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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Geometry optimization · DFTK.jl</title><meta name="title" content="Geometry optimization · DFTK.jl"/><meta property="og:title" content="Geometry optimization · DFTK.jl"/><meta property="twitter:title" content="Geometry optimization · DFTK.jl"/><meta name="description" content="Documentation for DFTK.jl."/><meta property="og:description" content="Documentation for DFTK.jl."/><meta property="twitter:description" content="Documentation for DFTK.jl."/><meta property="og:url" content="https://docs.dftk.org/stable/examples/geometry_optimization/"/><meta property="twitter:url" content="https://docs.dftk.org/stable/examples/geometry_optimization/"/><link rel="canonical" href="https://docs.dftk.org/stable/examples/geometry_optimization/"/><script data-outdated-warner src="../../assets/warner.js"></script><link 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class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Basic DFT calculations</a></li><li class="is-active"><a href>Geometry optimization</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Geometry optimization</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/JuliaMolSim/DFTK.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/JuliaMolSim/DFTK.jl/blob/master/examples/geometry_optimization.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Geometry-optimization"><a class="docs-heading-anchor" href="#Geometry-optimization">Geometry optimization</a><a id="Geometry-optimization-1"></a><a class="docs-heading-anchor-permalink" href="#Geometry-optimization" title="Permalink"></a></h1><p><a href="https://mybinder.org/v2/gh/JuliaMolSim/DFTK.jl/gh-pages?filepath=dev/examples/geometry_optimization.ipynb"><img src="https://mybinder.org/badge_logo.svg" alt/></a> <a href="https://nbviewer.jupyter.org/github/JuliaMolSim/DFTK.jl/blob/gh-pages/dev/examples/geometry_optimization.ipynb"><img src="https://img.shields.io/badge/show-nbviewer-579ACA.svg" alt/></a></p><p>We use the DFTK and Optim packages in this example to find the minimal-energy bond length of the <span>$H_2$</span> molecule. We setup <span>$H_2$</span> in an LDA model just like in the <a href="../../guide/tutorial/#Tutorial">Tutorial</a> for silicon.</p><pre><code class="language-julia hljs">using DFTK
using Optim
using LinearAlgebra
using Printf
kgrid = [1, 1, 1] # k-point grid
Ecut = 5 # kinetic energy cutoff in Hartree
tol = 1e-8 # tolerance for the optimization routine
a = 10 # lattice constant in Bohr
lattice = a * I(3)
H = ElementPsp(:H; psp=load_psp("hgh/lda/h-q1"));
atoms = [H, H];</code></pre><p>We define a Bloch wave and a density to be used as global variables so that we can transfer the solution from one iteration to another and therefore reduce the optimization time.</p><pre><code class="language-julia hljs">ψ = nothing
ρ = nothing</code></pre><p>First, we create a function that computes the solution associated to the position <span>$x ∈ ℝ^6$</span> of the atoms in reduced coordinates (cf. <a href="../../developer/conventions/#Reduced-and-Cartesian-coordinates">Reduced and Cartesian coordinates</a> for more details on the coordinates system). They are stored as a vector: <code>x[1:3]</code> represents the position of the first atom and <code>x[4:6]</code> the position of the second. We also update <code>ψ</code> and <code>ρ</code> for the next iteration.</p><pre><code class="language-julia hljs">function compute_scfres(x)
model = model_LDA(lattice, atoms, [x[1:3], x[4:6]])
basis = PlaneWaveBasis(model; Ecut, kgrid)
global ψ, ρ
if isnothing(ρ)
ρ = guess_density(basis)
end
is_converged = ScfConvergenceForce(tol / 10)
scfres = self_consistent_field(basis; ψ, ρ, is_converged, callback=identity)
ψ = scfres.ψ
ρ = scfres.ρ
scfres
end;</code></pre><p>Then, we create the function we want to optimize: <code>fg!</code> is used to update the value of the objective function <code>F</code>, namely the energy, and its gradient <code>G</code>, here computed with the forces (which are, by definition, the negative gradient of the energy).</p><pre><code class="language-julia hljs">function fg!(F, G, x)
scfres = compute_scfres(x)
if !isnothing(G)
grad = compute_forces(scfres)
G .= -[grad[1]; grad[2]]
end
scfres.energies.total
end;</code></pre><p>Now, we can optimize on the 6 parameters <code>x = [x1, y1, z1, x2, y2, z2]</code> in reduced coordinates, using <code>LBFGS()</code>, the default minimization algorithm in Optim. We start from <code>x0</code>, which is a first guess for the coordinates. By default, <code>optimize</code> traces the output of the optimization algorithm during the iterations. Once we have the minimizer <code>xmin</code>, we compute the bond length in Cartesian coordinates.</p><pre><code class="language-julia hljs">x0 = vcat(lattice \ [0., 0., 0.], lattice \ [1.4, 0., 0.])
xres = optimize(Optim.only_fg!(fg!), x0, LBFGS(),
Optim.Options(; show_trace=true, f_tol=tol))
xmin = Optim.minimizer(xres)
dmin = norm(lattice*xmin[1:3] - lattice*xmin[4:6])
@printf "\nOptimal bond length for Ecut=%.2f: %.3f Bohr\n" Ecut dmin</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Iter Function value Gradient norm
0 -1.061651e+00 6.219313e-01
* time: 5.507469177246094e-5
1 -1.064073e+00 2.919809e-01
* time: 0.8866548538208008
2 -1.066008e+00 4.821022e-02
* time: 1.592120885848999
3 -1.066049e+00 4.314828e-04
* time: 2.023360013961792
4 -1.066049e+00 6.954648e-09
* time: 2.3255929946899414
Optimal bond length for Ecut=5.00: 1.556 Bohr</code></pre><p>We used here very rough parameters to generate the example and setting <code>Ecut</code> to 10 Ha yields a bond length of 1.523 Bohr, which <a href="https://docs.abinit.org/tutorial/base1/">agrees with ABINIT</a>.</p><div class="admonition is-info"><header class="admonition-header">Degrees of freedom</header><div class="admonition-body"><p>We used here a very general setting where we optimized on the 6 variables representing the position of the 2 atoms and it can be easily extended to molecules with more atoms (such as <span>$H_2O$</span>). In the particular case of <span>$H_2$</span>, we could use only the internal degree of freedom which, in this case, is just the bond length.</p></div></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../graphene/">« Graphene band structure</a><a class="docs-footer-nextpage" href="../energy_cutoff_smearing/">Energy cutoff smearing »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.2.1 on <span class="colophon-date" title="Wednesday 10 April 2024 19:38">Wednesday 10 April 2024</span>. Using Julia version 1.10.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>