[QE-users] Compute the higer order commutator [r, [r, H]], or in other sense commutator between r and the commutator between r and nonlocal potential

[Tiwari, Vishal]

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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