Number of electrons for PARCHG for each k and n is 2?

[yu-chang_chen2]

Dear All,

I am afraid that the number of electrons for each quantum number, i.e., crystal momentum k and band-index n, is always 2 in my atomic chain calculation. It is different from the results of solid state text book. The detail information is as follows,

I want to calculate the PARCHG of each quantum number k and n for one-dimensional Aluminum atomic chain. My POSCAR, INCAR and KPOINTS are as follows,
POSCAR 2.50
1.0000
10.000 0.0000 0.0000
0.0000 10.0000 0.0000
0.0000 0.0000 2.419079874528345
Al
1
Direct
0.5000 0.5000 0.5000

INCAR
RWIGS = 1.402
ENCUT = 241
IALGO = 48
EDIFF = 1E-05
NELM = 999
ISMEAR = 0
SIGMA = 0.05
ISIF = 2
IBRION = -1
NSW = 0
LCHARG = .TRUE.
ISYM = -1

KPOINTS file
0
M
1 1 100
0 0 0

The PAW_PBE Al 04Jan2001 POTCAR was used. Furthermore, the unconventional settings, i.e., ISYM = -1 and 100 k-points are for some purposes. After the scf calculation was finished, I created another directory named PARCHG_1 and copied the POSCAR, KPOINTS, POTCAR and WAVECAR to the directory PARCHG_1. The INCAR was also modified to calculate the partial charge. The new INCAR is as follows,

RWIGS = 1.402
ENCUT = 241
IALGO = 48
EDIFF = 1E-05
NELM = 999
ISMEAR = 0
SIGMA = 0.05
ISIF = 2
IBRION = -1
NSW = 0
ISYM = -1
LPARD = .TRUE.
IBAND = 1
KPUSE = 1 2
LSEPK = .TRUE.
LSEPB = .TRUE.

After the calculations, there are two PARCHG files, i.e., PARCHG.0001.0001 and PARCHG.0001.0002. The number of the real space grids of CHGCAR and PARCHGs are 80*80*20 which was printed after the fractional coordinates in the CHGCAR and PARCHGs. The total number of electrons can be calculated by adding all the raw values of charge density in CHGCAR then dividing by the number of the real space grids (80*80*20). The result is 3.0000000341180426 which is consistent with the number of the valence electrons of Aluminum of the POTCAR. However, when the I did the same procedure for the PARCHG.0001.0001 (n = 1, k = 1) and PARCHG.0001.0002 (n = 1, k = 2); adding all raw charge density values in the PARCHG then divide the values by 80*80*20. The number of electrons are 2.00000091 and 1.99999981 for PARCHG.0001.0001 and PARCHG.0001.0002, respectively. According to the solid state physics, for the spin degenerate case, there are 2 electrons for each band for one-dimensional system, it means that for each band, summation for all k-points (ISYM = -1) will get 2. I did not expect that there are 2 electrons for each quantum number k and n. Would you please let me know which part is incorrect? Thank you very much in advanced.

[martin.schlipf]

Could you clarify what exactly you would like to do and what you expect?

The PARCHG file sets the weights (Fermi and k-point) for the included bands to 1 and thus you would expect 2 electrons for nonpolarized bands, right?

[yu-chang_chen2]

Dear Martin,

Thanks for your quick reply. I need the partial charge number for each quantum number crystal momentum k and band index n. I am sorry that I did not explain my question clearly. Please let me talk in detail.

The k-index (k) and band (n) decomposition charge density \rho_{k,n}(r) can be defined by
\rho_{k,n}(r) = \psi_{k,n}(r) \times \psi^*_{k,n}(r) (1)

where \psi_{k,n}(r) is the Kohn-Sham orbital for the quantum number k and n. The \psi^*_{k,n}(r) is its complex conjugate. The r is the position vector.

The k-index (k) and band (n) decomposition charge number can be defined by
N_{k,n} = \int \rho_{k,n}(r) d^3r (2)

For the degenerate case, summation for all k points (ISYM = -1) in the 1st Brillouin zone for each band will get 2, because each band can fill 2 electrons.
\sum_{n} N{k,n} = 2 (3)

However, in the VASP, for each PARCHG.nb.nk, if I sum for all values in the PARCHG.nb.nk (for example, for PARCHG.0001.0001) then divide by total grid number (NGX*NGY*NGZ), it will get 2. It seems that the result conflics with the Eq. (3). It seems that to get the number of charge for each k and n, I need to sum for all values in the PARCHG.nb.nk, then devide the grid number, then times the weighting (1/100 in my example). Would you please let me know whether it is correct?

Thanks and best wishes,

[martin.schlipf]

It seems that to get the number of charge for each k and n, I need to sum for all values in the PARCHG.nb.nk, then devide the grid number, then times the weighting (1/100 in my example). Would you please let me know whether it is correct?

Yes, the PARCHG files do not include the k point or Fermi weights. If you wanted to reconstruct the charge density from the PARCHG files you need to include those

CHGCAR = sum_nk f_nk w_k PARCHG_nk

where f_nk are the Fermi weights and w_k is the k point weight. In your specific case all k points have the same weight of 1/100 but you would still need to include the Fermi weights which are in the last column here (up to a factor of two)

Code: Select all

 k-point    13 :       0.0000    0.0000    0.1250
  band No.  band energies     occupation
      1      -9.3214      2.00000
      2      -3.3474      1.01837
      3      -3.3474      1.01837
      4       0.1975      0.00000
      5       1.1632      0.00000
      6       1.6677      0.00000
      7       2.1606      0.00000
      8       2.2286      0.00000

[yu-chang_chen2]

Dear Martin,

I appreciate for your helps.

Best regards,