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Performing Bader Analysis with FHI-aims code
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<h4>Introduction</h4>
<p>Bader analysis is a widely used technique to decompose the charge density into contributions from individual atoms <a href="https://pubs.acs.org/doi/10.1021/ar00109a003">[Bader]</a> <a href="https://iopscience.iop.org/article/10.1088/0953-8984/21/8/084204">[Tang]</a> <a href="https://doi.org/10.1063/1.3553716">[Yu]</a>. It belongs to the theory of atoms in molecules which helps explain the complex electronic interactions in molecules or materials. The electron density calculated by density functional theory is the most common object where Bader analysis working on. Especially, for the planewave pseudopotential codes such as VASP and Quantum Espresso, Bader analysis can be applied naturally on the electron density information saved in the <a href="http://paulbourke.net/dataformats/cube/">Cube files</a>, which are density values held on even distributed integration grids in periodic 3D space. </p>
<h4>Difficulty with Cube file in all-electron full-potential code FHI-aims</h4>
<p><a href="https://aimsclub.fhi-berlin.mpg.de/">FHI-aims</a> is an all-electron full-potential code based on the numerical atom-centered orbital basis functions <a href="https://www.sciencedirect.com/science/article/pii/S0010465509002033">[Blum]</a>. One advantage of FHI-aims implementation is the correct description of the cusp of the electron density in the core region near the nuclei. However, such cusps also raise a problem for the cube file output from the FHI-aims code. If we integrate the normal density information held in the cube file output by FHI-aims, the result can differ from the number of electrons in the system.</p>
<p>I plot a 1D example to explain the problem.
<img alt="Alt Text" src="./images/aims_bader_cusp_1d.png"></p>
<p>As plotted in the figure, we have a 1D electron density function f defined from 0.0 to 20.0. Its integral is 4.59931, as shown in blue color in the figure. To mimic the cube file information, we evenly separate the space into 20 small "voxel" with a length of 1.0 as plotted in yellow vertical lines. The information saved in the cube files are then the values of the red vertical lines, which are the values of the f function at the center of voxels. The integral of the f function calculated in
this way is then the summation of these values, which is 4.46412. It is close to but different from the exact integral.
Such summation will be a good approximation if the f function is smooth as the electron density in the planewave pseudopotential code. For the electron density in FHI-aims code, the error can be significant due to the cusps.</p>
<h4>Integral instead of value at the voxel center in FHI-aims with Cube file</h4>
<p>We could record the integral in each voxel instead of the value at the voxel center in the cube file to solve the problem. As plotted in black, the integrals in each voxel are slightly different from the value at the voxel center, and the summation of these integrals results in the correct integration of the whole space. Note, to calculate the integrals in these small voxels; we can evaluate electron density adaptively in theory. However, electron density calculation from a density matrix is computationally relatively expensive.
Note, the density difference between total density and free atom density at voxel center is a smooth quantity that can be captured in the normal cube file.
We could evaluate a computationally efficient quantity, which is the free atom electron density. It is represented as the summation of spline functions centered at each atom. We can evaluate the free atom electron density adaptively to high accuracy.
Thus, the output value for total density is</p>
<p><span class="math">\(\rho_{total\ density}(voxel) = \rho_{total\ density}(voxel_{center}) - \rho_{free\ atom}(voxel_{center}) + \int_{voxel} \rho_{free\ atom}\)</span></p>
<p>Based on my test, the output cube files integrals to a value much closer to the total number of electrons in the system. I call the quantity integrable total density.</p>
<h4>Inputs in FHI-aims</h4>
<p>The keyword in FHI-aims to output the integrable total density is <code>output cube total_density_integrable</code>. Let me give two complete examples. One is the crystalline NaCl with 8 atoms in the unit cell, the other is the crystalline silicon with 8 atoms in the unit cell.</p>
<h5>Example Crystalline NaCl</h5>
<p>FHI-aims needs two files as inputs <code>control.in</code> and <code>geometry.in</code>.</p>
<p><code>control.in</code> </p>
<div class="highlight"><pre><span></span><code># <span class="nv">YY</span>:<span class="nv">WARNING</span> <span class="nv">just</span> <span class="k">for</span> <span class="nv">test</span>
# <span class="nv">NEVER</span> <span class="nv">use</span> <span class="nv">the</span> <span class="nv">numerical</span> <span class="nv">setting</span> <span class="nv">here</span> <span class="k">for</span> <span class="nv">productive</span> <span class="nv">run</span>
#
# <span class="nv">Physical</span> <span class="nv">settings</span>
<span class="nv">xc</span> <span class="nv">pw91_gga</span>
<span class="nv">spin</span> <span class="nv">none</span>
# <span class="nv">k</span><span class="o">-</span><span class="nv">point</span> <span class="nv">grid</span>
<span class="nv">k_grid</span> <span class="mi">4</span> <span class="mi">4</span> <span class="mi">4</span>
<span class="nv">KS_method</span> <span class="nv">serial</span> <span class="o">!</span> <span class="nv">ELSI</span> <span class="nv">lapack</span>
<span class="nv">preconditioner</span> <span class="nv">none</span>
<span class="nv">output</span> <span class="nv">cube</span> <span class="nv">total_density_integrable</span>
<span class="nv">cube</span> <span class="nv">edge</span> <span class="mi">120</span> <span class="mi">0</span>.<span class="mi">04737105709166667</span> <span class="mi">0</span>.<span class="mi">0</span> <span class="mi">0</span>.<span class="mi">0</span> # <span class="nv">these</span> <span class="nv">are</span> <span class="nv">in</span> <span class="nv">unit</span> <span class="nv">Angstrom</span>
<span class="nv">cube</span> <span class="nv">edge</span> <span class="mi">120</span> <span class="mi">0</span>.<span class="mi">0</span> <span class="mi">0</span>.<span class="mi">04737105709166667</span> <span class="mi">0</span>.<span class="mi">0</span> # <span class="nv">these</span> <span class="nv">are</span> <span class="nv">in</span> <span class="nv">unit</span> <span class="nv">Angstrom</span>
<span class="nv">cube</span> <span class="nv">edge</span> <span class="mi">120</span> <span class="mi">0</span>.<span class="mi">0</span> <span class="mi">0</span>.<span class="mi">0</span> <span class="mi">0</span>.<span class="mi">04737105709166667</span> # <span class="nv">these</span> <span class="nv">are</span> <span class="nv">in</span> <span class="nv">unit</span> <span class="nv">Angstrom</span>
<span class="nv">cube</span> <span class="nv">divisor</span> <span class="mi">10</span>
<span class="nv">cube_content_unit</span> <span class="nv">bohr</span>
################################################################################
#
# <span class="nv">FHI</span><span class="o">-</span><span class="nv">aims</span> <span class="nv">code</span> <span class="nv">project</span>
# <span class="nv">Volker</span> <span class="nv">Blum</span>, <span class="nv">Fritz</span> <span class="nv">Haber</span> <span class="nv">Institute</span> <span class="nv">Berlin</span>, <span class="mi">2009</span>
#
# <span class="nv">Suggested</span> <span class="s2">"</span><span class="s">intermediate</span><span class="s2">"</span> <span class="nv">defaults</span> <span class="k">for</span> <span class="nv">Na</span> <span class="nv">atom</span> <span class="ss">(</span><span class="nv">to</span> <span class="nv">be</span> <span class="nv">pasted</span> <span class="nv">into</span> <span class="nv">control</span>.<span class="nv">in</span> <span class="nv">file</span><span class="ss">)</span>
#
# <span class="nv">See</span> <span class="nv">https</span>:<span class="o">//</span><span class="nv">aaltodoc</span>.<span class="nv">aalto</span>.<span class="nv">fi</span><span class="o">/</span><span class="nv">handle</span><span class="o">/</span><span class="mi">123456789</span><span class="o">/</span><span class="mi">33620</span> <span class="k">for</span> <span class="nv">the</span> <span class="nv">report</span> <span class="nv">these</span>
# <span class="nv">setting</span> <span class="nv">are</span> <span class="nv">based</span> <span class="nv">on</span>
################################################################################
<span class="nv">species</span> <span class="nv">Na</span>
# <span class="nv">global</span> <span class="nv">species</span> <span class="nv">definitions</span>
<span class="nv">nucleus</span> <span class="mi">11</span>
<span class="nv">mass</span> <span class="mi">22</span>.<span class="mi">98976928</span>
#
<span class="nv">l_hartree</span> <span class="mi">6</span>
#
<span class="nv">cut_pot</span> <span class="mi">4</span>.<span class="mi">5</span> <span class="mi">2</span>.<span class="mi">0</span> <span class="mi">1</span>.<span class="mi">0</span>
<span class="nv">basis_dep_cutoff</span> <span class="mi">1</span><span class="nv">e</span><span class="o">-</span><span class="mi">4</span>
#
<span class="nv">radial_base</span> <span class="mi">40</span> <span class="mi">7</span>.<span class="mi">0</span>
<span class="nv">radial_multiplier</span> <span class="mi">2</span>
<span class="nv">angular_grids</span> <span class="nv">specified</span>
<span class="nv">division</span> <span class="mi">0</span>.<span class="mi">5925</span> <span class="mi">110</span>
<span class="nv">division</span> <span class="mi">0</span>.<span class="mi">7843</span> <span class="mi">194</span>
<span class="nv">division</span> <span class="mi">1</span>.<span class="mi">0201</span> <span class="mi">302</span>
<span class="nv">division</span> <span class="mi">1</span>.<span class="mi">1879</span> <span class="mi">434</span>
# <span class="nv">division</span> <span class="mi">1</span>.<span class="mi">3799</span> <span class="mi">590</span>
# <span class="nv">division</span> <span class="mi">1</span>.<span class="mi">4503</span> <span class="mi">770</span>
# <span class="nv">division</span> <span class="mi">7</span>.<span class="mi">0005</span> <span class="mi">974</span>
# <span class="nv">outer_grid</span> <span class="mi">974</span>
<span class="nv">outer_grid</span> <span class="mi">434</span>
################################################################################
#
# <span class="nv">Definition</span> <span class="nv">of</span> <span class="s2">"</span><span class="s">minimal</span><span class="s2">"</span> <span class="nv">basis</span>
#
################################################################################
# <span class="nv">valence</span> <span class="nv">basis</span> <span class="nv">states</span>
<span class="nv">valence</span> <span class="mi">3</span> <span class="nv">s</span> <span class="mi">1</span>.
<span class="nv">valence</span> <span class="mi">2</span> <span class="nv">p</span> <span class="mi">6</span>.
# <span class="nv">ion</span> <span class="nv">occupancy</span>
<span class="nv">ion_occ</span> <span class="mi">2</span> <span class="nv">s</span> <span class="mi">2</span>.
<span class="nv">ion_occ</span> <span class="mi">2</span> <span class="nv">p</span> <span class="mi">6</span>.
################################################################################
#
# <span class="nv">Suggested</span> <span class="nv">additional</span> <span class="nv">basis</span> <span class="nv">functions</span>. <span class="k">For</span> <span class="nv">production</span> <span class="nv">calculations</span>,
# <span class="nv">uncomment</span> <span class="nv">them</span> <span class="nv">one</span> <span class="nv">after</span> <span class="nv">another</span> <span class="ss">(</span><span class="nv">the</span> <span class="nv">most</span> <span class="nv">important</span> <span class="nv">basis</span> <span class="nv">functions</span> <span class="nv">are</span>
# <span class="nv">listed</span> <span class="nv">first</span><span class="ss">)</span>.
#
# <span class="nv">Constructed</span> <span class="k">for</span> <span class="nv">dimers</span>: <span class="mi">2</span>.<span class="mi">0</span> <span class="nv">A</span>, <span class="mi">2</span>.<span class="mi">5</span> <span class="nv">A</span>, <span class="mi">3</span>.<span class="mi">0</span> <span class="nv">A</span>, <span class="mi">3</span>.<span class="mi">75</span> <span class="nv">A</span>, <span class="mi">4</span>.<span class="mi">5</span> <span class="nv">A</span>
#
################################################################################
# <span class="s2">"</span><span class="s">First tier</span><span class="s2">"</span> <span class="o">-</span> <span class="nv">improvements</span>: <span class="o">-</span><span class="mi">60</span>.<span class="mi">09</span> <span class="nv">meV</span> <span class="nv">to</span> <span class="o">-</span><span class="mi">10</span>.<span class="mi">02</span> <span class="nv">meV</span>
<span class="nv">hydro</span> <span class="mi">2</span> <span class="nv">p</span> <span class="mi">1</span>.<span class="mi">2</span>
<span class="nv">hydro</span> <span class="mi">3</span> <span class="nv">s</span> <span class="mi">1</span>.<span class="mi">8</span>
<span class="nv">hydro</span> <span class="mi">3</span> <span class="nv">d</span> <span class="mi">3</span>.<span class="mi">8</span>
# <span class="s2">"</span><span class="s">Second tier</span><span class="s2">"</span> <span class="o">-</span> <span class="nv">improvements</span>: <span class="o">-</span><span class="mi">2</span>.<span class="mi">94</span> <span class="nv">meV</span> <span class="nv">to</span> <span class="o">-</span><span class="mi">1</span>.<span class="mi">27</span> <span class="nv">meV</span>
<span class="nv">hydro</span> <span class="mi">4</span> <span class="nv">p</span> <span class="mi">3</span>.<span class="mi">1</span>
<span class="nv">hydro</span> <span class="mi">3</span> <span class="nv">s</span> <span class="mi">10</span>
<span class="nv">for_aux</span> <span class="nv">hydro</span> <span class="mi">4</span> <span class="nv">f</span> <span class="mi">6</span>.<span class="mi">2</span>
# <span class="nv">hydro</span> <span class="mi">4</span> <span class="nv">d</span> <span class="mi">1</span>.<span class="mi">3</span>
# <span class="s2">"</span><span class="s">Third tier</span><span class="s2">"</span> <span class="o">-</span> <span class="nv">improvements</span>: <span class="o">-</span><span class="mi">0</span>.<span class="mi">83</span> <span class="nv">meV</span> <span class="nv">to</span> <span class="o">-</span><span class="mi">0</span>.<span class="mi">07</span> <span class="nv">meV</span>
# <span class="nv">hydro</span> <span class="mi">3</span> <span class="nv">d</span> <span class="mi">7</span>.<span class="mi">8</span>
# <span class="nv">hydro</span> <span class="mi">3</span> <span class="nv">p</span> <span class="mi">2</span>.<span class="mi">3</span>
# <span class="nv">hydro</span> <span class="mi">5</span> <span class="nv">g</span> <span class="mi">9</span>.<span class="mi">6</span>
# <span class="nv">hydro</span> <span class="mi">4</span> <span class="nv">p</span> <span class="mi">0</span>.<span class="mi">85</span>
# <span class="nv">hydro</span> <span class="mi">5</span> <span class="nv">f</span> <span class="mi">1</span>.<span class="mi">8</span>
# <span class="nv">hydro</span> <span class="mi">2</span> <span class="nv">s</span> <span class="mi">0</span>.<span class="mi">6</span>
# <span class="nv">Further</span> <span class="nv">basis</span> <span class="nv">functions</span> <span class="nv">that</span> <span class="nv">fell</span> <span class="nv">out</span> <span class="nv">of</span> <span class="nv">the</span> <span class="nv">optimization</span> <span class="o">-</span> <span class="nv">noise</span> <span class="nv">level</span>...
# <span class="nv">hydro</span> <span class="mi">5</span> <span class="nv">g</span> <span class="mi">0</span>.<span class="mi">1</span>
# <span class="nv">hydro</span> <span class="mi">4</span> <span class="nv">d</span> <span class="mi">3</span>.<span class="mi">4</span>
# <span class="nv">hydro</span> <span class="mi">4</span> <span class="nv">s</span> <span class="mi">0</span>.<span class="mi">1</span>
################################################################################
#
# <span class="nv">FHI</span><span class="o">-</span><span class="nv">aims</span> <span class="nv">code</span> <span class="nv">project</span>
# <span class="nv">Volker</span> <span class="nv">Blum</span>, <span class="mi">2017</span>
#
# <span class="nv">Suggested</span> <span class="s2">"</span><span class="s">intermediate</span><span class="s2">"</span> <span class="nv">defaults</span> <span class="k">for</span> <span class="nv">Cl</span> <span class="nv">atom</span> <span class="ss">(</span><span class="nv">to</span> <span class="nv">be</span> <span class="nv">pasted</span> <span class="nv">into</span> <span class="nv">control</span>.<span class="nv">in</span> <span class="nv">file</span><span class="ss">)</span>
#
################################################################################
<span class="nv">species</span> <span class="nv">Cl</span>
# <span class="nv">global</span> <span class="nv">species</span> <span class="nv">definitions</span>
<span class="nv">nucleus</span> <span class="mi">17</span>
<span class="nv">mass</span> <span class="mi">35</span>.<span class="mi">453</span>
#
<span class="nv">l_hartree</span> <span class="mi">6</span>
#
<span class="nv">cut_pot</span> <span class="mi">4</span>.<span class="mi">0</span> <span class="mi">2</span>.<span class="mi">0</span> <span class="mi">1</span>.<span class="mi">0</span>
<span class="nv">basis_dep_cutoff</span> <span class="mi">1</span><span class="nv">e</span><span class="o">-</span><span class="mi">4</span>
#
<span class="nv">radial_base</span> <span class="mi">45</span> <span class="mi">7</span>.<span class="mi">0</span>
<span class="nv">radial_multiplier</span> <span class="mi">2</span>
<span class="nv">angular_grids</span> <span class="nv">specified</span>
<span class="nv">division</span> <span class="mi">0</span>.<span class="mi">4412</span> <span class="mi">110</span>
<span class="nv">division</span> <span class="mi">0</span>.<span class="mi">5489</span> <span class="mi">194</span>
<span class="nv">division</span> <span class="mi">0</span>.<span class="mi">6734</span> <span class="mi">302</span>
<span class="nv">division</span> <span class="mi">0</span>.<span class="mi">7794</span> <span class="mi">434</span>
# <span class="nv">division</span> <span class="mi">0</span>.<span class="mi">9402</span> <span class="mi">590</span>
# <span class="nv">division</span> <span class="mi">1</span>.<span class="mi">0779</span> <span class="mi">770</span>
# <span class="nv">division</span> <span class="mi">1</span>.<span class="mi">1792</span> <span class="mi">974</span>
# <span class="nv">outer_grid</span> <span class="mi">974</span>
<span class="nv">outer_grid</span> <span class="mi">434</span>
################################################################################
#
# <span class="nv">Definition</span> <span class="nv">of</span> <span class="s2">"</span><span class="s">minimal</span><span class="s2">"</span> <span class="nv">basis</span>
#
################################################################################
# <span class="nv">valence</span> <span class="nv">basis</span> <span class="nv">states</span>
<span class="nv">valence</span> <span class="mi">3</span> <span class="nv">s</span> <span class="mi">2</span>.
<span class="nv">valence</span> <span class="mi">3</span> <span class="nv">p</span> <span class="mi">5</span>.
# <span class="nv">ion</span> <span class="nv">occupancy</span>
<span class="nv">ion_occ</span> <span class="mi">3</span> <span class="nv">s</span> <span class="mi">1</span>.
<span class="nv">ion_occ</span> <span class="mi">3</span> <span class="nv">p</span> <span class="mi">4</span>.
################################################################################
#
# <span class="nv">Suggested</span> <span class="nv">additional</span> <span class="nv">basis</span> <span class="nv">functions</span>. <span class="k">For</span> <span class="nv">production</span> <span class="nv">calculations</span>,
# <span class="nv">uncomment</span> <span class="nv">them</span> <span class="nv">one</span> <span class="nv">after</span> <span class="nv">another</span> <span class="ss">(</span><span class="nv">the</span> <span class="nv">most</span> <span class="nv">important</span> <span class="nv">basis</span> <span class="nv">functions</span> <span class="nv">are</span>
# <span class="nv">listed</span> <span class="nv">first</span><span class="ss">)</span>.
#
# <span class="nv">Constructed</span> <span class="k">for</span> <span class="nv">dimers</span>: <span class="mi">1</span>.<span class="mi">65</span> <span class="nv">A</span>, <span class="mi">2</span>.<span class="mi">0</span> <span class="nv">A</span>, <span class="mi">2</span>.<span class="mi">5</span> <span class="nv">A</span>, <span class="mi">3</span>.<span class="mi">25</span> <span class="nv">A</span>, <span class="mi">4</span>.<span class="mi">0</span> <span class="nv">A</span>
#
################################################################################
# <span class="s2">"</span><span class="s">First tier</span><span class="s2">"</span> <span class="o">-</span> <span class="nv">improvements</span>: <span class="o">-</span><span class="mi">429</span>.<span class="mi">57</span> <span class="nv">meV</span> <span class="nv">to</span> <span class="o">-</span><span class="mi">15</span>.<span class="mi">03</span> <span class="nv">meV</span>
<span class="nv">ionic</span> <span class="mi">3</span> <span class="nv">d</span> <span class="nv">auto</span>
<span class="nv">hydro</span> <span class="mi">2</span> <span class="nv">p</span> <span class="mi">1</span>.<span class="mi">9</span>
<span class="nv">hydro</span> <span class="mi">4</span> <span class="nv">f</span> <span class="mi">7</span>.<span class="mi">4</span>
<span class="nv">ionic</span> <span class="mi">3</span> <span class="nv">s</span> <span class="nv">auto</span>
<span class="nv">for_aux</span> <span class="nv">hydro</span> <span class="mi">5</span> <span class="nv">g</span> <span class="mi">10</span>.<span class="mi">4</span>
# <span class="s2">"</span><span class="s">Second tier</span><span class="s2">"</span> <span class="o">-</span> <span class="nv">improvements</span>: <span class="o">-</span><span class="mi">7</span>.<span class="mi">84</span> <span class="nv">meV</span> <span class="nv">to</span> <span class="o">-</span><span class="mi">0</span>.<span class="mi">48</span> <span class="nv">meV</span>
# <span class="nv">hydro</span> <span class="mi">3</span> <span class="nv">d</span> <span class="mi">3</span>.<span class="mi">3</span>
# <span class="nv">hydro</span> <span class="mi">5</span> <span class="nv">f</span> <span class="mi">9</span>.<span class="mi">8</span>
# <span class="nv">hydro</span> <span class="mi">1</span> <span class="nv">s</span> <span class="mi">0</span>.<span class="mi">75</span>
# <span class="nv">hydro</span> <span class="mi">5</span> <span class="nv">g</span> <span class="mi">11</span>.<span class="mi">2</span>
# <span class="nv">hydro</span> <span class="mi">4</span> <span class="nv">p</span> <span class="mi">10</span>.<span class="mi">4</span>
# <span class="s2">"</span><span class="s">Third tier</span><span class="s2">"</span> <span class="o">-</span> <span class="nv">improvements</span>: <span class="o">-</span><span class="mi">1</span>.<span class="mi">00</span> <span class="nv">meV</span> <span class="nv">to</span> <span class="o">-</span><span class="mi">0</span>.<span class="mi">12</span> <span class="nv">meV</span>
# <span class="nv">hydro</span> <span class="mi">4</span> <span class="nv">d</span> <span class="mi">12</span>.<span class="mi">8</span>
# <span class="nv">hydro</span> <span class="mi">4</span> <span class="nv">f</span> <span class="mi">4</span>.<span class="mi">6</span>
# <span class="nv">hydro</span> <span class="mi">4</span> <span class="nv">d</span> <span class="mi">10</span>.<span class="mi">8</span>
# <span class="nv">hydro</span> <span class="mi">2</span> <span class="nv">s</span> <span class="mi">1</span>.<span class="mi">8</span>
# <span class="nv">hydro</span> <span class="mi">3</span> <span class="nv">p</span> <span class="mi">3</span>
# <span class="nv">Further</span> <span class="nv">functions</span> <span class="nv">that</span> <span class="nv">fell</span> <span class="nv">out</span> <span class="o">-</span> <span class="nv">improvements</span>: <span class="o">-</span><span class="mi">0</span>.<span class="mi">10</span> <span class="nv">meV</span> <span class="nv">and</span> <span class="nv">below</span>
# <span class="nv">hydro</span> <span class="mi">5</span> <span class="nv">f</span> <span class="mi">14</span>.<span class="mi">4</span>
# <span class="nv">hydro</span> <span class="mi">4</span> <span class="nv">s</span> <span class="mi">12</span>.<span class="mi">8</span>
# <span class="nv">hydro</span> <span class="mi">3</span> <span class="nv">d</span> <span class="mi">11</span>.<span class="mi">6</span>
# <span class="nv">hydro</span> <span class="mi">4</span> <span class="nv">s</span> <span class="mi">4</span>.<span class="mi">1</span>
</code></pre></div>
<p><code>geometry.in</code> </p>
<div class="highlight"><pre><span></span><code>lattice_vector 5.6845268510 0.0 0.0
lattice_vector 0.0 5.6845268510 0.0
lattice_vector 0.0 0.0 5.6845268510
atom_frac 0.75 0.25 0.25 Na
atom_frac 0.25 0.75 0.25 Na
atom_frac 0.25 0.25 0.75 Na
atom_frac 0.75 0.75 0.75 Na
atom_frac 0.25 0.25 0.25 Cl
atom_frac 0.75 0.75 0.25 Cl
atom_frac 0.75 0.25 0.75 Cl
atom_frac 0.25 0.75 0.75 Cl
</code></pre></div>
<h5>Example Crystalline Silicon</h5>
<p><code>control.in</code></p>
<div class="highlight"><pre><span></span><code># <span class="nv">YY</span>:<span class="nv">WARNING</span> <span class="nv">just</span> <span class="k">for</span> <span class="nv">test</span>
# <span class="nv">NEVER</span> <span class="nv">use</span> <span class="nv">the</span> <span class="nv">numerical</span> <span class="nv">setting</span> <span class="nv">here</span> <span class="k">for</span> <span class="nv">productive</span> <span class="nv">run</span>
#
# <span class="nv">Physical</span> <span class="nv">settings</span>
<span class="nv">xc</span> <span class="nv">pbe</span>
<span class="nv">spin</span> <span class="nv">none</span>
# <span class="nv">k</span><span class="o">-</span><span class="nv">point</span> <span class="nv">grid</span>
<span class="nv">k_grid</span> <span class="mi">4</span> <span class="mi">4</span> <span class="mi">4</span>
<span class="nv">KS_method</span> <span class="nv">serial</span> <span class="o">!</span> <span class="nv">ELSI</span> <span class="nv">lapack</span>
<span class="nv">output</span> <span class="nv">cube</span> <span class="nv">total_density_integrable</span>
<span class="nv">cube</span> <span class="nv">divisor</span> <span class="mi">10</span>
<span class="nv">cube_content_unit</span> <span class="nv">bohr</span>
################################################################################
#
# <span class="nv">FHI</span><span class="o">-</span><span class="nv">aims</span> <span class="nv">code</span> <span class="nv">project</span>
# <span class="nv">Volker</span> <span class="nv">Blum</span>, <span class="mi">2017</span>
#
# <span class="nv">Suggested</span> <span class="s2">"</span><span class="s">intermediate</span><span class="s2">"</span> <span class="nv">defaults</span> <span class="k">for</span> <span class="nv">Si</span> <span class="nv">atom</span> <span class="ss">(</span><span class="nv">to</span> <span class="nv">be</span> <span class="nv">pasted</span> <span class="nv">into</span> <span class="nv">control</span>.<span class="nv">in</span> <span class="nv">file</span><span class="ss">)</span>
#
################################################################################
<span class="nv">species</span> <span class="nv">Si</span>
# <span class="nv">global</span> <span class="nv">species</span> <span class="nv">definitions</span>
<span class="nv">nucleus</span> <span class="mi">14</span>
<span class="nv">mass</span> <span class="mi">28</span>.<span class="mi">0855</span>
#
<span class="nv">l_hartree</span> <span class="mi">6</span>
#
<span class="nv">cut_pot</span> <span class="mi">4</span>.<span class="mi">0</span> <span class="mi">2</span>.<span class="mi">0</span> <span class="mi">1</span>.<span class="mi">0</span>
<span class="nv">basis_dep_cutoff</span> <span class="mi">1</span><span class="nv">e</span><span class="o">-</span><span class="mi">4</span>
#
<span class="nv">radial_base</span> <span class="mi">42</span> <span class="mi">7</span>.<span class="mi">0</span>
<span class="nv">radial_multiplier</span> <span class="mi">2</span>
<span class="nv">angular_grids</span> <span class="nv">specified</span>
<span class="nv">division</span> <span class="mi">0</span>.<span class="mi">4121</span> <span class="mi">50</span>
<span class="nv">division</span> <span class="mi">0</span>.<span class="mi">7665</span> <span class="mi">110</span>
<span class="nv">division</span> <span class="mi">1</span>.<span class="mi">0603</span> <span class="mi">194</span>
<span class="nv">division</span> <span class="mi">1</span>.<span class="mi">2846</span> <span class="mi">302</span>
<span class="nv">division</span> <span class="mi">1</span>.<span class="mi">4125</span> <span class="mi">434</span>
# <span class="nv">division</span> <span class="mi">1</span>.<span class="mi">4810</span> <span class="mi">590</span>
# <span class="nv">division</span> <span class="mi">1</span>.<span class="mi">5529</span> <span class="mi">770</span>
# <span class="nv">division</span> <span class="mi">1</span>.<span class="mi">6284</span> <span class="mi">974</span>
# <span class="nv">division</span> <span class="mi">2</span>.<span class="mi">6016</span> <span class="mi">1202</span>
# <span class="nv">outer_grid</span> <span class="mi">974</span>
<span class="nv">outer_grid</span> <span class="mi">434</span>
################################################################################
#
# <span class="nv">Definition</span> <span class="nv">of</span> <span class="s2">"</span><span class="s">minimal</span><span class="s2">"</span> <span class="nv">basis</span>
#
################################################################################
# <span class="nv">valence</span> <span class="nv">basis</span> <span class="nv">states</span>
<span class="nv">valence</span> <span class="mi">3</span> <span class="nv">s</span> <span class="mi">2</span>.
<span class="nv">valence</span> <span class="mi">3</span> <span class="nv">p</span> <span class="mi">2</span>.
# <span class="nv">ion</span> <span class="nv">occupancy</span>
<span class="nv">ion_occ</span> <span class="mi">3</span> <span class="nv">s</span> <span class="mi">1</span>.
<span class="nv">ion_occ</span> <span class="mi">3</span> <span class="nv">p</span> <span class="mi">1</span>.
################################################################################
#
# <span class="nv">Suggested</span> <span class="nv">additional</span> <span class="nv">basis</span> <span class="nv">functions</span>. <span class="k">For</span> <span class="nv">production</span> <span class="nv">calculations</span>,
# <span class="nv">uncomment</span> <span class="nv">them</span> <span class="nv">one</span> <span class="nv">after</span> <span class="nv">another</span> <span class="ss">(</span><span class="nv">the</span> <span class="nv">most</span> <span class="nv">important</span> <span class="nv">basis</span> <span class="nv">functions</span> <span class="nv">are</span>
# <span class="nv">listed</span> <span class="nv">first</span><span class="ss">)</span>.
#
# <span class="nv">Constructed</span> <span class="k">for</span> <span class="nv">dimers</span>: <span class="mi">1</span>.<span class="mi">75</span> <span class="nv">A</span>, <span class="mi">2</span>.<span class="mi">0</span> <span class="nv">A</span>, <span class="mi">2</span>.<span class="mi">25</span> <span class="nv">A</span>, <span class="mi">2</span>.<span class="mi">75</span> <span class="nv">A</span>, <span class="mi">3</span>.<span class="mi">75</span> <span class="nv">A</span>
#
################################################################################
# <span class="s2">"</span><span class="s">First tier</span><span class="s2">"</span> <span class="o">-</span> <span class="nv">improvements</span>: <span class="o">-</span><span class="mi">571</span>.<span class="mi">96</span> <span class="nv">meV</span> <span class="nv">to</span> <span class="o">-</span><span class="mi">37</span>.<span class="mi">03</span> <span class="nv">meV</span>
<span class="nv">hydro</span> <span class="mi">3</span> <span class="nv">d</span> <span class="mi">4</span>.<span class="mi">2</span>
<span class="nv">hydro</span> <span class="mi">2</span> <span class="nv">p</span> <span class="mi">1</span>.<span class="mi">4</span>
<span class="nv">hydro</span> <span class="mi">4</span> <span class="nv">f</span> <span class="mi">6</span>.<span class="mi">2</span>
<span class="nv">ionic</span> <span class="mi">3</span> <span class="nv">s</span> <span class="nv">auto</span>
# <span class="s2">"</span><span class="s">Second tier</span><span class="s2">"</span> <span class="o">-</span> <span class="nv">improvements</span>: <span class="o">-</span><span class="mi">16</span>.<span class="mi">76</span> <span class="nv">meV</span> <span class="nv">to</span> <span class="o">-</span><span class="mi">3</span>.<span class="mi">03</span> <span class="nv">meV</span>
# <span class="nv">hydro</span> <span class="mi">3</span> <span class="nv">d</span> <span class="mi">9</span>
<span class="nv">for_aux</span> <span class="nv">hydro</span> <span class="mi">5</span> <span class="nv">g</span> <span class="mi">9</span>.<span class="mi">4</span>
# <span class="nv">hydro</span> <span class="mi">4</span> <span class="nv">p</span> <span class="mi">4</span>
# <span class="nv">hydro</span> <span class="mi">1</span> <span class="nv">s</span> <span class="mi">0</span>.<span class="mi">65</span>
# <span class="s2">"</span><span class="s">Third tier</span><span class="s2">"</span> <span class="o">-</span> <span class="nv">improvements</span>: <span class="o">-</span><span class="mi">3</span>.<span class="mi">89</span> <span class="nv">meV</span> <span class="nv">to</span> <span class="o">-</span><span class="mi">0</span>.<span class="mi">60</span> <span class="nv">meV</span>
# <span class="nv">ionic</span> <span class="mi">3</span> <span class="nv">d</span> <span class="nv">auto</span>
# <span class="nv">hydro</span> <span class="mi">3</span> <span class="nv">s</span> <span class="mi">2</span>.<span class="mi">6</span>
# <span class="nv">hydro</span> <span class="mi">4</span> <span class="nv">f</span> <span class="mi">8</span>.<span class="mi">4</span>
# <span class="nv">hydro</span> <span class="mi">3</span> <span class="nv">d</span> <span class="mi">3</span>.<span class="mi">4</span>
# <span class="nv">hydro</span> <span class="mi">3</span> <span class="nv">p</span> <span class="mi">7</span>.<span class="mi">8</span>
# <span class="s2">"</span><span class="s">Fourth tier</span><span class="s2">"</span> <span class="o">-</span> <span class="nv">improvements</span>: <span class="o">-</span><span class="mi">0</span>.<span class="mi">33</span> <span class="nv">meV</span> <span class="nv">to</span> <span class="o">-</span><span class="mi">0</span>.<span class="mi">11</span> <span class="nv">meV</span>
# <span class="nv">hydro</span> <span class="mi">2</span> <span class="nv">p</span> <span class="mi">1</span>.<span class="mi">6</span>
# <span class="nv">hydro</span> <span class="mi">5</span> <span class="nv">g</span> <span class="mi">10</span>.<span class="mi">8</span>
# <span class="nv">hydro</span> <span class="mi">5</span> <span class="nv">f</span> <span class="mi">11</span>.<span class="mi">2</span>
# <span class="nv">hydro</span> <span class="mi">3</span> <span class="nv">d</span> <span class="mi">1</span>
# <span class="nv">hydro</span> <span class="mi">4</span> <span class="nv">s</span> <span class="mi">4</span>.<span class="mi">5</span>
# <span class="nv">Further</span> <span class="nv">basis</span> <span class="nv">functions</span> <span class="nv">that</span> <span class="nv">fell</span> <span class="nv">out</span> <span class="nv">of</span> <span class="nv">the</span> <span class="nv">optimization</span> <span class="o">-</span> <span class="nv">noise</span>
# <span class="nv">level</span>... <span class="o"><</span> <span class="o">-</span><span class="mi">0</span>.<span class="mi">08</span> <span class="nv">meV</span>
# <span class="nv">hydro</span> <span class="mi">4</span> <span class="nv">d</span> <span class="mi">6</span>.<span class="mi">6</span>
# <span class="nv">hydro</span> <span class="mi">5</span> <span class="nv">g</span> <span class="mi">16</span>.<span class="mi">4</span>
# <span class="nv">hydro</span> <span class="mi">4</span> <span class="nv">d</span> <span class="mi">9</span>
</code></pre></div>
<p><code>geometry.in</code></p>
<div class="highlight"><pre><span></span><code>lattice_vector 5.43 0.0000000000000000 0.0000000000000000
lattice_vector 0.0000000000000000 5.43 0.0000000000000000
lattice_vector 0.0000000000000000 0.0000000000000000 5.43
atom 0.0000000000000000 0.0000000000000000 0.0000000000000000 Si
atom 1.3575000000000000 1.3575000000000000 1.3575000000000000 Si
atom 0.0000000000000000 2.7150000000000000 2.7150000000000000 Si
atom 1.3575000000000000 4.0725000000000000 4.0725000000000000 Si
atom 2.7150000000000000 0.0000000000000000 2.7150000000000000 Si
atom 4.0725000000000000 1.3575000000000000 4.0725000000000000 Si
atom 2.7150000000000000 2.7150000000000000 0.0000000000000000 Si
atom 4.0725000000000000 4.0725000000000000 1.3575000000000000 Si
</code></pre></div>
<h4>How to perform Bader analysis on the output cube files</h4>
<p>Codes exist to perform Bader analysis on cube files, here I have two recommendations. One is the <a href="http://theory.cm.utexas.edu/henkelman/code/bader/">Bader code</a> by Henkelman's Group. The other is the <a href="https://aoterodelaroza.github.io/critic2/">Critic2 code</a> by Alberto's Group.</p>
<h5>Bader code</h5>
<p>The way to run the Bader code is quite intuitive. Here is an example command line for the Bader analysis.</p>
<p><code>bader -vac 0 cube_001_total_density_integrable.cube</code>
The output file <code>ACF.dat</code> shows the number of electrons associate with each atom. Here is the example of crystalline NaCl.</p>
<div class="highlight"><pre><span></span><code> # X Y Z CHARGE MIN DIST ATOMIC VOL
--------------------------------------------------------------------------------
1 8.056649 2.685550 2.685550 10.134933 2.127317 66.679119
2 2.685550 8.056649 2.685550 10.134933 2.127317 66.679119
3 2.685550 2.685550 8.056649 10.134933 2.127317 66.679119
4 8.056649 8.056649 8.056649 10.134933 2.127313 66.679119
5 2.685550 2.685550 2.685550 17.863308 3.089009 242.798590
6 8.056649 8.056649 2.685550 17.865052 3.088990 243.331581
7 8.056649 2.685550 8.056649 17.866106 3.088990 243.652954
8 2.685550 8.056649 8.056649 17.864259 3.088990 243.081226
--------------------------------------------------------------------------------
VACUUM CHARGE: 0.0000
VACUUM VOLUME: 0.0000
NUMBER OF ELECTRONS: 111.9985
</code></pre></div>
<h5>Critic2 code</h5>
<p>The critic2 code is more versatile. It is basically a general code to perform all kinds of operators on some fields. Here we present a simple input for the same Bader analysis.</p>
<p><code>critic2.in</code></p>
<div class="highlight"><pre><span></span><code><span class="c1"># Read the structure from any of the cube files</span>
<span class="n">crystal</span> <span class="n">cube_001_total_density_integrable</span><span class="o">.</span><span class="n">cube</span>
<span class="c1"># Load the electron density</span>
<span class="nb">load</span> <span class="n">cube_001_total_density_integrable</span><span class="o">.</span><span class="n">cube</span> <span class="n">id</span> <span class="n">rho</span>
<span class="c1"># Use Bader.</span>
<span class="n">bader</span>
</code></pre></div>
<p>Then we run the code via commandline.</p>
<p><code>critic2 critic2.in > critic2.out</code></p>
<p>We can take a look at the <code>critic2.out</code> file. </p>
<div class="highlight"><pre><span></span><code><span class="o">*</span> <span class="nv">Integrated</span> <span class="nv">atomic</span> <span class="nv">properties</span>
# <span class="ss">(</span><span class="nv">See</span> <span class="nv">key</span> <span class="nv">above</span> <span class="k">for</span> <span class="nv">interpretation</span> <span class="nv">of</span> <span class="nv">column</span> <span class="nv">headings</span>.<span class="ss">)</span>
# <span class="nv">Integrable</span> <span class="nv">properties</span> <span class="mi">1</span> <span class="nv">to</span> <span class="mi">3</span>
# <span class="nv">Id</span> <span class="nv">cp</span> <span class="nv">ncp</span> <span class="nv">Name</span> <span class="nv">Z</span> <span class="nv">mult</span> <span class="nv">Volume</span> <span class="nv">Pop</span> <span class="nv">Lap</span>
<span class="mi">1</span> <span class="mi">1</span> <span class="mi">1</span> <span class="nv">Na</span> <span class="mi">11</span> <span class="o">--</span> <span class="mi">6</span>.<span class="mi">66791186</span><span class="nv">E</span><span class="o">+</span><span class="mi">01</span> <span class="mi">1</span>.<span class="mi">01349330</span><span class="nv">E</span><span class="o">+</span><span class="mi">01</span> <span class="mi">2</span>.<span class="mi">37946134</span><span class="nv">E</span><span class="o">-</span><span class="mi">02</span>
<span class="mi">2</span> <span class="mi">2</span> <span class="mi">1</span> <span class="nv">Na</span> <span class="mi">11</span> <span class="o">--</span> <span class="mi">6</span>.<span class="mi">66791186</span><span class="nv">E</span><span class="o">+</span><span class="mi">01</span> <span class="mi">1</span>.<span class="mi">01349330</span><span class="nv">E</span><span class="o">+</span><span class="mi">01</span> <span class="mi">2</span>.<span class="mi">37946061</span><span class="nv">E</span><span class="o">-</span><span class="mi">02</span>
<span class="mi">3</span> <span class="mi">3</span> <span class="mi">1</span> <span class="nv">Na</span> <span class="mi">11</span> <span class="o">--</span> <span class="mi">6</span>.<span class="mi">66791186</span><span class="nv">E</span><span class="o">+</span><span class="mi">01</span> <span class="mi">1</span>.<span class="mi">01349330</span><span class="nv">E</span><span class="o">+</span><span class="mi">01</span> <span class="mi">2</span>.<span class="mi">37946091</span><span class="nv">E</span><span class="o">-</span><span class="mi">02</span>
<span class="mi">4</span> <span class="mi">4</span> <span class="mi">1</span> <span class="nv">Na</span> <span class="mi">11</span> <span class="o">--</span> <span class="mi">6</span>.<span class="mi">66791186</span><span class="nv">E</span><span class="o">+</span><span class="mi">01</span> <span class="mi">1</span>.<span class="mi">01349330</span><span class="nv">E</span><span class="o">+</span><span class="mi">01</span> <span class="mi">2</span>.<span class="mi">37946131</span><span class="nv">E</span><span class="o">-</span><span class="mi">02</span>
<span class="mi">5</span> <span class="mi">5</span> <span class="mi">2</span> <span class="nv">Cl</span> <span class="mi">17</span> <span class="o">--</span> <span class="mi">2</span>.<span class="mi">42798590</span><span class="nv">E</span><span class="o">+</span><span class="mi">02</span> <span class="mi">1</span>.<span class="mi">78633076</span><span class="nv">E</span><span class="o">+</span><span class="mi">01</span> <span class="o">-</span><span class="mi">2</span>.<span class="mi">72005475</span><span class="nv">E</span><span class="o">-</span><span class="mi">02</span>
<span class="mi">6</span> <span class="mi">6</span> <span class="mi">2</span> <span class="nv">Cl</span> <span class="mi">17</span> <span class="o">--</span> <span class="mi">2</span>.<span class="mi">43331581</span><span class="nv">E</span><span class="o">+</span><span class="mi">02</span> <span class="mi">1</span>.<span class="mi">78650521</span><span class="nv">E</span><span class="o">+</span><span class="mi">01</span> <span class="o">-</span><span class="mi">2</span>.<span class="mi">28806825</span><span class="nv">E</span><span class="o">-</span><span class="mi">02</span>
<span class="mi">7</span> <span class="mi">7</span> <span class="mi">2</span> <span class="nv">Cl</span> <span class="mi">17</span> <span class="o">--</span> <span class="mi">2</span>.<span class="mi">43652954</span><span class="nv">E</span><span class="o">+</span><span class="mi">02</span> <span class="mi">1</span>.<span class="mi">78661064</span><span class="nv">E</span><span class="o">+</span><span class="mi">01</span> <span class="o">-</span><span class="mi">2</span>.<span class="mi">02729109</span><span class="nv">E</span><span class="o">-</span><span class="mi">02</span>
<span class="mi">8</span> <span class="mi">8</span> <span class="mi">2</span> <span class="nv">Cl</span> <span class="mi">17</span> <span class="o">--</span> <span class="mi">2</span>.<span class="mi">43081226</span><span class="nv">E</span><span class="o">+</span><span class="mi">02</span> <span class="mi">1</span>.<span class="mi">78642592</span><span class="nv">E</span><span class="o">+</span><span class="mi">01</span> <span class="o">-</span><span class="mi">2</span>.<span class="mi">48243008</span><span class="nv">E</span><span class="o">-</span><span class="mi">02</span>
<span class="o">--------------------------------------------------------------------------------</span>
<span class="nv">Sum</span> <span class="mi">1</span>.<span class="mi">23958083</span><span class="nv">E</span><span class="o">+</span><span class="mi">03</span> <span class="mi">1</span>.<span class="mi">11998457</span><span class="nv">E</span><span class="o">+</span><span class="mi">02</span> <span class="o">-</span><span class="mi">7</span>.<span class="mi">66990638</span><span class="nv">E</span><span class="o">-</span><span class="mi">13</span>
</code></pre></div>
<p>Clearly, the outputs from Bader and critic2 codes match with each other.</p>
<h4>a side note about critic2 code</h4>
<p>With critic2, we can calculate more quantities than the Bader integrals. For example, we can plot <a href="https://doi.org/10.1021/ct100641a">NCIPLOT</a> with critic2 with the input <code>critic2_nci.in</code></p>
<div class="highlight"><pre><span></span><code><span class="c1"># Read the structure from any of the cube files</span>
<span class="n">crystal</span> <span class="n">cube_001_total_density_integrable</span><span class="o">.</span><span class="n">cube</span>
<span class="c1"># Load the electron density</span>
<span class="nb">load</span> <span class="n">cube_001_total_density_integrable</span><span class="o">.</span><span class="n">cube</span> <span class="n">id</span> <span class="n">rho</span>
<span class="c1">#NCIplot</span>
<span class="n">NCIPLOT</span>
<span class="n">END</span>
</code></pre></div>
<p>The result can be visualized in VMD via command <code>vmd -e critic2_nci.vmd</code></p>
<p><img alt="Alt Text" src="./images/aims_bader_nciplot.png"></p>
<p>For the crystalline NaCl here, this plot doesn't tell much. But, for systems with vdW interaction, hybrogen bond, or steric clashes, this plot can provide useful information.</p>
<h4>Convergence of the integral in the integrable total density cube file output by FHI-aims.</h4>
<p>An interesting observation I have for the cube file output is the integral of the integrable total density cube file has a convergence behavior. With the more accurate numerical settings the more accurate the total number of electrons is got.
My suspicion is the normalization of the integration grid with different setting might play a role here. Without dense enough integration setting, the integration in the atom centered integration grid will introduce a normalization error which will pass to the total number of electrons calculated by the integration of the integrable total density cube file. Luckily, the error is relatively small and will reduce with a more strict numerical setting as I showed the results for both examples.</p>
<p>The convergence behavior for crystalline sillicon.</p>
<p><center></p>
<table>
<thead>
<tr>
<th align="center">Numerical Setting <code> </code></th>
<th align="center">Integral of integrable total density cube file</th>
</tr>
</thead>
<tbody>
<tr>
<td align="center">Minimal</td>
<td align="center">111.9851</td>
</tr>
<tr>
<td align="center">Light</td>
<td align="center">111.9909</td>
</tr>
<tr>
<td align="center">Intermediate</td>
<td align="center">111.9966</td>
</tr>
<tr>
<td align="center">Tight</td>
<td align="center">111.9965</td>
</tr>
<tr>
<td align="center">Really tight</td>
<td align="center">111.9998</td>
</tr>
</tbody>
</table>
<p></center></p>
<p>The convergence behavior for crystalline NaCl.</p>
<p><center></p>
<table>
<thead>
<tr>
<th align="center">Numerical Setting <code> </code></th>
<th align="center">Integral of integrable total density cube file <code> </code></th>
<th align="center">Charge on Na <code> </code></th>
<th align="center">Charge on Cl <code> </code></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center">Minimal</td>
<td align="center">111.9968</td>
<td align="center">0.8665</td>
<td align="center">-0.8657</td>
</tr>
<tr>
<td align="center">Light</td>
<td align="center">111.9970</td>
<td align="center">0.8645</td>
<td align="center">-0.8638</td>
</tr>
<tr>
<td align="center">Intermediate</td>
<td align="center">111.9985</td>
<td align="center">0.8651</td>
<td align="center">-0.8647</td>
</tr>
<tr>
<td align="center">Tight</td>
<td align="center">111.9984</td>
<td align="center">0.8653</td>
<td align="center">-0.8649</td>
</tr>
<tr>
<td align="center">Really tight</td>
<td align="center">111.9991</td>
<td align="center">0.8653</td>
<td align="center">-0.8651</td>
</tr>
</tbody>
</table>
<p></center></p>
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