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<p>Hello,</p>
<p>I have written a subroutine that computes the forces from force
constants as a function of atomic displacements from their
equilibrium positions as part of a TDEP implementation here:
<a class="moz-txt-link-freetext" href="https://github.com/anharmonic/d3q/blob/main/thermal2/harmonic.f90">https://github.com/anharmonic/d3q/blob/main/thermal2/harmonic.f90</a></p>
<p><br>
</p>
<p>It is a bit more complicated because it computes the force on
atoms in a supercell n×m×l starting from force constants for the
unit cell from a q-point grid n×m×l, which requires a bit of
mapping to decide which atom is which. I.e. the force acting on
atom i in cell R along direction x is </p>
<p>f_ixR = - \sum_jyR' F(ix0; jyR'-R) u_ijR'</p>
<p>Where u_jyR' is the displacement in Bohr f the atom y in cell R'
from it equilibrium position along direction y. One has to take
into account that the force constants are only stored with R=0, so
on has to use translational invariance to get the other elements.</p>
<p><br>
</p>
<p>You can find more information in Appendix A of this manuscript:
<a class="moz-txt-link-freetext" href="https://people.impmc.fr/lpaulatto/HDR.pdf">https://people.impmc.fr/lpaulatto/HDR.pdf</a><br>
</p>
<p><br>
</p>
<p>You could also probably find something similar, but in python, as
part of the SSCHA package <a class="moz-txt-link-freetext" href="https://sscha.eu/">https://sscha.eu/</a> (or it could also be
in Fortran, I'm not sure what they have re-implemented when moving
to python)<br>
</p>
<p><br>
</p>
<p>hth</p>
<p><br>
</p>
<p><br>
</p>
<div class="moz-cite-prefix">On 11/05/2025 19:29, Chirantan Pramanik
wrote:<br>
</div>
<blockquote type="cite"
cite="mid:VI1P191MB02865AD8553940FD76E122B8B994A@VI1P191MB0286.EURP191.PROD.OUTLOOK.COM">
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Dear Dr. Lorenzo,</div>
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<br>
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Thank you for your reply and the code. I am not familiar with
Fortran. I need to calculate the total force constant of the
atom numbered 205 (Mg). I can understand that each pair of atoms
has a force constant matrix with 9 elements, and for my case,
for one supercell, there are a total of {(205*2)*9}=3690 values
of matrix elements only related to the 205<sup>th</sup> atom. I
need the addition of all of these from each pair of root mean
square values. Is there any direct way or any calculation in QE
itself to find this?<br>
<br>
Thanks,</div>
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Chirantan</div>
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<span
style="font-family: garamond, "times new roman", serif;">Chirantan
Pramanik<br>
Postdoctoral Researcher<br>
Dept. of Earth and Planetary Sciences<br>
Weizmann Institute of Science<br>
Rehovot, Israel</span><br>
</div>
</div>
<hr style="display:inline-block;width:98%" tabindex="-1">
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style="font-size:11pt" color="#000000"><b>From:</b> users
<a class="moz-txt-link-rfc2396E" href="mailto:users-bounces@lists.quantum-espresso.org"><users-bounces@lists.quantum-espresso.org></a> on behalf of
Lorenzo Paulatto <a class="moz-txt-link-rfc2396E" href="mailto:lorenzo.paulatto@cnrs.fr"><lorenzo.paulatto@cnrs.fr></a><br>
<b>Sent:</b> Tuesday, May 6, 2025 11:09 AM<br>
<b>To:</b> <a class="moz-txt-link-abbreviated" href="mailto:users@lists.quantum-espresso.org">users@lists.quantum-espresso.org</a>
<a class="moz-txt-link-rfc2396E" href="mailto:users@lists.quantum-espresso.org"><users@lists.quantum-espresso.org></a><br>
<b>Subject:</b> Re: [QE-users] Format of the interatomic force
constant .fc file</font>
<div> </div>
</div>
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<div class="x_moz-cite-prefix">On 05/05/2025 12:52, Chirantan
Pramanik wrote:<br>
</div>
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Dear All, </div>
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I am stuck with the format of the interatomic force constant
file obtained by the q2r.x code. Below is a preview of the
.fc file. Please help me to understand the entries. I need
the interatomic force constant, suppose, only for the atom
'205'.</div>
</blockquote>
<p>Hello,</p>
<p>please note that force constants always act between two
atoms, assuming you know what you are doing, here is the
subroutine that reads the force constants stripped of all the
parallel distribution and ssafety checks, it is only a couple
dozen liens of code. The subroutine reads the force constants
for a supercell of size nr1×nr2×nr3, because of translational
invariance, the first atom (i,na) is always in the first unit
cell (i.e. that with coordinates 0,0,0) while the second atom
(j,nb) is in cell (nr1-1)*a1 + (nr2-1)*a2 + (nr3-1)*a3. (Or
maybe it is the second atom which is in cell 0,0,0, I never
remember).</p>
<p>Because in your case the cell is 1×1×1, it won't make any
difference.</p>
<p><br>
</p>
<p>hth<br>
</p>
<p><br>
</p>
<br>
<p><br>
</p>
<div
style="color:#333333; background-color:#ffffff; font-family:'Nimbus Mono L','Droid Sans Mono','monospace',monospace,'Droid Sans Fallback','Droid Sans Mono','monospace',monospace; font-weight:normal; font-size:14px; line-height:19px; white-space:pre">
<div><span style="color:#008000">!-----------------------------------------------------------------------</span></div>
<div><span style="color:#0000ff">SUBROUTINE</span><span
style="color:#333333"> </span>
<span style="color:#795e26">readfc</span><span style="color:#333333"> ( </span><span
style="color:#001080">flfrc</span><span style="color:#333333">,
</span><span style="color:#001080">nr1</span><span style="color:#333333">, </span>
<span style="color:#001080">nr2</span><span style="color:#333333">, </span><span
style="color:#001080">nr3</span><span style="color:#333333">,
</span><span style="color:#001080">epsil</span><span
style="color:#333333">, </span>
<span style="color:#001080">nat</span><span style="color:#333333">, </span><span
style="color:#000000">&</span></div>
<div><span style="color:#333333"></span><span style="color:#001080">ibrav</span><span
style="color:#333333">,
</span><span style="color:#001080">alat</span><span
style="color:#333333">, </span>
<span style="color:#001080">at</span><span style="color:#333333">, </span><span
style="color:#001080">ntyp</span><span style="color:#333333">,
</span><span style="color:#001080">amass</span><span
style="color:#333333">, </span>
<span style="color:#001080">omega</span><span style="color:#333333">, </span><span
style="color:#000000">&</span></div>
<div><span style="color:#333333"></span><span style="color:#001080">has_zstar</span><span
style="color:#333333">,
</span><span style="color:#001080">alph</span><span
style="color:#333333">, </span>
<span style="color:#001080">read_lr</span><span style="color:#333333"> )</span></div>
<div><span style="color:#333333"></span><span style="color:#008000">!-----------------------------------------------------------------------</span></div>
<div><span style="color:#333333"></span><span style="color:#008000">!</span></div>
<div><span style="color:#333333"></span><span style="color:#0000ff">IMPLICIT</span><span
style="color:#333333">
</span><span style="color:#0000ff">NONE</span></div>
<div><span style="color:#333333"></span><span style="color:#008000">! I/O variable</span></div>
<div><span style="color:#333333"></span><span style="color:#0000ff">CHARACTER</span><span
style="color:#333333">(</span><span style="color:#001080">LEN</span><span
style="color:#000000">=</span><span style="color:#098658">256</span><span
style="color:#333333">)
</span><span style="color:#000000">::</span><span style="color:#333333"> flfrc</span></div>
<div><span style="color:#333333"></span><span style="color:#0000ff">CHARACTER</span><span
style="color:#333333">(</span><span style="color:#001080">LEN</span><span
style="color:#000000">=</span><span style="color:#098658">80</span><span
style="color:#333333">)
</span><span style="color:#000000">::</span><span style="color:#333333"> line</span></div>
<div><span style="color:#333333"></span><span style="color:#0000ff">INTEGER</span><span
style="color:#333333">
</span><span style="color:#000000">::</span><span style="color:#333333"> ibrav, nr1,nr2,nr3,nat, ntyp</span></div>
<div><span style="color:#333333"></span><span style="color:#0000ff">REAL</span><span
style="color:#333333">(DP)
</span><span style="color:#000000">::</span><span style="color:#333333"> alat, at(</span><span
style="color:#098658">3</span><span style="color:#333333">,</span><span
style="color:#098658">3</span><span style="color:#333333">), epsil(</span><span
style="color:#098658">3</span><span style="color:#333333">,</span><span
style="color:#098658">3</span><span style="color:#333333">),
alph</span></div>
<div><span style="color:#333333"></span><span style="color:#0000ff">LOGICAL</span><span
style="color:#333333">
</span><span style="color:#000000">::</span><span style="color:#333333"> has_zstar, read_lr</span></div>
<div><span style="color:#333333"></span><span style="color:#008000">! local variables</span></div>
<div><span style="color:#333333"></span><span style="color:#0000ff">INTEGER</span><span
style="color:#333333">
</span><span style="color:#000000">::</span><span style="color:#333333"> i, j, na, nb, m1,m2,m3</span></div>
<div><span style="color:#333333"></span><span style="color:#0000ff">INTEGER</span><span
style="color:#333333">
</span><span style="color:#000000">::</span><span style="color:#333333"> ios, ibid, jbid, nabid, nbbid, m1bid,m2bid,m3bid</span></div>
<div><span style="color:#333333"></span><span style="color:#0000ff">REAL</span><span
style="color:#333333">(DP)
</span><span style="color:#000000">::</span><span style="color:#333333"> amass(ntyp), amass_from_file, omega</span></div>
<div><span style="color:#333333"></span><span style="color:#0000ff">INTEGER</span><span
style="color:#333333">
</span><span style="color:#000000">::</span><span style="color:#333333"> nt</span></div>
<div><span style="color:#333333"></span><span style="color:#008000">!</span></div>
<div>
<div
style="color:#333333; background-color:#ffffff; font-family:'Nimbus Mono L','Droid Sans Mono','monospace',monospace,'Droid Sans Fallback','Droid Sans Mono','monospace',monospace; font-weight:normal; font-size:14px; line-height:19px; white-space:pre">
<div><span style="color:#af00db">OPEN</span><span style="color:#333333"> (</span><span
style="color:#001080">unit</span><span style="color:#000000">=</span><span
style="color:#098658">1</span><span style="color:#333333">,</span><span
style="color:#001080">file</span><span style="color:#000000">=</span><span
style="color:#333333">flfrc,</span><span style="color:#001080">status</span><span
style="color:#000000">=</span><span style="color:#a31515">'old'</span><span
style="color:#333333">,</span><span style="color:#001080">form</span><span
style="color:#000000">=</span><span style="color:#a31515">'formatted'</span><span
style="color:#333333">)</span></div>
<div><span style="color:#333333">!</span></div>
<div><span style="color:#333333"></span><span style="color:#008000">! Read cell parameters</span><span
style="color:#333333"></span></div>
</div>
</div>
<div><span style="color:#333333"></span><span style="color:#af00db">READ</span><span
style="color:#333333">(</span><span style="color:#098658">1</span><span
style="color:#333333">,</span><span style="color:#000000">*</span><span
style="color:#333333">) ntyp,nat,ibrav,(celldm(i),i</span><span
style="color:#000000">=</span><span style="color:#098658">1</span><span
style="color:#333333">,</span><span style="color:#098658">6</span><span
style="color:#333333">)</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">if</span><span
style="color:#333333"> (ibrav</span><span style="color:#0000ff">==</span><span
style="color:#098658">0</span><span style="color:#333333">)
</span><span style="color:#af00db">then</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">read</span><span
style="color:#333333">(</span><span style="color:#098658">1</span><span
style="color:#333333">,</span><span style="color:#000000">*</span><span
style="color:#333333">) ((at(i,j),i</span><span
style="color:#000000">=</span><span style="color:#098658">1</span><span
style="color:#333333">,</span><span style="color:#098658">3</span><span
style="color:#333333">),</span><span style="color:#001080">j</span><span
style="color:#000000">=</span><span style="color:#098658">1</span><span
style="color:#333333">,</span><span style="color:#098658">3</span><span
style="color:#333333">)</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">end if</span></div>
<div><span style="color:#333333"></span><span style="color:#008000">!</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">CALL</span><span
style="color:#333333">
</span><span style="color:#795e26">latgen</span><span
style="color:#333333">(ibrav,celldm,at(</span><span
style="color:#098658">1</span><span style="color:#333333">,</span><span
style="color:#098658">1</span><span style="color:#333333">),at(</span><span
style="color:#098658">1</span><span style="color:#333333">,</span><span
style="color:#098658">2</span><span style="color:#333333">),at(</span><span
style="color:#098658">1</span><span style="color:#333333">,</span><span
style="color:#098658">3</span><span style="color:#333333">),omega)</span></div>
<div><span style="color:#333333">alat </span><span style="color:#000000">=</span><span
style="color:#333333"> celldm(</span><span style="color:#098658">1</span><span
style="color:#333333">)</span></div>
<div><span style="color:#333333">at </span><span style="color:#000000">=</span><span
style="color:#333333"> at / alat
</span><span style="color:#008000">! bring at in units of alat</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">CALL</span><span
style="color:#333333">
</span><span style="color:#795e26">volume</span><span
style="color:#333333">(alat,at(</span><span
style="color:#098658">1</span><span style="color:#333333">,</span><span
style="color:#098658">1</span><span style="color:#333333">),at(</span><span
style="color:#098658">1</span><span style="color:#333333">,</span><span
style="color:#098658">2</span><span style="color:#333333">),at(</span><span
style="color:#098658">1</span><span style="color:#333333">,</span><span
style="color:#098658">3</span><span style="color:#333333">),omega)</span></div>
<div><span style="color:#333333"></span><span style="color:#008000">!</span></div>
<div><span style="color:#333333"></span><span style="color:#008000">! read atomic types, positions and masses</span></div>
<div><span style="color:#333333"></span><span style="color:#008000">!</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">ALLOCATE</span><span
style="color:#333333"> (atm(ntyp))</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">DO</span><span
style="color:#333333"> nt
</span><span style="color:#000000">=</span><span style="color:#333333"> </span><span
style="color:#098658">1</span><span style="color:#333333">,ntyp</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">READ</span><span
style="color:#333333">(</span><span style="color:#098658">1</span><span
style="color:#333333">,</span><span style="color:#000000">*</span><span
style="color:#333333">) i,atm(nt),amass_from_file
! type and mass</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">END DO</span></div>
<div><span style="color:#333333"></span><span style="color:#008000">!</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">ALLOCATE</span><span
style="color:#333333"> (tau(</span><span style="color:#098658">3</span><span
style="color:#333333">,nat), ityp(nat), zeu(</span><span
style="color:#098658">3</span><span style="color:#333333">,</span><span
style="color:#098658">3</span><span style="color:#333333">,nat))</span></div>
<div><span style="color:#333333"></span><span style="color:#008000">!</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">DO</span><span
style="color:#333333"> na</span><span style="color:#000000">=</span><span
style="color:#098658">1</span><span style="color:#333333">,nat</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">READ</span><span
style="color:#333333">(</span><span style="color:#098658">1</span><span
style="color:#333333">,</span><span style="color:#000000">*</span><span
style="color:#333333">) i,ityp(na),(tau(j,na),j</span><span
style="color:#000000">=</span><span style="color:#098658">1</span><span
style="color:#333333">,</span><span style="color:#098658">3</span><span
style="color:#333333">)
! positions of atoms in the cell</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">END DO</span></div>
<div><span style="color:#333333"></span><span style="color:#008000">!</span></div>
<div><span style="color:#333333"></span><span style="color:#008000">! read macroscopic variables</span></div>
<div><span style="color:#008000">! (dielectric constant and effective charges) </span>
</div>
<div><span style="color:#333333"></span><span style="color:#008000">!</span></div>
<div><span style="color:#333333">alph </span><span style="color:#000000">=</span><span
style="color:#333333">
</span><span style="color:#098658">1.0_dp</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">READ</span><span
style="color:#333333"> (</span><span style="color:#098658">1</span><span
style="color:#333333">,</span><span style="color:#a31515">'(a)'</span><span
style="color:#333333">)
line</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">READ</span><span
style="color:#333333">(line,</span><span style="color:#000000">*</span><span
style="color:#333333">,</span><span style="color:#001080">iostat</span><span
style="color:#000000">=</span><span style="color:#333333">ios)
has_zstar, alph</span></div>
<div><span style="color:#333333"></span><span style="color:#008000">!</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">IF</span><span
style="color:#333333"> (has_zstar)
</span><span style="color:#af00db">THEN</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">READ</span><span
style="color:#333333">(</span><span style="color:#098658">1</span><span
style="color:#333333">,</span><span style="color:#000000">*</span><span
style="color:#333333">) ((epsil(i,j),j</span><span
style="color:#000000">=</span><span style="color:#098658">1</span><span
style="color:#333333">,</span><span style="color:#098658">3</span><span
style="color:#333333">),</span><span style="color:#001080">i</span><span
style="color:#000000">=</span><span style="color:#098658">1</span><span
style="color:#333333">,</span><span style="color:#098658">3</span><span
style="color:#333333">)</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">DO</span><span
style="color:#333333"> na</span><span style="color:#000000">=</span><span
style="color:#098658">1</span><span style="color:#333333">,nat</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">READ</span><span
style="color:#333333">(</span><span style="color:#098658">1</span><span
style="color:#333333">,</span><span style="color:#000000">*</span><span
style="color:#333333">)</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">READ</span><span
style="color:#333333">(</span><span style="color:#098658">1</span><span
style="color:#333333">,</span><span style="color:#000000">*</span><span
style="color:#333333">) ((zeu(i,j,na),j</span><span
style="color:#000000">=</span><span style="color:#098658">1</span><span
style="color:#333333">,</span><span style="color:#098658">3</span><span
style="color:#333333">),</span><span style="color:#001080">i</span><span
style="color:#000000">=</span><span style="color:#098658">1</span><span
style="color:#333333">,</span><span style="color:#098658">3</span><span
style="color:#333333">)</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">END DO</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">ELSE</span></div>
<div><span style="color:#333333">zeu (:,:,:) </span><span
style="color:#000000">=</span><span style="color:#333333">
</span><span style="color:#098658">0.d0</span></div>
<div><span style="color:#333333">epsil(:,:) </span><span
style="color:#000000">=</span><span style="color:#333333">
</span><span style="color:#098658">0.d0</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">END IF</span></div>
<div><span style="color:#333333"></span><span style="color:#008000">!</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">READ</span><span
style="color:#333333"> (</span><span style="color:#098658">1</span><span
style="color:#333333">,</span><span style="color:#000000">*</span><span
style="color:#333333">) nr1,nr2,nr3</span></div>
<div><span style="color:#333333"></span><span style="color:#008000">!</span></div>
<div><span style="color:#333333"></span><span style="color:#008000">! read real-space interatomic force constants</span></div>
<div><span style="color:#333333"></span><span style="color:#008000">!</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">ALLOCATE</span><span
style="color:#333333"> ( frc(nr1,nr2,nr3,</span><span
style="color:#098658">3</span><span style="color:#333333">,</span><span
style="color:#098658">3</span><span style="color:#333333">,nat,nat)
)</span></div>
<div><span style="color:#333333">frc(:,:,:,:,:,:,:) </span><span
style="color:#000000">=</span><span style="color:#333333">
</span><span style="color:#098658">0.d0</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">ALLOCATE</span><span
style="color:#333333"> ( frc_lr(nr1,nr2,nr3,</span><span
style="color:#098658">3</span><span style="color:#333333">,</span><span
style="color:#098658">3</span><span style="color:#333333">,nat,nat)
)</span></div>
<div><span style="color:#333333">frc_lr(:,:,:,:,:,:,:) </span><span
style="color:#000000">=</span><span style="color:#333333">
</span><span style="color:#098658">0.d0</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">DO</span><span
style="color:#333333"> i</span><span style="color:#000000">=</span><span
style="color:#098658">1</span><span style="color:#333333">,</span><span
style="color:#098658">3</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">DO</span><span
style="color:#333333"> j</span><span style="color:#000000">=</span><span
style="color:#098658">1</span><span style="color:#333333">,</span><span
style="color:#098658">3</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">DO</span><span
style="color:#333333"> na</span><span style="color:#000000">=</span><span
style="color:#098658">1</span><span style="color:#333333">,nat</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">DO</span><span
style="color:#333333"> nb</span><span style="color:#000000">=</span><span
style="color:#098658">1</span><span style="color:#333333">,nat</span></div>
<div><span style="color:#333333"></span><span style="color:#333333"></span><span
style="color:#af00db">READ</span><span style="color:#333333"> (</span><span
style="color:#098658">1</span><span style="color:#333333">,</span><span
style="color:#000000">*</span><span style="color:#333333">)
ibid, jbid, nabid, nbbid</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">READ</span><span
style="color:#333333"> (</span><span style="color:#098658">1</span><span
style="color:#333333">,</span><span style="color:#000000">*</span><span
style="color:#333333">) (((m1bid,
m2bid, m3bid, </span><span style="color:#000000">&</span></div>
<div><span style="color:#333333">frc(m1,m2,m3,i,j,na,nb), </span><span
style="color:#000000">&</span></div>
<div><span style="color:#333333">m1</span><span style="color:#000000">=</span><span
style="color:#098658">1</span><span style="color:#333333">,nr1),m2</span><span
style="color:#000000">=</span><span style="color:#098658">1</span><span
style="color:#333333">,nr2),</span><span style="color:#001080">m3</span><span
style="color:#000000">=</span><span style="color:#098658">1</span><span
style="color:#333333">,nr3)</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">END IF</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">END DO</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">END DO</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">END DO</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">END DO</span></div>
<div><span style="color:#333333"></span><span style="color:#008000">!</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">CLOSE</span><span
style="color:#333333">(</span><span style="color:#001080">unit</span><span
style="color:#000000">=</span><span style="color:#098658">1</span><span
style="color:#333333">)</span></div>
<div><span style="color:#333333"></span><span style="color:#008000">!</span></div>
<div><span style="color:#333333"></span><span style="color:#af00db">RETURN</span></div>
<div><span style="color:#0000ff">END SUBROUTINE</span><span
style="color:#333333">
</span><span style="color:#795e26">readfc</span></div>
</div>
<p><br>
</p>
<br>
<div class="x_moz-signature">-- <br>
<small>Dr. Lorenzo Paulatto<br>
IdR @ IMPMC - CNRS UMR 7590 & Sorbonne Université<br>
phone: +33 (0)1 442 79822 / skype: paulatz<br>
<a href="http://people.impmc.fr/lpaulatto/"
class="x_moz-txt-link-freetext moz-txt-link-freetext"
moz-do-not-send="true">http://people.impmc.fr/lpaulatto/</a>
-
<a href="https://anharmonic.github.io/"
class="x_moz-txt-link-freetext moz-txt-link-freetext"
moz-do-not-send="true">https://anharmonic.github.io/</a><br>
23-24/423 B115, 4 place Jussieu 75252 Paris CX 05</small></div>
</div>
<br>
<fieldset class="moz-mime-attachment-header"></fieldset>
<pre wrap="" class="moz-quote-pre">_______________________________________________________________________________
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</blockquote>
<div class="moz-signature">-- <br>
<small>Dr. Lorenzo Paulatto<br>
IdR @ IMPMC - CNRS UMR 7590 & Sorbonne Université<br>
phone: +33 (0)1 442 79822 / telegram: lpaulatto<br>
<a href="http://www.impmc.upmc.fr/~paulatto/"
class="moz-txt-link-freetext">http://www.impmc.upmc.fr/~paulatto/</a>
- <a href="https://anharmonic.github.io/"
class="moz-txt-link-freetext">https://anharmonic.github.io/</a><br>
23-24/423 B115, 4 place Jussieu 75252 Paris CX 05<small></small></small></div>
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