chemps2-doc 1.8.3-2 / usr / share / doc / libchemps2 / html / symmetry.html

This file is indexed.

/usr/share/doc/libchemps2/html/symmetry.html is in chemps2-doc 1.8.3-2.

This file is owned by root:root, with mode 0o644.

The actual contents of the file can be viewed below. 

  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">


<html xmlns="http://www.w3.org/1999/xhtml">
  <head>
    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
    
    <title>4. Symmetries &#8212; CheMPS2 1.8.3 (2016-11-15) documentation</title>
    
    <link rel="stylesheet" href="_static/classic.css" type="text/css" />
    <link rel="stylesheet" href="_static/pygments.css" type="text/css" />
    
    <script type="text/javascript">
      var DOCUMENTATION_OPTIONS = {
        URL_ROOT:    './',
        VERSION:     '1.8.3 (2016-11-15)',
        COLLAPSE_INDEX: false,
        FILE_SUFFIX: '.html',
        HAS_SOURCE:  true
      };
    </script>
    <script type="text/javascript" src="/usr/share/javascript/jquery/jquery.js"></script>
    <script type="text/javascript" src="/usr/share/javascript/underscore/underscore.js"></script>
    <script type="text/javascript" src="_static/doctools.js"></script>
    <script type="text/javascript" src="/usr/share/javascript/mathjax/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
    <link rel="index" title="Index" href="genindex.html" />
    <link rel="search" title="Search" href="search.html" />
    <link rel="top" title="CheMPS2 1.8.3 (2016-11-15) documentation" href="index.html" />
    <link rel="next" title="5. CheMPS2::Hamiltonian" href="matrixelements.html" />
    <link rel="prev" title="3. DMRG algorithm" href="method.html" /> 
  </head>
  <body role="document">
    <div class="related" role="navigation" aria-label="related navigation">
      <h3>Navigation</h3>
      <ul>
        <li class="right" style="margin-right: 10px">
          <a href="genindex.html" title="General Index"
             accesskey="I">index</a></li>
        <li class="right" >
          <a href="matrixelements.html" title="5. CheMPS2::Hamiltonian"
             accesskey="N">next</a> |</li>
        <li class="right" >
          <a href="method.html" title="3. DMRG algorithm"
             accesskey="P">previous</a> |</li>
        <li class="nav-item nav-item-0"><a href="index.html">CheMPS2 1.8.3 (2016-11-15) documentation</a> &#187;</li> 
      </ul>
    </div>  

    <div class="document">
      <div class="documentwrapper">
        <div class="bodywrapper">
          <div class="body" role="main">
            
  <div class="section" id="symmetries">
<span id="index-0"></span><h1>4. Symmetries<a class="headerlink" href="#symmetries" title="Permalink to this headline">ΒΆ</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>&#8216;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>
</div>


          </div>
        </div>
      </div>
      <div class="sphinxsidebar" role="navigation" aria-label="main navigation">
        <div class="sphinxsidebarwrapper">
            <p class="logo"><a href="index.html">
              <img class="logo" src="_static/CheMPS2logo.png" alt="Logo"/>
            </a></p>
  <h4>Previous topic</h4>
  <p class="topless"><a href="method.html"
                        title="previous chapter">3. DMRG algorithm</a></p>
  <h4>Next topic</h4>
  <p class="topless"><a href="matrixelements.html"
                        title="next chapter">5. <code class="docutils literal"><span class="pre">CheMPS2::Hamiltonian</span></code></a></p>
  <div role="note" aria-label="source link">
    <h3>This Page</h3>
    <ul class="this-page-menu">
      <li><a href="_sources/symmetry.txt"
            rel="nofollow">Show Source</a></li>
    </ul>
   </div>
<div id="searchbox" style="display: none" role="search">
  <h3>Quick search</h3>
    <form class="search" action="search.html" method="get">
      <div><input type="text" name="q" /></div>
      <div><input type="submit" value="Go" /></div>
      <input type="hidden" name="check_keywords" value="yes" />
      <input type="hidden" name="area" value="default" />
    </form>
</div>
<script type="text/javascript">$('#searchbox').show(0);</script>
        </div>
      </div>
      <div class="clearer"></div>
    </div>
    <div class="related" role="navigation" aria-label="related navigation">
      <h3>Navigation</h3>
      <ul>
        <li class="right" style="margin-right: 10px">
          <a href="genindex.html" title="General Index"
             >index</a></li>
        <li class="right" >
          <a href="matrixelements.html" title="5. CheMPS2::Hamiltonian"
             >next</a> |</li>
        <li class="right" >
          <a href="method.html" title="3. DMRG algorithm"
             >previous</a> |</li>
        <li class="nav-item nav-item-0"><a href="index.html">CheMPS2 1.8.3 (2016-11-15) documentation</a> &#187;</li> 
      </ul>
    </div>
    <div class="footer" role="contentinfo">
        &#169; Copyright 2013-2016, Sebastian Wouters.
      Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.4.9.
    </div>
  </body>
</html>

APT Browse - Built by Thomas Orozco.