{ "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: 0:00:08\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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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", " 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": "1.8.4" }, "kernelspec": { "name": "julia-1.8", "display_name": "Julia 1.8.4", "language": "julia" } }, "nbformat": 4 }