Dear QE users,
I am trying to compute the full matrix elements of the commutator operator [r,
[r, H]] in the eigenvector basis, where r is the position operator and H is the
Kohn-Sham Hamiltonian. I need these (and even further nested commutators) to
calculate the non-linear response properties of materials. I am aware of the
code that outputs the [r,H]. I am also aware of using Wannier functions to
obtain the matrix elements but I specifically want them using DFT. I was
wondering if there is a code or step in a code that does this? If not, I have
two questions when trying to implement this on my own:
1.
In the code PP/src/compute_ppsi.f90 it says:
! commutator_Hx_psi calculates [H, x], here we need i/2 [H, x]
! ppsi = ppsi * (0.0_DP, 0.5_DP)
! ppsi contains p - i/2 [x, V_{nl}-eS] psi_v for the ipol polarization
My question is, since the momentum operator p = i [H,r] (in a.u.) then why is
there a i/2 factor in ppsi instead of just being i (and the discrepancy of the
factor in the two terms in ppsi)?
2.
My second question is, can the required matrix elements be calculated by doing
the following?
First, calculate the non-diagonal part in matrix(r) which can be obtained from
the matrix( [H,r] ) using the code PW/src/commutator_Hx_psi.f90. This is
because, <psi v | r | psi c > = <psi v | [H,r] | psi c > / (Ev - Ec). And the
diagonal of matrix( r ) is zero. Then, the required matrix elements of [r, [r,
H]] can be computed using: matrix( r ) * matrix( [r, H] ) - matrix( [r, H] ) *
matrix( r ). Am I missing a factor of ½ here somewhere?
Thank you very much. Any advice or insights are appreciated.
Best regards,
Vishal Tiwari
University of Rochester
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