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<div class="elementToProof"><span style="font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">Dear
</span><span style="letter-spacing: normal; font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0); background-color: rgb(255, 255, 255); font-weight: 400;">Eduardo,</span></div>
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</span></div>
<div class="elementToProof"><span style="letter-spacing: normal; font-family: Arial, Helvetica, sans-serif; font-size: 12pt; color: black; font-weight: 400;">> The atom indexes are relative to the atoms in the unit cell and include the neighbor atoms in the
eight surrounding unit cells. </span></div>
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<div class="elementToProof"><span style="letter-spacing: normal; font-family: Arial, Helvetica, sans-serif; font-size: 12pt; color: black; font-weight: 400;">Actually, the pw.x code generates a virtual 3x3x3 supercell with your real unit cell inside of it.
So in total there are 27 unit cells. </span></div>
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<div class="elementToProof"><span style="letter-spacing: normal; font-family: Arial, Helvetica, sans-serif; font-size: 12pt; color: black; font-weight: 400;">> For defect calculations, I need to use a supercell with a different shape. How can I transfer the
parameters to the supercell? I think this just needs a small code to generate </span></div>
<div class="elementToProof" style="text-align: left; margin: 0px;"><span style="letter-spacing: normal; font-family: Arial, Helvetica, sans-serif; font-size: 12pt; color: black; font-weight: 400;">the parameter file for the supercell. I can do it if this is
not available. Is it?</span></div>
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Yes, unfortunately the I and J couple indices will change if you change the shape of the original real cell. The algorithm can be found in PW/src/intersite_V.f90. </div>
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<div class="elementToProof"><span style="font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">You can run the HP code by setting
</span><span style="letter-spacing: normal; font-family: arial, helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0); background-color: rgb(255, 255, 153); font-weight: 700;">determine_num_pert_only</span><span style="font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);"> =
.true. For DFT+U+V, it will only determine the indices of couples without running heavy linear-response calculations. So then you can use this new file with the new indices and add there the U and V values that you previously computed using a smaller cell. </span></div>
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<span style="letter-spacing: normal; font-family: Arial, Helvetica, sans-serif; font-size: 16px; color: rgb(0, 0, 0); background-color: rgb(255, 255, 255); font-weight: 400;">Another practical question. I could refine the calculations recomputing the parameters
for the atoms and pairs close to the defects. How does the computation time scale? Considering that for a ten-atom unit cell, the HP calculation took 2 days with 12 cores, what can I expect for a supercell with 120 atoms?</span><span style="font-family: Arial, Helvetica, sans-serif; font-size: 16px; color: rgb(0, 0, 0);"><br>
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The scaling is cubic w.r.t. the number of atoms. But you can reduce the size of the k and q meshes. So overall it will be much more expensive than for the 10-atoms cell and you would need to use a HPC cluster. </div>
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HTH</div>
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Iurii</div>
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<div><span style="font-family: Cambria, Georgia, serif; font-size: 12pt; color: rgb(102, 102, 102);">Dr. Iurii TIMROV</span></div>
<div><span style="font-family: Cambria, Georgia, serif; font-size: 12pt; color: rgb(102, 102, 102);">Tenure-track scientist</span></div>
<div><span style="font-family: Cambria, Georgia, serif; font-size: 12pt; color: rgb(102, 102, 102);">Laboratory for Materials Simulations (LMS)</span></div>
<div><span style="font-family: Cambria, Georgia, serif; font-size: 12pt; color: rgb(102, 102, 102);">Paul Scherrer Institut (PSI)</span></div>
<div><span style="font-family: Cambria, Georgia, serif; font-size: 12pt; color: rgb(102, 102, 102);">CH-5232 Villigen, Switzerland</span></div>
<div><span style="font-family: Cambria, Georgia, serif; font-size: 12pt; color: rgb(102, 102, 102);">+41 56 310 62 14</span></div>
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<div id="divRplyFwdMsg" dir="ltr"><font face="Calibri, sans-serif" style="font-size:11pt" color="#000000"><b>From:</b> users <users-bounces@lists.quantum-espresso.org> on behalf of EDUARDO ARIEL MENENDEZ PROUPIN <emenendez@us.es><br>
<b>Sent:</b> Friday, January 19, 2024 11:09<br>
<b>To:</b> users@lists.quantum-espresso.org <users@lists.quantum-espresso.org><br>
<b>Subject:</b> Re: [QE-users] Thermodynamics with DFT+U</font>
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Dear Iurii,</div>
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Your explanations were quite useful and wide. I am still reading papers, but in fact I think I may have solved my problem for Fe2O3 with just U(Fe-d). I confirm that for Fe2O3, using the U(O-2p) computed with HP code cause a too large band gap (~4 ev). Using
U(Fe-d) and V(Fe-O) gives a slightly large bangap (2.74 eV vs experimental range 2-2.6 eV). Using just U(Fe-d) I got a gap of 2.43 eV, which is inside the experimental range. Anyway, I still wish to have DFT+U+V as an option. The experimental gap may be corrected
by future measurements, or it may be affected by zero-point motion, or maybe other property may need the V, e.g., magnetic moments.</div>
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Then I have a practical problem. The parameter file, that contains the indexes of every pair of Fe and O atoms, was computed with a unit cell. The atom indexes are relative to the atoms in the unit cell and include the neighbor atoms in the eight surrounding
unit cells. For defect calculations, I need to use a supercell with a different shape. How can I transfer the parameters to the supercell? I think this just needs a small code to generate </div>
<div class="x_elementToProof" style="font-family:Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">
the parameter file for the supercell. I can do it if this is not available. Is it?</div>
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Another practical question. I could refine the calculations recomputing the parameters for the atoms and pairs close to the defects. How does the computation time scale? Considering that for a ten-atom unit cell, the HP calculation took 2 days with 12 cores,
what can I expect for a supercell with 120 atoms?</div>
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<div style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">
Eduardo A. Menéndez Proupin</div>
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Departamento de Física Aplicada I</div>
<div style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">
Universidad de Sevilla</div>
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Teléfono: +34 9554 20231</div>
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<a href="https://personal.us.es/emenendez/" id="OWA74c1c5a4-004e-749b-e56c-f86083909f6e" class="x_OWAAutoLink" data-loopstyle="linkonly">https://personal.us.es/emenendez/</a></div>
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<a href="https://personal.us.es/emenendez/docencia/" id="OWA6834b263-1cae-ef78-3cc6-f5e24bdb3714" class="x_OWAAutoLink" data-loopstyle="linkonly">https://personal.us.es/emenendez/docencia/</a><br>
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<div id="x_divRplyFwdMsg" dir="ltr"><span style="font-family:Calibri,sans-serif; font-size:11pt; color:rgb(0,0,0)"><b>De:</b> Timrov Iurii <iurii.timrov@psi.ch><br>
<b>Enviado:</b> lunes, 11 de diciembre de 2023 12:03<br>
<b>Para:</b> users@lists.quantum-espresso.org <users@lists.quantum-espresso.org><br>
<b>Asunto:</b> Re: [QE-users] Thermodynamics with DFT+U</span>
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<div style="direction:ltr"><span style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">Dear
</span><span style="letter-spacing:normal; font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0); background-color:rgb(255,255,255); font-weight:400">Eduardo</span><span style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">,</span></div>
<div style="direction:ltr"><span style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)"><br>
</span></div>
<div style="direction:ltr"><span style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">Your questions are tricky. There is a lot one can say. Please see my comments below. Maybe someone
else can have a different viewpoint and comment as well.</span></div>
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<span style="letter-spacing:normal; font-family:Arial,Helvetica,sans-serif; font-size:16px; color:rgb(0,0,0); background-color:rgb(255,255,255); font-weight:400">Should we choose one average value, or use the computed value for each system?</span><span style="font-family:Arial,Helvetica,sans-serif; font-size:16px; color:rgb(0,0,0)"><br>
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Both options are used in the literature. From my experience, it is better to use the second one. </div>
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<span style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">In DFT+U with empirical U people often use one value and compare the total energies. Why? One reason is because how would you
choose different U values for different systems (e.g. FM vs AFM)? Maybe this can be done, but it is easier to use one empirical value. And it is claimed that the total energies must be compared with the same U value. But why? Is there a theorem or a proof?
See below for the discussion why I would not use the same U value.</span></li><li style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0); direction:ltr; list-style-type:"- "">
<span style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">In the second case, one uses different U values for different structures, provided that these U value are computed ab initio.
Does this make sense? At least to me, yes. Why? Because different structures require different corrections. And, indeed, if one computes U e.g. for the Co-3d states in LiCoO2 and CoO2, the U values appear to be different. Why? Because the electronic screening
is different, and the magnitude of self-interaction errors is different in LiCoO2 and CoO2. One can make an approximation and use an average U value for these two systems, but why doing so? From our experience using different ab initio U values and comparing
total energies gives results in good agreement with experiments (e.g. voltages for batteries). But we do not have a (mathematical) justification for doing so, as well as we do not have a proof why one should not do it. Hence, at present there is no consensus
in the literature on this topic. More investigations for various systems is needed to see trends. But for me, comparing total energies with different U values obtained from linear-response theory makes sense and it provides reasonable results. </span></li></ul>
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<span style="letter-spacing:normal; font-family:Arial,Helvetica,sans-serif; font-size:16px; color:rgb(0,0,0); background-color:rgb(255,255,255); font-weight:400">Concerning the advantage of self consistency, let me rise the example LiCoO2 that comes with the
HP code. The example produces U for Co and also for O, as well as V(Co-O). U(O-2p)=8.0439 eV. Is this parameter useful? As the example is not converged w.r.t. to k-points and cutoffs the number may change, but U(O-2p) is still there. I read PRB101, 064305
(2020) by Floris et al, and it seems that U(O-2p) is discarded. I am curious why, but I couldn't find a discussion. Maybe there is another article. My point here is that using self consistent parameters for some elements and shells, and discarding others is
just a partial self-consistency.</span><span style="font-family:Arial,Helvetica,sans-serif; font-size:16px; color:rgb(0,0,0)"><br>
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We did not apply the U correction to O-2p states. The question of whether to apply or not the U correction to O-2p is another big question. Many things can be said here, and you will possibly receive different answers from different people. A few comments from
my side:</div>
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<span style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">We generally do not apply U to O-2p, when U is computed from linear-response theory, because it is large (8-9 eV) and from
our experience the accuracy of some properties (e.g. voltages) are worsened. </span></li><li style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0); direction:ltr; list-style-type:"- "">
<span style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">If you use ACBN0 to compute U, you might get 2-3 eV, and applying this correction to O-2p might improve the results. So you
see that it matters which value of U to apply to O-2 states and how it was computed. If one tunes U by hand, then of course you can get whatever you want. E.g. people apply empirical U to O-2p states in ZnO to get the right band gap. But this touches on another
topic: DFT+U for band gaps. U generally improves the band gaps if the correction is applied to the edge states. Have a look at this paper:
<a href="https://www.mdpi.com/2076-3417/11/5/2395" id="OWA67d9f7ef-e9b2-eb3f-e497-f383980ba651" class="x_OWAAutoLink" data-auth="NotApplicable" data-loopstyle="linkonly">
https://www.mdpi.com/2076-3417/11/5/2395</a></span></li><li style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0); direction:ltr; list-style-type:"- "">
<span style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">In some works, even U from linear-response theory is applied to O-2p to get better band gaps.</span></li><li style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0); direction:ltr; list-style-type:"- "">
<span style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">Applying U to O-2p localizes these states more. Is it good or bad? It depends on the system. E.g. in systems with strong covalency,
this is not good as you will kill the hybridization between TM-3d and O-2p states. E.g. in the case of BaTiO3 applying U to O-2p does exactly that and one gets the cubic phase instead of the rhombohedral one, in contradiction to experiments. While not applying
U to O-2p is ok, because the inter-site hybridization is there and the DFT+U+V approach preserves the rhombohedral symmetry:
<a href="https://arxiv.org/abs/2309.04348" id="OWAf998e80b-cc12-9388-ab18-2778e057c27d" class="x_OWAAutoLink" data-auth="NotApplicable" data-loopstyle="linkonly">
https://arxiv.org/abs/2309.04348</a> </span></li></ul>
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<span style="letter-spacing:normal; font-family:Arial,Helvetica,sans-serif; font-size:12pt; color:black; font-weight:400">A related question is whether the forces and energies are consistent with variable U and V. That is, Let us move the Fe impurity atom inside
a crystal, and recompute the U and V for each position. Force is the gradient of energy obtained in the Hellman-Feynman way, I guess with constant U,V.</span><span style="font-family:Arial,Helvetica,sans-serif; font-size:12pt; color:black"><br>
</span></li></ul>
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<span style="letter-spacing:normal; font-family:Arial,Helvetica,sans-serif; font-size:12pt; color:black; font-weight:400">Pressure is the negative of the derivative of the energy with respect to volume, which implies a variation of U and V. I guess the stress
is computed with constant U, V. I think that self-consistency could be implemented, but first we must be sure that comparing energies with variable, self-consistent parameters is correct.</span><span style="font-family:Arial,Helvetica,sans-serif; font-size:12pt; color:black"><br>
</span></li></ul>
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<div style="direction:ltr"><span style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)"><br>
</span></div>
<div style="direction:ltr"><span style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">Another excellent question. In Quantum ESPRESSO, U is constant and its derivative dU/dR is set to
zero when computing Hubbard forces (and same for Hubbard stresses): <a href="https://journals.aps.org/prb/abstract/10.1103/PhysRevB.102.235159" id="OWA09f4c964-b4fe-5ff7-0343-2435add72b44" class="x_OWAAutoLink" data-auth="NotApplicable" data-loopstyle="linkonly">
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.102.235159</a></span></div>
<div style="direction:ltr"><span style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">In order to circumvent this problem, we perform the calculation of U in a self-consistent fashion,
by performing cyclic calculations (recalculation of U and structural optimization with DFT+U), thus pushing the system to the energy extremum:
<a href="https://journals.aps.org/prb/abstract/10.1103/PhysRevB.103.045141" id="OWAe51c5337-3d90-bf1c-8ed5-0e3eb70fdcc8" class="x_OWAAutoLink" data-auth="NotApplicable" data-loopstyle="linkonly">
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.103.045141</a><br>
<br>
</span></div>
<div style="direction:ltr"><span style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">HTH</span></div>
<div style="direction:ltr"><span style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)"><br>
</span></div>
<div style="direction:ltr"><span style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">Greetings,</span></div>
<div style="direction:ltr"><span style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">Iurii</span></div>
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<div><span style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(102,102,102)">----------------------------------------------------------</span></div>
<div><span style="font-family:Cambria,Georgia,serif; font-size:12pt; color:rgb(102,102,102)">Dr. Iurii TIMROV</span></div>
<div><span style="font-family:Cambria,Georgia,serif; font-size:12pt; color:rgb(102,102,102)">Tenure-track scientist</span></div>
<div><span style="font-family:Cambria,Georgia,serif; font-size:12pt; color:rgb(102,102,102)">Laboratory for Materials Simulations (LMS)</span></div>
<div><span style="font-family:Cambria,Georgia,serif; font-size:12pt; color:rgb(102,102,102)">Paul Scherrer Institut (PSI)</span></div>
<div><span style="font-family:Cambria,Georgia,serif; font-size:12pt; color:rgb(102,102,102)">CH-5232 Villigen, Switzerland</span></div>
<div><span style="font-family:Cambria,Georgia,serif; font-size:12pt; color:rgb(102,102,102)">+41 56 310 62 14</span></div>
<div><span style="font-family:Aptos,Aptos_EmbeddedFont,Aptos_MSFontService,Calibri,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)"><a href="https://www.psi.ch/en/lms/people/iurii-timrov" id="OWA908e7075-f2bc-16a9-4561-400fdfaec01b" class="x_OWAAutoLink" data-auth="NotApplicable" data-linkindex="0" data-loopstyle="linkonly">https://www.psi.ch/en/lms/people/iurii-timrov</a></span></div>
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<div id="x_x_divRplyFwdMsg" dir="ltr"><span style="font-family:Calibri,sans-serif; font-size:11pt; color:rgb(0,0,0)"><b>From:</b> users <users-bounces@lists.quantum-espresso.org> on behalf of EDUARDO ARIEL MENENDEZ PROUPIN <emenendez@us.es><br>
<b>Sent:</b> Wednesday, December 6, 2023 10:24<br>
<b>To:</b> users@lists.quantum-espresso.org <users@lists.quantum-espresso.org><br>
<b>Subject:</b> Re: [QE-users] Thermodynamics with DFT+U</span>
<div> </div>
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Hello!</div>
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I have read this thread, which is from three years ago, and I would like to know if there is any update, consensus, or a study about this issue. </div>
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The topic of the thread was how to compare the energies of two systems when there is at least one element subject to Hubbard correction, in the case that the Hubbard parameters are computed self-consistently via the HP code, and have different values in the
two systems compared. Should we choose one average value, or use the computed value for each system? The two systems may be either:</div>
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<span style="font-family:Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">Two phases of a material</span></li><li style="font-family:Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0); direction:ltr; list-style-type:"b) "">
<span style="font-family:Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">Two antiferromagnetic configurations</span></li><li style="font-family:Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0); direction:ltr; list-style-type:"c) "">
<span style="font-family:Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">Crystal with a transition metal impurity vs clean crystal and impurity in bulk metal.<br>
</span></li></ol>
<div style="direction:ltr"><span style="font-family:Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)"><br>
</span></div>
<div style="direction:ltr"><span style="font-family:Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">I may have a case of type (b), with certain energy order when using the self-consistent U values for each AFM configuration, and the opposite order
when the same U is used for both configurations. The same U was computed for one configuration, I am waiting for the queue to finish calculations with the other U, but this is published (Naveas et al,
<a href="https://doi.org/10.1016/j.isci.2023.106033" id="OWAcbb13323-2a58-b27e-3136-0cf874d29e9a" class="x_OWAAutoLink" data-auth="NotApplicable" data-loopstyle="linkonly">
https://doi.org/10.1016/j.isci.2023.106033</a>).</span></div>
<div style="direction:ltr"><span style="font-family:Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)"><br>
</span></div>
<div style="direction:ltr"><span style="font-family:Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">Concerning the advantage of self consistency, let me rise the example LiCoO2 that comes with the HP code. The example produces U for Co and also
for O, as well as V(Co-O). U(O-2p)=8.0439 eV. Is this parameter useful? As the example is not converged w.r.t. to k-points and cutoffs the number may change, but U(O-2p) is still there. I read PRB101, 064305 (2020) by Floris et al, and it seems that U(O-2p)
is discarded. I am curious why, but I couldn't find a discussion. Maybe there is another article. My point here is that using self consistent parameters for some elements and shells, and discarding others is just a partial self-consistency.</span></div>
<div style="direction:ltr"><span style="font-family:Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)"><br>
</span></div>
<div style="direction:ltr"><span style="font-family:Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">A related question is whether the forces and energies are consistent with variable U and V. That is, Let us move the Fe impurity atom inside a
crystal, and recompute the U and V for each position. Force is the gradient of energy obtained in the Hellman-Feynman way, I guess with constant U,V.</span></div>
<div style="direction:ltr"><span style="font-family:Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">Pressure is the negative of the derivative of the energy with respect to volume, which implies a variation of U and V. I guess the stress is computed
with constant U, V. I think that self-consistency could be implemented, but first we must be sure that comparing energies with variable, self-consistent parameters is correct.</span></div>
<div style="direction:ltr"><span style="font-family:Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)"><br>
</span></div>
<div style="direction:ltr"><span style="font-family:Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">Best regards,</span></div>
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</span></div>
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<div style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">
Eduardo A. Menéndez Proupin</div>
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Departamento de Física Aplicada I</div>
<div style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">
Universidad de Sevilla</div>
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Teléfono: +34 9554 20231</div>
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<a href="https://personal.us.es/emenendez/" id="OWAd3d4c599-edb3-51cd-d5d7-c6c27c40a712" class="x_OWAAutoLink" data-auth="NotApplicable" data-loopstyle="linkonly">https://personal.us.es/emenendez/</a></div>
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<a href="https://personal.us.es/emenendez/docencia/" id="OWA79a53eb7-4de9-07fd-9981-8ec0fb705788" class="x_OWAAutoLink" data-auth="NotApplicable" data-loopstyle="linkonly">https://personal.us.es/emenendez/docencia/</a></div>
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