[Pw_forum] questions about computing the third order coupling tensor implemented by d3.x code

[Tian Lan]

Dear all,

I have a few questions about computing the third order coupling
tensor implemented by d3.x code. I have gone through all previous messages
about this similar content, as well as the suggested papers. Basically,
from example14, I have known how to use pw.x, ph.x and d3.x to get the
result, but the output is still confusing to me.

Question1: Take the example14 for example, I think si. anh_G file contains
the third order derivative, in the example output it looks like this
------------------------------------------------------------------------------
Third derivative in Cartesian axes
     q = (    0.000000000   0.000000000   0.000000000 )

            modo:    1
  1  1
      0.000000000000E+00      0.000000000000E+00      0.000000000000E+00
      0.000000000000E+00     -0.138777878078E-16      0.000000000000E+00
      0.416333634234E-16      0.000000000000E+00      0.277555756156E-16
      0.000000000000E+00      0.383235688562E+00      0.000000000000E+00
     -0.693889390391E-16      0.000000000000E+00      0.383235688562E+00
      0.000000000000E+00     -0.693889390391E-16      0.000000000000E+00
  1  2
      0.555111512313E-16      0.000000000000E+00     -0.138777878078E-16
      0.000000000000E+00      0.000000000000E+00      0.000000000000E+00
      0.416333634234E-16      0.000000000000E+00     -0.693889390391E-16
      0.000000000000E+00     -0.385638438801E+00      0.000000000000E+00
      0.138777878078E-16      0.000000000000E+00     -0.385638438801E+00
      0.000000000000E+00      0.000000000000E+00      0.000000000000E+00
  2  1
     -0.832667268469E-16      0.000000000000E+00     -0.416333634234E-16
      0.000000000000E+00      0.277555756156E-16      0.000000000000E+00
     -0.277555756156E-16      0.000000000000E+00      0.138777878078E-16
      0.000000000000E+00     -0.385638438801E+00      0.000000000000E+00
     -0.971445146547E-16      0.000000000000E+00     -0.385638438801E+00
      0.000000000000E+00      0.277555756156E-16      0.000000000000E+00
  2  2
     -0.138777878078E-16      0.000000000000E+00      0.000000000000E+00
      0.000000000000E+00     -0.138777878078E-16      0.000000000000E+00
     -0.277555756156E-16      0.000000000000E+00     -0.277555756156E-16
      0.000000000000E+00      0.385638438801E+00      0.000000000000E+00
      0.138777878078E-16      0.000000000000E+00      0.385638438801E+00
      0.000000000000E+00     -0.138777878078E-16      0.000000000000E+00
            modo:    2
........................
......................
---------------------------------------------------------------

According to the theory, the anharmonic tensor should be something like
C_xyz(q1=0,j1;q2=0,j2;q3=0,j3|s1,s2,s3) in this particular case, where xyz
is cartesian indices, j's branch number, s1,s2,s3 atom basis index in a
cell.

I guess modo1 means j1=1, i.e, the first mode for q1=0, and every q2 and q3
should each have 6 modes. Then why in the above matrices, their
entries are 6 by 3? Also how do I understand the 1 1, 1 2, 2 1, 2 2
indices on top of these matrices, I bet they correspond to the basis index
of a unit cell.

Question2: This question is similar to the first one. I read the README in
example14 as below, I am not sure about the arguments of C's, such
as  C_{x,y,z} (0,X,-X|1,1,1), does 1 1 1 means atom basis index?

----------------------------------------------------------
By displacing one atom, one can get also these coefficients by a
finite-difference mathod. We give first the values obtained by
the 2n+1 method, then the values by the finite-differences.
All units are in Ryd/(a_b)^3.
  tensor                 |    2n+1       |  fin. dif.
------------------------------------------------------------
C_{x,y,z} (0,0,0|1,1,1)  |   0.38314     |    0.38446
------------------------------------------------------------
C_{x,y,z} (0,X,-X|1,1,1) |   0.34043     |    0.34109
C_{x,x,z} (0,X,-X|1,1,2) |  -0.25316     |   -0.25296
C_{z,x,y} (0,X,-X|1,1,1) |   0.35781     |    0.35767
C_{z,x,x} (0,X,-X|1,1,2) |  -0.25706     |   -0.25491
C_{z,z,z} (0,X,-X|1,1,2) |  -0.13133     |   -0.12813
----------------------------------------------------------
Question3: using DFPT, we can get the third order derivative of energy
w.r.t phonon displacement u(q), but I think in order to compute the
lifetime, the polarization vectors e1, e2, e3 must be multiplied, as
mentioned in many papers including Dr. Debernardi's thesis, Eq 2.9.
However, I don't see any comment in the code. So have those polarization
vectors already been added in the d3.x output?

In all, I am confused at this point, because I cannot match those output to
my general understanding of phonon anharmonic coupling. I know this is a
specialized topic, and I only understand the minimum, hopefully I can get
some help from some professionals. Thank you in advance !

-- 
Lan, Tian
Ph.D. Candidate, Department of Applied Physics and Materials Science
California Institute of Technology,
Caltech M/C 138-78, Pasadena, CA, 91125
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