( e^{i k⋅x} f \\right)\n",
    "   = -i\\frac{d}{dx} \\left( e^{i k⋅x} f \\right)\n",
    "   = k e^{i k⋅x} f + e^{i k⋅x} (-i∇) f = e^{i k⋅x} (-i∇ + k) f.\n",
    "$$\n",
    "Using this result twice one shows that applying $-Δ$ yields\n",
    "$$\n",
    "\\begin{aligned}\n",
    "   -\\Delta \\left(e^{i k⋅x} u_{kn}(x)\\right)\n",
    "   &= -i∇ ⋅ \\left[-i∇ \\left(u_{kn}(x) e^{i k⋅x} \\right) \\right] \\\\\n",
    "   &= -i∇ ⋅ \\left[e^{i k⋅x} (-i∇ + k) u_{kn}(x) \\right] \\\\\n",
    "   &= e^{i k⋅x} (-i∇ + k)^2 u_{kn}(x) \\\\\n",
    "   &= e^{i k⋅x} 2H_k u_{kn}(x),\n",
    "\\end{aligned}\n",
    "$$\n",
    "where we defined\n",
    "$$\n",
    "    H_k = \\frac12 (-i∇ + k)^2.\n",
    "$$\n",
    "The action of this operator on a function $u_{kn}$ is given by\n",
    "$$\n",
    "    H_k u_{kn} = e^{-i k⋅x} H e^{i k⋅x} u_{kn},\n",
    "$$\n",
    "which in particular implies that\n",
    "$$\n",
    "   H_k u_{kn} = ε_{kn} u_{kn} \\quad ⇔ \\quad H (e^{i k⋅x} u_{kn}) = ε_{kn} (e^{i k⋅x} u_{kn}).\n",
    "$$\n",
    "To seek the eigenpairs of $H$ we may thus equivalently\n",
    "find the eigenpairs of *all* $H_k$.\n",
    "The point of this is that the eigenfunctions $u_{kn}$ of $H_k$\n",
    "are periodic (unlike the eigenfunctions $ψ_{kn}$ of $H$).\n",
    "In contrast to $ψ_{kn}$ the functions $u_{kn}$ can thus be fully\n",
    "represented considering the eigenproblem only on the unit cell.\n",
    "\n",
    "A detailed mathematical analysis shows that the transformation from $H$\n",
    "to the set of all $H_k$ for a suitable set of values for $k$ (details below)\n",
    "is actually a unitary transformation, the so-called **Bloch transform**.\n",
    "This transform brings the Hamiltonian into the symmetry-adapted basis for\n",
    "translational symmetry, which are exactly the Bloch functions.\n",
    "Similar to the case of choosing a symmetry-adapted basis for other kinds of symmetries\n",
    "(like the point group symmetry in molecules), the Bloch transform also makes\n",
    "the Hamiltonian $H$ block-diagonal[^1]:\n",
    "$$\n",
    "    T_B H T_B^{-1} ⟶ \\left( \\begin{array}{cccc} H_1&&&0 \\\\ &H_2\\\\&&H_3\\\\0&&&\\ddots \\end{array} \\right)\n",
    "$$\n",
    "with each block $H_k$ taking care of one value $k$.\n",
    "This block-diagonal structure under the basis of Bloch functions lets us\n",
    "completely describe the spectrum of $H$ by looking only at the spectrum\n",
    "of all $H_k$ blocks.\n",
    "\n",
    "[^1]: Notice that block-diagonal is a bit an abuse of terms here, since the Hamiltonian\n",
    "      is not a matrix but an operator and the number of blocks is infinite.\n",
    "      The mathematically precise term is that the Bloch transform reveals the fibers\n",
    "      of the Hamiltonian.\n",
    "\n",
    "## The Brillouin zone\n",
    "\n",
    "We now consider the parameter $k$ of the Hamiltonian blocks in detail.\n",
    "\n",
    "- As discussed $k$ is a real number. It turns out, however, that some of\n",
    "  these $k∈\\mathbb{R}$ give rise to operators related by unitary transformations\n",
    "  (again due to translational symmetry).\n",
    "- Since such operators have the same eigenspectrum, only one version needs to be considered.\n",
    "- The smallest subset from which $k$ is chosen is the **Brillouin zone** (BZ).\n",
    "\n",
    "- The BZ is the unit cell of the **reciprocal lattice**, which may be constructed from\n",
    "  the **real-space lattice** by a Fourier transform.\n",
    "- In our simple 1D case the reciprocal lattice is just\n",
    "  ```\n",
    "    ... |--------|--------|--------| ...\n",
    "           2π/a     2π/a     2π/a\n",
    "  ```\n",
    "  i.e. like the real-space lattice, but just with a different lattice constant\n",
    "  $b = 2π / a$.\n",
    "- The BZ in our example is thus $B = [-π/a, π/a)$. The members of $B$\n",
    "  are typically called $k$-points.\n",
    "\n",
    "## Discretization and plane-wave basis sets\n",
    "\n",
    "With what we discussed so far the strategy to find all eigenpairs of a periodic\n",
    "Hamiltonian $H$ thus reduces to finding the eigenpairs of all $H_k$ with $k ∈ B$.\n",
    "This requires *two* discretisations:\n",
    "\n",
    "  - $B$ is infinite (and not countable). To discretize we first only pick a finite number\n",
    "    of $k$-points. Usually this **$k$-point sampling** is done by picking $k$-points\n",
    "    along a regular grid inside the BZ, the **$k$-grid**.\n",
    "  - Each $H_k$ is still an infinite-dimensional operator.\n",
    "    Following a standard Ritz-Galerkin ansatz we project the operator into a finite basis\n",
    "    and diagonalize the resulting matrix.\n",
    "\n",
    "For the second step multiple types of bases are used in practice (finite differences,\n",
    "finite elements, Gaussians, ...). In DFTK we currently support only plane-wave\n",
    "discretizations.\n",
    "\n",
    "For our 1D example normalized plane waves are defined as the functions\n",
    "$$\n",
    "e_{G}(x) = \\frac{e^{i G x}}{\\sqrt{a}}  \\qquad G \\in b\\mathbb{Z}\n",
    "$$\n",
    "and typically one forms basis sets from these by specifying a\n",
    "**kinetic energy cutoff** $E_\\text{cut}$:\n",
    "$$\n",
    "\\left\\{ e_{G} \\, \\big| \\, (G + k)^2 \\leq 2E_\\text{cut} \\right\\}\n",
    "$$\n",
    "\n",
    "## Correspondence of theory to DFTK code\n",
    "\n",
    "Before solving a few example problems numerically in DFTK, a short overview\n",
    "of the correspondence of the introduced quantities to data structures inside DFTK.\n",
    "\n",
    "- $H$ is represented by a `Hamiltonian` object and variables for hamiltonians are usually called `ham`.\n",
    "- $H_k$ by a `HamiltonianBlock` and variables are `hamk`.\n",
    "- $ψ_{kn}$ is usually just called `ψ`.\n",
    "  $u_{kn}$ is not stored (in favor of $ψ_{kn}$).\n",
    "- $ε_{kn}$ is called `eigenvalues`.\n",
    "- $k$-points are represented by `Kpoint` and respective variables called `kpt`.\n",
    "- The basis of plane waves is managed by `PlaneWaveBasis` and variables usually just called `basis`.\n",
    "\n",
    "## Solving the free-electron Hamiltonian\n",
    "\n",
    "One typical approach to get physical insight into a Hamiltonian $H$ is to plot\n",
    "a so-called **band structure**, that is the eigenvalues of $H_k$ versus $k$.\n",
    "In DFTK we achieve this using the following steps:\n",
    "\n",
    "Step 1: Build the 1D lattice. DFTK is mostly tailored for 3D problems.\n",
    "Therefore quantities related to the problem space are have a fixed\n",
    "dimension 3. The convention is that for 1D / 2D problems the\n",
    "trailing entries are always zero and ignored in the computation.\n",
    "For the lattice we therefore construct a 3x3 matrix with only one entry."
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "using DFTK\n",
    "\n",
    "lattice = zeros(3, 3)\n",
    "lattice[1, 1] = 20.;"
   ],
   "metadata": {},
   "execution_count": 1
  },
  {
   "cell_type": "markdown",
   "source": [
    "Step 2: Select a model. In this case we choose a free-electron model,\n",
    "which is the same as saying that there is only a Kinetic term\n",
    "(and no potential) in the model."
   ],
   "metadata": {}
  },
  {
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "Model(custom, 1D):\n    lattice (in Bohr)    : [20        , 0         , 0         ]\n                           [0         , 0         , 0         ]\n                           [0         , 0         , 0         ]\n    unit cell volume     : 20 Bohr\n\n    num. electrons       : 0\n    spin polarization    : none\n    temperature          : 0 Ha\n\n    terms                : Kinetic()"
     },
     "metadata": {},
     "execution_count": 2
    }
   ],
   "cell_type": "code",
   "source": [
    "model = Model(lattice; terms=[Kinetic()])"
   ],
   "metadata": {},
   "execution_count": 2
  },
  {
   "cell_type": "markdown",
   "source": [
    "Step 3: Define a plane-wave basis using this model and a cutoff $E_\\text{cut}$\n",
    "of 300 Hartree. The $k$-point grid is given as a regular grid in the BZ\n",
    "(a so-called **Monkhorst-Pack** grid). Here we select only one $k$-point (1x1x1),\n",
    "see the note below for some details on this choice."
   ],
   "metadata": {}
  },
  {
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "PlaneWaveBasis discretization:\n    architecture         : DFTK.CPU()\n    num. mpi processes   : 1\n    num. julia threads   : 1\n    num. DFTK  threads   : 1\n    num. blas  threads   : 2\n    num. fft   threads   : 1\n\n    Ecut                 : 300.0 Ha\n    fft_size             : (320, 1, 1), 320 total points\n    kgrid                : MonkhorstPack([1, 1, 1])\n    num.   red. kpoints  : 1\n    num. irred. kpoints  : 1\n\n    Discretized Model(custom, 1D):\n        lattice (in Bohr)    : [20        , 0         , 0         ]\n                               [0         , 0         , 0         ]\n                               [0         , 0         , 0         ]\n        unit cell volume     : 20 Bohr\n    \n        num. electrons       : 0\n        spin polarization    : none\n        temperature          : 0 Ha\n    \n        terms                : Kinetic()"
     },
     "metadata": {},
     "execution_count": 3
    }
   ],
   "cell_type": "code",
   "source": [
    "basis = PlaneWaveBasis(model; Ecut=300, kgrid=(1, 1, 1))"
   ],
   "metadata": {},
   "execution_count": 3
  },
  {
   "cell_type": "markdown",
   "source": [
    "Step 4: Plot the bands! Select a density of $k$-points for the $k$-grid to use\n",
    "for the bandstructure calculation, discretize the problem and diagonalize it.\n",
    "Afterwards plot the bands."
   ],
   "metadata": {}
  },
  {
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "┌ Warning: Calling plot_bandstructure without first computing the band data is deprecated and will be removed in the next minor version bump.\n",
      "└ @ DFTKPlotsExt ~/work/DFTK.jl/DFTK.jl/ext/DFTKPlotsExt.jl:15\n"
     ]
    },
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "Plot{Plots.GRBackend() n=6}",
      "image/png": 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       "</svg>\n"
      ]
     },
     "metadata": {},
     "execution_count": 4
    }
   ],
   "cell_type": "code",
   "source": [
    "using Unitful\n",
    "using UnitfulAtomic\n",
    "using Plots\n",
    "\n",
    "plot_bandstructure(basis; n_bands=6, kline_density=100)"
   ],
   "metadata": {},
   "execution_count": 4
  },
  {
   "cell_type": "markdown",
   "source": [
    "> **Selection of k-point grids in `PlaneWaveBasis` construction**\n",
    ">\n",
    "> You might wonder why we only selected a single $k$-point (clearly a very crude\n",
    "> and inaccurate approximation). In this example the `kgrid` parameter specified\n",
    "> in the construction of the `PlaneWaveBasis`\n",
    "> is not actually used for plotting the bands. It is only used when solving more\n",
    "> involved models like density-functional theory (DFT) where the Hamiltonian is\n",
    "> non-linear. In these cases before plotting the bands the self-consistent field\n",
    "> equations (SCF) need to be solved first. This is typically done on\n",
    "> a different $k$-point grid than the grid used for the bands later on.\n",
    "> In our case we don't need this extra step and therefore the `kgrid` value passed\n",
    "> to `PlaneWaveBasis` is actually arbitrary."
   ],
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "source": [
    "## Adding potentials\n",
    "So far so good. But free electrons are actually a little boring,\n",
    "so let's add a potential interacting with the electrons.\n",
    "\n",
    "- The modified problem we will look at consists of diagonalizing\n",
    "  $$\n",
    "  H_k = \\frac12 (-i \\nabla + k)^2 + V\n",
    "  $$\n",
    "  for all $k \\in B$ with a periodic potential $V$ interacting with the electrons.\n",
    "\n",
    "- A number of \"standard\" potentials are readily implemented in DFTK and\n",
    "  can be assembled using the `terms` kwarg of the model.\n",
    "  This allows to seamlessly construct\n",
    "\n",
    "  * density-functial theory (DFT) models for treating electronic structures\n",
    "    (see the Tutorial).\n",
    "  * Gross-Pitaevskii models for bosonic systems\n",
    "    (see Gross-Pitaevskii equation in one dimension)\n",
    "  * even some more unusual cases like anyonic models.\n",
    "\n",
    "We will use `ElementGaussian`, which is an analytic potential describing a Gaussian\n",
    "interaction with the electrons to DFTK. See Custom potential for\n",
    "how to create a custom potential.\n",
    "\n",
    "A single potential looks like:"
   ],
   "metadata": {}
  },
  {
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "Plot{Plots.GRBackend() n=1}",
      "image/png": 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      ]
     },
     "metadata": {},
     "execution_count": 5
    }
   ],
   "cell_type": "code",
   "source": [
    "using Plots\n",
    "using LinearAlgebra\n",
    "nucleus = ElementGaussian(0.3, 10.0)\n",
    "plot(r -> DFTK.local_potential_real(nucleus, norm(r)), xlims=(-50, 50))"
   ],
   "metadata": {},
   "execution_count": 5
  },
  {
   "cell_type": "markdown",
   "source": [
    "With this element at hand we can easily construct a setting\n",
    "where two potentials of this form are located at positions\n",
    "$20$ and $80$ inside the lattice $[0, 100]$:"
   ],
   "metadata": {}
  },
  {
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "Plot{Plots.GRBackend() n=1}",
      "image/png": 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911.932,1108.98 913.115,1104.49 914.298,1099.98 915.481,1095.44 916.664,1090.89 917.847,1086.31 919.03,1081.72 920.213,1077.1 921.395,1072.47 922.578,1067.82 923.761,1063.15 924.944,1058.46 926.127,1053.76 927.31,1049.03 928.493,1044.3 929.676,1039.54 930.859,1034.78 932.042,1029.99 933.225,1025.19 934.408,1020.38 935.591,1015.56 936.774,1010.72 937.957,1005.87 939.14,1001 940.323,996.129 941.506,991.243 942.689,986.345 943.872,981.438 945.055,976.52 946.238,971.592 947.421,966.656 948.604,961.711 949.787,956.757 950.97,951.796 952.153,946.827 953.336,941.852 954.519,936.869 955.702,931.881 956.885,926.887 958.068,921.887 959.251,916.883 960.434,911.874 961.617,906.862 962.8,901.845 963.982,896.826 965.165,891.803 966.348,886.778 967.531,881.752 968.714,876.723 969.897,871.694 971.08,866.663 972.263,861.633 973.446,856.602 974.629,851.572 975.812,846.542 976.995,841.514 978.178,836.487 979.361,831.462 980.544,826.44 981.727,821.42 982.91,816.403 984.093,811.39 985.276,806.38 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1135.51,291.182 1136.7,288.474 1137.88,285.788 1139.06,283.125 1140.25,280.484 1141.43,277.866 1142.61,275.27 1143.79,272.697 1144.98,270.146 1146.16,267.618 1147.34,265.111 1148.53,262.627 1149.71,260.166 1150.89,257.726 1152.08,255.308 1153.26,252.912 1154.44,250.538 1155.62,248.186 1156.81,245.855 1157.99,243.546 1159.17,241.259 1160.36,238.993 1161.54,236.748 1162.72,234.525 1163.9,232.323 1165.09,230.142 1166.27,227.982 1167.45,225.843 1168.64,223.725 1169.82,221.628 1171,219.551 1172.19,217.496 1173.37,215.46 1174.55,213.445 1175.73,211.451 1176.92,209.476 1178.1,207.522 1179.28,205.588 1180.47,203.674 1181.65,201.78 1182.83,199.905 1184.02,198.05 1185.2,196.215 1186.38,194.399 1187.56,192.602 1188.75,190.825 1189.93,189.067 1191.11,187.328 1192.3,185.608 1193.48,183.906 1194.66,182.224 1195.84,180.56 1197.03,178.914 1198.21,177.287 1199.39,175.679 1200.58,174.088 1201.76,172.516 1202.94,170.962 1204.13,169.425 1205.31,167.906 1206.49,166.405 1207.67,164.922 1208.86,163.456 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1731.73,871.694 1732.91,876.723 1734.1,881.752 1735.28,886.778 1736.46,891.803 1737.65,896.826 1738.83,901.845 1740.01,906.862 1741.19,911.874 1742.38,916.883 1743.56,921.887 1744.74,926.887 1745.93,931.881 1747.11,936.869 1748.29,941.852 1749.48,946.827 1750.66,951.796 1751.84,956.757 1753.02,961.711 1754.21,966.656 1755.39,971.592 1756.57,976.52 1757.76,981.438 1758.94,986.345 1760.12,991.243 1761.31,996.129 1762.49,1001 1763.67,1005.87 1764.85,1010.72 1766.04,1015.56 1767.22,1020.38 1768.4,1025.19 1769.59,1029.99 1770.77,1034.78 1771.95,1039.54 1773.14,1044.3 1774.32,1049.03 1775.5,1053.76 1776.68,1058.46 1777.87,1063.15 1779.05,1067.82 1780.23,1072.47 1781.42,1077.1 1782.6,1081.72 1783.78,1086.31 1784.96,1090.89 1786.15,1095.44 1787.33,1099.98 1788.51,1104.49 1789.7,1108.98 1790.88,1113.45 1792.06,1117.9 1793.25,1122.33 1794.43,1126.73 1795.61,1131.1 1796.79,1135.46 1797.98,1139.79 1799.16,1144.09 1800.34,1148.37 1801.53,1152.62 1802.71,1156.85 1803.89,1161.04 1805.08,1165.22 1806.26,1169.36 1807.44,1173.48 1808.62,1177.56 1809.81,1181.62 1810.99,1185.65 1812.17,1189.65 1813.36,1193.62 1814.54,1197.56 1815.72,1201.46 1816.91,1205.34 1818.09,1209.18 1819.27,1212.99 1820.45,1216.77 1821.64,1220.52 1822.82,1224.23 1824,1227.91 1825.19,1231.55 1826.37,1235.16 1827.55,1238.74 1828.73,1242.28 1829.92,1245.78 1831.1,1249.25 1832.28,1252.68 1833.47,1256.07 1834.65,1259.43 1835.83,1262.74 1837.02,1266.03 1838.2,1269.27 1839.38,1272.47 1840.56,1275.63 1841.75,1278.76 1842.93,1281.84 1844.11,1284.89 1845.3,1287.89 1846.48,1290.86 1847.66,1293.78 1848.85,1296.66 1850.03,1299.5 1851.21,1302.3 1852.39,1305.05 1853.58,1307.76 1854.76,1310.43 1855.94,1313.06 1857.13,1315.64 1858.31,1318.18 1859.49,1320.67 1860.68,1323.12 1861.86,1325.53 1863.04,1327.89 1864.22,1330.2 1865.41,1332.47 1866.59,1334.69 1867.77,1336.87 1868.96,1339 1870.14,1341.08 1871.32,1343.12 1872.5,1345.11 1873.69,1347.06 1874.87,1348.95 1876.05,1350.8 1877.24,1352.6 1878.42,1354.35 1879.6,1356.06 1880.79,1357.71 1881.97,1359.32 1883.15,1360.88 1884.33,1362.38 1885.52,1363.84 1886.7,1365.25 1887.88,1366.62 1889.07,1367.93 1890.25,1369.19 1891.43,1370.4 1892.62,1371.56 1893.8,1372.67 1894.98,1373.73 1896.16,1374.74 1897.35,1375.7 1898.53,1376.61 1899.71,1377.47 1900.9,1378.28 1902.08,1379.04 1903.26,1379.74 1904.45,1380.39 1905.63,1381 1906.81,1381.55 1907.99,1382.05 1909.18,1382.5 1910.36,1382.9 1911.54,1383.24 1912.73,1383.54 1913.91,1383.78 1915.09,1383.97 1916.27,1384.11 1917.46,1384.2 1918.64,1384.24 1919.82,1384.22 1921.01,1384.16 1922.19,1384.04 1923.37,1383.87 1924.56,1383.65 1925.74,1383.38 1926.92,1383.05 1928.1,1382.68 1929.29,1382.25 1930.47,1381.77 1931.65,1381.25 1932.84,1380.67 1934.02,1380.03 1935.2,1379.35 1936.39,1378.62 1937.57,1377.83 1938.75,1377 1939.93,1376.12 1941.12,1375.18 1942.3,1374.19 1943.48,1373.16 1944.67,1372.07 1945.85,1370.94 1947.03,1369.75 1948.22,1368.51 1949.4,1367.23 1950.58,1365.89 1951.76,1364.51 1952.95,1363.08 1954.13,1361.6 1955.31,1360.06 1956.5,1358.49 1957.68,1356.86 1958.86,1355.18 1960.04,1353.46 1961.23,1351.69 1962.41,1349.87 1963.59,1348.01 1964.78,1346.09 1965.96,1344.14 1967.14,1342.13 1968.33,1340.08 1969.51,1337.98 1970.69,1335.84 1971.87,1333.65 1973.06,1331.41 1974.24,1329.13 1975.42,1326.81 1976.61,1324.44 1977.79,1322.03 1978.97,1319.57 1980.16,1317.07 1981.34,1314.53 1982.52,1311.95 1983.7,1309.32 1984.89,1306.65 1986.07,1303.93 1987.25,1301.18 1988.44,1298.38 1989.62,1295.55 1990.8,1292.67 1991.98,1289.75 1993.17,1286.79 1994.35,1283.8 1995.53,1280.76 1996.72,1277.69 1997.9,1274.57 1999.08,1271.42 2000.27,1268.23 2001.45,1265 2002.63,1261.74 2003.81,1258.44 2005,1255.1 2006.18,1251.73 2007.36,1248.32 2008.55,1244.88 2009.73,1241.4 2010.91,1237.88 2012.1,1234.34 2013.28,1230.76 2014.46,1227.15 2015.64,1223.5 2016.83,1219.82 2018.01,1216.11 2019.19,1212.37 2020.38,1208.6 2021.56,1204.8 2022.74,1200.97 2023.93,1197.1 2025.11,1193.21 2026.29,1189.29 2027.47,1185.35 2028.66,1181.37 2029.84,1177.37 2031.02,1173.34 2032.21,1169.28 2033.39,1165.2 2034.57,1161.09 2035.75,1156.96 2036.94,1152.8 2038.12,1148.62 2039.3,1144.41 2040.49,1140.18 2041.67,1135.93 2042.85,1131.66 2044.04,1127.36 2045.22,1123.04 2046.4,1118.7 2047.58,1114.35 2048.77,1109.97 2049.95,1105.57 2051.13,1101.15 2052.32,1096.72 2053.5,1092.26 2054.68,1087.79 2055.87,1083.3 2057.05,1078.8 2058.23,1074.28 2059.41,1069.74 2060.6,1065.19 2061.78,1060.63 2062.96,1056.05 2064.15,1051.45 2065.33,1046.85 2066.51,1042.23 2067.7,1037.6 2068.88,1032.95 2070.06,1028.3 2071.24,1023.63 2072.43,1018.96 2073.61,1014.27 2074.79,1009.58 2075.98,1004.88 2077.16,1000.17 2078.34,995.446 2079.52,990.72 2080.71,985.987 2081.89,981.248 2083.07,976.503 2084.26,971.753 2085.44,966.997 2086.62,962.238 2087.81,957.474 2088.99,952.708 2090.17,947.938 2091.35,943.166 2092.54,938.392 2093.72,933.616 2094.9,928.839 2096.09,924.062 2097.27,919.285 2098.45,914.508 2099.64,909.732 2100.82,904.958 2102,900.185 2103.18,895.415 2104.37,890.647 2105.55,885.883 2106.73,881.122 2107.92,876.365 2109.1,871.613 2110.28,866.867 2111.47,862.125 2112.65,857.39 2113.83,852.661 2115.01,847.939 2116.2,843.224 2117.38,838.517 2118.56,833.818 2119.75,829.128 2120.93,824.447 2122.11,819.775 2123.29,815.114 2124.48,810.462 2125.66,805.822 2126.84,801.193 2128.03,796.576 2129.21,791.97 2130.39,787.377 2131.58,782.797 2132.76,778.231 2133.94,773.678 2135.12,769.14 2136.31,764.616 2137.49,760.107 2138.67,755.613 2139.86,751.135 2141.04,746.673 2142.22,742.228 2143.41,737.8 2144.59,733.39 2145.77,728.997 2146.95,724.622 2148.14,720.266 2149.32,715.928 2150.5,711.61 2151.69,707.312 2152.87,703.033 2154.05,698.775 2155.24,694.538 2156.42,690.321 2157.6,686.127 2158.78,681.954 2159.97,677.803 2161.15,673.675 2162.33,669.569 2163.52,665.487 2164.7,661.428 2165.88,657.394 2167.06,653.383 2168.25,649.397 2169.43,645.436 2170.61,641.5 2171.8,637.589 2172.98,633.705 2174.16,629.846 2175.35,626.014 2176.53,622.209 2177.71,618.431 2178.89,614.68 2180.08,610.957 2181.26,607.262 2182.44,603.595 2183.63,599.957 2184.81,596.348 2185.99,592.767 2187.18,589.216 2188.36,585.695 2189.54,582.204 2190.72,578.743 2191.91,575.312 2193.09,571.913 2194.27,568.544 2195.46,565.207 2196.64,561.901 2197.82,558.626 2199.01,555.384 2200.19,552.175 2201.37,548.997 2202.55,545.853 2203.74,542.741 2204.92,539.663 2206.1,536.618 2207.29,533.607 2208.47,530.629 2209.65,527.686 2210.83,524.777 2212.02,521.903 2213.2,519.064 2214.38,516.259 2215.57,513.49 2216.75,510.756 2217.93,508.057 2219.12,505.395 2220.3,502.768 2221.48,500.178 2222.66,497.623 2223.85,495.106 2225.03,492.625 2226.21,490.181 2227.4,487.774 2228.58,485.404 2229.76,483.071 2230.95,480.777 2232.13,478.519 2233.31,476.3 2234.49,474.119 2235.68,471.976 2236.86,469.871 2238.04,467.805 2239.23,465.777 2240.41,463.788 2241.59,461.838 2242.78,459.927 2243.96,458.055 2245.14,456.222 2246.32,454.429 2247.51,452.675 2248.69,450.961 2249.87,449.287 2251.06,447.653 2252.24,446.058 2253.42,444.504 2254.6,442.99 2255.79,441.516 2256.97,440.083 2258.15,438.69 2259.34,437.337 2260.52,436.026 2261.7,434.755 2262.89,433.525 2264.07,432.336 2265.25,431.188 2266.43,430.081 2267.62,429.015 2268.8,427.99 2269.98,427.007 2271.17,426.065 2272.35,425.165 2273.53,424.306 2274.72,423.488 2275.9,422.713 2277.08,421.978 2278.26,421.286 2279.45,420.635 2280.63,420.026 2281.81,419.459 2283,418.934 2284.18,418.45 2285.36,418.009 2286.54,417.609 2287.73,417.252 2288.91,416.936 2290.09,416.663 2291.28,416.431 2292.46,416.242 2293.64,416.095 2294.83,415.989 2296.01,415.926 \"/>\n",
       "</svg>\n"
      ]
     },
     "metadata": {},
     "execution_count": 6
    }
   ],
   "cell_type": "code",
   "source": [
    "using LinearAlgebra\n",
    "\n",
    "# Define the 1D lattice [0, 100]\n",
    "lattice = diagm([100., 0, 0])\n",
    "\n",
    "# Place them at 20 and 80 in *fractional coordinates*,\n",
    "# that is 0.2 and 0.8, since the lattice is 100 wide.\n",
    "nucleus   = ElementGaussian(0.3, 10.0)\n",
    "atoms     = [nucleus, nucleus]\n",
    "positions = [[0.2, 0, 0], [0.8, 0, 0]]\n",
    "\n",
    "# Assemble the model, discretize and build the Hamiltonian\n",
    "model = Model(lattice, atoms, positions; terms=[Kinetic(), AtomicLocal()])\n",
    "basis = PlaneWaveBasis(model; Ecut=300, kgrid=(1, 1, 1));\n",
    "ham   = Hamiltonian(basis)\n",
    "\n",
    "# Extract the total potential term of the Hamiltonian and plot it\n",
    "potential = DFTK.total_local_potential(ham)[:, 1, 1]\n",
    "rvecs = collect(r_vectors_cart(basis))[:, 1, 1]  # slice along the x axis\n",
    "x = [r[1] for r in rvecs]                        # only keep the x coordinate\n",
    "plot(x, potential, label=\"\", xlabel=\"x\", ylabel=\"V(x)\")"
   ],
   "metadata": {},
   "execution_count": 6
  },
  {
   "cell_type": "markdown",
   "source": [
    "This potential is the sum of two \"atomic\" potentials (the two \"Gaussian\" elements).\n",
    "Due to the periodic setting we are considering interactions naturally also occur\n",
    "across the unit cell boundary (i.e. wrapping from `100` over to `0`).\n",
    "The required periodization of the atomic potential is automatically taken care,\n",
    "such that the potential is smooth across the cell boundary at `100`/`0`.\n",
    "\n",
    "With this setup, let's look at the bands:"
   ],
   "metadata": {}
  },
  {
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "┌ Warning: Calling plot_bandstructure without first computing the band data is deprecated and will be removed in the next minor version bump.\n",
      "└ @ DFTKPlotsExt ~/work/DFTK.jl/DFTK.jl/ext/DFTKPlotsExt.jl:15\n"
     ]
    },
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "Plot{Plots.GRBackend() n=6}",
      "image/png": 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       "</svg>\n"
      ]
     },
     "metadata": {},
     "execution_count": 7
    }
   ],
   "cell_type": "code",
   "source": [
    "using Unitful\n",
    "using UnitfulAtomic\n",
    "\n",
    "plot_bandstructure(basis; n_bands=6, kline_density=500)"
   ],
   "metadata": {},
   "execution_count": 7
  },
  {
   "cell_type": "markdown",
   "source": [
    "The bands are noticeably different.\n",
    " - The bands no longer overlap, meaning that the spectrum of $H$ is no longer continuous\n",
    "   but has gaps.\n",
    "\n",
    " - The two lowest bands are almost flat. This is because they represent\n",
    "   two tightly bound and localized electrons inside the two Gaussians.\n",
    "\n",
    " - The higher the bands are in energy, the more free-electron-like they are.\n",
    "   In other words the higher the kinetic energy of the electrons, the less they feel\n",
    "   the effect of the two Gaussian potentials. As it turns out the curvature of the bands,\n",
    "   (the degree to which they are free-electron-like) is highly related to the delocalization\n",
    "   of electrons in these bands: The more curved the more delocalized. In some sense\n",
    "   \"free electrons\" correspond to perfect delocalization.\n",
    "\n",
    " - Try playing with the parameters of the Gaussian potentials by setting\n",
    "   ```julia\n",
    "   nucleus   = ElementGaussian(α, L)\n",
    "   ```\n",
    "   with different $α$ and $L$. You should notice the influence on the bands. Pay particular\n",
    "   attention to the relation between the depth of the potential and the shape of the bands."
   ],
   "metadata": {}
  }
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