# [Pw_forum] symmetry of the electron-phonon coupling coefficients

Stefano de Gironcoli degironc at sissa.it
Thu May 10 22:56:49 CEST 2012

```dear Alex

I don't think that the two matrix element should be equal and opposite.
suppose you apply both displacement together which means you displace
the molecule as a whole.

the eigenvalue will not change, hence < i | dV | i > = 0
but the new eigen-wavefunctions | i' > will be the translated of the
original ones, hence different from them.
to first order |i'> = |i> + sum |j> <j|dv|i>/delta e_IJ
since |i'> =/= |i> no reason to believe that < j | d V | i > = 0.

stefano

On 05/10/2012 06:53 PM, Alex Kutana wrote:
> Dear PW forum users and developers:
>
> Could please somebody with the knowledge of inner workings of the
> electron-phonon code help me with this issue:
> I am looking at the symmetry of the electron-phonon coupling coefficients
> with respect to displacement patterns.
> In particular, if one displacement pattern is the negative of the other,
> the coefficients should be identical but with opposite signs.
> However, when I look at these coefficients in, an e.g. a CO molecule, there
> is no such symmetry.
> I have a C atom at the origin, and an O atom on the x axis at 1.133999556
> 0.0 0.0 A.
> The first displacement pattern is
> -1.000000000000000E+000,-0.000000000000000E+000
> -0.000000000000000E+000,-0.000000000000000E+000
> -0.000000000000000E+000,-0.000000000000000E+000
> -0.000000000000000E+000,-0.000000000000000E+000
> -0.000000000000000E+000,-0.000000000000000E+000
> -0.000000000000000E+000,-0.000000000000000E+000
>
> and the second one is
> -0.000000000000000E+000,-0.000000000000000E+000
> -0.000000000000000E+000,-0.000000000000000E+000
> -0.000000000000000E+000,-0.000000000000000E+000
> -1.000000000000000E+000,-0.000000000000000E+000
> -0.000000000000000E+000,-0.000000000000000E+000
> -0.000000000000000E+000,-0.000000000000000E+000
>
> i.e. the second is the negative of the first.
>
> However, the electron-phonon matrix elements for these patterns differ by
> more than just a sign. Below is a list of matrix elements for transitions
> among 9 electronic bands. As expected, the diagonal elements<i | dV | i>
> are same and of opposite signs, but the off-diagonal elements are very
> different. Is there an explanation of this that I was missing?
> Thank you.
>
> Alex Kutana
> Naval Research Laboratory
>
>
> displacement pattern #1:
>     0->0     0.286566684793000 -0.000000000000000 |g|^2= 0.082120464833251
>     1->0    -0.034559192981500  0.000031395207500 |g|^2= 0.001194338805192
>     2->0    -0.000000009558500  0.000000000406000 |g|^2= 0.000000000000000
>     3->0     0.000000003843000 -0.000000000408500 |g|^2= 0.000000000000000
>     4->0     0.209537778542500  0.001560705053000 |g|^2= 0.043908516436788
>     5->0    -0.000000245218500  0.000000483888500 |g|^2= 0.000000000000294
>     6->0    -0.000000151028000 -0.000000094130500 |g|^2= 0.000000000000032
>     7->0     0.015210336449500  0.009035433075000 |g|^2= 0.000312993385760
>     8->0    -0.018916187765500  0.000692293887500 |g|^2= 0.000358301430406
>     0->1    -0.065753284810500 -0.000059732881500 |g|^2= 0.004323498031388
>     1->1    -0.010470695547000  0.000000000000000 |g|^2= 0.000109635465238
>     2->1     0.000000003980500  0.000000000330000 |g|^2= 0.000000000000000
>     3->1    -0.000000001742000  0.000000000782000 |g|^2= 0.000000000000000
>     4->1    -0.116488042493500 -0.000973469004500 |g|^2= 0.013570411685870
>     5->1     0.000000058681500 -0.000000183316500 |g|^2= 0.000000000000037
>     6->1     0.000000093539000  0.000000028930500 |g|^2= 0.000000000000010
>     7->1    -0.007170125204000 -0.004268243132000 |g|^2= 0.000069628594875
>     8->1     0.006675139863000 -0.000242290788500 |g|^2= 0.000044616197017
>     0->2    -0.000000006649000 -0.000000000795500 |g|^2= 0.000000000000000
>     1->2     0.000000003757500 -0.000000000281000 |g|^2= 0.000000000000000
>     2->2     0.172144600342000 -0.000000000000000 |g|^2= 0.029633763426907
>     3->2    -0.000000007757000  0.000000000424000 |g|^2= 0.000000000000000
>     4->2    -0.000000007901500 -0.000000000563000 |g|^2= 0.000000000000000
>     5->2     0.030940253556500 -0.030452295289500 |g|^2= 0.001884641578539
>     6->2    -0.001917560140000  0.001237350092000 |g|^2= 0.000005208072141
>     7->2    -0.000005383433500 -0.000006566545000 |g|^2= 0.000000000072101
>     8->2     0.001206036691000  0.000501228329000 |g|^2= 0.000001705754338
>     0->3     0.000000002888000 -0.000000000118000 |g|^2= 0.000000000000000
>     1->3    -0.000000001701000 -0.000000000735000 |g|^2= 0.000000000000000
>     2->3    -0.000000007757000 -0.000000000424000 |g|^2= 0.000000000000000
>     3->3     0.172144582271500 -0.000000000000000 |g|^2= 0.029633757205429
>     4->3     0.000000004962500  0.000000000048500 |g|^2= 0.000000000000000
>     5->3    -0.002270121070500  0.000239348528500 |g|^2= 0.000005210737393
>     6->3    -0.043196420568000 -0.004313967793500 |g|^2= 0.001884541068011
>     7->3    -0.000000730034000 -0.000003790831000 |g|^2= 0.000000000014903
>     8->3    -0.000008416957500  0.000059829663500 |g|^2= 0.000000003650434
>     0->4     0.214778021593500 -0.001599735961000 |g|^2= 0.046132157714763
>     1->4    -0.124602739906000  0.001041282029500 |g|^2= 0.015526927060347
>     2->4    -0.000000007138000  0.000000000629000 |g|^2= 0.000000000000000
>     3->4     0.000000004680500  0.000000000120000 |g|^2= 0.000000000000000
>     4->4    -0.044695925543000 -0.000000000000000 |g|^2= 0.001997725760145
>     5->4    -0.000000242546000  0.000000487089500 |g|^2= 0.000000000000296
>     6->4    -0.000000143113500 -0.000000102335000 |g|^2= 0.000000000000031
>     7->4    -0.004399490592000 -0.002570431486500 |g|^2= 0.000025962635496
>     8->4    -0.042523842474500  0.001638507220500 |g|^2= 0.001810961884708
>     0->5    -0.000000167311000 -0.000000343222500 |g|^2= 0.000000000000146
>     1->5     0.000000058893000  0.000000158656000 |g|^2= 0.000000000000029
>     2->5     0.032613474298500  0.032099121156000 |g|^2= 0.002093992284806
>     3->5    -0.002392872109500 -0.000252279387500 |g|^2= 0.000005789481822
>     4->5    -0.000000180652000 -0.000000410655000 |g|^2= 0.000000000000201
>     5->5    -0.185897249768000  0.000000000000000 |g|^2= 0.034557787471306
>     6->5    -0.000000087123000 -0.000000071728500 |g|^2= 0.000000000000013
>     7->5     0.000000490662500 -0.000011219521500 |g|^2= 0.000000000126118
>     8->5     0.000649413466000  0.001510073388500 |g|^2= 0.000002702059488
>     0->6    -0.000000102127000  0.000000060209500 |g|^2= 0.000000000000014
>     1->6     0.000000074069000 -0.000000025468000 |g|^2= 0.000000000000006
>     2->6    -0.002021278101500 -0.001304265009000 |g|^2= 0.000005786672377
>     3->6    -0.045532542731000  0.004547271821500 |g|^2= 0.002093890128569
>     4->6    -0.000000144484000  0.000000080179500 |g|^2= 0.000000000000027
>     5->6    -0.000000087123000  0.000000071728500 |g|^2= 0.000000000000013
>     6->6    -0.185898399888000 -0.000000000000000 |g|^2= 0.034558215080919
>     7->6     0.000000984953000  0.000004942462000 |g|^2= 0.000000000025398
>     8->6    -0.000038006325000 -0.000142788293500 |g|^2= 0.000000021832978
>     0->7     0.015372187389000 -0.009132041345000 |g|^2= 0.000319698324249
>     1->7    -0.009727469332000  0.005790740403500 |g|^2= 0.000128156334026
>     2->7    -0.000004934450500  0.000006008964000 |g|^2= 0.000000000060456
>     3->7    -0.000000667739000  0.000003460446000 |g|^2= 0.000000000012421
>     4->7    -0.009877533487000  0.005769648051000 |g|^2= 0.000130854506419
>     5->7     0.000000496943500  0.000011202097500 |g|^2= 0.000000000125734
>     6->7     0.000000981650000 -0.000004929340500 |g|^2= 0.000000000025262
>     7->7    -0.000633793442000  0.000000000000000 |g|^2= 0.000000401694127
>     8->7     0.000030518449500 -0.000027055324500 |g|^2= 0.000000001663366
>     0->8    -0.013941745790500 -0.000533958928500 |g|^2= 0.000194657387824
>     1->8     0.004172651103000  0.000183619549500 |g|^2= 0.000017444733366
>     2->8     0.001106454865000 -0.000459738505500 |g|^2= 0.000001435601862
>     3->8    -0.000007736827500 -0.000054991042000 |g|^2= 0.000000003083873
>     4->8    -0.042832546265500 -0.001607772328500 |g|^2= 0.001837211951446
>     5->8     0.000648997548500 -0.001509835303000 |g|^2= 0.000002700800460
>     6->8    -0.000038017739500  0.000142907934000 |g|^2= 0.000000021868026
>     7->8     0.000025705638000  0.000024838755500 |g|^2= 0.000000001277744
>     8->8    -0.003734049495500  0.000000000000000 |g|^2= 0.000013943125635
>
> displacement pattern #2:
>     0->0    -0.286565847377500 -0.000000000000000 |g|^2= 0.082119984883185
>     1->0     0.379851516091000 -0.000345068730000 |g|^2= 0.144287293349060
>     2->0     0.000000006163500 -0.000000002258000 |g|^2= 0.000000000000000
>     3->0     0.000000001618500 -0.000000001134500 |g|^2= 0.000000000000000
>     4->0    -0.133458411497500 -0.000994042731000 |g|^2= 0.017812135720387
>     5->0     0.000000190857000 -0.000000051579000 |g|^2= 0.000000000000039
>     6->0    -0.000000069523000 -0.000000017790500 |g|^2= 0.000000000000005
>     7->0    -0.019890115224000 -0.011817220800500 |g|^2= 0.000535263391072
>     8->0    -0.039430161288000  0.001065942700500 |g|^2= 0.001555873853038
>     0->1     0.462246877696500  0.000419919480500 |g|^2= 0.213672352272533
>     1->1     0.010494926539500  0.000000000000000 |g|^2= 0.000110143483070
>     2->1    -0.000000003602000 -0.000000000104000 |g|^2= 0.000000000000000
>     3->1    -0.000000004409500  0.000000001875500 |g|^2= 0.000000000000000
>     4->1     0.065269015154500  0.000545440362000 |g|^2= 0.004260341844427
>     5->1    -0.000000404569000  0.000000095724000 |g|^2= 0.000000000000173
>     6->1     0.000000304352500 -0.000000014508500 |g|^2= 0.000000000000093
>     7->1     0.018784573101500  0.011183159264000 |g|^2= 0.000477923237730
>     8->1     0.042649049483000 -0.001095308735500 |g|^2= 0.001820141123029
>     0->2     0.000000009800000  0.000000002671000 |g|^2= 0.000000000000000
>     1->2    -0.000000003294500  0.000000000339500 |g|^2= 0.000000000000000
>     2->2    -0.172141736923500  0.000000000000000 |g|^2= 0.029632777591039
>     3->2     0.000000095818500 -0.000000007298000 |g|^2= 0.000000000000009
>     4->2     0.000000001027500  0.000000000264500 |g|^2= 0.000000000000000
>     5->2     0.058175921580500 -0.057258304602500 |g|^2= 0.006662951297693
>     6->2    -0.003606567884000  0.002326495942500 |g|^2= 0.000018419915272
>     7->2     0.000017034603500  0.000020332543000 |g|^2= 0.000000000703590
>     8->2    -0.003455200640000 -0.001431532813500 |g|^2= 0.000013987697659
>     0->3     0.000000001244500  0.000000000964500 |g|^2= 0.000000000000000
>     1->3    -0.000000004262500 -0.000000001807000 |g|^2= 0.000000000000000
>     2->3     0.000000095818500  0.000000007298000 |g|^2= 0.000000000000009
>     3->3    -0.172141573131500  0.000000000000000 |g|^2= 0.029632721200188
>     4->3     0.000000002276500 -0.000000000110500 |g|^2= 0.000000000000000
>     5->3    -0.004267707540000  0.000449461611500 |g|^2= 0.000018415343387
>     6->3    -0.081224513696500 -0.008111670337000 |g|^2= 0.006663220820889
>     7->3     0.000001861979500  0.000010745158500 |g|^2= 0.000000000118925
>     8->3     0.000028586479000 -0.000167502870000 |g|^2= 0.000000028874398
>     0->4    -0.231003310734000  0.001720588928000 |g|^2= 0.053365489996328
>     1->4     0.047001759819500 -0.000392784013500 |g|^2= 0.002209319705411
>     2->4     0.000000001059500 -0.000000000150500 |g|^2= 0.000000000000000
>     3->4     0.000000002186000  0.000000000118000 |g|^2= 0.000000000000000
>     4->4     0.044659534793500  0.000000000000000 |g|^2= 0.001994474047972
>     5->4     0.000000150192500  0.000000032648000 |g|^2= 0.000000000000024
>     6->4    -0.000000156776000 -0.000000002718000 |g|^2= 0.000000000000025
>     7->4    -0.001310064441500 -0.000765519326000 |g|^2= 0.000002302288679
>     8->4    -0.020407897616500  0.000742934507000 |g|^2= 0.000417034236807
>     0->5     0.000000235769000  0.000000210453500 |g|^2= 0.000000000000100
>     1->5    -0.000000265807500 -0.000000046707500 |g|^2= 0.000000000000073
>     2->5     0.056219406513000  0.055332648710500 |g|^2= 0.006222323681994
>     3->5    -0.004124172441500 -0.000434330422000 |g|^2= 0.000017197441243
>     4->5     0.000000122701500 -0.000000029810500 |g|^2= 0.000000000000016
>     5->5     0.185880014452500  0.000000000000000 |g|^2= 0.034551379772862
>     6->5     0.000000140947000  0.000000125297000 |g|^2= 0.000000000000036
>     7->5    -0.000000941527500  0.000011971521000 |g|^2= 0.000000000144204
>     8->5    -0.000798785488000 -0.001869578039500 |g|^2= 0.000004133380302
>     0->6     0.000000026588500  0.000000024247000 |g|^2= 0.000000000000001
>     1->6     0.000000266009000  0.000000033201000 |g|^2= 0.000000000000072
>     2->6    -0.003485291586500 -0.002248257310000 |g|^2= 0.000017201918375
>     3->6    -0.078492855467500  0.007838873909000 |g|^2= 0.006222576303603
>     4->6    -0.000000139482000 -0.000000000856000 |g|^2= 0.000000000000019
>     5->6     0.000000140947000 -0.000000125297000 |g|^2= 0.000000000000036
>     6->6     0.185880943275000 -0.000000000000000 |g|^2= 0.034551725072804
>     7->6    -0.000001321611500 -0.000005771953000 |g|^2= 0.000000000035062
>     8->6     0.000048260043000  0.000176321833000 |g|^2= 0.000000033418421
>     0->7    -0.026071694374000  0.015489281608000 |g|^2= 0.000919651092263
>     1->7     0.001697741814000 -0.001010730029500 |g|^2= 0.000003903902460
>     2->7     0.000015106160500 -0.000018083799000 |g|^2= 0.000000000555220
>     3->7     0.000001672337500 -0.000009663610500 |g|^2= 0.000000000096182
>     4->7     0.002315940634500 -0.001352389584500 |g|^2= 0.000007192538611
>     5->7    -0.000000927570500 -0.000011708792000 |g|^2= 0.000000000137956
>     6->7    -0.000001291943000  0.000005661201000 |g|^2= 0.000000000033718
>     7->7     0.000818543483500 -0.000000000000000 |g|^2= 0.000000670013434
>     8->7    -0.000425351836000  0.000280356039500 |g|^2= 0.000000259523693
>     0->8    -0.026410868906000 -0.000617126400500 |g|^2= 0.000697914841364
>     1->8     0.008979264003000  0.000226470303000 |g|^2= 0.000080678470834
>     2->8    -0.003085839551500  0.001279050530500 |g|^2= 0.000011158375997
>     3->8     0.000025569678000  0.000149732451000 |g|^2= 0.000000023073615
>     4->8    -0.010002857794000 -0.000375228864000 |g|^2= 0.000100197960747
>     5->8    -0.000775808756500  0.001816292782500 |g|^2= 0.000003900798698
>     6->8     0.000046818670500 -0.000171407224000 |g|^2= 0.000000031572424
>     7->8    -0.000239755938500 -0.000163419629500 |g|^2= 0.000000084188885
>     8->8     0.003793073996500  0.000000000000000 |g|^2= 0.000014387410343
>
>
>
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