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  <div class="section" id="symmetries">
<span id="index-0"></span><h1>4. Symmetries<a class="headerlink" href="#symmetries" title="Permalink to this headline">¶</a></h1>
<p>CheMPS2 exploits the <span class="math">\(\mathsf{SU(2)}\)</span> spin symmetry, <span class="math">\(\mathsf{U(1)}\)</span> particle number symmetry, and abelian point group symmetries <span class="math">\(\{ \mathsf{C_1}, \mathsf{C_i}, \mathsf{C_2}, \mathsf{C_s}, \mathsf{D_2}, \mathsf{C_{2v}}, \mathsf{C_{2h}}, \mathsf{D_{2h}}  \}\)</span> of ab initio quantum chemistry Hamiltonians. Thereto the orbital occupation and virtual indices have to be represented by states which transform according to a particular row of one of the irreps of the symmetry group of the Hamiltonian. For example for orbital <span class="math">\(k\)</span>:</p>
<div class="math">
\[\begin{split}\left|-\right\rangle &amp; \rightarrow &amp; \left|s = 0;s^z=0;N=0; I=I_0\right\rangle\\
\left|\uparrow\right\rangle &amp; \rightarrow &amp; \left|s = \frac{1}{2};s^z=\frac{1}{2};N=1; I=I_k\right\rangle\\
\left|\downarrow\right\rangle &amp; \rightarrow &amp; \left|s = \frac{1}{2};s^z=-\frac{1}{2};N=1; I=I_k\right\rangle\\
\left|\uparrow\downarrow\right\rangle &amp; \rightarrow &amp; \left|s = 0;s^z=0;N=2; I=I_k \otimes I_k = I_0\right\rangle.\end{split}\]</div>
<p>Then the MPS tensors factorize into Clebsch-Gordan coefficients and reduced tensors due to the Wigner-Eckart theorem:</p>
<div class="math">
\[A[i]^{(ss^zNI)}_{(j_L j_L^z N_L I_L \alpha_L);(j_R j_R^z N_R I_R \alpha_R)} = \left\langle j_L j_L^z s s^z \mid j_R j_R^z \right\rangle \delta_{N_L+N,N_R} \delta_{I_L\otimes I, I_R} T[i]^{(sNI)}_{(j_L N_L I_L \alpha_L);(j_R N_R I_R \alpha_R)}.\]</div>
<p>This has three important consequences:</p>
<ol class="arabic simple">
<li>There is block-sparsity due to the Clebsch-Gordan coefficients. Remember that the Clebsch-Gordan coefficients of abelian groups are Kronecker <span class="math">\(\delta\)</span>‘s. The block-sparsity results in both memory and CPU time savings.</li>
<li>There is information compression for spin symmetry sectors other than singlets, as the tensor <span class="math">\(\mathbf{T[i]}\)</span> does not contain spin projection indices. The virtual dimension associated with <span class="math">\(\mathbf{T[i]}\)</span> is called the reduced virtual dimension <span class="math">\(D_{\mathsf{SU(2)}}\)</span>. This also results in both memory and CPU time savings.</li>
<li>Excited states in different symmetry sectors can be obtained by ground-state calculations.</li>
</ol>
<p>The operators</p>
<div class="math">
\[\begin{split}\hat{b}^{\dagger}_{k\sigma} &amp; = &amp; \hat{a}^{\dagger}_{k\sigma}\\
\hat{b}_{k\sigma} &amp; = &amp; (-1)^{\frac{1}{2}-\sigma}\hat{a}_{k-\sigma}\end{split}\]</div>
<p>of orbital <span class="math">\(c\)</span> transform according to row <span class="math">\((s = \frac{1}{2}; s^z=\sigma; N=\pm 1; I_c)\)</span> of irrep <span class="math">\((s = \frac{1}{2}; N=\pm 1; I_c)\)</span>. <span class="math">\(\hat{b}^{\dagger}\)</span> and <span class="math">\(\hat{b}\)</span> are hence both doublet irreducible tensor operators, and the Wigner-Eckart theorem allows to factorize corresponding matrix elements into Clebsch-Gordan coefficients and reduced matrix elements. Together with the Wigner-Eckart theorem for the MPS tensors, this allows to work with reduced quantities only in CheMPS2. Only Wigner 6-j and 9-j symbols are needed, but never Wigner 3-j symbols or Clebsch-Gordan coefficients.</p>
<p>For more information on the exploitation of symmetry in the DMRG method, please read Ref. <a class="reference internal" href="#symm1" id="id1">[SYMM1]</a>.</p>
<table class="docutils citation" frame="void" id="symm1" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id1">[SYMM1]</a></td><td><ol class="first last upperalpha simple" start="19">
<li>Wouters and D. Van Neck, <em>European Physical Journal D</em> <strong>68</strong>, 272 (2014), doi: <a class="reference external" href="http://dx.doi.org/10.1140/epjd/e2014-50500-1">10.1140/epjd/e2014-50500-1</a></li>
</ol>
</td></tr>
</tbody>
</table>
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