{ "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. 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:14\n" ] }, { "output_type": "execute_result", "data": { "text/plain": "Plot{Plots.GRBackend() n=6}", "image/png": 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`PlaneWaveBasis` construction\"\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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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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"\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:14\n" ] }, { "output_type": "execute_result", "data": { "text/plain": "Plot{Plots.GRBackend() n=6}", "image/png": 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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": {} } ], "nbformat_minor": 3, "metadata": { "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.10.3" }, "kernelspec": { "name": "julia-1.10", "display_name": "Julia 1.10.3", "language": "julia" } }, "nbformat": 4 }