{ "cells": [ { "cell_type": "markdown", "source": [ "# 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." ], "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 Ecut : 300.0 Ha\n fft_size : (320, 1, 1), 320 total points\n kgrid type : Monkhorst-Pack\n kgrid : [1, 1, 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": [ "Computing bands along kpath:\n", " Γ -> X\n", "\rDiagonalising Hamiltonian kblocks: 12%|██ | ETA: 0:00:09\u001b[K\rDiagonalising Hamiltonian kblocks: 100%|████████████████| Time: 0:00:01\u001b[K\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": [ "Computing bands along kpath:\n", " Γ -> X\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." ], "metadata": {} } ], "nbformat_minor": 3, "metadata": { "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.8.5" }, "kernelspec": { "name": "julia-1.8", "display_name": "Julia 1.8.5", "language": "julia" } }, "nbformat": 4 }