{ "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 Bohr³\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 Bohr³\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", " -½ -> ½\n", "\rDiagonalising Hamiltonian kblocks: 6%|█ | ETA: 0:00:18\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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"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": [ "!!! note \"Selection of k-point grids in `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", " -½ -> ½\n", "\rDiagonalising Hamiltonian kblocks: 67%|██████████▋ | ETA: 0:00:00\u001b[K\rDiagonalising Hamiltonian kblocks: 100%|████████████████| Time: 0:00:00\u001b[K\n" ] }, { "output_type": "execute_result", "data": { "text/plain": "Plot{Plots.GRBackend() n=6}", "image/png": 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"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." ], "metadata": {} } ], "nbformat_minor": 3, "metadata": { "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.8.1" }, "kernelspec": { "name": "julia-1.8", "display_name": "Julia 1.8.1", "language": "julia" } }, "nbformat": 4 }