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  <div class="section" id="internally-contracted-caspt2">
<span id="index-0"></span><h1>10. Internally contracted CASPT2<a class="headerlink" href="#internally-contracted-caspt2" title="Permalink to this headline"></a></h1>
<div class="section" id="the-required-4-rdm-elements">
<h2>10.1. The required 4-RDM elements<a class="headerlink" href="#the-required-4-rdm-elements" title="Permalink to this headline"></a></h2>
<p>As DMRG is a wavefunction method, higher order reduced density matrices can be evaluated.
In CheMPS2, up to the 3-RDM is implemented with the usual sweep algorithm.
For internally contracted CASPT2, the contraction of the 4-RDM with the generalized Fock operator is needed. The generalized Fock operator</p>
<div class="math">
\[\hat{F} &amp; = &amp; \sum\limits_{pq} F_{pq} \hat{E}_{pq}\]</div>
<p>has matrix elements</p>
<div class="math">
\[F_{pq}  =  \frac{1}{2} \sum\limits_{\sigma} \left\langle \hat{a}_{p\sigma} \left[ \hat{H}, \hat{a}_{q \sigma}^{\dagger} \right] - \hat{a}_{p\sigma}^{\dagger} \left[ \hat{H}, \hat{a}_{q \sigma} \right]  \right\rangle  =  t_{pq} + \sum\limits_{rs} \left\langle \hat{E}_{rs} \right\rangle \left( \left( pq | rs \right) - \frac{1}{2} \left( pr | qs \right) \right).\]</div>
<p>These matrix elements are symmetric and diagonal in the spatial orbital irreps. The required contraction of the 4-RDM with the generalized Fock operator</p>
<div class="math">
\[\left( F . \Gamma^4 \right)_{pqr; wxy} &amp; = &amp; \sum_{\sigma \tau \upsilon} \left\langle \hat{a}^{\dagger}_{p \sigma} \hat{a}^{\dagger}_{q \tau} \hat{a}^{\dagger}_{r \upsilon} \hat{F} \hat{a}_{y \upsilon} \hat{a}_{ x \tau} \hat{a}_{w \sigma} \right\rangle\]</div>
<p>can therefore be obtained from excited wavefunctions, formed by symmetry-conserving single-particle excitations op top of the reference wavefunction. For the matrix element <span class="math">\(F_{sz} = F_{zs}\)</span> the following sum of 4-RDM elements is needed:</p>
<div class="math">
\[\Gamma^4_{pqrs; wxyz} + \Gamma^4_{pqrz; wxys}.\]</div>
<p>It can be obtained by calculating</p>
<div class="math">
\[\left| sz, \alpha, \beta \right\rangle = \left[ \alpha \left( \hat{E}_{sz} + {E}_{zs} \right) + \beta \right]  \left| \Psi_0 \right\rangle\]</div>
<p>in which the spatial orbitals <span class="math">\(s\)</span> and <span class="math">\(z\)</span> have equal irreps. This excited wavefunction is decomposed into an MPS, with a sweep algorithm with negligible cost <a class="reference internal" href="#caspt2" id="id1">[CASPT2]</a>. Denote the 3-RDM of the (unnormalized) excited wavefunctions as</p>
<div class="math">
\[\Gamma(sz,\alpha,\beta)^3_{pqr; wxy} = \sum\limits_{ \sigma \tau \upsilon } \left\langle sz, \alpha, \beta \mid \hat{a}_{p\sigma}^{\dagger} \hat{a}_{q\tau}^{\dagger} \hat{a}_{r\upsilon}^{\dagger} \hat{a}_{y\upsilon} \hat{a}_{x\tau} \hat{a}_{w\sigma} \mid sz, \alpha, \beta \right\rangle.\]</div>
<p>In this notation <span class="math">\(\Gamma( sz, 0, 1 )^3_{pqr; wxy} = \Gamma^3_{pqr; wxy}\)</span>, the 3-RDM of the reference wavefunction. The following identity holds:</p>
<div class="math">
\[\begin{split}&amp; &amp; 2 \left[ \Gamma^4_{pqrs; wxyz} + \Gamma^4_{pqrz; wxys} \right] + \Gamma^3_{pqr; wxy} \\
&amp; = &amp; \Gamma( sz, 1, 1 )^3_{pqr; wxy} - \Gamma( sz, 1, 0 )^3_{pqr; wxy} \\
&amp; - &amp; \delta_{s,p} \Gamma^3_{zqr; wxy} - \delta_{s,q} \Gamma^3_{pzr; wxy} - \delta_{s,r} \Gamma^3_{pqz; wxy} \\
&amp; - &amp; \delta_{s,w} \Gamma^3_{pqr; zxy} - \delta_{s,x} \Gamma^3_{pqr; wzy} - \delta_{s,y} \Gamma^3_{pqr; wxz} \\
&amp; - &amp; \delta_{z,p} \Gamma^3_{sqr; wxy} - \delta_{z,q} \Gamma^3_{psr; wxy} - \delta_{z,r} \Gamma^3_{pqs; wxy} \\
&amp; - &amp; \delta_{z,w} \Gamma^3_{pqr; sxy} - \delta_{z,x} \Gamma^3_{pqr; wsy} - \delta_{z,y} \Gamma^3_{pqr; wxs},\end{split}\]</div>
<p>which allows to obtain the contraction <span class="math">\(\left( F . \Gamma^4 \right)\)</span> by calculating the 3-RDM of several excited wavefunctions! This algorithm has in practice the same computational cost as the regular 4-RDM evaluation during the usual sweep algorithm.</p>
</div>
<div class="section" id="cas-perturbation-theory">
<h2>10.2. CAS perturbation theory<a class="headerlink" href="#cas-perturbation-theory" title="Permalink to this headline"></a></h2>
<p>The full Hilbert space <span class="math">\(\mathcal{H}\)</span> is split up into four parts <a class="reference internal" href="#roos1" id="id2">[ROOS1]</a> <a class="reference internal" href="#roos2" id="id3">[ROOS2]</a>:</p>
<div class="math">
\[\mathcal{H} = \mathcal{V}_0 \oplus \mathcal{V}_{\text{K}} \oplus \mathcal{V}_{\text{SD}} \oplus \mathcal{V}_{\text{TQ..}}.\]</div>
<ol class="arabic">
<li><p class="first"><span class="math">\(\mathcal{V}_0\)</span> contains only the CASSCF solution <span class="math">\(\left| \Psi_0 \right\rangle\)</span>.</p>
</li>
<li><p class="first"><span class="math">\(\mathcal{V}_{\text{K}}\)</span> is the space spanned by all possible active space excitations on top of <span class="math">\(\left| \Psi_0 \right\rangle\)</span> which are orthogonal to <span class="math">\(\mathcal{V}_0\)</span>. Wavefunctions in <span class="math">\(\mathcal{V}_{\text{K}}\)</span> have the same core and virtual orbitals as <span class="math">\(\left| \Psi_0 \right\rangle\)</span>, with the same occupation.</p>
</li>
<li><p class="first"><span class="math">\(\mathcal{V}_{{\text{SD}}}\)</span> contains all single and double particle excitations on top of <span class="math">\(\left| \Psi_0 \right\rangle\)</span> which are orthogonal to <span class="math">\(\mathcal{V}_0 \oplus \mathcal{V}_{\text{K}}\)</span>. With the indices <span class="math">\(ij\)</span> for core orbitals, <span class="math">\(tuv\)</span> for active orbitals, and <span class="math">\(ab\)</span> for virtual orbitals, <span class="math">\(\mathcal{V}_{{\text{SD}}}\)</span> is spanned by the following excitation types:</p>
<blockquote>
<div><div class="math">
\[\begin{split}\text{A} &amp; : &amp; \quad \hat{E}_{ti} \hat{E}_{uv} \left| \Psi_0 \right\rangle, \\
\text{B} &amp; : &amp; \quad \hat{E}_{ti} \hat{E}_{uj} \left| \Psi_0 \right\rangle, \\
\text{C} &amp; : &amp; \quad \hat{E}_{at} \hat{E}_{uv} \left| \Psi_0 \right\rangle, \\
\text{D} &amp; : &amp; \quad \hat{E}_{ai} \hat{E}_{tu} \left| \Psi_0 \right\rangle,~\hat{E}_{ti}\hat{E}_{au} \left| \Psi_0 \right\rangle, \\
\text{E} &amp; : &amp; \quad \hat{E}_{ti} \hat{E}_{aj} \left| \Psi_0 \right\rangle, \\
\text{F} &amp; : &amp; \quad \hat{E}_{at} \hat{E}_{bu} \left| \Psi_0 \right\rangle, \\
\text{G} &amp; : &amp; \quad \hat{E}_{ai} \hat{E}_{bt} \left| \Psi_0 \right\rangle, \\
\text{H} &amp; : &amp; \quad \hat{E}_{ai} \hat{E}_{bj} \left| \Psi_0 \right\rangle.\end{split}\]</div>
</div></blockquote>
</li>
<li><p class="first">And <span class="math">\(\mathcal{V}_{{\text{TQ..}}}\)</span> is the remainder of <span class="math">\(\mathcal{H}\)</span>.</p>
</li>
</ol>
<p>The zeroth order Hamiltonian for internally contracted CASPT2 is</p>
<div class="math">
\[\hat{H}_0 = \hat{P}_0 \hat{F} \hat{P}_0 + \hat{P}_{\text{K}} \hat{F} \hat{P}_{\text{K}} + \hat{P}_{{\text{SD}}} \hat{F} \hat{P}_{{\text{SD}}} + \hat{P}_{{\text{TQ..}}} \hat{F} \hat{P}_{{\text{TQ..}}},\]</div>
<p>where <span class="math">\(\hat{P}_{\text{X}}\)</span> is the projector onto <span class="math">\(\mathcal{V}_{\text{X}}\)</span>.
The first order wavefunction <span class="math">\(\left| \Psi_1 \right\rangle\)</span> for internally contracted CASPT2 is spanned by a linear combination over <span class="math">\(\mathcal{V}_{\text{SD}}\)</span>:</p>
<div class="math">
\[\left| \Psi_1 \right\rangle = \sum_{pq;rs \in \mathcal{V}_{\text{SD}}} C_{pq;rs} \hat{E}_{pq} \hat{E}_{rs} \left| \Psi_0 \right\rangle = \sum_{pq;rs \in \mathcal{V}_{\text{SD}}} C_{pq;rs} \left| \Psi_{pq;rs} \right\rangle.\]</div>
<p>The coefficients can be found by solving</p>
<div class="math">
\[\sum_{pq;rs \in \mathcal{V}_{\text{SD}}} \left\langle \Psi_{wx;yz} \mid \hat{H}_0 - E_0 \mid \Psi_{pq;rs} \right\rangle C_{pq;rs} = - \left\langle \Psi_{wx;yz} \mid \hat{H} \mid \Psi_0 \right\rangle.\]</div>
<p>The overlap matrix <span class="math">\(\left\langle \Psi_{wx;yz} \mid \Psi_{pq;rs} \right\rangle\)</span> is block-diagonal in the different excitation types (A to H). It is diagonalized, small eigenvalues are discarded, and the linear equation is transformed to</p>
<div class="math">
\[\sum\limits_{ \beta } \left( \mathcal{F}_{\alpha\beta} - E_0 \delta_{\alpha,\beta} \right) \mathcal{C}_{\beta} = - \mathcal{V}_{\alpha},\]</div>
<p>with <span class="math">\(\mathcal{F}\)</span> diagonal for two excitations of the same type (A to H). The following initial guess is used to solve this linear equation with either the conjugate gradient or Davidson algorithm:</p>
<div class="math">
\[\mathcal{C}_{\alpha}^{\text{ini}} = - \frac{ \mathcal{V}_{\alpha} }{ \mathcal{F}_{\alpha\alpha} - E_0 }.\]</div>
<p>If the active space orbitals in the DMRG algorithm are not pseudocanonical, <span class="math">\(\Gamma^1\)</span>, <span class="math">\(\Gamma^2\)</span>, <span class="math">\(\Gamma^3\)</span>, and <span class="math">\(\left(F.\Gamma^4\right)\)</span> are rotated to the pseudocanonical orbital basis before building the required intermediates to solve the CASPT2 linear equation.</p>
<p>In order to mitigate intruder state problems, CheMPS2 allows to specify an imaginary level shift <a class="reference internal" href="#imag" id="id4">[IMAG]</a> and/or ionization potential - electron affinity shift <a class="reference internal" href="#ipea" id="id5">[IPEA]</a>. For the latter, the left-hand side matrix of the CASPT2 linear equation is shifted with</p>
<div class="math">
\[\left\langle \Psi_{ wx;yz } \mid \hat{F} \mid \Psi_{pq;rs } \right\rangle \mathrel{+}= \delta_{p,w} \delta_{q,x} \delta_{r,y} \delta_{s,z} \frac{ \epsilon^{\text{IPEA}}}{2} \left\langle \Psi_{wx;yz} \mid \Psi_{pq;rs} \right\rangle \left( 4 + \left\langle \hat{E}_{pp} \right\rangle - \left\langle \hat{E}_{qq} \right\rangle + \left\langle \hat{E}_{rr} \right\rangle - \left\langle \hat{E}_{ss} \right\rangle \right).\]</div>
</div>
<div class="section" id="caspt2-calculations">
<h2>10.3. CASPT2 calculations<a class="headerlink" href="#caspt2-calculations" title="Permalink to this headline"></a></h2>
<p>In order to calculate the CASPT2 variational second order perturbation correction energy, the following call should be made:</p>
<div class="highlight-c++"><div class="highlight"><pre><span></span><span class="kt">double</span> <span class="n">CheMPS2</span><span class="o">::</span><span class="n">CASSCF</span><span class="o">::</span><span class="n">caspt2</span><span class="p">(</span> <span class="k">const</span> <span class="kt">int</span> <span class="n">Nelectrons</span><span class="p">,</span> <span class="k">const</span> <span class="kt">int</span> <span class="n">TwoS</span><span class="p">,</span> <span class="k">const</span> <span class="kt">int</span> <span class="n">Irrep</span><span class="p">,</span> <span class="n">ConvergenceScheme</span> <span class="o">*</span> <span class="n">OptScheme</span><span class="p">,</span> <span class="k">const</span> <span class="kt">int</span> <span class="n">rootNum</span><span class="p">,</span> <span class="n">DMRGSCFoptions</span> <span class="o">*</span> <span class="n">scf_options</span><span class="p">,</span> <span class="k">const</span> <span class="kt">double</span> <span class="n">IPEA</span><span class="p">,</span> <span class="k">const</span> <span class="kt">double</span> <span class="n">IMAG</span><span class="p">,</span> <span class="k">const</span> <span class="kt">bool</span> <span class="n">PSEUDOCANONICAL</span><span class="p">,</span> <span class="k">const</span> <span class="kt">bool</span> <span class="n">CHECKPOINT</span><span class="p">,</span> <span class="k">const</span> <span class="kt">bool</span> <span class="n">CUMULANT</span> <span class="p">)</span>
</pre></div>
</div>
<p><strong>after</strong> the CASSCF orbital rotations are converged.</p>
<ol class="arabic">
<li><p class="first">The first six parameters are the same as for <code class="docutils literal"><span class="pre">CheMPS2::CASSCF::solve</span></code> in <a class="reference internal" href="dmrgscfcalcs.html#label-casscf-calculations-api"><span class="std std-ref">CheMPS2::CASSCF</span></a>.</p>
</li>
<li><p class="first">IPEA and IMAG are the ionization potential - electron affinity and imaginary level shifts.</p>
</li>
<li><p class="first">PSEUDOCANONICAL allows to change the converged CASSCF orbitals (localized, natural, ...) to pseudocanonical orbitals <strong>before</strong> the DMRG calculation to obtain the contraction of the 4-RDM with the generalized Fock operator. This has the advantage that the Fock operator matrix elements are diagonal, which leads to a significant reduction in computational cost <strong>if</strong> it not requires a significantly larger virtual dimension <span class="math">\(D\)</span>.</p>
</li>
<li><p class="first">CHECKPOINT allows to switch on the creation of checkpoints to calculate the required contraction during multiple runs. If CHECKPOINT is true, then after the initial run</p>
<blockquote>
<div><div class="highlight-c++"><div class="highlight"><pre><span></span><span class="n">scf_options</span><span class="o">-&gt;</span><span class="n">setDoDIIS</span><span class="p">(</span> <span class="nb">false</span> <span class="p">)</span>
<span class="n">scf_options</span><span class="o">-&gt;</span><span class="n">setWhichActiveSpace</span><span class="p">(</span> <span class="mi">0</span> <span class="p">)</span>
</pre></div>
</div>
</div></blockquote>
<p>should be set in order to use exactly the same orbitals in consecutive runs!</p>
</li>
<li><p class="first">It is advised to leave CUMULANT = false.</p>
</li>
</ol>
<table class="docutils citation" frame="void" id="caspt2" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id1">[CASPT2]</a></td><td><ol class="first last upperalpha simple" start="19">
<li>Wouters, V. Van Speybroeck and D. Van Neck, <em>Journal of Chemical Physics</em> <strong>145</strong>, 054120 (2016), doi: <a class="reference external" href="http://dx.doi.org/10.1063/1.4959817">10.1063/1.4959817</a></li>
</ol>
</td></tr>
</tbody>
</table>
<table class="docutils citation" frame="void" id="roos1" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id2">[ROOS1]</a></td><td><ol class="first last upperalpha simple" start="11">
<li>Andersson, P.-A. Malmqvist, B.O. Roos, A.J. Sadlej and K. Wolinski, <em>Journal of Physical Chemistry</em> <strong>94</strong>, 5483-5488 (1990). doi: <a class="reference external" href="http://dx.doi.org/10.1021/j100377a012">10.1021/j100377a012</a></li>
</ol>
</td></tr>
</tbody>
</table>
<table class="docutils citation" frame="void" id="roos2" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id3">[ROOS2]</a></td><td><ol class="first last upperalpha simple" start="11">
<li>Andersson, P.‐A. Malmqvist and B.O. Roos, <em>Journal of Chemical Physics</em> <strong>96</strong>, 1218-1226 (1992). doi: <a class="reference external" href="http://dx.doi.org/10.1063/1.462209">10.1063/1.462209</a></li>
</ol>
</td></tr>
</tbody>
</table>
<table class="docutils citation" frame="void" id="imag" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id4">[IMAG]</a></td><td><ol class="first last upperalpha simple" start="14">
<li>Forsberg and P.-A. Malmqvist, <em>Chemical Physics Letters</em> <strong>274</strong>, 196-204 (1997). doi: <a class="reference external" href="http://dx.doi.org/10.1016/S0009-2614(97)00669-6">10.1016/S0009-2614(97)00669-6</a></li>
</ol>
</td></tr>
</tbody>
</table>
<table class="docutils citation" frame="void" id="ipea" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id5">[IPEA]</a></td><td><ol class="first last upperalpha simple" start="7">
<li>Ghigo, B.O. Roos and P.-A. Malmqvist, <em>Chemical Physics Letters</em> <strong>396</strong>, 142-149 (2004). doi: <a class="reference external" href="http://dx.doi.org/10.1016/j.cplett.2004.08.032">10.1016/j.cplett.2004.08.032</a></li>
</ol>
</td></tr>
</tbody>
</table>
</div>
</div>


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  <h3><a href="index.html">Table Of Contents</a></h3>
  <ul>
<li><a class="reference internal" href="#">10. Internally contracted CASPT2</a><ul>
<li><a class="reference internal" href="#the-required-4-rdm-elements">10.1. The required 4-RDM elements</a></li>
<li><a class="reference internal" href="#cas-perturbation-theory">10.2. CAS perturbation theory</a></li>
<li><a class="reference internal" href="#caspt2-calculations">10.3. CASPT2 calculations</a></li>
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