[Wien] Any additional phase factor for the complex spherical harmonics used on wien2k
Yongxin Yao
ykent at iastate.edu
Sun Jul 2 19:06:05 CEST 2017
Dear Peter, developers and users:
Are there any additional phase factors associated with the complex
spherical harmonics (besides the commonly used Condon-Shortley Phase) when
wien2k calculates the local one-body density matrix in SRC_lapwdm, which is
needed, for instance, for LDA+U?
I did some calculation for FeO (struct file attached), where I got the
the following density matrix of Fe-d orbitals using "x lapwdm" (case.indm
attached.) for a lapw-lda calculation.
nks.real in comp_sph_harm basis:
1.2815 0.0000 -0.0000 0.3136 0.0000
0.0000 1.0857 -0.0000 0.0000 -0.3136
-0.0000 -0.0000 1.3175 0.0000 -0.0000
0.3136 0.0000 0.0000 1.0857 -0.0000
0.0000 -0.3136 -0.0000 -0.0000 1.2815
nks.imag in comp_sph_harm basis:
0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000
which are all real numbers up to numerical accuracy not shown.
The puzzle is that this density matrix does not commute with all of the the
local site symmetries, which are attached in the end.
Then I cross-checked my analysis using VASP, where I (of course) did a
basis transformation from real Harmonics to complex harmonics, and I
obtained the following dm:
nks.real in comp_sph_harm basis:
0.6611 -0.0000 -0.0000 0.0000 0.0000
-0.0000 0.5653 0.0000 -0.0000 -0.0000
-0.0000 0.0000 0.6790 -0.0000 -0.0000
0.0000 -0.0000 -0.0000 0.5653 0.0000
0.0000 -0.0000 -0.0000 0.0000 0.6611
nks.imag in comp_sph_harm basis:
0.0000 0.0000 0.0000 -0.1554 0.0000
-0.0000 0.0000 -0.0000 0.0000 0.1554
-0.0000 0.0000 0.0000 0.0000 0.0000
0.1554 -0.0000 -0.0000 0.0000 -0.0000
0.0000 -0.1554 -0.0000 0.0000 0.0000
Note that there is a difference of spin-factor of two. I should say that
the numbers are quite close, but what I got from VASP has the off-diagonal
elements in imaginary part instead of real part. And this density matrix
does commute with all of the site symmetry operations!
To summarize, besides a commonly used "Condon-Shortley Phase" as stated in
ylm.F file, does wien2k "secretly" add some additional phase factors to the
complex spherical Harmonics?
Note that here the locmat is identity and the parameter isplit doesn't seem
to be used in SRC_lapwdm.
Thanks for your help.
Regards, Yongxin
Appendix:
12 site point group operations for Fe site of the FeO struct
------------------------------------------------------------------------------------------------
Site = 0
Operation number = 1 / 12
agroup
type: unity I
Hermann_Mauguin: 1
Schoenflies: 1
1.0000e+00 0.0000e+00 0.0000e+00
0.0000e+00 1.0000e+00 0.0000e+00
0.0000e+00 0.0000e+00 1.0000e+00 Uc
1 0 0
0 1 0
0 0 1 Uf
0.0000e+00 0.0000e+00 0.0000e+00
0.0000e+00 0.0000e+00 0.0000e+00
0.0000e+00 0.0000e+00 0.0000e+00 A=generator U=+-exp(A) [not Uc
and -1 if inversion]
0 angle
0.0000e+00 0.0000e+00 0.0000e+00 axis
0 inversion
- 0 1 2 3 basis_atoms_map - 0 0 1 1 basis_types_map
------------------------------------------------------------------------------------------------
Site = 0
Operation number = 2 / 12
agroup
type: rotation
Hermann_Mauguin: 2
Schoenflies: C2
1.0000e+00 0.0000e+00 0.0000e+00
0.0000e+00 -1.0000e+00 0.0000e+00
0.0000e+00 0.0000e+00 -1.0000e+00 Uc
1 -1 0
0 -1 0
0 0 -1 Uf
0.0000e+00 -0.0000e+00 0.0000e+00
0.0000e+00 0.0000e+00 3.1416e+00
-0.0000e+00 -3.1416e+00 0.0000e+00 A=generator U=+-exp(A) [not Uc
and -1 if inversion]
180 angle
-1.0000e+00 0.0000e+00 0.0000e+00 axis
0 inversion
- 0 1 3 2 basis_atoms_map - 0 0 1 1 basis_types_map
------------------------------------------------------------------------------------------------
Site = 0
Operation number = 3 / 12
agroup
type: rotation
Hermann_Mauguin: 2
Schoenflies: C2
-5.0000e-01 8.6603e-01 0.0000e+00
8.6603e-01 5.0000e-01 0.0000e+00
0.0000e+00 0.0000e+00 -1.0000e+00 Uc
0 1 0
1 0 0
0 0 -1 Uf
0.0000e+00 -0.0000e+00 2.7207e+00
0.0000e+00 0.0000e+00 -1.5708e+00
-2.7207e+00 1.5708e+00 0.0000e+00 A=generator U=+-exp(A) [not Uc
and -1 if inversion]
180 angle
5.0000e-01 8.6603e-01 0.0000e+00 axis
0 inversion
- 0 1 3 2 basis_atoms_map - 0 0 1 1 basis_types_map
------------------------------------------------------------------------------------------------
Site = 0
Operation number = 4 / 12
agroup
type: rotation
Hermann_Mauguin: 3
Schoenflies: C3
-5.0000e-01 8.6603e-01 0.0000e+00
-8.6603e-01 -5.0000e-01 0.0000e+00
0.0000e+00 0.0000e+00 1.0000e+00 Uc
-1 1 0
-1 0 0
0 0 1 Uf
0.0000e+00 -4.1888e+00 0.0000e+00
4.1888e+00 0.0000e+00 -0.0000e+00
-0.0000e+00 0.0000e+00 0.0000e+00 A=generator U=+-exp(A) [not Uc
and -1 if inversion]
240 angle
0.0000e+00 0.0000e+00 1.0000e+00 axis
0 inversion
- 0 1 2 3 basis_atoms_map - 0 0 1 1 basis_types_map
------------------------------------------------------------------------------------------------
Site = 0
Operation number = 5 / 12
agroup
type: rotation
Hermann_Mauguin: 3
Schoenflies: C3
-5.0000e-01 -8.6603e-01 0.0000e+00
8.6603e-01 -5.0000e-01 0.0000e+00
0.0000e+00 0.0000e+00 1.0000e+00 Uc
0 -1 0
1 -1 0
0 0 1 Uf
0.0000e+00 -2.0944e+00 0.0000e+00
2.0944e+00 0.0000e+00 -0.0000e+00
-0.0000e+00 0.0000e+00 0.0000e+00 A=generator U=+-exp(A) [not Uc
and -1 if inversion]
120 angle
0.0000e+00 0.0000e+00 1.0000e+00 axis
0 inversion
- 0 1 2 3 basis_atoms_map - 0 0 1 1 basis_types_map
------------------------------------------------------------------------------------------------
Site = 0
Operation number = 6 / 12
agroup
type: rotation
Hermann_Mauguin: 2
Schoenflies: C2
-5.0000e-01 -8.6603e-01 0.0000e+00
-8.6603e-01 5.0000e-01 0.0000e+00
0.0000e+00 0.0000e+00 -1.0000e+00 Uc
-1 0 0
-1 1 0
0 0 -1 Uf
0.0000e+00 -0.0000e+00 2.7207e+00
0.0000e+00 0.0000e+00 1.5708e+00
-2.7207e+00 -1.5708e+00 0.0000e+00 A=generator U=+-exp(A) [not Uc
and -1 if inversion]
180 angle
-5.0000e-01 8.6603e-01 0.0000e+00 axis
0 inversion
- 0 1 3 2 basis_atoms_map - 0 0 1 1 basis_types_map
------------------------------------------------------------------------------------------------
Site = 0
Operation number = 7 / 12
agroup
type: inversion -I
Hermann_Mauguin: -1
Schoenflies: i
-1.0000e+00 0.0000e+00 0.0000e+00
0.0000e+00 -1.0000e+00 0.0000e+00
0.0000e+00 0.0000e+00 -1.0000e+00 Uc
-1 0 0
0 -1 0
0 0 -1 Uf
0.0000e+00 0.0000e+00 0.0000e+00
0.0000e+00 0.0000e+00 0.0000e+00
0.0000e+00 0.0000e+00 0.0000e+00 A=generator U=+-exp(A) [not Uc
and -1 if inversion]
0 angle
0.0000e+00 0.0000e+00 0.0000e+00 axis
1 inversion
- 0 1 3 2 basis_atoms_map - 0 0 1 1 basis_types_map
------------------------------------------------------------------------------------------------
Site = 0
Operation number = 8 / 12
agroup
type: rotoinversion
Hermann_Mauguin: m
Schoenflies: s
-1.0000e+00 0.0000e+00 0.0000e+00
0.0000e+00 1.0000e+00 0.0000e+00
0.0000e+00 0.0000e+00 1.0000e+00 Uc
-1 1 0
0 1 0
0 0 1 Uf
0.0000e+00 -0.0000e+00 0.0000e+00
0.0000e+00 0.0000e+00 3.1416e+00
-0.0000e+00 -3.1416e+00 0.0000e+00 A=generator U=+-exp(A) [not Uc
and -1 if inversion]
180 angle
-1.0000e+00 0.0000e+00 0.0000e+00 axis
1 inversion
- 0 1 2 3 basis_atoms_map - 0 0 1 1 basis_types_map
------------------------------------------------------------------------------------------------
Site = 0
Operation number = 9 / 12
agroup
type: rotoinversion
Hermann_Mauguin: -2
Schoenflies: S2
5.0000e-01 -8.6603e-01 0.0000e+00
-8.6603e-01 -5.0000e-01 0.0000e+00
0.0000e+00 0.0000e+00 1.0000e+00 Uc
0 -1 0
-1 0 0
0 0 1 Uf
0.0000e+00 -0.0000e+00 2.7207e+00
0.0000e+00 0.0000e+00 -1.5708e+00
-2.7207e+00 1.5708e+00 0.0000e+00 A=generator U=+-exp(A) [not Uc
and -1 if inversion]
180 angle
5.0000e-01 8.6603e-01 0.0000e+00 axis
1 inversion
- 0 1 2 3 basis_atoms_map - 0 0 1 1 basis_types_map
------------------------------------------------------------------------------------------------
Site = 0
Operation number = 10 / 12
agroup
type: rotoinversion
Hermann_Mauguin: -3
Schoenflies: S3
5.0000e-01 -8.6603e-01 0.0000e+00
8.6603e-01 5.0000e-01 0.0000e+00
0.0000e+00 0.0000e+00 -1.0000e+00 Uc
1 -1 0
1 0 0
0 0 -1 Uf
0.0000e+00 -4.1888e+00 0.0000e+00
4.1888e+00 0.0000e+00 -0.0000e+00
-0.0000e+00 0.0000e+00 0.0000e+00 A=generator U=+-exp(A) [not Uc
and -1 if inversion]
240 angle
0.0000e+00 0.0000e+00 1.0000e+00 axis
1 inversion
- 0 1 3 2 basis_atoms_map - 0 0 1 1 basis_types_map
------------------------------------------------------------------------------------------------
Site = 0
Operation number = 11 / 12
agroup
type: rotoinversion
Hermann_Mauguin: -3
Schoenflies: S3
5.0000e-01 8.6603e-01 0.0000e+00
-8.6603e-01 5.0000e-01 0.0000e+00
0.0000e+00 0.0000e+00 -1.0000e+00 Uc
0 1 0
-1 1 0
0 0 -1 Uf
0.0000e+00 -2.0944e+00 0.0000e+00
2.0944e+00 0.0000e+00 -0.0000e+00
-0.0000e+00 0.0000e+00 0.0000e+00 A=generator U=+-exp(A) [not Uc
and -1 if inversion]
120 angle
0.0000e+00 0.0000e+00 1.0000e+00 axis
1 inversion
- 0 1 3 2 basis_atoms_map - 0 0 1 1 basis_types_map
------------------------------------------------------------------------------------------------
Site = 0
Operation number = 12 / 12
agroup
type: rotoinversion
Hermann_Mauguin: -2
Schoenflies: S2
5.0000e-01 8.6603e-01 0.0000e+00
8.6603e-01 -5.0000e-01 0.0000e+00
0.0000e+00 0.0000e+00 1.0000e+00 Uc
1 0 0
1 -1 0
0 0 1 Uf
0.0000e+00 -0.0000e+00 2.7207e+00
0.0000e+00 0.0000e+00 1.5708e+00
-2.7207e+00 -1.5708e+00 0.0000e+00 A=generator U=+-exp(A) [not Uc
and -1 if inversion]
180 angle
-5.0000e-01 8.6603e-01 0.0000e+00 axis
1 inversion
- 0 1 2 3 basis_atoms_map - 0 0 1 1 basis_types_map
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