[Pw_forum] orthogonality of phonon eigenvectors?
Stefano Baroni
baroni at sissa.it
Fri Dec 30 09:35:17 CET 2011
Phonon eigenvectors are solution of a *generalized* eigenvalue equation (see the definition in any textbook of classical mechanics, dealing with "small oscillations"). As such, they are only orthogonal if all the nuclear masses are the same (simple self-evaluation exercise: explain why). HTH - SB
--
Stefano Baroni, Trieste -- swift message written and sent on the go
On 29/dic/2011, at 10:41, Sonu Kumar <1009ukumar at gmail.com> wrote:
> Dear all QE users,
>
> I am not getting the orthogonality of phonon eigenvectors
> obtained in xx.dyn file. However normalization condition is satisfied.
>
> For example:
> 1. for four atom unit cell, eigen vectors are:
>
> omega( 1) = -0.330081 [THz] = -11.010398 [cm-1]
> ( -0.000126 0.000000 -0.494472 0.000000 0.000000 0.000000 )
> ( -0.000128 0.000000 -0.501752 0.000000 0.000000 0.000000 )
> ( -0.000128 0.000000 -0.501868 0.000000 0.000000 0.000000 )
> ( -0.000128 0.000000 -0.501868 0.000000 0.000000 0.000000 )
> omega( 2) = -0.330081 [THz] = -11.010398 [cm-1]
> ( 0.494472 0.000000 -0.000126 0.000000 0.000000 0.000000 )
> ( 0.501752 0.000000 -0.000128 0.000000 0.000000 0.000000 )
> ( 0.501868 0.000000 -0.000128 0.000000 0.000000 0.000000 )
> ( 0.501868 0.000000 -0.000128 0.000000 0.000000 0.000000 )
> omega( 3) = -0.293090 [THz] = -9.776500 [cm-1]
> ( 0.000000 0.000000 0.000000 0.000000 0.499253 0.000000 )
> ( 0.000000 0.000000 0.000000 0.000000 0.500310 0.000000 )
> ( 0.000000 0.000000 0.000000 0.000000 0.500218 0.000000 )
> ( 0.000000 0.000000 0.000000 0.000000 0.500218 0.000000 )
> omega( 4) = 2.476743 [THz] = 82.615796 [cm-1]
> ( -0.506652 0.000000 -0.000002 0.000000 0.000000 0.000000 )
> ( 0.559568 0.000000 0.000002 0.000000 0.000000 0.000000 )
> ( 0.463782 0.000000 0.000002 0.000000 0.000000 0.000000 )
> ( 0.463782 0.000000 0.000002 0.000000 0.000000 0.000000 )
> omega( 5) = 2.476743 [THz] = 82.615796 [cm-1]
> ( -0.000002 0.000000 0.506652 0.000000 0.000000 0.000000 )
> ( 0.000002 0.000000 -0.559568 0.000000 0.000000 0.000000 )
> ( 0.000002 0.000000 -0.463782 0.000000 0.000000 0.000000 )
> ( 0.000002 0.000000 -0.463782 0.000000 0.000000 0.000000 )
> omega( 6) = 9.011016 [THz] = 300.577143 [cm-1]
> ( 0.000000 0.000000 0.000000 0.000000 -0.562468 0.000000 )
> ( 0.000000 0.000000 0.000000 0.000000 0.797241 0.000000 )
> ( 0.000000 0.000000 0.000000 0.000000 0.154980 0.000000 )
> ( 0.000000 0.000000 0.000000 0.000000 0.154980 0.000000 )
> omega( 7) = 10.988751 [THz] = 366.547723 [cm-1]
> ( 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 )
> ( 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 )
> ( 0.707107 0.000000 0.000001 0.000000 0.000000 0.000000 )
> ( -0.707107 0.000000 -0.000001 0.000000 0.000000 0.000000 )
> omega( 8) = 10.988751 [THz] = 366.547724 [cm-1]
> ( 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 )
> ( 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 )
> ( -0.000001 0.000000 0.707107 0.000000 0.000000 0.000000 )
> ( 0.000001 0.000000 -0.707107 0.000000 0.000000 0.000000 )
> omega( 9) = 11.334621 [THz] = 378.084770 [cm-1]
> ( 0.018232 0.000000 0.000000 0.000000 0.000000 0.000000 )
> ( 0.283432 0.000000 0.000000 0.000000 0.000000 0.000000 )
> ( -0.677987 0.000000 -0.000001 0.000000 0.000000 0.000000 )
> ( -0.677987 0.000000 -0.000001 0.000000 0.000000 0.000000 )
> omega(10) = 11.334621 [THz] = 378.084771 [cm-1]
> ( 0.000000 0.000000 -0.018232 0.000000 0.000000 0.000000 )
> ( 0.000000 0.000000 -0.283432 0.000000 0.000000 0.000000 )
> ( -0.000001 0.000000 0.677987 0.000000 0.000000 0.000000 )
> ( -0.000001 0.000000 0.677987 0.000000 0.000000 0.000000 )
> omega(11) = 17.157192 [THz] = 572.306135 [cm-1]
> ( 0.000000 0.000000 0.000000 0.000000 -0.097293 0.000000 )
> ( 0.000000 0.000000 0.000000 0.000000 -0.168080 0.000000 )
> ( 0.000000 0.000000 0.000000 0.000000 0.693644 0.000000 )
> ( 0.000000 0.000000 0.000000 0.000000 0.693644 0.000000 )
> omega(12) = 20.428646 [THz] = 681.430811 [cm-1]
> ( 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 )
> ( 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 )
> ( 0.000000 0.000000 0.000000 0.000000 -0.707107 0.000000 )
> ( 0.000000 0.000000 0.000000 0.000000 0.707107 0.000000 )
>
> My Output:
>
> mode# mode# sum of prod of modes
> 4 6 0.00000
> 5 7 0.00000
> 6 8 0.00000
> 7 9 0.00000
> 8 10 0.00000
> 9 11 0.00000
> 10 12 0.00000
> 12 5 0.00000
> 1 3 0.00000
> 2 4 0.49575 <-----why not zero.
>
> Similarly :
> mode# mode# sum of prod of modes
> 1 4 -0.00013
> 2 5 0.00013
> 3 6 0.27310 <---
> 4 7 0.00000
> 5 8 0.00000
> 6 9 0.00000
> 7 10 0.00000
> 8 11 0.00000
> 9 12 0.00000
> 10 1 -0.52929<--
> 11 2 0.00000
> 12 3 0.00000
>
> However all modes follow the normalization conditon:
> mode# sum of sqr of modes
> 1 1.00000
> 2 1.00000
> 3 1.00000
> 4 1.00000
> 5 1.00000
> 6 1.00000
> 7 1.00000
> 8 1.00000
> 9 1.00000
> 10 1.00000
> 11 1.00000
> 12 1.00000
>
> Thank you !!
>
> regards,
> ==========================================
> Sonu Kumar
> Phd Student,Physics Department
> Indian Institute of Technology ,Delhi-110016, India
> web:-http://www.iitd.ac.in/
> ==========================================
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