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It’s the integral all over kpoints that needs to be correct; at any given kpoint, in a metal, sometimes it will be higher, sometimes lower.
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<div>Nicola</div>
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<blockquote type="cite">On 27 May 2026, at 04:27, 林俊宏 <d10222004@ntu.edu.tw> wrote:<br>
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<p>Dear All,</p>
<p>I performed the spin-polarized SCF calculation of NiO using Fermi-Dirac smearing and focused on its occupation numbers. In the scf output file, the total number of electrons is 32. But the sum of occupation numbers for an arbitrary k-point is not equal to
32 (spin up + spin down); in fact, it exceeds it. Why the sum is not equal to the total number of electrons?</p>
<p>Following shows the truncated output:</p>
<p> bravais-lattice index = 0<br>
lattice parameter (alat) = 9.6512 a.u.<br>
unit-cell volume = 244.6666 (a.u.)^3<br>
number of atoms/cell = 4<br>
number of atomic types = 4<br>
number of electrons = 32.00<br>
number of Kohn-Sham states= 20<br>
kinetic-energy cutoff = 40.0000 Ry<br>
charge density cutoff = 160.0000 Ry</p>
<p>...</p>
<p> ------ SPIN UP ------------<br>
...</p>
<div><span> k = 0.0000 0.0000 0.5539 ( 1050 PWs) bands (ev):</span></div>
<div><span> </span></div>
<div><span> -8.0602 -7.1852 4.2164 5.1908 5.3054 6.0558 6.8531 7.1036</span></div>
<div><span> 9.2355 9.8018 9.9846 10.2167 10.2600 10.6732 10.8402 10.8480</span></div>
<div><span> 11.7452 12.2874 16.2553 19.9253</span></div>
<div><span> </span></div>
<div><span> occupation numbers</span></div>
<div><span> 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000</span></div>
<div><span> 0.9998 0.9981 0.9964 0.9915 0.9901 0.9563 0.9222 0.9201</span></div>
<div><span> 0.2986 0.0549 0.0000 0.0000</span></div>
<p> ------ SPIN DOWN ----------<br>
...<span></span></p>
<div><span> k = 0.0000 0.0000 0.5539 ( 1050 PWs) bands (ev):</span></div>
<div><span> </span></div>
<div><span> -8.0602 -7.1852 4.2164 5.1908 5.3055 6.0559 6.8531 7.1036</span></div>
<div><span> 9.2355 9.8019 9.9846 10.2167 10.2601 10.6733 10.8402 10.8480</span></div>
<div><span> 11.7452 12.2874 16.2553 19.9253</span></div>
<div><span> </span></div>
<div><span> occupation numbers</span></div>
<div><span> 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000</span></div>
<div><span> 0.9998 0.9981 0.9964 0.9915 0.9901 0.9563 0.9222 0.9201</span></div>
<div><span> 0.2986 0.0549 0.0000 0.0000</span></div>
<p>...</p>
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<p>The input file used for calculation is at <a href="https://docs.google.com/document/d/1f_hVjxORLTCa8xgCy3AkEkZuR7bnoIftWH1WdsEs5OA/edit?usp=sharing" target="_blank" rel="noopener noreferrer">
https://docs.google.com/document/d/1f_hVjxORLTCa8xgCy3AkEkZuR7bnoIftWH1WdsEs5OA/edit?usp=sharing</a></p>
<p>Best Regards,<br>
Chun Linn</p>
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