# [Pw_forum] LO-TO splitting

Fethi SOYALP fsoyalp at yyu.edu.tr
Fri Feb 4 22:15:25 CET 2005

```> You guess? The acoustic mode is 3-fold degenerate. The other three =20
> modes are split in a singlet (LO) plus a doublet (TO). Why LT and TO =20
> modes are split would require some more explanation, and I would like =20=
>
> to urge you to consult any text-book in solid-state physics to =20
> understant why. If after studying your favorite text you still do not =20=
>
> understand, please revert to us and we will give further help.

Dear Stefano,

Thank you very much for your help about LO-TO splitting in example06.
Now I calculate vibration modes for a CaF2 structure.I am confused
because there isn’t degenerate some direcion, for egzample (110). How can
I  determine which modes are LO-TO and which modes are LA-TA when
degeneracy disapear. according to the output file, first two and last five
eigenvectors are below. for first three eigenvectors, because of  the
small modes, I can say first three modes are acoustic,  last six modes are
optic .Near the X point (last four eigenvectors) acoustic and optic
branches are crossing. I can’t determine acouistic  and optic modes from

q =       0.0250      0.0250      0.0000
**************************************************************************
omega( 1) =       0.512971 [THz] =      17.110993 [cm-1]
( -0.406694   0.025294     0.406694  -0.025294     0.000000   0.000000   )
( -0.404596   0.057287     0.404596  -0.057287     0.000000   0.000000   )
( -0.408577  -0.006711     0.408577   0.006711     0.000000   0.000000   )
omega( 2) =       0.542446 [THz] =      18.094187 [cm-1]
(  0.000000   0.000000     0.000000   0.000000    -0.574443   0.045210   )
( -0.002253  -0.014223    -0.002253  -0.014223    -0.570445   0.090350   )
(  0.000000   0.014401     0.000000   0.014401    -0.577556   0.000000   )
omega( 3) =       0.671957 [THz] =      22.414210 [cm-1]
(  0.031989   0.406461     0.031989   0.406461     0.000000   0.000000   )
(  0.063866   0.403232     0.063866   0.403232    -0.020143   0.003190   )
(  0.000000   0.408258     0.000000   0.408258     0.020394   0.000000   )
omega( 4) =       9.419107 [THz] =     314.189680 [cm-1]
(  0.000000   0.000000     0.000000   0.000000    -0.129018   0.010154   )
( -0.075525  -0.476846    -0.075525  -0.476846     0.157578  -0.024958   )
(  0.000000   0.482790     0.000000   0.482790     0.159542   0.000000   )
omega( 5) =       9.420889 [THz] =     314.249133 [cm-1]
( -0.007168  -0.091078    -0.007168  -0.091078     0.000000   0.000000   )
(  0.017643   0.111392     0.017643   0.111392     0.674393  -0.106813   )
(  0.000000   0.112780     0.000000   0.112780    -0.682799   0.000000   )
omega( 6) =       9.448169 [THz] =     315.159105 [cm-1]
(  0.000000   0.000000     0.000000   0.000000     0.000000   0.000000   )
(  0.097607   0.490380    -0.097607  -0.490380     0.000000   0.000000   )
( -0.019693  -0.499612     0.019693   0.499612     0.000000   0.000000   )
omega( 7) =       9.751590 [THz] =     325.280207 [cm-1]
(  0.246460   0.309902    -0.246460  -0.309902     0.000000   0.000000   )
( -0.282494  -0.302995     0.282494   0.302995     0.000000   0.000000   )
( -0.231618  -0.343456     0.231618   0.343456     0.000000   0.000000   )
omega( 8) =       9.773977 [THz] =     326.026956 [cm-1]
(  0.000000   0.000000     0.000000   0.000000     0.531826  -0.041856   )
( -0.025955  -0.163873    -0.025955  -0.163873    -0.543362   0.086060   )
(  0.000000   0.165916     0.000000   0.165916    -0.550135   0.000000   )
omega( 9) =       9.775480 [THz] =     326.077074 [cm-1]
( -0.029580  -0.375850    -0.029580  -0.375850     0.000000   0.000000   )
(  0.060880   0.384382     0.060880   0.384382    -0.231528   0.036670   )
(  0.000000   0.389173     0.000000   0.389173     0.234414   0.000000   )
**************************************************************************
diagonalizing the dynamical matrix ...

17.111000   TA (because all atoms vibrate same direction)
18.094200   LA (L because the vibrational eigenvector has non-vanishing z
component; the wavevector is directed along xy, hence the mode is
transverse)

22.414200  ?A
314.189697  O (O because small atoms vibrations are bigger than small atom)
314.249115  ?O
315.159088  TO
325.280212  TO
326.027008  ?O
326.077087  ?O

q =       0.0500      0.0500      0.0000
**************************************************************************
omega( 1) =       0.622850 [THz] =      20.776190 [cm-1]
(  0.405477  -0.048461    -0.405477   0.048461     0.000000   0.000000   )
(  0.392739  -0.111248    -0.392739   0.111248     0.000000   0.000000   )
(  0.407894   0.015560    -0.407894  -0.015560     0.000000   0.000000   )
omega( 2) =       0.715136 [THz] =      23.854512 [cm-1]
(  0.000000   0.000000     0.000000   0.000000    -0.570236   0.090317   )
( -0.008890  -0.027362    -0.008890  -0.027362    -0.547730   0.177968   )
(  0.000000   0.028770     0.000000   0.028770    -0.575918   0.000000   )
omega( 3) =       1.066602 [THz] =      35.578241 [cm-1]
( -0.064032  -0.404285    -0.064032  -0.404285     0.000000   0.000000   )
( -0.125672  -0.386777    -0.125672  -0.386777     0.038912  -0.012643   )
(  0.000000  -0.406682     0.000000  -0.406682    -0.040915   0.000000   )
omega( 4) =       9.355051 [THz] =     312.052975 [cm-1]
(  0.000000   0.000000     0.000000   0.000000     0.204219  -0.032345   )
(  0.139849   0.430410     0.139849   0.430410    -0.249827   0.081174   )
(  0.000000  -0.452560     0.000000  -0.452560    -0.262684   0.000000   )
omega( 5) =       9.362934 [THz] =     312.315930 [cm-1]
(  0.022737   0.143558     0.022737   0.143558     0.000000   0.000000   )
( -0.057382  -0.176604    -0.057382  -0.176604    -0.608907   0.197846   )
(  0.000000  -0.185692     0.000000  -0.185692     0.640243   0.000000   )
omega( 6) =       9.454900 [THz] =     315.383604 [cm-1]
(  0.000000   0.000000     0.000000   0.000000     0.000000   0.000000   )
(  0.068326   0.495310    -0.068326  -0.495310     0.000000   0.000000   )
(  0.088078  -0.492181    -0.088078   0.492181     0.000000   0.000000   )
omega( 7) =       9.758643 [THz] =     325.515459 [cm-1]
(  0.101694   0.381763    -0.101694  -0.381763     0.000000   0.000000   )
( -0.168110  -0.379074     0.168110   0.379074     0.000000   0.000000   )
( -0.042742  -0.412469     0.042742   0.412469     0.000000   0.000000   )
omega( 8) =       9.831911 [THz] =     327.959444 [cm-1]
(  0.000000   0.000000     0.000000   0.000000     0.486764  -0.077096   )
( -0.080649  -0.248212    -0.080649  -0.248212    -0.468179   0.152121   )
(  0.000000   0.260985     0.000000   0.260985    -0.492272   0.000000   )
omega( 9) =       9.837634 [THz] =     328.150317 [cm-1]
( -0.054427  -0.343638    -0.054427  -0.343638     0.000000   0.000000   )
(  0.107811   0.331810     0.107811   0.331810    -0.350098   0.113754   )
(  0.000000   0.348885     0.000000   0.348885     0.368115   0.000000   )
**************************************************************************
diagonalizing the dynamical matrix ...

20.776199
23.854500
35.578201
312.053009
312.315887
315.383606
325.515503
327.959412
328.150299

last five eigenvector.

q =       0.9000      0.9000      0.0000
**************************************************************************
omega( 1) =       5.112475 [THz] =     170.534951 [cm-1]
(  0.453049  -0.311934    -0.453049   0.311934     0.000000   0.000000   )
( -0.301186   0.089491     0.301186  -0.089491     0.000000   0.000000   )
( -0.191063   0.249433     0.191063  -0.249433     0.000000   0.000000   )
omega( 2) =       5.481749 [THz] =     182.852665 [cm-1]
( -0.172532   0.530999    -0.172532   0.530999     0.000000   0.000000   )
( -0.175797   0.241963    -0.175797   0.241963     0.078318   0.056902   )
(  0.000000   0.299083     0.000000   0.299083    -0.096807   0.000000   )
omega( 3) =       7.581087 [THz] =     252.879509 [cm-1]
(  0.000000   0.000000     0.000000   0.000000     0.078334   0.025452   )
(  0.093007  -0.128014     0.093007  -0.128014     0.540610   0.392776   )
(  0.000000   0.158234     0.000000   0.158234     0.668231   0.000000   )
omega( 4) =       7.745696 [THz] =     258.370336 [cm-1]
(  0.000000   0.000000     0.000000   0.000000    -0.493523  -0.160355   )
( -0.228501   0.314505    -0.228501   0.314505     0.203237   0.147660   )
(  0.000000  -0.388749     0.000000  -0.388749     0.251214   0.000000   )
omega( 5) =       7.974611 [THz] =     266.006149 [cm-1]
(  0.000000   0.000000     0.000000   0.000000     0.000000   0.000000   )
( -0.360316  -0.346659     0.360316   0.346659     0.000000   0.000000   )
(  0.495263   0.068665    -0.495263  -0.068665     0.000000   0.000000   )
omega( 6) =       8.411441 [THz] =     280.577334 [cm-1]
(  0.000000   0.000000     0.000000   0.000000    -0.576853  -0.187431   )
(  0.229436  -0.315791     0.229436  -0.315791    -0.086098  -0.062554   )
(  0.000000   0.390339     0.000000   0.390339    -0.106424   0.000000   )
omega( 7) =      10.869387 [THz] =     362.566124 [cm-1]
( -0.022070   0.067924    -0.022070   0.067924     0.000000   0.000000   )
(  0.140863  -0.193881     0.140863  -0.193881     0.498735   0.362352   )
(  0.000000  -0.239650     0.000000  -0.239650    -0.616470   0.000000   )
omega( 8) =      11.043721 [THz] =     368.381318 [cm-1]
(  0.123427  -0.222049    -0.123427   0.222049     0.000000   0.000000   )
(  0.341637  -0.317828    -0.341637   0.317828     0.000000   0.000000   )
(  0.089575  -0.457937    -0.089575   0.457937     0.000000   0.000000   )
omega( 9) =      11.085726 [THz] =     369.782482 [cm-1]
( -0.071758   0.220847    -0.071758   0.220847     0.000000   0.000000   )
(  0.232682  -0.320260     0.232682  -0.320260    -0.294668  -0.214089   )
(  0.000000  -0.395863     0.000000  -0.395863     0.364229   0.000000   )
**************************************************************************
diagonalizing the dynamical matrix ...

170.535004
182.852707
252.879501
258.370300
266.006104
280.577301
362.566101
368.381287
369.782501

q =       0.9250      0.9250      0.0000
**************************************************************************
omega( 1) =       5.167330 [THz] =     172.364732 [cm-1]
(  0.442339  -0.329097    -0.442339   0.329097     0.000000   0.000000   )
( -0.287867   0.123077     0.287867  -0.123077     0.000000   0.000000   )
( -0.200616   0.240351     0.200616  -0.240351     0.000000   0.000000   )
omega( 2) =       5.378558 [THz] =     179.410579 [cm-1]
( -0.129809   0.540692    -0.129809   0.540692     0.000000   0.000000   )
( -0.138284   0.271397    -0.138284   0.271397     0.064520   0.032874   )
(  0.000000   0.304596     0.000000   0.304596    -0.072412   0.000000   )
omega( 3) =       7.584358 [THz] =     252.988617 [cm-1]
(  0.000000   0.000000     0.000000   0.000000    -0.042364  -0.010171   )
( -0.054594   0.107147    -0.054594   0.107147    -0.610927  -0.311283   )
(  0.000000  -0.120254     0.000000  -0.120254    -0.685660   0.000000   )
omega( 4) =       7.808341 [THz] =     260.459944 [cm-1]
(  0.000000   0.000000     0.000000   0.000000    -0.440308  -0.105709   )
( -0.193961   0.380669    -0.193961   0.380669     0.160424   0.081740   )
(  0.000000  -0.427235     0.000000  -0.427235     0.180048   0.000000   )
omega( 5) =       7.946584 [THz] =     265.071270 [cm-1]
(  0.000000   0.000000     0.000000   0.000000     0.000000   0.000000   )
( -0.405802   0.292104     0.405802  -0.292104     0.000000   0.000000   )
(  0.228960  -0.444497    -0.228960   0.444497     0.000000   0.000000   )
omega( 6) =       8.319704 [THz] =     277.517305 [cm-1]
(  0.000000   0.000000     0.000000   0.000000     0.662283   0.159000   )
( -0.164136   0.322134    -0.164136   0.322134     0.072537   0.036959   )
(  0.000000  -0.361540     0.000000  -0.361540     0.081410   0.000000   )
omega( 7) =      10.871653 [THz] =     362.641727 [cm-1]
( -0.013202   0.054992    -0.013202   0.054992     0.000000   0.000000   )
(  0.084463  -0.165768     0.084463  -0.165768     0.582623   0.296861   )
(  0.000000  -0.186045     0.000000  -0.186045    -0.653893   0.000000   )
omega( 8) =      11.050990 [THz] =     368.623808 [cm-1]
(  0.097862  -0.233025    -0.097862   0.233025     0.000000   0.000000   )
(  0.276325  -0.376438    -0.276325   0.376438     0.000000   0.000000   )
(  0.075308  -0.460858    -0.075308   0.460858     0.000000   0.000000   )
omega( 9) =      11.075187 [THz] =     369.430936 [cm-1]
( -0.055930   0.232966    -0.055930   0.232966     0.000000   0.000000   )
(  0.192932  -0.378651     0.192932  -0.378651    -0.254209  -0.129526   )
(  0.000000  -0.424970     0.000000  -0.424970     0.285306   0.000000   )
**************************************************************************
diagonalizing the dynamical matrix ...

172.364700
179.410599
252.988602
260.459900
265.071289
277.517303
362.641693
368.623810
369.430908

q =       0.9500      0.9500      0.0000
**************************************************************************
omega( 1) =       5.207227 [THz] =     173.695562 [cm-1]
(  0.393952  -0.387044    -0.393952   0.387044     0.000000   0.000000   )
( -0.254228   0.181294     0.254228  -0.181294     0.000000   0.000000   )
( -0.185762   0.250982     0.185762  -0.250982     0.000000   0.000000   )
omega( 2) =       5.302235 [THz] =     176.864690 [cm-1]
(  0.086726  -0.547564     0.086726  -0.547564     0.000000   0.000000   )
(  0.095328  -0.293390     0.095328  -0.293390    -0.045832  -0.014892   )
(  0.000000  -0.308488     0.000000  -0.308488     0.048190   0.000000   )
omega( 3) =       7.587883 [THz] =     253.106224 [cm-1]
(  0.000000   0.000000     0.000000   0.000000    -0.019256  -0.003050   )
( -0.025701   0.079099    -0.025701   0.079099    -0.663000  -0.215422   )
(  0.000000  -0.083170     0.000000  -0.083170    -0.697119   0.000000   )
omega( 4) =       7.859171 [THz] =     262.155459 [cm-1]
(  0.000000   0.000000     0.000000   0.000000    -0.345835  -0.054775   )
( -0.142325   0.438032    -0.142325   0.438032     0.114290   0.037135   )
(  0.000000  -0.460575     0.000000  -0.460575     0.120172   0.000000   )
omega( 5) =       7.926161 [THz] =     264.390041 [cm-1]
(  0.000000   0.000000     0.000000   0.000000     0.000000   0.000000   )
(  0.264399  -0.424374    -0.264399   0.424374     0.000000   0.000000   )
( -0.120319   0.485307     0.120319  -0.485307     0.000000   0.000000   )
omega( 6) =       8.244308 [THz] =     275.002344 [cm-1]
(  0.000000   0.000000     0.000000   0.000000     0.776467   0.122980   )
( -0.094861   0.291952    -0.094861   0.291952     0.047725   0.015507   )
(  0.000000  -0.306977     0.000000  -0.306977     0.050181   0.000000   )
omega( 7) =      10.872635 [THz] =     362.674492 [cm-1]
( -0.006135   0.038735    -0.006135   0.038735     0.000000   0.000000   )
(  0.039328  -0.121039     0.039328  -0.121039     0.649278   0.210963   )
(  0.000000  -0.127268     0.000000  -0.127268    -0.682692   0.000000   )
omega( 8) =      11.055931 [THz] =     368.788605 [cm-1]
(  0.067935  -0.242448    -0.067935   0.242448     0.000000   0.000000   )
(  0.194892  -0.424640    -0.194892   0.424640     0.000000   0.000000   )
(  0.054132  -0.464082    -0.054132   0.464082     0.000000   0.000000   )
omega( 9) =      11.066863 [THz] =     369.153260 [cm-1]
( -0.038426   0.242610    -0.038426   0.242610     0.000000   0.000000   )
(  0.138374  -0.425871     0.138374  -0.425871    -0.186942  -0.060741   )
(  0.000000  -0.447787     0.000000  -0.447787     0.196562   0.000000   )
**************************************************************************
diagonalizing the dynamical matrix ...

173.695602
176.864700
253.106201
262.155487
264.390015
275.002289
362.674500
368.788605
369.153290

q =       0.9750      0.9750      0.0000
**************************************************************************
omega( 1) =       5.231455 [THz] =     174.503706 [cm-1]
(  0.243465  -0.496342    -0.243465   0.496342     0.000000   0.000000   )
( -0.158826   0.268252     0.158826  -0.268252     0.000000   0.000000   )
( -0.114907   0.289795     0.114907  -0.289795     0.000000   0.000000   )
omega( 2) =       5.255377 [THz] =     175.301680 [cm-1]
(  0.043417  -0.551666     0.043417  -0.551666     0.000000   0.000000   )
(  0.048621  -0.306979     0.048621  -0.306979    -0.023775  -0.003766   )
(  0.000000  -0.310806     0.000000  -0.310806     0.024071   0.000000   )
omega( 3) =       7.590739 [THz] =     253.201497 [cm-1]
(  0.000000   0.000000     0.000000   0.000000     0.004963   0.000391   )
(  0.006713  -0.042387     0.006713  -0.042387     0.695815   0.110206   )
(  0.000000   0.042915     0.000000   0.042915     0.704489   0.000000   )
omega( 4) =       7.895472 [THz] =     263.366337 [cm-1]
(  0.000000   0.000000     0.000000   0.000000    -0.198742  -0.015641   )
( -0.076350   0.482056    -0.076350   0.482056     0.060190   0.009533   )
(  0.000000  -0.488065     0.000000  -0.488065     0.060941   0.000000   )
omega( 5) =       7.913739 [THz] =     263.975668 [cm-1]
(  0.000000   0.000000     0.000000   0.000000     0.000000   0.000000   )
(  0.131005  -0.482533    -0.131005   0.482533     0.000000   0.000000   )
( -0.053908   0.497085     0.053908  -0.497085     0.000000   0.000000   )
omega( 6) =       8.191441 [THz] =     273.238867 [cm-1]
(  0.000000   0.000000     0.000000   0.000000    -0.916819  -0.072155   )
(  0.030660  -0.193581     0.030660  -0.193581    -0.016851  -0.002669   )
(  0.000000   0.195994     0.000000   0.195994    -0.017061   0.000000   )
omega( 7) =      10.872943 [THz] =     362.684763 [cm-1]
( -0.001575   0.020019    -0.001575   0.020019     0.000000   0.000000   )
(  0.010114  -0.063855     0.010114  -0.063855     0.692254   0.109642   )
(  0.000000  -0.064651     0.000000  -0.064651    -0.700883   0.000000   )
omega( 8) =      11.058790 [THz] =     368.883976 [cm-1]
( -0.036765   0.248500     0.036765  -0.248500     0.000000   0.000000   )
( -0.104468   0.455560     0.104468  -0.455560     0.000000   0.000000   )
( -0.031917   0.466293     0.031917  -0.466293     0.000000   0.000000   )
omega( 9) =      11.061549 [THz] =     368.976011 [cm-1]
( -0.019585   0.248848    -0.019585   0.248848     0.000000   0.000000   )
(  0.072335  -0.456708     0.072335  -0.456708    -0.099078  -0.015692   )
(  0.000000  -0.462400     0.000000  -0.462400     0.100313   0.000000   )
**************************************************************************
diagonalizing the dynamical matrix ...

174.503693
175.301697
253.201508
263.366302
263.975708
273.238892
362.684814
368.884003
368.976013

q =       1.0000      1.0000      0.0000
**************************************************************************
omega( 1) =       5.239579 [THz] =     174.774706 [cm-1]
(  0.104210  -0.002667    -0.774879   0.019464     0.000000   0.000000   )
( -0.436565   0.010966     0.058711  -0.001502     0.000000   0.000000   )
( -0.436565   0.010966     0.058711  -0.001502     0.000000   0.000000   )
omega( 2) =       5.239579 [THz] =     174.774706 [cm-1]
( -0.774540  -0.030066    -0.104163  -0.004093     0.000000   0.000000   )
( -0.058685  -0.002306    -0.436374  -0.016939     0.000000   0.000000   )
( -0.058685  -0.002306    -0.436374  -0.016939     0.000000   0.000000   )
omega( 3) =       7.591846 [THz] =     253.238404 [cm-1]
(  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   )
omega( 4) =       7.909569 [THz] =     263.836578 [cm-1]
(  0.000000   0.000000     0.000000   0.000000     0.000000   0.000000   )
( -0.340127  -0.006494     0.619793   0.011327     0.000000   0.000000   )
(  0.340127   0.006494    -0.619793  -0.011327     0.000000   0.000000   )
omega( 5) =       7.909569 [THz] =     263.836578 [cm-1]
(  0.000000   0.000000     0.000000   0.000000     0.000000   0.000000   )
(  0.616368  -0.066041     0.338223  -0.036518     0.000000   0.000000   )
( -0.616368   0.066041    -0.338223   0.036518     0.000000   0.000000   )
omega( 6) =       8.171481 [THz] =     272.573075 [cm-1]
(  0.000000   0.000000     0.000000   0.000000     1.000000   0.000296   )
(  0.000000   0.000000     0.000000   0.000000     0.000000   0.000000   )
(  0.000000   0.000000     0.000000   0.000000     0.000000   0.000000   )
omega( 7) =      10.872996 [THz] =     362.686515 [cm-1]
(  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   )
omega( 8) =      11.059725 [THz] =     368.915167 [cm-1]
( -0.150757  -0.065210     0.289515   0.123338     0.000000   0.000000   )
( -0.539143  -0.229683     0.280745   0.121436     0.000000   0.000000   )
( -0.539143  -0.229683     0.280745   0.121436     0.000000   0.000000   )
omega( 9) =      11.059725 [THz] =     368.915167 [cm-1]
(  0.306189  -0.072661     0.159606  -0.038807     0.000000   0.000000   )
( -0.297223   0.072268    -0.570194   0.135311     0.000000   0.000000   )
( -0.297223   0.072268    -0.570194   0.135311     0.000000   0.000000   )
**************************************************************************

174.774704
174.774704
253.238403
263.836609
263.836609
272.573090
362.686493
368.915192
368.915192

Best Regarts

Fethi Soyalp

```