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Dear QE users,</div>
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I am trying to compute the full matrix elements of the commutator operator <i>[r, [r, H]]</i> 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
<i>[r,H]</i>. 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:</div>
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<div class="elementToProof"><span style="font-family: "Segoe UI", "Segoe UI Web (West European)", "Helvetica Neue", sans-serif; font-size: 12pt; color: rgb(0, 0, 0);"> In the code
</span><span style="letter-spacing: normal; font-family: "Segoe UI", "Segoe UI Web (West European)", "Helvetica Neue", sans-serif; font-size: 16px; font-weight: 400; color: rgb(0, 0, 0); background-color: rgb(255, 255, 255);">PP/src/compute_ppsi.f90 it says: </span></div>
<div class="elementToProof"><span style="font-family: "Segoe UI", "Segoe UI Web (West European)", "Helvetica Neue", sans-serif; font-size: 12pt; color: rgb(0, 0, 0);"><i>! commutator_Hx_psi calculates [H, x], here we need i/2 [H, x]</i></span></div>
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<div class="elementToProof"><span style="font-family: "Segoe UI", "Segoe UI Web (West European)", "Helvetica Neue", sans-serif; font-size: 12pt; color: rgb(0, 0, 0);"><i>! ppsi = ppsi * (0.0_DP, 0.5_DP)</i></span></div>
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<div class="elementToProof"><span style="font-family: "Segoe UI", "Segoe UI Web (West European)", "Helvetica Neue", sans-serif; font-size: 12pt; color: rgb(0, 0, 0);"><i>! ppsi contains p - i/2 [x, V_{nl}-eS] psi_v for the ipol polarization</i></span></div>
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<div class="elementToProof"><span style="font-family: "Segoe UI", "Segoe UI Web (West European)", "Helvetica Neue", sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">My question is, since
</span><span style="letter-spacing: normal; font-family: "Segoe UI", "Segoe UI Web (West European)", "Helvetica Neue", sans-serif; font-size: 16px; font-weight: 400; color: rgb(0, 0, 0); background-color: rgb(255, 255, 255);">the momentum operator
<i>p = i [H,r] </i>(in a.u.) then </span><span style="font-family: "Segoe UI", "Segoe UI Web (West European)", "Helvetica Neue", sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">why is there a
<i>i/2</i> factor in ppsi instead of just being <i>i </i>(and the discrepancy of the factor in the two terms in ppsi)?</span></div>
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</li><li style="letter-spacing: normal; font-family: "Segoe UI", "Segoe UI Web (West European)", "Helvetica Neue", sans-serif; font-size: 12pt; font-weight: 400; list-style-type: "2) "; color: rgb(0, 0, 0);">
<div class="elementToProof" style="margin: 0px;"><span style="letter-spacing: normal; font-family: "Segoe UI", "Segoe UI Web (West European)", "Helvetica Neue", sans-serif; font-size: 12pt; font-weight: 400; color: rgb(0, 0, 0);">My second question is, can
the required matrix elements be calculated by doing the following? </span></div>
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<div class="elementToProof" style="margin: 0px;"><span style="letter-spacing: normal; font-family: "Segoe UI", "Segoe UI Web (West European)", "Helvetica Neue", sans-serif; font-size: 12pt; font-weight: 400; color: rgb(0, 0, 0);">First, calculate the non-diagonal
part in matrix(r) which can be obtained from the </span><span style="letter-spacing: normal; font-family: "Segoe UI", "Segoe UI Web (West European)", "Helvetica Neue", sans-serif; font-size: 16px; font-weight: 400; color: rgb(0, 0, 0); background-color: rgb(255, 255, 255);">matrix(
<i>[H,r] </i>) using the </span><span style="letter-spacing: normal; font-family: "Segoe UI", "Segoe UI Web (West European)", "Helvetica Neue", sans-serif; font-size: 12pt; font-weight: 400; color: rgb(0, 0, 0);">code
</span><span style="letter-spacing: normal; font-family: "Segoe UI", "Segoe UI Web (West European)", "Helvetica Neue", sans-serif; font-size: 16px; font-weight: 400; color: rgb(0, 0, 0); background-color: rgb(255, 255, 255);">PW/src/commutator_Hx_psi.f90</span><span style="letter-spacing: normal; font-family: "Segoe UI", "Segoe UI Web (West European)", "Helvetica Neue", sans-serif; font-size: 12pt; font-weight: 400; color: rgb(0, 0, 0);">.
This is because, </span><span style="letter-spacing: normal; font-family: "Segoe UI", "Segoe UI Web (West European)", "Helvetica Neue", sans-serif; font-size: 16px; font-weight: 400; color: rgb(0, 0, 0); background-color: rgb(255, 255, 255);"><i><psi v | r
| psi c > </i></span><span style="letter-spacing: normal; font-family: "Segoe UI", "Segoe UI Web (West European)", "Helvetica Neue", sans-serif; font-size: 12pt; font-weight: 400; color: rgb(0, 0, 0);"><i>= <psi v | [H,r] | psi c > / (Ev - Ec)</i>. And the
diagonal of matrix( <i>r</i> ) is zero. Then, the required matrix elements of <i>
[r, [r, H]]</i> can be computed using: matrix( <i>r </i>) * matrix( <i>[r, H]</i> ) - matrix(
<i>[r, H] </i>) * matrix( <i>r </i>). Am I missing a factor of ½ here somewhere?</span></div>
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Thank you very much. Any advice or insights are appreciated.</div>
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Best regards,</div>
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<div class="elementToProof"><span style="font-family: "Segoe UI", "Segoe UI Web (West European)", "Helvetica Neue", sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">Vishal Tiwari</span></div>
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University of Rochester</div>
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