Sigma Bond can not be correctly decomposed

[Chengcheng-Xiao]

Sigma bond in N2 example cannot be correctly decomposed. This may be a problem for all sigma bonds constructed by p-orbitals. The problems are:

• C_mats for one sigma bond are not symmetrical with respect to two p-orbitals.

  MO=     1 | AO=     1 | C_nt=   0.662683
  MO=     1 | AO=     4 | C_nt=  -0.748900
  

• WOOP matrixes are not symmetrical as well.

  MO=  1 | AO_m=  1 [ 0, 0, 0] | AO_l=  1 [ 0, 0, 0] | B_iml=   0.878296
  MO=  1 | AO_m=  4 [ 0, 0, 0] | AO_l=  4 [ 0, 0, 0] | B_iml=   1.121704
  

The origin of this problem does not belong to the WOOPs code. Instead, It's originated from a non symmetrical U matrix produced by Wannier90.

Using this symmetrical U matrix for AO fixes this problem.

 written on 11Dec2018 at 19:39:22 
           1           6           6

   0.0000000000  +0.0000000000  +0.0000000000
  -0.6451639475  -0.0747564771
  -0.2730002632  +0.0603789107
  -0.0000000036  +0.0000000002
  +0.0325655349  +0.2240466950
  +0.6642261507  -0.0868698949
  +0.0000000909  +0.0000001684
  +0.0000002705  +0.0000000314
  -0.0000001345  +0.0000000296
  +0.7071067812  +0.0000000000
  -0.0000000007  -0.0000000051
  -0.0000000020  +0.0000000003
  -0.7071067812  -0.0000000000
  -0.2777241678  -0.0323116569
  +0.6342211169  -0.1399593686
  -0.0000000044  +0.0000000003
  +0.0963146453  +0.6629225045
  -0.2244911467  +0.0293455389
  +0.0000001087  +0.0000002015
  -0.6451641971  -0.0747565061
  -0.2730003689  +0.0603789340
  -0.0000000032  +0.0000000002
  -0.0325655223  -0.2240466083
  -0.6642258936  +0.0868698613
  +0.0000000875  +0.0000001622
  +0.0000002483  +0.0000000288
  -0.0000001235  +0.0000000272
  +0.7071067812  +0.0000000000
  -0.0000000007  -0.0000000051
  -0.0000000020  +0.0000000003
  +0.7071067812  +0.0000000000
  -0.2777242753  -0.0323116694
  +0.6342213623  -0.1399594228
  -0.0000000038  +0.0000000002
  -0.0963146080  -0.6629222480
  +0.2244910598  -0.0293455275
  +0.0000001048  +0.0000001941

However, this problem may not only happen to the U matrix for AOs, It may as well happen for MOs since we don't yet have a symmetrical initial projection for p-orbital Sigma bond. To formally deal with this problem, a modified version of DFT2Wannier90 interface is needed to:

• Symmetrize the U matrixes under site symmetry operation.

• Add user-defined initial projections. Or, since a perfect p-orbital sigma is like a ellipsoid, it can be easily employed into DFT2wannier90 code, just change the definition of the Spherical Harmonics to a ellipsoid like function.

[Chengcheng-Xiao]

I’ll write a patch file for bonding orbitals (MOs) for VASP2WANNIER90 interface,as a guideline for other DFT2WANNIER90 interfaces. Specifically, add ellipsoid-like orbital initial projections for head to head sigma bond.

As for atomic-like basis orbitals (AOs), using spglib to get site symmetry and then averaging corresponding u_mat elements.