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    <p><tt>Dear Phani,</tt></p>
    <p><tt>   there is no Journal reference for the local-TF scheme. <br>
      </tt></p>
    <p><tt>   if one assumes that the error in self-consistency is small
        enough that the linear regime applies then</tt></p>
    <p><tt>   \delta n_inp = \chi \chi_0^-1 \Delta n_I/O     or   
        chi^-1 \delta n_inp = chi_0^-1 \Delta n_I/O<br>
      </tt></p>
    <p><tt>   where \delta n_inp is the error in the input density (the
        error that you want to correct) and \Delta n_I/O = n_inp - n_out
        is the difference between input and output densities (that one
        has access to in the calculation).</tt></p>
    <p><tt>  chi_0 is the non-interacting electron density response
        function of the system and chi the interacting electron one,
        linked to the previous by the Dyson equation \chi = \chi_0 +
        \chi_0 ( v_c + f_xc ) \chi  or, equivalently, </tt><tt><tt>\chi^-1
          = \chi_0^-1 - (v_c + f_xc)  </tt>where v_c is the Coulomb
        kernel and f_xc the exchange-correlation one (neglected in the
        following).<br>
      </tt></p>
    <p><tt>  in the TF context chi_0^-1 =  \partial V_KS/ \partial n = -
        \partial^2 T[n]_TF / \partial n^2  \propto   n(r)^(-1/3)</tt></p>
    <p><tt>  so using this approximation for \chi_0^-1 and iteratively
        solving (with some not too strict threshold) the linear system
        one gets an estimate for \delta n_inp from \Delta n_I/O</tt></p>
    <p><tt>  best</tt></p>
    <p><tt>stefano</tt><br>
    </p>
    <div class="moz-cite-prefix">On 03/04/19 09:04, Phani Motamarri
      wrote:<br>
    </div>
    <blockquote type="cite"
cite="mid:CALJ=y3-XYijXb31__wYRwq1W0HvZfEu9ZEcLb2V2j1K56UZcLw@mail.gmail.com">
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        <div>Hi,</div>
        <div><br>
        </div>
        <div>Many thanks for the reply. I already saw the reference, but
          I am talking of the reference for the mixing scheme which
          applies local-density dependent TF preconditioning. (Key word:
          '<span
style="color:rgb(0,128,0);font-family:monospace;font-size:14px;font-weight:700">local-TF').</span></div>
        <div><span style="font-family:monospace;font-size:14px"><font
              style="" color="#000000">The reference</font></span> D.D.
          Johnson PRB 38, 12807 (1988) does not talk about this local
          T-F.</div>
        <div><br>
        </div>
        <div>Regards</div>
        <div>Phani</div>
        <div><br>
        </div>
        <br>
      </div>
      <br>
      <div class="gmail_quote">
        <div dir="ltr" class="gmail_attr">On Wed, Apr 3, 2019 at 2:54 AM
          Paolo Giannozzi <<a href="mailto:p.giannozzi@gmail.com"
            moz-do-not-send="true">p.giannozzi@gmail.com</a>> wrote:<br>
        </div>
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          0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
          <div dir="ltr">
            <div dir="ltr">
              <div>The reference is in the header of file
                PW/src/mix_rho.f90:</div>
              <div>  ! ... Modified Broyden's method for charge density
                mixing<br>
                  ! ...         D.D. Johnson PRB 38, 12807 (1988)<br>
              </div>
              <div>Paolo<br>
              </div>
            </div>
          </div>
          <br>
          <div class="gmail_quote">
            <div dir="ltr" class="gmail_attr">On Wed, Apr 3, 2019 at
              7:36 AM Phani Motamarri <<a
                href="mailto:phanim@umich.edu" target="_blank"
                moz-do-not-send="true">phanim@umich.edu</a>> wrote:<br>
            </div>
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              <div dir="ltr">
                <div dir="ltr">Hello Quantum Espresso Developers,
                  <div><br>
                  </div>
                  <div>We are developing an in-house real-space DFT code
                    and looking to implement better mixing schemes
                    beyond Anderson and Broyden specifically for
                    in-homogeneous systems. To this end, I came across
                    local-density-dependent Thomas Fermi screening
                    mixing implemented in Quantum expresso which is
                    performing really well for  in-homogeneous systems.
                    I would like to know if you  have any suitable
                    journal article reference for understanding the
                    mixing method implemented in Quantum espresso.</div>
                  <div><br>
                  </div>
                  <div>Looking forward to hearing from you</div>
                  <div><br>
                  </div>
                  <div>Thanks</div>
                  <div>Phani</div>
                </div>
              </div>
              _______________________________________________<br>
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            </blockquote>
          </div>
          <br clear="all">
          <br>
          -- <br>
          <div dir="ltr"
            class="gmail-m_-7903780632649496780gmail_signature">
            <div dir="ltr">
              <div>
                <div dir="ltr">
                  <div>Paolo Giannozzi, Dip. Scienze Matematiche
                    Informatiche e Fisiche,<br>
                    Univ. Udine, via delle Scienze 208, 33100 Udine,
                    Italy<br>
                    Phone +39-0432-558216, fax +39-0432-558222<br>
                    <br>
                  </div>
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