# [Pw_forum] orthogonality of phonon eigenvectors?

Suza W suza.rri at gmail.com
Fri Feb 10 00:32:04 CET 2012

```     Dear All ,

>
>
>> > IIRC the phonon code, and all other auxiliary codes, always print
>> > displacement patterns, not phonon eigenvectors
>>
>>   Not always. In case of a homo-atomic system, the phonon eigen-vectots
> (v)
>   can directly be  presented as atomic displacements (u) . However, for a
> hetero-
>   atomic system, it is not so straight-forward. We need  some more
> algebra, because
>
>  u = v / sqrt (M)   and  SUM_i  ( u_i . u_i . M_i )  = constant  .
>
>  After that one can obtain correct atomic displacement  from the
> eigen-vectors in case of a
>  hetero-atomic system.
>

For example, after the above algebra, we get, these well-arranged eigen
displacements

( 0.0   0.0    2.30182985E-05 )
( 0.0   0.0    1.65666412E-03 )
( 0.0   0.0   -1.75258026E-04 )
( 0.0   0.0   -3.12875743E-03 )
( 0.0   0.0   -1.75258026E-04 )

( 0.0    2.30182985E-05   0.0 )
( 0.0    1.65666412E-03   0.0 )
( 0.0   -1.75258026E-04   0.0 )
( 0.0   -3.12875743E-03   0.0 )
( 0.0   -1.75258026E-04   0.0 )

(  2.30182985E-05   0.0   0.0 )
(  1.65666412E-03   0.0   0.0 )
( -1.75258026E-04   0.0   0.0 )
( -3.12875743E-03   0.0   0.0 )
( -1.75258026E-04   0.0   0.0 )

,whereas the initial eigen-vectors that QE code prints directly (which is
indeed messy)
after diagonalising the dynamical matrix were

omega( 1) =      2.395159 [THz] =     79.894433 [cm-1]
(  0.004885  0.000000  0.000925  0.000000  0.000021  0.000000 )
(  0.453433  0.000000  0.085812  0.000000  0.001953  0.000000 )
( -0.869153  0.000000 -0.008829  0.000000 -0.000201  0.000000 )
( -0.046653  0.000000 -0.164486  0.000000 -0.000201  0.000000 )
( -0.046653  0.000000 -0.008829  0.000000 -0.003743  0.000000 )
omega( 2) =      2.395159 [THz] =     79.894433 [cm-1]
(  0.000050  0.000000 -0.000153  0.000000 -0.004969  0.000000 )
(  0.004666  0.000000 -0.014157  0.000000 -0.461245  0.000000 )
( -0.008943  0.000000  0.001457  0.000000  0.047457  0.000000 )
( -0.000480  0.000000  0.027136  0.000000  0.047457  0.000000 )
( -0.000480  0.000000  0.001457  0.000000  0.884127  0.000000 )
omega( 3) =      2.395159 [THz] =     79.894433 [cm-1]
( -0.000923  0.000000  0.004883  0.000000 -0.000159  0.000000 )
( -0.085707  0.000000  0.453217  0.000000 -0.014777  0.000000 )
(  0.164285  0.000000 -0.046631  0.000000  0.001520  0.000000 )
(  0.008818  0.000000 -0.868738  0.000000  0.001520  0.000000 )
(  0.008818  0.000000 -0.046631  0.000000  0.028326  0.000000 )

Thanking you all,

Yours sincerely,
Suza

>
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