# Output format of Effective charges tensor using DFPT

Hello,

I’m trying to calculate the Born effective charge tensor of BiFeO3 using the DFPT method, while the output formats of the Z*_ij from different responses (electric field vs phonon) are different. For each atom, the Born effective charge tensors calculated from the two responses appear to be transposed. Can anyone tell me which one is the correct format, from the electric field or from phonon?

Best regards,
Yali

Hi Yali,

this is just a convention - the two are equivalent, and neither is “more correct”. One of the indices only runs to 3, for the E field, and the other to 3natom, for the displacement. Depending on the symmetry of your phase of BFO it may be diagonal (I presume not since you noticed something), but in general the 3x3 matrices are not symmetric.

An example from the abinit output file of test v4/t75

Effective charges, in cartesian coordinates,
(from electric field response)
if specified in the inputs, asr has been imposed
j1 j2 matrix element
dir pert dir pert real part imaginary part

1 1 1 4 2.2483998615 0.0000000000
2 1 1 4 -0.0000000000 0.0000000000
3 1 1 4 0.0000000000 0.0000000000
1 2 1 4 -2.2484037957 0.0000000000
2 2 1 4 0.0000000000 0.0000000000
3 2 1 4 -0.0000000000 0.0000000000

here the derivatives are indexed with dir for spatial directions and pert=1…natom for atoms and pert = natom+2 = 4 for the electric field (natom+1 is reserved for the d/dk perturbation).

In the finite field case you will have to see yourself which force or polarization term is being used, and the corresponding BEC component.

Hi mverstra,

Thanks.
The matrices are not symmetric in my results:
Effective charges, in cartesian coordinates,
(from phonon response)
if specified in the inputs, asr has been imposed
j1 j2 matrix element
dir pert dir pert real part imaginary part
1 22 1 1 5.1976623311 0.0000000000
2 22 1 1 0.1544269535 0.0000000000
3 22 1 1 0.1961986587 0.0000000000

1 22 2 1 0.0189906052 0.0000000000
2 22 2 1 4.9664438720 0.0000000000
3 22 2 1 -0.8812701513 0.0000000000

1 22 3 1 0.2494595460 0.0000000000
2 22 3 1 -0.7277099824 0.0000000000
3 22 3 1 3.4949936545 0.0000000000
(…),

Thus, Z*_Bi1 = 5.1976623311 0.0189906052 0.2494595460
0.1544269535 4.9664438720 -0.7277099824
0.1961986587 -0.8812701513 3.4949936545

Effective charges, in cartesian coordinates,
(from electric field response)
if specified in the inputs, asr has been imposed
j1 j2 matrix element
dir pert dir pert real part imaginary part
1 1 1 22 5.1976630586 0.0000000000
2 1 1 22 0.0189836550 0.0000000000
3 1 1 22 0.2494692469 0.0000000000
(…)
1 1 2 22 0.1544188192 0.0000000000
2 1 2 22 4.9664641468 0.0000000000
3 1 2 22 -0.7276831295 0.0000000000
(…)
1 1 3 22 0.1962104405 0.0000000000
2 1 3 22 -0.8812975291 0.0000000000
3 1 3 22 3.4949886509 0.0000000000
(…),

Thus, Z*_Bi1 = 5.1976630586 0.1544188192 0.1962104405
0.0189836550 4.9664641468 -0.8812975291
0.2494692469 -0.7276831295 3.4949886509

Here, I show the tensor for the first atom Z*_Bi1, we can see that the tensor from phonon response is formally a transpose of the tensor from electric field response. So, I want to ask which is the correct format.

Best regards,
Yali

As I mentioned before there is no “correct” format. You see from the output both perturbations are calculated R,E and E,R and they correspond very well (to within 0.04%!!). You can present them as you want, but have to specify which index is which.

Perhaps a note of clarification, the symmetry effect is not about
\frac{\partial E}{\partial \mathcal{E} \partial R} = \frac{\partial E}{\partial R \partial \mathcal{E}}
which is always true. It’s about
\frac{\partial E}{\partial R_y \partial \mathcal{E_x}} = \frac{\partial E}{\partial \mathcal{E_x} \partial R_y} \neq \frac{\partial E}{\partial \mathcal{E_y} \partial R_x}
Which shows why the BEC is not a symmetric 3x3 matrix.