<div dir="ltr">Dear Peter, developers and users:<div><br></div><div> 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? </div><div> 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. </div><div><br></div><div><div>nks.real in comp_sph_harm basis:</div><div> 1.2815 0.0000 -0.0000 0.3136 0.0000</div><div> 0.0000 1.0857 -0.0000 0.0000 -0.3136</div><div> -0.0000 -0.0000 1.3175 0.0000 -0.0000</div><div> 0.3136 0.0000 0.0000 1.0857 -0.0000</div><div> 0.0000 -0.3136 -0.0000 -0.0000 1.2815</div><div>nks.imag in comp_sph_harm basis:</div><div> 0.0000 0.0000 0.0000 0.0000 0.0000</div><div> 0.0000 0.0000 0.0000 0.0000 0.0000</div><div> 0.0000 0.0000 0.0000 0.0000 0.0000</div><div> 0.0000 0.0000 0.0000 0.0000 0.0000</div><div> 0.0000 0.0000 0.0000 0.0000 0.0000</div></div><div><br></div><div>which are all real numbers up to numerical accuracy not shown.</div><div><br></div><div>The puzzle is that this density matrix does not commute with all of the the local site symmetries, which are attached in the end. </div><div><br></div><div>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:</div><div><br></div><div><div>nks.real in comp_sph_harm basis:</div><div> 0.6611 -0.0000 -0.0000 0.0000 0.0000</div><div> -0.0000 0.5653 0.0000 -0.0000 -0.0000</div><div> -0.0000 0.0000 0.6790 -0.0000 -0.0000</div><div> 0.0000 -0.0000 -0.0000 0.5653 0.0000</div><div> 0.0000 -0.0000 -0.0000 0.0000 0.6611</div><div>nks.imag in comp_sph_harm basis:</div><div> 0.0000 0.0000 0.0000 -0.1554 0.0000</div><div> -0.0000 0.0000 -0.0000 0.0000 0.1554</div><div> -0.0000 0.0000 0.0000 0.0000 0.0000</div><div> 0.1554 -0.0000 -0.0000 0.0000 -0.0000</div><div> 0.0000 -0.1554 -0.0000 0.0000 0.0000</div></div><div><br></div><div>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! </div><div><br></div>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? <div><br><div>Note that here the locmat is identity and the parameter isplit doesn't seem to be used in SRC_lapwdm. </div><div><br></div><div> Thanks for your help.</div><div><br></div><div> Regards, Yongxin</div><div><br></div><div><br></div><div>Appendix:</div><div><pre style="color:rgb(0,0,0)">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 </pre></div><div><br></div><div><br></div></div></div>