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<div class="section" id="dmrg-workshop-12-jul-2016-hands-on-session">
<span id="index-0"></span><h1>12. DMRG workshop (12-jul-2016): hands-on session<a class="headerlink" href="#dmrg-workshop-12-jul-2016-hands-on-session" title="Permalink to this headline">¶</a></h1>
<div class="section" id="introduction">
<h2>12.1. Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline">¶</a></h2>
<p>The geometry of tetracene was optimized at the restricted B3LYP/6-31G* level of theory, and can be found in the file <a class="reference external" href="https://github.com/sebwouters/chemps2/raw/master/sphinx/tetracene.fcidump.in">tetracene.fcidump.in</a>:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="n">memory</span> <span class="mi">28</span> <span class="n">Gb</span>
<span class="n">molecule</span> <span class="n">tetracene</span> <span class="p">{</span>
<span class="mi">0</span> <span class="mi">1</span>
<span class="n">symmetry</span> <span class="n">csz</span>
<span class="n">C</span> <span class="mf">4.888883611380</span> <span class="o">-</span><span class="mf">0.715374463486</span> <span class="o">-</span><span class="mf">0.000000000000</span>
<span class="n">C</span> <span class="mf">4.888883611380</span> <span class="mf">0.715374463486</span> <span class="mf">0.000000000000</span>
<span class="n">C</span> <span class="o">-</span><span class="mf">4.888883611380</span> <span class="o">-</span><span class="mf">0.715374463486</span> <span class="mf">0.000000000000</span>
<span class="n">C</span> <span class="o">-</span><span class="mf">4.888883611380</span> <span class="mf">0.715374463486</span> <span class="o">-</span><span class="mf">0.000000000000</span>
<span class="n">C</span> <span class="mf">3.711144499602</span> <span class="o">-</span><span class="mf">1.409316610825</span> <span class="mf">0.000000000000</span>
<span class="n">C</span> <span class="mf">3.711144499602</span> <span class="mf">1.409316610825</span> <span class="mf">0.000000000000</span>
<span class="n">C</span> <span class="o">-</span><span class="mf">3.711144499602</span> <span class="o">-</span><span class="mf">1.409316610825</span> <span class="mf">0.000000000000</span>
<span class="n">C</span> <span class="o">-</span><span class="mf">3.711144499602</span> <span class="mf">1.409316610825</span> <span class="mf">0.000000000000</span>
<span class="n">C</span> <span class="mf">2.450542389320</span> <span class="o">-</span><span class="mf">0.725895641808</span> <span class="mf">0.000000000000</span>
<span class="n">C</span> <span class="mf">2.450542389320</span> <span class="mf">0.725895641808</span> <span class="mf">0.000000000000</span>
<span class="n">C</span> <span class="o">-</span><span class="mf">2.450542389320</span> <span class="o">-</span><span class="mf">0.725895641808</span> <span class="mf">0.000000000000</span>
<span class="n">C</span> <span class="o">-</span><span class="mf">2.450542389320</span> <span class="mf">0.725895641808</span> <span class="mf">0.000000000000</span>
<span class="n">C</span> <span class="mf">1.235393613403</span> <span class="o">-</span><span class="mf">1.406341384439</span> <span class="o">-</span><span class="mf">0.000000000000</span>
<span class="n">C</span> <span class="mf">1.235393613403</span> <span class="mf">1.406341384439</span> <span class="o">-</span><span class="mf">0.000000000000</span>
<span class="n">C</span> <span class="o">-</span><span class="mf">1.235393613403</span> <span class="o">-</span><span class="mf">1.406341384439</span> <span class="mf">0.000000000000</span>
<span class="n">C</span> <span class="o">-</span><span class="mf">1.235393613403</span> <span class="mf">1.406341384439</span> <span class="o">-</span><span class="mf">0.000000000000</span>
<span class="n">C</span> <span class="mf">0.000000000000</span> <span class="o">-</span><span class="mf">0.726150477978</span> <span class="mf">0.000000000000</span>
<span class="n">C</span> <span class="o">-</span><span class="mf">0.000000000000</span> <span class="mf">0.726150477978</span> <span class="o">-</span><span class="mf">0.000000000000</span>
<span class="n">H</span> <span class="o">-</span><span class="mf">5.836431028249</span> <span class="mf">1.247257941939</span> <span class="mf">0.000000000000</span>
<span class="n">H</span> <span class="o">-</span><span class="mf">5.836431028249</span> <span class="o">-</span><span class="mf">1.247257941939</span> <span class="o">-</span><span class="mf">0.000000000000</span>
<span class="n">H</span> <span class="mf">5.836431028249</span> <span class="o">-</span><span class="mf">1.247257941939</span> <span class="mf">0.000000000000</span>
<span class="n">H</span> <span class="mf">5.836431028249</span> <span class="mf">1.247257941939</span> <span class="o">-</span><span class="mf">0.000000000000</span>
<span class="n">H</span> <span class="mf">3.708923113951</span> <span class="o">-</span><span class="mf">2.496817544333</span> <span class="mf">0.000000000000</span>
<span class="n">H</span> <span class="mf">3.708923113951</span> <span class="mf">2.496817544333</span> <span class="mf">0.000000000000</span>
<span class="n">H</span> <span class="o">-</span><span class="mf">3.708923113951</span> <span class="o">-</span><span class="mf">2.496817544333</span> <span class="o">-</span><span class="mf">0.000000000000</span>
<span class="n">H</span> <span class="o">-</span><span class="mf">3.708923113951</span> <span class="mf">2.496817544333</span> <span class="mf">0.000000000000</span>
<span class="n">H</span> <span class="mf">1.235230564126</span> <span class="o">-</span><span class="mf">2.494555353711</span> <span class="mf">0.000000000000</span>
<span class="n">H</span> <span class="mf">1.235230564126</span> <span class="mf">2.494555353711</span> <span class="mf">0.000000000000</span>
<span class="n">H</span> <span class="o">-</span><span class="mf">1.235230564126</span> <span class="mf">2.494555353711</span> <span class="mf">0.000000000000</span>
<span class="n">H</span> <span class="o">-</span><span class="mf">1.235230564126</span> <span class="o">-</span><span class="mf">2.494555353711</span> <span class="mf">0.000000000000</span>
<span class="p">}</span>
<span class="n">sys</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span> <span class="mi">0</span><span class="p">,</span> <span class="s1">'./..'</span> <span class="p">)</span>
<span class="kn">import</span> <span class="nn">fcidump</span>
<span class="nb">set</span> <span class="n">basis</span> <span class="mi">6</span><span class="o">-</span><span class="mi">31</span><span class="n">g</span><span class="o">*</span>
<span class="nb">set</span> <span class="n">reference</span> <span class="n">rhf</span>
<span class="nb">set</span> <span class="n">scf_type</span> <span class="n">DIRECT</span>
<span class="nb">set</span> <span class="n">e_convergence</span> <span class="mi">1</span><span class="n">e</span><span class="o">-</span><span class="mi">12</span>
<span class="nb">set</span> <span class="n">d_convergence</span> <span class="mi">1</span><span class="n">e</span><span class="o">-</span><span class="mi">10</span>
<span class="nb">set</span> <span class="n">ints_tolerance</span> <span class="mf">0.0</span>
<span class="nb">set</span> <span class="n">fcidump</span> <span class="n">dumpfilename</span> <span class="n">tetracene</span><span class="o">.</span><span class="n">fcidump</span>
<span class="n">E</span><span class="p">,</span> <span class="n">wfn</span> <span class="o">=</span> <span class="n">energy</span><span class="p">(</span> <span class="s1">'fcidump'</span><span class="p">,</span> <span class="n">return_wfn</span><span class="o">=</span><span class="kc">True</span> <span class="p">)</span>
<span class="n">molden</span><span class="p">(</span> <span class="n">wfn</span><span class="p">,</span> <span class="s1">'tetracene.molden'</span> <span class="p">)</span>
</pre></div>
</div>
<p>The goal of this afternoon is to calculate the vertical singlet-triplet gap with DMRG(18, 18)-CASPT2/6-31G*.</p>
<p><a class="reference external" href="https://github.com/sebwouters/chemps2">chemps2</a> is a C++ library for spin-adapted DMRG calculations which can be incorporated in quantum chemistry packages. This has been done for <a class="reference external" href="http://www.psicode.org/">psi4</a>. Alternatively, the same functionality can be used with the binary, when the required matrix elements have been generated in <code class="docutils literal"><span class="pre">FCIDUMP</span></code> format. We will follow the second route this afternoon. The advantage of the latter route is that you are not tied to <a class="reference external" href="http://www.psicode.org/">psi4</a> to obtain matrix elements. In the future you can use <a class="reference external" href="http://www.molcas.org/">molcas</a>, <a class="reference external" href="https://www.molpro.net/">molpro</a>, <a class="reference external" href="http://www.daltonprogram.org/">dalton</a>... The disadvantage is that a full-rank <code class="docutils literal"><span class="pre">FCIDUMP</span></code> file is required, and that less virtual (secondary) orbitals can be used than with density-fitted DMRG-SCF and DMRG-CASPT2.</p>
<p>Please read an entire section before starting the instructions. Then you will have all the useful information you need!</p>
</div>
<div class="section" id="ugent-hpc">
<h2>12.2. UGent HPC<a class="headerlink" href="#ugent-hpc" title="Permalink to this headline">¶</a></h2>
<p>Follow the <a class="reference external" href="http://hpc.ugent.be/userwiki/index.php/User:VscConnect">instructions</a> to log in to the UGent HPC.</p>
<p>Submit an interactive job in the XYZ queue:</p>
<div class="highlight-bash"><div class="highlight"><pre><span></span>$ qsub -I -W <span class="nv">x</span><span class="o">=</span>FLAGS:ADVRES:dmrg.198 -l <span class="nv">walltime</span><span class="o">=</span>06:00:00 -l <span class="nv">nodes</span><span class="o">=</span>1:ppn<span class="o">=</span>8
</pre></div>
</div>
<p>Once you are on the node, change directory to, for example, the following folder:</p>
<div class="highlight-bash"><div class="highlight"><pre><span></span>$ <span class="nb">cd</span> <span class="nv">$VSC_SCRATCH_NODE</span>
$ mkdir dmrg_workshop
$ <span class="nb">cd</span> dmrg_workshop/
</pre></div>
</div>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p class="last">Please keep in mind that you will need about 20 GB of disk for the <code class="docutils literal"><span class="pre">FCIDUMP</span></code> file and 1 GB of disk for the <a class="reference external" href="https://github.com/sebwouters/chemps2">chemps2</a> checkpoints!</p>
</div>
</div>
<div class="section" id="fcidump-and-molden">
<h2>12.3. <code class="docutils literal"><span class="pre">FCIDUMP</span></code> and <code class="docutils literal"><span class="pre">MOLDEN</span></code><a class="headerlink" href="#fcidump-and-molden" title="Permalink to this headline">¶</a></h2>
<p>We will first use a plugin to <a class="reference external" href="http://www.psicode.org/">psi4</a> to generate the RHF matrix elements in <code class="docutils literal"><span class="pre">FCIDUMP</span></code> format, as well as the corresponding <code class="docutils literal"><span class="pre">MOLDEN</span></code> file. As said before, any other program which is able to generate these two types of files can be used as well. Load the <a class="reference external" href="http://www.psicode.org/">psi4</a> module and generate a new plugin called <code class="docutils literal"><span class="pre">fcidump</span></code>:</p>
<div class="highlight-bash"><div class="highlight"><pre><span></span>$ module load PSI4/1.0-intel-2016a-mt-Python-2.7.11
$ psi4 --new-plugin fcidump
</pre></div>
</div>
<p>Overwrite the dummy file <code class="docutils literal"><span class="pre">fcidump.cc</span></code> and compile:</p>
<div class="highlight-bash"><div class="highlight"><pre><span></span>$ <span class="nb">cd</span> fcidump/
$ rm fcidump.cc
$ wget <span class="s1">'https://github.com/sebwouters/chemps2/raw/master/integrals/psi4plugins/fcidump.cc'</span>
$ make
$ <span class="nb">cd</span> ../
</pre></div>
</div>
<p>The required <code class="docutils literal"><span class="pre">FCIDUMP</span></code> file and the corresponding <code class="docutils literal"><span class="pre">MOLDEN</span></code> file can now be generated with <a class="reference external" href="http://www.psicode.org/">psi4</a>:</p>
<div class="highlight-bash"><div class="highlight"><pre><span></span>$ wget <span class="s1">'https://github.com/sebwouters/chemps2/raw/master/sphinx/tetracene.fcidump.in'</span>
$ <span class="nv">OMP_NUM_THREADS</span><span class="o">=</span><span class="m">8</span> psi4 -n <span class="m">8</span> tetracene.fcidump.in <span class="p">&</span>
$ tail -n <span class="m">3000</span> -f tetracene.fcidump.out
$ ls -alh TETRACENE.FCIDUMP
$ ls -alh tetracene.molden
</pre></div>
</div>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p class="last">The specified symmetry group in <code class="docutils literal"><span class="pre">tetracene.fcidump.in</span></code> was <code class="docutils literal"><span class="pre">csz</span></code>, a subgroup of <code class="docutils literal"><span class="pre">d2h</span></code>. In the <code class="docutils literal"><span class="pre">csz</span></code> symmetry group, the 18 active space <span class="math">\(\pi\)</span>-orbitals can be localized to the carbon atoms. This is not the case for the <code class="docutils literal"><span class="pre">d2h</span></code> symmetry group.</p>
</div>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p class="last">While you are waiting for the <code class="docutils literal"><span class="pre">FCIDUMP</span></code> file of size 20 GB, you can already proceed with the next section.</p>
</div>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p>You can also use the precreated files from the folder <code class="docutils literal"><span class="pre">/apps/gent/tutorials/DMRG/</span></code> instead:</p>
<div class="last highlight-bash"><div class="highlight"><pre><span></span>$ ls -al /apps/gent/tutorials/DMRG/tetracene.fcidump.in
$ ls -al /apps/gent/tutorials/DMRG/tetracene.fcidump.out
$ ls -al /apps/gent/tutorials/DMRG/TETRACENE.FCIDUMP
$ ls -al /apps/gent/tutorials/DMRG/tetracene.molden
</pre></div>
</div>
</div>
</div>
<div class="section" id="basis-choice">
<h2>12.4. Basis choice<a class="headerlink" href="#basis-choice" title="Permalink to this headline">¶</a></h2>
<p>Now that you have the required matrix elements in <code class="docutils literal"><span class="pre">FCIDUMP</span></code> format and the corresponding <code class="docutils literal"><span class="pre">MOLDEN</span></code> file, we can perform calculations with <a class="reference external" href="https://github.com/sebwouters/chemps2">chemps2</a> v1.7.2. This module should have been loaded together with the <a class="reference external" href="http://www.psicode.org/">psi4</a> module. If this was not the case, you can load it with:</p>
<div class="highlight-bash"><div class="highlight"><pre><span></span>$ module load CheMPS2/1.7.2-intel-2016a
</pre></div>
</div>
<p>Study the options of the binary:</p>
<div class="highlight-bash"><div class="highlight"><pre><span></span>$ chemps2 --version
$ chemps2 --help
</pre></div>
</div>
<p>Perform each calculation in a separate folder. This way checkpoint files will not get mixed up. Create a folder <code class="docutils literal"><span class="pre">ci_input_orbs/</span></code> and in that folder an input file <code class="docutils literal"><span class="pre">ci_input_orbs.in</span></code> for <a class="reference external" href="https://github.com/sebwouters/chemps2">chemps2</a> with the following options:</p>
<ul class="simple">
<li>Target the singlet ground state</li>
<li>Use an (18, 18) active space</li>
<li>Switch off the CASPT2 calculation</li>
<li>Overwrite the tmp folder with the existing path <code class="docutils literal"><span class="pre">/local/NUMBER.master15.delcatty.gent.vsc/</span></code>, where <code class="docutils literal"><span class="pre">NUMBER</span></code> is the job number which you see with</li>
</ul>
<div class="highlight-bash"><div class="highlight"><pre><span></span>$ qstat -n
</pre></div>
</div>
<ul class="simple">
<li>Perform one DMRG-SCF iteration, which corresponds to DMRG-CI</li>
<li>The active space orbitals should be the RHF molecular orbitals (i.e. the input orbitals)</li>
<li>Use the convergence scheme</li>
</ul>
<blockquote>
<div><table border="1" class="docutils">
<colgroup>
<col width="20%" />
<col width="19%" />
<col width="18%" />
<col width="25%" />
<col width="18%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head"><span class="math">\(D_{SU(2)}\)</span></th>
<th class="head"><span class="math">\(E_{conv}\)</span></th>
<th class="head"><span class="math">\(N_{max}\)</span></th>
<th class="head"><span class="math">\(\gamma_{noise}\)</span></th>
<th class="head"><span class="math">\(r_{tol}\)</span></th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td>200</td>
<td>1e-6</td>
<td>10</td>
<td>0.05</td>
<td>1e-5</td>
</tr>
<tr class="row-odd"><td>400</td>
<td>1e-6</td>
<td>10</td>
<td>0.05</td>
<td>1e-5</td>
</tr>
<tr class="row-even"><td>600</td>
<td>1e-6</td>
<td>10</td>
<td>0.05</td>
<td>1e-5</td>
</tr>
<tr class="row-odd"><td>600</td>
<td>1e-8</td>
<td>3</td>
<td>0.0</td>
<td>1e-5</td>
</tr>
<tr class="row-even"><td>400</td>
<td>1e-8</td>
<td>3</td>
<td>0.0</td>
<td>1e-5</td>
</tr>
<tr class="row-odd"><td>200</td>
<td>1e-8</td>
<td>3</td>
<td>0.0</td>
<td>1e-5</td>
</tr>
</tbody>
</table>
</div></blockquote>
<ul class="simple">
<li>Set the option <code class="docutils literal"><span class="pre">SCF_MOLDEN</span></code> to the corresponding molden file</li>
</ul>
<p>When you have created the input file, you can double check with the <a class="reference internal" href="#first-ptr-solution"><span class="std std-ref">solution</span></a>.</p>
<p>Run the calculation:</p>
<div class="highlight-bash"><div class="highlight"><pre><span></span>$ <span class="nb">cd</span> ci_input_orbs/
$ <span class="nv">OMP_NUM_THREADS</span><span class="o">=</span><span class="m">4</span> chemps2 --file<span class="o">=</span>ci_input_orbs.in <span class="p">&</span>> ci_input_orbs.out <span class="p">&</span>
$ <span class="nb">cd</span> ../
</pre></div>
</div>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p class="last">You have now only used 4 of the 8 available cores. Proceed with the instructions below while waiting for the calculation to finish.</p>
</div>
<p>Create a folder <code class="docutils literal"><span class="pre">ci_local_orbs/</span></code> and in that folder an input file <code class="docutils literal"><span class="pre">ci_local_orbs.in</span></code> for <a class="reference external" href="https://github.com/sebwouters/chemps2">chemps2</a>, which is identical to <code class="docutils literal"><span class="pre">ci_input_orbs.in</span></code>, except for the active space orbitals. These should now be localized orbitals. When you have created the input file, you can double check with the <a class="reference internal" href="#second-ptr-solution"><span class="std std-ref">solution</span></a>.</p>
<p>Run the calculation:</p>
<div class="highlight-bash"><div class="highlight"><pre><span></span>$ <span class="nb">cd</span> ci_local_orbs/
$ <span class="nv">OMP_NUM_THREADS</span><span class="o">=</span><span class="m">4</span> chemps2 --file<span class="o">=</span>ci_local_orbs.in <span class="p">&</span>> ci_local_orbs.out <span class="p">&</span>
$ <span class="nb">cd</span> ../
$ tail -n <span class="m">300</span> ci_input_orbs/ci_input_orbs.out
$ tail -n <span class="m">300</span> ci_local_orbs/ci_local_orbs.out
</pre></div>
</div>
<p>When the calculations are finished, take a look at the files</p>
<blockquote>
<div><ul class="simple">
<li><code class="docutils literal"><span class="pre">ci_input_orbs/tetracene.molden.rotated</span></code></li>
<li><code class="docutils literal"><span class="pre">ci_local_orbs/tetracene.molden.rotated</span></code></li>
</ul>
</div></blockquote>
<p>with your favourite visualization software. Do the first 18 <code class="docutils literal"><span class="pre">App</span></code> or <code class="docutils literal"><span class="pre">A"</span></code> orbitals have the desired shape? How are they ordered? Once you have formulated your own answer, you can double check with the <a class="reference internal" href="#third-ptr-solution"><span class="std std-ref">solution</span></a>.</p>
<p>Compare the energies of the last three sweep instructions as a function of <span class="math">\(D_{SU(2)}\)</span> for both calculations. Thereto you can grep for:</p>
<div class="highlight-bash"><div class="highlight"><pre><span></span>$ grep <span class="s2">"Minimum energy encountered during the last sweep"</span> ci_input_orbs/ci_input_orbs.out
$ grep <span class="s2">"Minimum energy encountered during the last sweep"</span> ci_local_orbs/ci_local_orbs.out
</pre></div>
</div>
<p>What do you observe? Can you explain it? Once you have formulated your own answer, you can double check with the <a class="reference internal" href="#fourth-ptr-solution"><span class="std std-ref">solution</span></a>.</p>
</div>
<div class="section" id="dmrg-scf">
<h2>12.5. DMRG-SCF<a class="headerlink" href="#dmrg-scf" title="Permalink to this headline">¶</a></h2>
<p>Use localized orbitals for the active space from now on. Perform the DMRG-SCF orbital optimization for the singlet and the triplet. Also put DIIS on when the update norm is smaller than 1e-2, switch <code class="docutils literal"><span class="pre">PRINT_CORR</span></code> to <code class="docutils literal"><span class="pre">TRUE</span></code>, and remove the <code class="docutils literal"><span class="pre">SCF_MOLDEN</span></code> line. Use the following convergence scheme:</p>
<blockquote>
<div><table border="1" class="docutils">
<colgroup>
<col width="20%" />
<col width="19%" />
<col width="18%" />
<col width="25%" />
<col width="18%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head"><span class="math">\(D_{SU(2)}\)</span></th>
<th class="head"><span class="math">\(E_{conv}\)</span></th>
<th class="head"><span class="math">\(N_{max}\)</span></th>
<th class="head"><span class="math">\(\gamma_{noise}\)</span></th>
<th class="head"><span class="math">\(r_{tol}\)</span></th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td>250</td>
<td>1e-6</td>
<td>8</td>
<td>0.05</td>
<td>1e-5</td>
</tr>
<tr class="row-odd"><td>500</td>
<td>1e-8</td>
<td>8</td>
<td>0.05</td>
<td>1e-5</td>
</tr>
<tr class="row-even"><td>750</td>
<td>1e-10</td>
<td>8</td>
<td>0.0</td>
<td>1e-8</td>
</tr>
</tbody>
</table>
</div></blockquote>
<p>Why is the reduced virtual dimension not lowered at the end of the DMRG calculation? Why is the last <span class="math">\(r_{tol}\)</span> smaller? When you have created the input files, you can double check with the solution for the <a class="reference internal" href="#fifth-ptr-solution"><span class="std std-ref">singlet</span></a> and the <a class="reference internal" href="#sixth-ptr-solution"><span class="std std-ref">triplet</span></a>.</p>
<p>Run the calculation in separate folders:</p>
<div class="highlight-bash"><div class="highlight"><pre><span></span>$ <span class="nb">cd</span> scf_singlet/
$ <span class="nv">OMP_NUM_THREADS</span><span class="o">=</span><span class="m">4</span> chemps2 --file<span class="o">=</span>scf_singlet.in <span class="p">&</span>> scf_singlet.out <span class="p">&</span>
$ <span class="nb">cd</span> ../scf_triplet/
$ <span class="nv">OMP_NUM_THREADS</span><span class="o">=</span><span class="m">4</span> chemps2 --file<span class="o">=</span>scf_triplet.in <span class="p">&</span>> scf_triplet.out <span class="p">&</span>
$ <span class="nb">cd</span> ../
$ tail -n <span class="m">300</span> scf_singlet/scf_singlet.out
$ tail -n <span class="m">300</span> scf_triplet/scf_triplet.out
</pre></div>
</div>
<p>What is the DMRG-SCF singlet-triplet gap you obtain? Double check with the <a class="reference internal" href="#seventh-ptr-solution"><span class="std std-ref">solution</span></a>.</p>
<p>Do you see polyradical character in the natural orbital occupation numbers for the singlet and/or triplet? How can you observe this in the correlation functions? Tip: It might be interesting to read</p>
<ol class="upperalpha simple" start="10">
<li>Hachmann, J. J. Dorando, Michael Avilés and Garnet Kin-Lic Chan, <em>Journal of Chemical Physics</em> <strong>127</strong>, 134309 (2007): <a class="reference external" href="http://dx.doi.org/10.1063/1.2768362">doi link</a> or <a class="reference external" href="https://arxiv.org/abs/0707.3120">arXiv</a></li>
</ol>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p>If you are in a hurry or immediately want to start with the DMRG-CASPT2 calculations, you can also use the precreated checkpoints from the folder <code class="docutils literal"><span class="pre">/apps/gent/tutorials/DMRG/</span></code>:</p>
<div class="last highlight-bash"><div class="highlight"><pre><span></span>$ cp /apps/gent/tutorials/DMRG/CheMPS2_CASSCF.h5.singlet scf_singlet/.
$ cp /apps/gent/tutorials/DMRG/CheMPS2_CASSCF.h5.triplet scf_triplet/.
</pre></div>
</div>
</div>
</div>
<div class="section" id="dmrg-caspt2">
<h2>12.6. DMRG-CASPT2<a class="headerlink" href="#dmrg-caspt2" title="Permalink to this headline">¶</a></h2>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p>DMRG-CASPT2 checkpoints can be used when you kill a DMRG-CASPT2 calculation before it is finished, or to redo the DMRG-CASPT2 calculation with another IPEA or IMAG shift. In case you would like to use checkpoints for the DMRG-CASPT2 calculations, it is important that for subsequent runs <strong>exactly</strong> the same orbitals are used. Therefore, start from the converged DMRG-SCF checkpoint <code class="docutils literal"><span class="pre">CheMPS2_CASSCF.h5</span></code> and do the following things:</p>
<blockquote>
<div><ul class="simple">
<li>Put <code class="docutils literal"><span class="pre">SCF_DIIS_THR</span></code> to <code class="docutils literal"><span class="pre">0.0</span></code></li>
<li>Delete any checkpoints named <code class="docutils literal"><span class="pre">CheMPS2_DIIS.h5</span></code></li>
<li>Switch <code class="docutils literal"><span class="pre">SCF_ACTIVE_SPACE</span></code> to <code class="docutils literal"><span class="pre">I</span></code></li>
</ul>
</div></blockquote>
<p class="last">This ensures that for the subsequent DMRG-CASPT2 runs, <strong>exactly</strong> the orbitals from <code class="docutils literal"><span class="pre">CheMPS2_CASSCF.h5</span></code> are used.</p>
</div>
<p>How large is the singlet-triplet gap with DMRG-CASPT2 when an IPEA shift of 0.0 and an IMAG shift of 0.0 are used? Is it best to use <code class="docutils literal"><span class="pre">A</span></code> or <code class="docutils literal"><span class="pre">P</span></code> for the option <code class="docutils literal"><span class="pre">CASPT2_ORBS</span></code>, and why? In your input file, also switch on the DMRG-CASPT2 checkpoint, because later we will redo the calculation with an IPEA shift of 0.25. Use the same convergence scheme as for the DMRG-SCF calculations.</p>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p>Sometimes a larger virtual dimension can be required for DMRG-CASPT2 as compared to DMRG-SCF, because the excited wavefunctions</p>
<div class="math">
\[\left| sz, \alpha, \beta \right\rangle = \left[ \alpha \left( \hat{E}_{sz} + \hat{E}_{zs} \right) + \beta \right] \left| \Psi_0 \right\rangle\]</div>
<p class="last">are a linear combination over three matrix product states: <span class="math">\(\left| \Psi_0 \right\rangle\)</span>, <span class="math">\(\hat{E}_{sz} \left| \Psi_0 \right\rangle\)</span>, and <span class="math">\(\hat{E}_{zs} \left| \Psi_0 \right\rangle\)</span>. In practice, you should therefore check how the DMRG-CASPT2 second order energy in <a class="reference external" href="https://github.com/sebwouters/chemps2">chemps2</a> varies with <span class="math">\(D_{SU(2)}\)</span>!</p>
</div>
<p>When you have created the input files, you can double check with the solution for the <a class="reference internal" href="#eigth-ptr-solution"><span class="std std-ref">singlet</span></a> and the <a class="reference internal" href="#nineth-ptr-solution"><span class="std std-ref">triplet</span></a>.</p>
<p>Run the calculations, but please remember to copy over the converged DMRG-SCF orbitals:</p>
<div class="highlight-bash"><div class="highlight"><pre><span></span>$ <span class="nb">cd</span> pt2_singlet/
$ cp ../scf_singlet/CheMPS2_CASSCF.h5 .
$ <span class="nv">OMP_NUM_THREADS</span><span class="o">=</span><span class="m">4</span> chemps2 --file<span class="o">=</span>pt2_singlet.in <span class="p">&</span>> pt2_singlet.out <span class="p">&</span>
$ <span class="nb">cd</span> ../pt2_triplet/
$ cp ../scf_triplet/CheMPS2_CASSCF.h5 .
$ <span class="nv">OMP_NUM_THREADS</span><span class="o">=</span><span class="m">4</span> chemps2 --file<span class="o">=</span>pt2_triplet.in <span class="p">&</span>> pt2_triplet.out <span class="p">&</span>
$ <span class="nb">cd</span> ../
$ tail -n <span class="m">300</span> pt2_singlet/pt2_singlet.out
$ tail -n <span class="m">300</span> pt2_triplet/pt2_triplet.out
</pre></div>
</div>
<p>How large is the singlet-triplet gap with DMRG-CASPT2 when an IPEA shift of 0.0 and an IMAG shift of 0.0 are used?</p>
<p>And with an IPEA shift of 0.25 and an IMAG shift of 0.0?</p>
<p>You can double check with the <a class="reference internal" href="#tenth-ptr-solution"><span class="std std-ref">solution</span></a>.</p>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p>You will see</p>
<div class="highlight-bash"><div class="highlight"><pre><span></span>CheMPS2::DMRG::Symm4RDM<span class="o">(</span> X , Y <span class="o">)</span> : Elapsed wall <span class="nb">time</span> <span class="o">=</span> Z seconds.
</pre></div>
</div>
<p>appear in the output, with X and Y integers, and Z a floating point number. An estimate for the total wall time for the contraction of the 4-RDM with the Fock matrix is <span class="math">\(\frac{18 (18 + 1)}{2} Z\)</span> seconds.</p>
<p><strong>So this last exercise is homework!</strong></p>
<p>Compile <a class="reference external" href="https://github.com/sebwouters/chemps2">chemps2</a> on your institution’s HPC (or ask your admin or Sebastian to), and submit a non-interactive job for the DMRG-CASPT2 calculations.</p>
<p class="last">Yes, I have tricked you into using <a class="reference external" href="https://github.com/sebwouters/chemps2">chemps2</a> in the future!</p>
</div>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p>You can also find the precreated DMRG-CASPT2 checkpoints and the corresponding output in the folder <code class="docutils literal"><span class="pre">/apps/gent/tutorials/DMRG/</span></code>:</p>
<div class="last highlight-bash"><div class="highlight"><pre><span></span>$ cp /apps/gent/tutorials/DMRG/CheMPS2_f4rdm.h5.singlet pt2_singlet/.
$ cp /apps/gent/tutorials/DMRG/CheMPS2_MPS0.h5.singlet pt2_singlet/.
$ less /apps/gent/tutorials/DMRG/pt2_singlet.out.0.0
$ less /apps/gent/tutorials/DMRG/pt2_singlet.out.0.25
$ cp /apps/gent/tutorials/DMRG/CheMPS2_f4rdm.h5.triplet pt2_triplet/.
$ cp /apps/gent/tutorials/DMRG/CheMPS2_MPS0.h5.triplet pt2_triplet/.
$ less /apps/gent/tutorials/DMRG/pt2_triplet.out.0.0
$ less /apps/gent/tutorials/DMRG/pt2_triplet.out.0.25
</pre></div>
</div>
</div>
<p>To study an example DMRG-CASPT2 output during this workshop, perform a small active space calculation. For example:</p>
<div class="highlight-bash"><div class="highlight"><pre><span></span>$ mkdir small_caspt2/
$ <span class="nb">cd</span> small_caspt2/
$ wget <span class="s1">'https://github.com/sebwouters/chemps2/raw/master/tests/matrixelements/N2.CCPVDZ.FCIDUMP'</span>
$ wget <span class="s1">'https://github.com/sebwouters/chemps2/raw/master/tests/test14.input'</span>
$ sed -i <span class="s2">"s/\/path\/to/./"</span> test14.input
$ sed -i <span class="s2">"s/\/tmp/\/local\/NUMBER.master15.delcatty.gent.vsc\//"</span> test14.input
$ cat test14.input
$ chemps2 --file<span class="o">=</span>test14.input <span class="p">&</span>> test14.output <span class="p">&</span>
$ tail -n <span class="m">3000</span> -f test14.output
</pre></div>
</div>
<p>Do you know the difference between the diagonal, non-variational, and variational second order perturbation energies? How is the reference weight calculated and what does it mean?</p>
<div class="highlight-bash"><div class="highlight"><pre><span></span>$ grep <span class="s2">"E2"</span> test14.output
$ grep <span class="s2">"Reference weight"</span> test14.output
</pre></div>
</div>
</div>
<div class="section" id="solutions">
<h2>12.7. Solutions<a class="headerlink" href="#solutions" title="Permalink to this headline">¶</a></h2>
<div class="section" id="ci-input-orbs-in">
<span id="first-ptr-solution"></span><h3>12.7.1. ci_input_orbs.in<a class="headerlink" href="#ci-input-orbs-in" title="Permalink to this headline">¶</a></h3>
<div class="highlight-bash"><div class="highlight"><pre><span></span><span class="nv">FCIDUMP</span> <span class="o">=</span> /path/to/TETRACENE.FCIDUMP
<span class="nv">GROUP</span> <span class="o">=</span> 3
<span class="nv">MULTIPLICITY</span> <span class="o">=</span> 1
<span class="nv">NELECTRONS</span> <span class="o">=</span> 120
<span class="nv">IRREP</span> <span class="o">=</span> 0
<span class="nv">EXCITATION</span> <span class="o">=</span> 0
<span class="nv">SWEEP_STATES</span> <span class="o">=</span> 200, 400, 600, 600, 400, 200
<span class="nv">SWEEP_ENERGY_CONV</span> <span class="o">=</span> 1e-6, 1e-6, 1e-6, 1e-8, 1e-8, 1e-8
<span class="nv">SWEEP_MAX_SWEEPS</span> <span class="o">=</span> 10, 10, 10, 3, 3, 3
<span class="nv">SWEEP_NOISE_PREFAC</span> <span class="o">=</span> 0.05, 0.05, 0.05, 0.0, 0.0, 0.0
<span class="nv">SWEEP_DVDSON_RTOL</span> <span class="o">=</span> 1e-5, 1e-5, 1e-5, 1e-5, 1e-5, 1e-5
<span class="nv">NOCC</span> <span class="o">=</span> 51, 0
<span class="nv">NACT</span> <span class="o">=</span> 0, 18
<span class="nv">NVIR</span> <span class="o">=</span> 171, 54
<span class="nv">SCF_STATE_AVG</span> <span class="o">=</span> FALSE
<span class="nv">SCF_DIIS_THR</span> <span class="o">=</span> 0.0
<span class="nv">SCF_GRAD_THR</span> <span class="o">=</span> 1e-6
<span class="nv">SCF_MAX_ITER</span> <span class="o">=</span> 1
<span class="nv">SCF_ACTIVE_SPACE</span> <span class="o">=</span> I
<span class="nv">SCF_MOLDEN</span> <span class="o">=</span> /path/to/tetracene.molden
<span class="nv">CASPT2_CALC</span> <span class="o">=</span> FALSE
<span class="nv">CASPT2_ORBS</span> <span class="o">=</span> A
<span class="nv">CASPT2_IPEA</span> <span class="o">=</span> 0.0
<span class="nv">CASPT2_IMAG</span> <span class="o">=</span> 0.0
<span class="nv">CASPT2_CHECKPT</span> <span class="o">=</span> FALSE
<span class="nv">CASPT2_CUMUL</span> <span class="o">=</span> FALSE
<span class="nv">PRINT_CORR</span> <span class="o">=</span> TRUE
<span class="nv">TMP_FOLDER</span> <span class="o">=</span> /local/NUMBER.master15.delcatty.gent.vsc/
</pre></div>
</div>
</div>
<div class="section" id="ci-local-orbs-in">
<span id="second-ptr-solution"></span><h3>12.7.2. ci_local_orbs.in<a class="headerlink" href="#ci-local-orbs-in" title="Permalink to this headline">¶</a></h3>
<p>Difference with <a class="reference internal" href="#first-ptr-solution"><span class="std std-ref">input orbitals</span></a>:</p>
<div class="highlight-bash"><div class="highlight"><pre><span></span><span class="nv">SCF_ACTIVE_SPACE</span> <span class="o">=</span> L
</pre></div>
</div>
</div>
<div class="section" id="tetracene-molden-rotated-for-the-localized-active-space-orbitals">
<span id="third-ptr-solution"></span><h3>12.7.3. tetracene.molden.rotated for the localized active space orbitals<a class="headerlink" href="#tetracene-molden-rotated-for-the-localized-active-space-orbitals" title="Permalink to this headline">¶</a></h3>
<img alt="_images/handson_orbitals.png" src="_images/handson_orbitals.png" />
<p>For <code class="docutils literal"><span class="pre">ci_local_orbs/tetracene.molden.rotated</span></code>, the active space orbitals are localized on the carbon atoms, and are ordered according to the one-dimensional topology of the molecule.</p>
</div>
<div class="section" id="molecular-vs-localized-orbitals">
<span id="fourth-ptr-solution"></span><h3>12.7.4. Molecular vs. localized orbitals<a class="headerlink" href="#molecular-vs-localized-orbitals" title="Permalink to this headline">¶</a></h3>
<img alt="_images/handson_comparison.png" src="_images/handson_comparison.png" />
</div>
<div class="section" id="scf-singlet-in">
<span id="fifth-ptr-solution"></span><h3>12.7.5. scf_singlet.in<a class="headerlink" href="#scf-singlet-in" title="Permalink to this headline">¶</a></h3>
<div class="highlight-bash"><div class="highlight"><pre><span></span><span class="nv">FCIDUMP</span> <span class="o">=</span> /path/to/TETRACENE.FCIDUMP
<span class="nv">GROUP</span> <span class="o">=</span> 3
<span class="nv">MULTIPLICITY</span> <span class="o">=</span> 1
<span class="nv">NELECTRONS</span> <span class="o">=</span> 120
<span class="nv">IRREP</span> <span class="o">=</span> 0
<span class="nv">EXCITATION</span> <span class="o">=</span> 0
<span class="nv">SWEEP_STATES</span> <span class="o">=</span> 250, 500, 750
<span class="nv">SWEEP_ENERGY_CONV</span> <span class="o">=</span> 1e-6, 1e-8, 1e-10
<span class="nv">SWEEP_MAX_SWEEPS</span> <span class="o">=</span> 8, 8, 8
<span class="nv">SWEEP_NOISE_PREFAC</span> <span class="o">=</span> 0.05, 0.05, 0.0
<span class="nv">SWEEP_DVDSON_RTOL</span> <span class="o">=</span> 1e-5, 1e-5, 1e-8
<span class="nv">NOCC</span> <span class="o">=</span> 51, 0
<span class="nv">NACT</span> <span class="o">=</span> 0, 18
<span class="nv">NVIR</span> <span class="o">=</span> 171, 54
<span class="nv">SCF_STATE_AVG</span> <span class="o">=</span> FALSE
<span class="nv">SCF_DIIS_THR</span> <span class="o">=</span> 1e-2
<span class="nv">SCF_GRAD_THR</span> <span class="o">=</span> 1e-6
<span class="nv">SCF_MAX_ITER</span> <span class="o">=</span> 100
<span class="nv">SCF_ACTIVE_SPACE</span> <span class="o">=</span> L
<span class="nv">CASPT2_CALC</span> <span class="o">=</span> FALSE
<span class="nv">CASPT2_ORBS</span> <span class="o">=</span> A
<span class="nv">CASPT2_IPEA</span> <span class="o">=</span> 0.0
<span class="nv">CASPT2_IMAG</span> <span class="o">=</span> 0.0
<span class="nv">CASPT2_CHECKPT</span> <span class="o">=</span> FALSE
<span class="nv">CASPT2_CUMUL</span> <span class="o">=</span> FALSE
<span class="nv">PRINT_CORR</span> <span class="o">=</span> TRUE
<span class="nv">TMP_FOLDER</span> <span class="o">=</span> /local/NUMBER.master15.delcatty.gent.vsc/
</pre></div>
</div>
</div>
<div class="section" id="scf-triplet-in">
<span id="sixth-ptr-solution"></span><h3>12.7.6. scf_triplet.in<a class="headerlink" href="#scf-triplet-in" title="Permalink to this headline">¶</a></h3>
<p>Difference with <a class="reference internal" href="#fifth-ptr-solution"><span class="std std-ref">singlet</span></a>:</p>
<div class="highlight-bash"><div class="highlight"><pre><span></span><span class="nv">MULTIPLICITY</span> <span class="o">=</span> 3
</pre></div>
</div>
</div>
<div class="section" id="dmrg-scf-singlet-triplet-gap">
<span id="seventh-ptr-solution"></span><h3>12.7.7. DMRG-SCF singlet-triplet gap<a class="headerlink" href="#dmrg-scf-singlet-triplet-gap" title="Permalink to this headline">¶</a></h3>
<p>Both DMRG-SCF calculations are converged with 8 macro-iterations. The gap is</p>
<div class="math">
\[\Delta E = E_{triplet} - E_{singlet} = -688.803387 - (-688.867150) ~ E_{h} = 63.763 ~ mE_{h} = 40.012 ~ kcal/mol\]</div>
</div>
<div class="section" id="pt2-singlet-in">
<span id="eigth-ptr-solution"></span><h3>12.7.8. pt2_singlet.in<a class="headerlink" href="#pt2-singlet-in" title="Permalink to this headline">¶</a></h3>
<div class="highlight-bash"><div class="highlight"><pre><span></span><span class="nv">FCIDUMP</span> <span class="o">=</span> /path/to/TETRACENE.FCIDUMP
<span class="nv">GROUP</span> <span class="o">=</span> 3
<span class="nv">MULTIPLICITY</span> <span class="o">=</span> 1
<span class="nv">NELECTRONS</span> <span class="o">=</span> 120
<span class="nv">IRREP</span> <span class="o">=</span> 0
<span class="nv">EXCITATION</span> <span class="o">=</span> 0
<span class="nv">SWEEP_STATES</span> <span class="o">=</span> 250, 500, 750
<span class="nv">SWEEP_ENERGY_CONV</span> <span class="o">=</span> 1e-6, 1e-8, 1e-10
<span class="nv">SWEEP_MAX_SWEEPS</span> <span class="o">=</span> 8, 8, 8
<span class="nv">SWEEP_NOISE_PREFAC</span> <span class="o">=</span> 0.05, 0.05, 0.0
<span class="nv">SWEEP_DVDSON_RTOL</span> <span class="o">=</span> 1e-5, 1e-5, 1e-8
<span class="nv">NOCC</span> <span class="o">=</span> 51, 0
<span class="nv">NACT</span> <span class="o">=</span> 0, 18
<span class="nv">NVIR</span> <span class="o">=</span> 171, 54
<span class="nv">SCF_STATE_AVG</span> <span class="o">=</span> FALSE
<span class="nv">SCF_DIIS_THR</span> <span class="o">=</span> 0.0
<span class="nv">SCF_GRAD_THR</span> <span class="o">=</span> 1e-6
<span class="nv">SCF_MAX_ITER</span> <span class="o">=</span> 100
<span class="nv">SCF_ACTIVE_SPACE</span> <span class="o">=</span> I
<span class="nv">CASPT2_CALC</span> <span class="o">=</span> TRUE
<span class="nv">CASPT2_ORBS</span> <span class="o">=</span> A
<span class="nv">CASPT2_IPEA</span> <span class="o">=</span> 0.0
<span class="nv">CASPT2_IMAG</span> <span class="o">=</span> 0.0
<span class="nv">CASPT2_CHECKPT</span> <span class="o">=</span> TRUE
<span class="nv">CASPT2_CUMUL</span> <span class="o">=</span> FALSE
<span class="nv">PRINT_CORR</span> <span class="o">=</span> TRUE
<span class="nv">TMP_FOLDER</span> <span class="o">=</span> /local/NUMBER.master15.delcatty.gent.vsc/
</pre></div>
</div>
</div>
<div class="section" id="pt2-triplet-in">
<span id="nineth-ptr-solution"></span><h3>12.7.9. pt2_triplet.in<a class="headerlink" href="#pt2-triplet-in" title="Permalink to this headline">¶</a></h3>
<p>Difference with <a class="reference internal" href="#eigth-ptr-solution"><span class="std std-ref">singlet</span></a>:</p>
<div class="highlight-bash"><div class="highlight"><pre><span></span><span class="nv">MULTIPLICITY</span> <span class="o">=</span> 3
</pre></div>
</div>
</div>
<div class="section" id="dmrg-caspt2-singlet-triplet-gaps">
<span id="tenth-ptr-solution"></span><h3>12.7.10. DMRG-CASPT2 singlet-triplet gaps<a class="headerlink" href="#dmrg-caspt2-singlet-triplet-gaps" title="Permalink to this headline">¶</a></h3>
<p>For IPEA = 0.0 a.u. the gap is</p>
<div class="math">
\[\Delta E = E_{triplet} - E_{singlet} = -690.945663 - (-691.000608) ~ E_{h} = 54.945 ~ mE_{h} = 34.479 ~ kcal/mol\]</div>
<p>and for IPEA = 0.25 a.u. the gap is</p>
<div class="math">
\[\Delta E = E_{triplet} - E_{singlet} = -690.923604 - (-690.987560) ~ E_{h} = 63.956 ~ mE_{h} = 40.133 ~ kcal/mol\]</div>
</div>
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<h3><a href="index.html">Table Of Contents</a></h3>
<ul>
<li><a class="reference internal" href="#">12. DMRG workshop (12-jul-2016): hands-on session</a><ul>
<li><a class="reference internal" href="#introduction">12.1. Introduction</a></li>
<li><a class="reference internal" href="#ugent-hpc">12.2. UGent HPC</a></li>
<li><a class="reference internal" href="#fcidump-and-molden">12.3. <code class="docutils literal"><span class="pre">FCIDUMP</span></code> and <code class="docutils literal"><span class="pre">MOLDEN</span></code></a></li>
<li><a class="reference internal" href="#basis-choice">12.4. Basis choice</a></li>
<li><a class="reference internal" href="#dmrg-scf">12.5. DMRG-SCF</a></li>
<li><a class="reference internal" href="#dmrg-caspt2">12.6. DMRG-CASPT2</a></li>
<li><a class="reference internal" href="#solutions">12.7. Solutions</a><ul>
<li><a class="reference internal" href="#ci-input-orbs-in">12.7.1. ci_input_orbs.in</a></li>
<li><a class="reference internal" href="#ci-local-orbs-in">12.7.2. ci_local_orbs.in</a></li>
<li><a class="reference internal" href="#tetracene-molden-rotated-for-the-localized-active-space-orbitals">12.7.3. tetracene.molden.rotated for the localized active space orbitals</a></li>
<li><a class="reference internal" href="#molecular-vs-localized-orbitals">12.7.4. Molecular vs. localized orbitals</a></li>
<li><a class="reference internal" href="#scf-singlet-in">12.7.5. scf_singlet.in</a></li>
<li><a class="reference internal" href="#scf-triplet-in">12.7.6. scf_triplet.in</a></li>
<li><a class="reference internal" href="#dmrg-scf-singlet-triplet-gap">12.7.7. DMRG-SCF singlet-triplet gap</a></li>
<li><a class="reference internal" href="#pt2-singlet-in">12.7.8. pt2_singlet.in</a></li>
<li><a class="reference internal" href="#pt2-triplet-in">12.7.9. pt2_triplet.in</a></li>
<li><a class="reference internal" href="#dmrg-caspt2-singlet-triplet-gaps">12.7.10. DMRG-CASPT2 singlet-triplet gaps</a></li>
</ul>
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