PROCAR projections for degenerate bands

[inigo_robredo]

Dear VASP community,

I have a question regarding the calculation of orbital/spin projections in the PROCAR file in the case of degenerate bands. Looking at the PROCAR file for different systems, it seems that the projections blocks of degenerate bands are exactly the same. Is this because VASP averages the projections of degenerate bands? Does it take into account the degeneracy in any way?

Best,

Iñigo

[martin.schlipf]

I don't think that degeneracies are accounted like this. Could you provide the input files for your calculation?

[inigo_robredo]

I am analyzing this material:
1.0
4.9100000000000001 0.0000000000000000 0.0000000000000000
0.0000000000000000 4.9100000000000001 0.0000000000000000
0.0000000000000000 0.0000000000000000 4.9100000000000001
Pt Ga
4 4
Direct
0.3650000000000000 0.8650000000000000 0.6350000000000000
0.8650000000000000 0.6350000000000000 0.3650000000000000
0.6350000000000000 0.3650000000000000 0.8650000000000000
0.1350000000000000 0.1350000000000000 0.1350000000000000
0.6550000000000000 0.1550000000000000 0.3450000000000000
0.1550000000000000 0.3450000000000000 0.6550000000000000
0.3450000000000000 0.6550000000000000 0.1550000000000000
0.8450000000000000 0.8450000000000000 0.8450000000000000

at the Gamma point. I just rechecked the PROCAR file and it turns out that when SOC is turned on, there are some projections that are the same and some that are opposite. Below, I attach the lines of the PROCAR file regarding two degenerate bands. How does VASP distinguish between the two degenerate bands to perform the projections?

k-point 1 : 0.00000000 0.00000000 0.00000000 weight = 0.00166667

band 49 # energy 7.63964460 # occ. 1.00000000

ion s py pz px dxy dyz dz2 dxz x2-y2 tot
1 0.003 0.006 0.006 0.006 0.009 0.009 0.065 0.009 0.065 0.178
2 0.003 0.006 0.006 0.006 0.009 0.009 0.065 0.009 0.065 0.178
3 0.003 0.006 0.006 0.006 0.009 0.009 0.065 0.009 0.065 0.178
4 0.003 0.006 0.006 0.006 0.009 0.009 0.065 0.009 0.065 0.178
5 0.028 0.007 0.007 0.007 0.000 0.000 0.002 0.000 0.002 0.053
6 0.028 0.007 0.007 0.007 0.000 0.000 0.002 0.000 0.002 0.053
7 0.028 0.007 0.007 0.007 0.000 0.000 0.002 0.000 0.002 0.053
8 0.028 0.007 0.007 0.007 0.000 0.000 0.002 0.000 0.002 0.053
tot 0.125 0.053 0.053 0.053 0.036 0.036 0.268 0.036 0.268 0.926
1 0.001 0.001 0.004 0.005 -0.002 0.008 0.008 -0.005 0.007 0.027
2 0.000 -0.004 0.002 0.002 0.001 0.005 0.003 0.004 0.057 0.069
3 0.001 0.001 0.004 0.005 -0.002 0.008 0.009 -0.005 0.007 0.028
4 0.000 -0.003 0.002 0.002 0.000 0.005 0.002 0.004 0.057 0.068
5 0.016 0.000 0.002 0.000 -0.000 0.000 0.002 0.000 -0.001 0.019
6 -0.001 0.006 -0.004 0.005 -0.000 0.000 0.001 0.000 0.001 0.009
7 0.016 0.000 0.002 -0.000 -0.000 0.000 0.002 0.000 -0.001 0.019
8 -0.000 0.006 -0.004 0.005 -0.000 0.000 0.001 0.000 0.001 0.009
tot 0.034 0.008 0.008 0.025 -0.003 0.026 0.028 -0.002 0.126 0.249
1 0.001 -0.002 -0.002 0.000 0.006 -0.002 0.052 -0.001 -0.038 0.015
2 0.002 0.004 0.006 0.004 -0.008 -0.006 -0.059 0.003 -0.001 -0.054
3 -0.001 0.002 0.002 -0.000 -0.006 0.002 -0.052 0.001 0.037 -0.016
4 -0.002 -0.004 -0.006 -0.004 0.008 0.006 0.059 -0.004 0.000 0.053
5 -0.003 0.006 0.005 -0.007 0.000 -0.000 -0.000 0.000 0.001 0.002
6 0.024 -0.003 -0.006 0.004 -0.000 -0.000 -0.002 0.000 -0.000 0.018
7 0.003 -0.006 -0.005 0.007 -0.000 0.000 0.000 -0.000 -0.001 -0.002
8 -0.024 0.003 0.006 -0.004 0.000 0.000 0.002 -0.000 0.000 -0.018
tot -0.000 -0.000 -0.000 -0.000 0.000 0.000 0.000 -0.000 -0.002 -0.003
1 0.003 0.006 0.004 0.003 0.006 -0.002 0.038 -0.007 -0.053 -0.003
2 -0.002 -0.002 0.000 -0.004 0.002 0.002 0.028 0.006 0.032 0.061
3 0.003 0.006 0.004 0.003 0.006 -0.002 0.038 -0.007 -0.053 -0.004
4 -0.002 -0.002 0.000 -0.004 0.002 0.002 0.028 0.006 0.031 0.061
5 0.023 -0.003 0.005 0.001 0.000 -0.000 0.001 -0.000 -0.001 0.025
6 -0.014 0.002 -0.002 0.003 0.000 0.000 -0.000 0.000 0.002 -0.008
7 0.023 -0.003 0.005 0.001 0.000 -0.000 0.001 -0.000 -0.001 0.025
8 -0.013 0.002 -0.002 0.003 0.000 0.000 -0.000 0.000 0.002 -0.008
tot 0.020 0.005 0.015 0.005 0.015 -0.001 0.133 -0.002 -0.042 0.148

band 50 # energy 7.63964460 # occ. 1.00000000

ion s py pz px dxy dyz dz2 dxz x2-y2 tot
1 0.003 0.006 0.006 0.006 0.009 0.009 0.065 0.009 0.065 0.178
2 0.003 0.006 0.006 0.006 0.009 0.009 0.065 0.009 0.065 0.178
3 0.003 0.006 0.006 0.006 0.009 0.009 0.065 0.009 0.065 0.178
4 0.003 0.006 0.006 0.006 0.009 0.009 0.065 0.009 0.065 0.178
5 0.028 0.007 0.007 0.007 0.000 0.000 0.002 0.000 0.002 0.053
6 0.028 0.007 0.007 0.007 0.000 0.000 0.002 0.000 0.002 0.053
7 0.028 0.007 0.007 0.007 0.000 0.000 0.002 0.000 0.002 0.053
8 0.028 0.007 0.007 0.007 0.000 0.000 0.002 0.000 0.002 0.053
tot 0.125 0.053 0.053 0.053 0.036 0.036 0.268 0.036 0.268 0.926
1 -0.001 -0.001 -0.004 -0.005 0.002 -0.008 -0.008 0.005 -0.007 -0.027
2 -0.000 0.004 -0.002 -0.002 -0.001 -0.005 -0.003 -0.004 -0.057 -0.069
3 -0.001 -0.001 -0.004 -0.005 0.002 -0.008 -0.009 0.005 -0.007 -0.028
4 -0.000 0.003 -0.002 -0.002 -0.000 -0.005 -0.002 -0.004 -0.057 -0.068
5 -0.016 -0.000 -0.002 -0.000 0.000 -0.000 -0.002 -0.000 0.001 -0.019
6 0.001 -0.006 0.004 -0.005 0.000 -0.000 -0.001 -0.000 -0.001 -0.009
7 -0.016 -0.000 -0.002 0.000 0.000 -0.000 -0.002 -0.000 0.001 -0.019
8 0.000 -0.006 0.004 -0.005 0.000 -0.000 -0.001 -0.000 -0.001 -0.009
tot -0.034 -0.008 -0.008 -0.025 0.003 -0.026 -0.028 0.002 -0.126 -0.249
1 -0.001 0.002 0.002 -0.000 -0.006 0.002 -0.052 0.001 0.038 -0.015
2 -0.002 -0.004 -0.006 -0.004 0.008 0.006 0.059 -0.003 0.001 0.054
3 0.001 -0.002 -0.002 0.000 0.006 -0.002 0.052 -0.001 -0.037 0.016
4 0.002 0.004 0.006 0.004 -0.008 -0.006 -0.059 0.004 -0.000 -0.053
5 0.003 -0.006 -0.005 0.007 -0.000 0.000 0.000 -0.000 -0.001 -0.002
6 -0.024 0.003 0.006 -0.004 0.000 0.000 0.002 -0.000 0.000 -0.018
7 -0.003 0.006 0.005 -0.007 0.000 -0.000 -0.000 0.000 0.001 0.002
8 0.024 -0.003 -0.006 0.004 -0.000 -0.000 -0.002 0.000 -0.000 0.018
tot 0.000 0.000 0.000 0.000 -0.000 -0.000 -0.000 0.000 0.002 0.003
1 -0.003 -0.006 -0.004 -0.003 -0.006 0.002 -0.038 0.007 0.053 0.003
2 0.002 0.002 -0.000 0.004 -0.002 -0.002 -0.028 -0.006 -0.032 -0.061
3 -0.003 -0.006 -0.004 -0.003 -0.006 0.002 -0.038 0.007 0.053 0.004
4 0.002 0.002 -0.000 0.004 -0.002 -0.002 -0.028 -0.006 -0.031 -0.061
5 -0.023 0.003 -0.005 -0.001 -0.000 0.000 -0.001 0.000 0.001 -0.025
6 0.014 -0.002 0.002 -0.003 -0.000 -0.000 0.000 -0.000 -0.002 0.008
7 -0.023 0.003 -0.005 -0.001 -0.000 0.000 -0.001 0.000 0.001 -0.025
8 0.013 -0.002 0.002 -0.003 -0.000 -0.000 0.000 -0.000 -0.002 0.008
tot -0.020 -0.005 -0.015 -0.005 -0.015 0.001 -0.133 0.002 0.042 -0.148

[martin.schlipf]

But for the special case of Gamma and SOC you would expect two identical states, wouldn't you? Time-reversal symmetry dictates that psi(k) = psi*(-k) so that for k=0 the state is degenerate. The spin of the two orbitals would be different, though. If you switch of SOC and calculate with ISPIN=2, you should get more or less the orbital projections, but one for spin up and one for spin down.

[inigo_robredo]

Yes, I would expect that there is a degeneracy due to TRS+Kramers theorem. Here's my question then: How does VASP compute the orbital/spin projections of the degenerate states? If I understand it correctly, the independent projections are ill-defined due to the degeneracy; only the trace of the projections on the degenerate space is well defined. Then, does VASP compute the trace of the projections or does it just blindly compute the orbital/spin projections?

[martin.schlipf]

VASP does some symmetrization on the DOS, but there is nothing done to the degenerate bands for the band projection. So the results should be exactly the same whether you switch off symmetry or not. Of course everything is only well defined up to a unitary transformation in this degenerate subspace.

[inigo_robredo]

Thank you for your answer. Then, to get the physically meaningful results of the projections one would have to sum up all degenerate bands.

It still puzzles me why the projections are so similar between degenerate bands, maybe it can be tracked down to the diagonalization procedure? When computing the eigenstates of degemnerate bands, does VASP choose a specific criteria for the eigenstate basis?

[martin.schlipf]

Well you could change ALGO to see if the specific diagonalization routine has an impact. Generally speaking, what VASP does is an iterative diagonalization to get NBANDS orbitals Ψn(r). Then you construct a Hamiltonian in the subspace of these orbitals, i.e., compute matrix elements Hnm = ∫d3r Ψn*(r) H Ψm(r). Diagonalizing this matrix and rotating the orbitals in this subspace gives then the eigenstates. Its possible that some symmetrization in setting up this matrix leads to your observed behavior, but you can explore switching off the symmetry (ISYM = -1) to check that.