Cubic Harmonics as Linear Combinations of Spherical Harmonics. By Jiirg Muggli, Lab. fiir Festk6rperphysik ETH, Ziirich. 1. Introduction. Many calculations in ...
Journal of Applied Mathematics and Physics (ZAM P) Vol. 23, 1972
Birkh~iuser Verlag Basel
Cubic Harmonics as Linear Combinations of Spherical Harmonics By Jiirg Muggli, Lab. fiir Festk6rperphysik ETH, Ziirich
1. Introduction
Many calculations in solid state physics can be simplified appreciably by using orthonormal functions adapted to the symmetry of the material. Since every point group is a subgroup of the hexagonal group D6h or the cubic group Oh it suffices to find a set of orthonormal functions belonging to irreducible representations of the group D 6h or o f O h. The decomposition of these sets under the operations of a subgroup is then a simple task. The orthonormal functions belonging to irreducible representations of D6h are simple combinations of spherical harmonics, which are given for all orders by Altmann and Bradley [1]. Thus only the cubic group has to be investigated.' In the past the cubic harmonics were calculated to twelfth order by Altmann and coworkers [2-4] (unfortunately with some errors) and by Muggli [5] as linear combinations of spherical harmonics (numerically as well as expressed in exact numbers). Functions transforming as the representation Alg were calculated numerically to 30th order by Muller and Priestley [6] and expressed in general formulae by Puff [7]. However, Puffs functions are not orthogonal. Since there is sometimes a need of higher order functions of all representations the numerical expansion of cubic harmonics in spherical harmonics is presented to 30th order in the following. 2. The Irreducible Representations of
Oh
The group Oh is a subgroup of the rotation group. Thus the spherical harmonics of different number I are not mixed. The parity is determined by I. The number n m of equivalent representations R into which the representation of the rotation group D l is split is found by the recursion formula [3] nRl+l 2 =nRl+dimension R. Similar arguments as used, e.g. by Muller and Priestley [6] yield the following results for the different representations. A I ~ I , A 2 ~ x y z and E~(x2--y2,2z2--x2--y2) a r e constituted by spherical harmonics Y/m of even m linearly combined to 1 Y/Cm=~-~ (Y/m-[- Y/-m) (for I even) Vz
or
i Y/~= - - -7 _ (y/m- Y/_~) (for l odd).
1/2
312
Jiirg Muggli
ZAMP
A 1 and E 2 consist of spherical harmonics with m = 0 (mod 4), E 1 and A 2 ofm = 2 (mod 4). The functions belonging to A~ are found easiest. They span the space of those functions which are invariant under a rotation of ~z/2 around the y-axis in addition to the above mentioned restrictions. E 2 functions are then orthogonal to that space, E~ can be found by rotating the E 2 functions and A 2 is orthogonal to E 1. This procedure is carried out in the next section. The T~,-, (x, y, z) and T2 ~ (y z, z x, x y) representations are found in the following manner. The third component of both representations is simple: l even:
T1 : Yl~, T2: Yl~
(m = 4 mod 4) ( m = 2 mod4)
/odd:
T1 : Yl~ T2 : Yl~
( m = 0 mod4) ( m = 2 mod4).
The two remaining components are linear combinations of all Y~mwith m = 1 rood 2. It is much simpler to use only the third component and to turn the reference frame in order to express the first and second component. Thus the first and second component of T~ and T2 are not given as linear combinations of spherical harmonics in the following, since the very large table would be of no use.
3. Numerical Calculation The first task is to find the functions transforming as A 1. The transformation matrix for the spherical harmonics corresponding to a certain rotation is defined, e.g. by Edmonds [-8]. The calculation of the matrix elements d~,, m for a rc/2-rotation around the y-axis (designed in the following by S) is simplified by using the recursion relations found by Altmann and Bradley [-4]. The A 1 functions are expanded in the appropriate spherical harmonics l/4 m=0
(~c= c and s for 1even and odd, respectively). The invariance under S is then expressed by l/4
F, atm(Dm,m--C~m,m)__O, z _
(1)
m--0
where D ~ is the matrix for the spherical harmonics Y~4munder the rotation S. The l for matrix D ' = D ~ - 6 has the dimension s x s with s = + + 1 for even 1 and s 4 odd l. It is singular with rank r = s - p where p is the number nal z of linearly independent A1 functions. Equation (1) means that the vector a~ is orthogonal to all row vectors of D'. In general the last r row vectors of D' are linearly independent (but not orthogonal). The p Al-functions were then found successively. The first function was
Cubic Harmonics as Linear Combinations of Spherical Harmonics
Vol. 23, 1972
313
chosen to have a ~ = 0 for m=0, ..., p - 1 . Choosing the sign of a~p as positive the remaining numbers a~m are unambigous defining the first normalized A~ function. The second A~ function is orthogonal to the last r row vectors of D' and to the first A 1 function. With a , ~ = 0 for m = 0 . . . . . p - 2 and azv_~>O the second A t function is determined too. In the same manner the remaining A~ function was found. The functions transforming as the second component of the E representation are determined as the functions orthonormal to the A~ functions constituted by Y~,m just as A~. The analogous convention of zeros and the sign was used for the different functions. The functions transforming as the first component of the E representation corresponding to the different E 2 functions are found using the transformation property of the spherical harmonics and of the two components of the E representation. The following equation determines the first component
Y~,=
1
Y~=+ ~
2
SYE=
(2)
V~ The transformed function SY~ expressed in spherical harmonics was found with the help of the already used matrix d. Thus all functions YE~ are determined. YE~ Representation A~ L
N
0 4 8 8 9
1 1 1 I I 1 1 2 i I 1 1 2 1 & 2 i 1 2 i 2 I 2 i 1 2 3 1 2 1 2 1 2 1 2 3 1 2 I 2 3
10
12 12 13 14 15 16 16 17 18 18 19 20 21 21 21 22 22 23 24 24 24 25 25 26 26 27 27 28 28 28 29 29 30 30 38
H=
0 1.30000000 .76376262 ,35355339 .71807033 .41142537 .OOOOOOUO .69558267 .44009646 .00000000 .68136168 .00~00000 .45791514 .OOO00OOg .67141495
.00000000 .47032748 .00000000 .00000000 .66391780
.0OOOOOOO .47959635
.00000000 .00000000 .65799999
.OOO00000 ,OOOOZO00 *48685079
4
.64549122 --,93541435 .38188151 .84162541 -.58630197 .55897937 .31412557 .78644109 -.45768183 .29973947 .63704821 .275~6502 .73675368 .14872753 -.3B645598 .38905936 .66299559 .24982619 .00000000 .70210429 .21497472 -.33386008 .43823181 .OOOo000O .67270179 .23043629 .00000000 .67780754 .26049909 --.30642627 .00000000 .46901677 .000~0000 .67603415 .21519894 .00000000 .b6023935 .OOOOOOOO .29408113 -.28097713
8
.58184334 -.54006172 -.69783893 -.80o26751 .34844954 .22821773 -.49113230 -.81009259 -.32~99035 .29J48987 .35031608 -.63774601 -.40209403 -.65102083 -.i1295259 .25782847 .4o890021 .34235861 -.62045553 -.34871263 -.51269686 .26563038 -.00361069 .23~42094 .59489564 .30777385 -.63956246 -.31202490 .$39535~4 -.41197835 .37778608 .05638341 .21856660 .64932864 .27103172 .06998603 -.59F01924 -,28478708
id
16
26
24
28
.19358400 .54422798 -.57395739 -.59634848 .50389111 -.64798075 .32756975 .83622328 .72334167 -.4374o593 -.32224392 -.42738441 .27469053 -,74378409 .25016831 .1512693u -.36650300 -.45529183 -.68d361i3 -.2449~919 .24510495 -.45171996 .30759150 -.11681330 --.32263185 -.52503418 -.44563997 -.63788874 -.&251~217 .22482529 -.2104?231 .32020593 -.36082392 -.24375552 -.29176029
.25572816 .51164543 -.57706829 -.21894516 -.53657149 .56653644 -.52810433 .31248919 .46023523 -.09512844 .62745986 -.40183161 -.07760242 ~ -.44669340 *26292778 -~ .14423142 ~ -.34116420 *71145951 -.29223013 -~ -.31376508 ~ -.43891613 .24294484 .67571877 .13917955 -*30317407
.29347436 .4971995b -.12288760 -.56409799 -.296i1988 -.49587667 .57600523 -.25204068 -.43171390 .30074488 .52429486 -.18506558 ,50757374 -.37590093 -.43261528 .06713397 .56861802 -.43318~03 .25360378 -.10662495 .05296981 -.59060739 .42377~01 -.32189928
.~2759743 .~1864944 .48006030 -,19272710 -.54423128 -.34119025 -.46577346 .i0660~15 .56727211 -.35209583 -.3~3~0400 .29110413 .51884445 -.24514241 .24146891 .40271379 -.35589280
.04830505 .33626926 .46682222 -.24069397 -.52199165 -*03464090 -.369~0004 -.44227251
314
Jtirg Muggli
ZAMP
Representation E, first component L
N
2 4 5 6 7 8 8 9 10 10 11 ii 12 12 13 13 14 14 14 15 15 16 16 16 17 17 17 18 18 18 19 19 i9 28 20 20 20 21 21 21 22 22 22 22 23 25 25 23 24 24 24 24 25 25 25 25 26 26 26 26 25 27 27 27 27 2& 28 28 28 28 29 29 29 29 29 30 38 30 30 30
1 I 1 1 1 I 2 1 1 2 i 2 1 2 1 2 i 2 3 1 2 1 2 3 I 2 3 I 2 3 i 2 3 1 2 3 4 1 2 3 1 2 3 4 1 2 3 4 i 2 3 4 1 2 3 4 i 2 3 4 5 1 2 3 4 1 2 3 4 5 i 2 3 4 5 i 2 3 4 5
M=
2
1.00000000 -I.O000uOJO -l,JOO00000 .95901699 .67700320 -.75935032 -,66068202 -.9013878Z ,39845247 ,44497917 ,64851993" ,36975499 -,83033957 -,51600908 -,72679302 -,47368351 ,31287505 .38861700 ,38276947 .73089700 .25026391 -.36008422 -.77436940 -,43975013 -.59135394 -.57773240 -.29510613 .41500599 ,36708868 ,34169076 ,65531572 ,39208952 .i6448565 -,2008978* -.46404058 -,700990b9 -,35918265 -.66079003 -,47971454 -.20642578 .27282080 .39155379 .34655030 .31183787 ,50313713 ,53537726 .26267184 ,14324947 -.29126909 -,51618229 -.639481~5 -,35255961 -.59992129 -.40467219 -,38990285 -.15504812 .14944227 .358129~2 .34622407 .32850253 .28882858 .61244216 .45620930 ,20045499 .11532286 -.21353187 -,29336004 -,53846670 -.58099662 -,32448811 -,47403164 -.4879~463 -.34238943 -.31353785 -,12211450 .227/8449 ,39~85321 ,31114309 ,31282803 ,27036926
6
10
,82915620 -.73598~? ,65068202 -,75935032 -.43301270 -,85957253 ,48620518 -,74608352 ,48210152 ,13051364 -,54714937 ,21206834 -.64352678 -,86005276 -,19799798 .39917223 -.45472583 .5208i636 .4757249b -.22196134 -,45359219 ,50510226 -.43162962 -.67354181 -.80969425 ,04096368 ,35026484 -,44667171 ,16600846 ,44717806 .51299153 ,2336330? -.36530761 -.39102810 ,44865701 -,65572422 -,52169246 -.57508826 -.48025778 ,1~3658i5 .31700421 -,45462153 -,02630946 ,30450564 ,37316022 .61961117 ,03810065 -.41753434 -,35744867 .47101681 -.11778437 -,68934966 -,41321107 -.39140980 -,54122355 -.24038353 ,1922*457 ,29223232 -~46434536 .20?54015 .328871S7 .31279089 ,50546674 .40698291 -,10075592 -,43132609 -.32727859 ,43126171 .15957023 -.34930250 -.63992767 -.335~448i -,5254620T -.43336120 -.09810110 ,21612295 ,27225417
.31995420 ,75206254 -.15095164 -.79426656 .54175861 -,65906160 -,65329649 .25315311 .39772861 -,80094175 .44109682 =.47903665 .27356454 -,06531451 .48752039 -,49098594 -.62336186 .37478466 I146544957 -.18693539 -,44005336 ,37132634 ,02113631 -,81425776 ,4099266i -,77856912 .15210191 ,16391674 -,4154F903 -.26590621 .1~656110 -,55805017 ,28008200 -,65171100 -,19423949 ,32878801 ,40201485 -,65491926 -.35316330 .43991878 -.5727~03~ .33717277 -,04332157 -.36835215 -,20669659 .33356279 -.13612183 -,52061321 ,52032282 -,23048685 -.63433829 -,04845491 ,29969432 .26403386 -.74781778 -.05242501 ,41346808 -,49850078 -.03498404 .30526700 -.16244468 -,33489777 -,36790690 .26758619 ,22821217 -.34158866 -,46023963 ,51580626 -.5167~695 -~ .03942421 .27786009
14
18
22
26
,08987434 ,46151613 ,6675~069 -.05399448 -.2b060063 -.47184650 .13527946 .55~34725 .16573312 -,55872618 -.24065859 -.51452380 ,58270038 ,45957766 .05971382 -,55572886 .40122432 -,28270978 -.44838904 ,54008208 -,19194877 .27898570 -.49123383 .54113413 -.42705712 -.57740210 .30776617 -,07007275 .00368900 .17888576 -.47775527 .58279635 -.52081048 .33654627 -.47473026 ~ -,25269742 ~ -.49077806 .37932367 -.3439~900 ,43434699 -.3679~478 -.46482508 .20443274 -,51248059 .3917~968 -.24888i46 -.37188884 .00979060 .18375234 -.39279445 .30054789
.Q2317602 .11494014 ,49333682 .63995109 -.00944455 -.08191843 -.34154948 -,73633197 .18165272 .5943b380 ,03876226 -.52817908 -.09821514 -.35235321 -,39339887 .61447862 .18613593 ,49718220 -.184204?5 -.45247100 .38573321 -.28605441 -~ .60242226 -.28592069 .43816716 -.06946570 -.32194033 .48196423 -.40620561 -.33474378 -.35471523 .49653777 -.2970168~ .13055~34 .22084254 -,35064~94 .53888538 -,50390128 ,32440173
.u0329216 .03929225 .14~09719 .5i393019 .61754393 -.01257622 -.i/413023 -.40618793 -.69692320 .Ob603369 .23429919 ,59188493 -*05436739 -.50411589 -.02573736 -.14755818 -.44221622 -.25298852 .62625517 .241971~9 ,46902901 -.33950du2 -,3o615923 .372~9082
30
,06504153 .4102~232 -,17165976 -,79749079 .44876575 ,336(4944 -.5996119B -.22~69310 -.58248169 .49255506 .35936421 -o67~9~Z44 ,42081052 -,60636914 .29621464 --.04~0s 92 6 8 9 4 2 1 3 --,63041627 .56041526 -.45399U95 --.48324494 .20069781 -.18510982 -.5519~893 .41777254 -,51896095 .35163272 -,5437f~07 ,27361926 -.55245864 ,26014332 -.31223193 -,15440108 ,35~43243 -,38e45585 -,06350556 -,70159128 ,h4010669 -,39570317 -,69811000 .219~7254 -,1636948& -.32618507 93 1 2 ~ b 5 5 9 -,17952Z14 -,0~d68364 -.50210583 ,35290237 .114849~3 -.T37~d594 91 8 ~ 6 6 U F b ,16158934 -134722261 .3260d529 -,6895*0i0 -,06272251 92 6 4 2 5 1 9 6 -,4435909~ -.40070331 ,20853002 -.41794048 -.18243461 .28551761
.d05438~5 ,0520299b ,16460288 ,52755106 .5~875489
Vol. 23, 1972
315
Cubic Harmonics as Linear Combinations of Spherical Harmonics
Representation E, second component
L
N
2 4 5 6 7 8 8 9 I0 I0 11 11 12 12 13 13 14 14 14 15 15 16 16 16 17 17 17 18 18 18 19 19 19 20 20 20 28 21 21 21 22 22 22 22 23 23 23 23 24 24 24 24 29 25 29 29 26 26 26 26 26 27 27 27 27 28 28 28 28 28 29 29 29 29 29 30 38 30 88 38
i i i i 1 1 2 I I 2 1 2 1 2 1 2 i 2 3 1 2 I 2 ~ 1 2 3 1 2 3 1 2 3 i 2 8 4 i 2 3 1 2 3 4 1 ~ 3 4 ~ 2 3 4 1 2 3 4 i 2 3 4 5 I 2 3 4 I 2 3 4 5 1 2 3 4 5 i 2 3 4 5
M=
0
4
8
12
16
i,QO000000 ,64549722 -.76376262 I.O000OGbO .93541435 ~ 1.000uOOaO .00000000 . 8 3 5 0 1 7 1 8 -,5487032b .09597~55 -,39400753 -.6U031914 .540U5172 .84162541 ,OOO00OO0 ,76564149 -,64326752 ,9114434~ ,20465658 ,315U0433 ,O00000OO l,O0000DO0 1,000GODOO ,00000000 .O0000OOG .70455149 .33875374 -,62356392 ,71852352 -.304o6127 -,33728553 -.52579140 .00000000 .92923691 .36946447 .61756530 -.29d57777 ,73079U~8 ,00000000 ,OOOOQOO0 .77191640 -.63572406 . O 0000D O 0 ,86035454 - . 3 2 4 0 2 b ~ 4 -.39344274 .89795050 .22431543 ,24070991 .29221764 .OOOOOOOD .52817607 .84913488 ,95402110 .25451924 -.158315~1 .000~0 .0008~000 .85174236 -.49601831 -.15282728 ,000000Oo .67239780 ,09018452 - ,363Jb/58 -.b3869729 ,73194689 -,25680265 -,27041397 -,30495124 -.48187069 .O0000OuO .00~00000 .99780782 ,06617826 .OOOUO00O .85932279 -.0342866b .51695972 ,67616123 -,38170875 -.04170304 .62878078 .00000000 ,O00000OO ,61284128 .32188801 -,72167779 ,O000DOOD ,88820425 -.11457952 -,36196353 -.25874143 ,88899591 ,19906058 .2~111594 .22533543 ,27638396 .O0000UO0 .0~000006 ,86922699 .49441323 .ODO00000 .7~751352 -.3494~174 .61428255 ,92121289 .27494818 ,13b09454 -.23926756 ,00000000 ,~O000OO0 .O000OO00 ,77U~594 -~ ,00000000 ,OOO00O08 .92858568 -.17~68500 -.20097930 .0000~88d .66847587 -.06342719 .23697268 .31115274 .74108162 -.2263488h -,23359085 -.24887031 -.28311318 .00008800 ,O00OO000 .51423512 .85351396 ,OOO00000 .7409270b ,2427350b -.20457332 ,71ZOT413 -.33756528 -.24666568 *09379654 .O0080UO0 .OOOOOO00 ,00080800 .864908~4 -,38906037 ,00000800 ,88000000 ,62790863 -,I0 2 8 9 4 8 9 .37849583 .80000000 .89748148 -.00896399 -,214~4313 -.34568385 .88249196 ,18112973 .18584768 .195~2918 ,21415770 .00080000 .00800800 .O00000bU .99104627 ,00800800 .80888800 ,78722064 -,06233730 ,OOOOOOO0 ,82137789 -,35173977 -,05995244 .89886199 .24996059 .2219733~ ,03783434 .0000~000 .08U~O000 .OOOO0~O0 .59218725 .34866015 ,00~80000 .DO0000uO .90030539 .89995196 -,2672598g .80880000 .67269300 -.14060135 ,09482817 .78563726 ,7478055G -,2045862f -,20981175 -,21760942 -,23343291 .00000000 ,O000000O .O~oOJOUO .81981804 .00080000 .000800~0 ,77609365 -,34953065 ,08080080 .68619953 ,13640499 ,19293583 ,73523937 -,28373268 -.2835b~57 -,13296506 ,80080000 ,00000000 ,8000U800 *OOOUUOuO .78278537 ,O008000U ,00880000 .O00OO000 ,86~47977 -,89296195 ,08000000 ,O00~OODO ,67992253 -.29462196 ,16864901 .O00OO000 .90~10777 ,04714009 -,10863774 -,25733625 ,87748923 ,16747889 .17~53885 ,17535613 .18646509 ,DOOOOUO0 ,OOOO0000 ,SO~UOUbO ,49645830 ,OD08DOSg .00008000 .659f6759 .23143949 ,80808000 ,87346380 -.18559306 -.29035925 .88318979 .218780~9 ,23665601 .15518889 .800~DOOO ,~00008~ .0000~8~8 .DgObOOb~ .85447882 ,O00OO000 ,OOO880~O .00000080 .6731216~ -~ ,00000080 ,O00UOO00 .85913103 ,18761551 -~ .800000~0 .67919549 -.17819017 -.00105645 ~ .75301794 -.18804452 -.1989~725 -.19645620 -.20565895 .~OOaOOuO .00000880 .88000008 ,OOOO080O tOOOOOO00 ~ .OOubOGdb ,75967950 ,00008000 ,00000008 ,88587141 -,24984739 ,O0080DDO .65943759 -.01849073 ,25136084 ,75105525 -,23825918 -,27269654 -,21358850 ,80000000 .OOSOQO00 .UOOggOOO .O0~GOObO ,58302941 .800dO000 ,00800080 .OOOOOOOO .77k66845 .16467622 ,88000000 ,88088~00 .7~540176 -,311854Z8 -,U8824147 .00000808 .90002357 .07854776 -.03973461 -.16952704 ,8734851b .15068706 ,15873892 .162bt889 .16897887
20
24
28
-,02503213 -,25867735 -.62945382 -,45845945 .08411992 ,5g182558 ,55619999 -,34018819 -,67222340 -.17018652 ,26427937 .13351891 ,61114993 ,44499799 -~ -,72542714 -,82934651 ,24749189 -,26708774 ,57217402 .49374893 -.30098288 ,17860954 -,61778152 -,05906320 .27911452 -,31309258 .20545063 ,85417119 -.25542611 ,I0973617 -.~3565133 -.25858662 ,38614222 -,27538320 ,28946503 -,22160833 ,97914803 -,13197219 -,85684963 ,06540661 -,84656482 .38437078 -.20814038 ,26933861 -.27018089 .17941567
-,03880488 -,32737009 -.613~9~27 -.42674851 .02269954 .17810b50 ,61863457 ,50171909 -,07476876 -.48512726 -,59110612 -,1071819& .2545F095 ,15466329 ,66775244 .325984• -.38124846 -.45045665 -,59559652 .08695532 ,18915388 -.254~2474 .20310918 .63559358 ,25861677 -.32731096 .21558034 -.70721659 .11166539 ,15505164 -,2772874u .19836250
-,0u936524 -,Q6274037 -,3~625101 -,5955512~ -,4u791726 .00397441 ,04177799 .27824775 ~ .4~887360 -.11033396 -,58374127 -,52181707 -,~bd23213 ~
316
Jiirg Muggli
ZAMP
Representation A 2 L
N
3 6 7 9 10 11 12 13 14 15 15 16 17 18 18 19 19 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 27 28 28 29 29 38 3@ 30
i 1 1 i 1 i i 1 i 1 2 1 1 1 2 1 2 1 I 2 I 2 1 2 1 2 I 2 i 2 1 2 3 1 2 I 2 1 2 3
M=
2
6
18
14
18
22
26
30
-.57622153 ,58851862 .52166600 -.71352266 ,02017179 -.71210268 ,48301854 -.28838060 -.50429471 -.78288923 t14785472 -.17481526 .37139917 -,4137o758 -.66594800 -.37360689 -.560232&8 .19270766 .07507765 ,29253146 -.48332128 --.43251159 -.66637149 -.28981125 -.37102416 .2~959821 .32825247 .183~7711 .230819~6 -.58345561 -.4220k254 -.5985U707 -,23291022 .1897b909 -,23845383 ,21497381
-.57727979 .26863441 .51216086 .578606s -.60481736 .48308496 -.fl5899941 -.5681756l .46~61285 -.057542~6 ,67613551 -,48444765 -.34922~52 .07678104 -.328~9003 .$8589346 ,73864&09 -.237180~I .06180799 -.38678599 -.46374610 .13515480 -,68293857 -,0925~162 .31821474 .414/4Z~1 -.300~9621 -.21958018 -,31541553 -.55/76881 -,410U164U ,16336895
-.872~0861 -.57277461 .35871602 .45249056 .57665830 -.23914904 -.52972813 .55098666 -.10912824 -,43627564 .44418580 -.44972726 .06S17301 .59696880 -.4579458S -.094850?8 .02531884 ~ -.58553723 .3841i275 .39682070 -.12426999 .31157642 -.38428206 ~68364784 -~ .08982531
-.11239107 -.56630492 .41079848 .40569879 .08784205 .57056296 -.33072264 -.47378992 .5570256U -.14294666 -.26004417 -.31732338 ,~18510~0 -.53405936 .13048575 .47351166 -.43136982 -.41683651 .06927U55 -.013~b488
-.14352963 -.55922496 ,05372767 .44157026 .366292o5 .~3822832 .56855302 -.38518153 -.4301430J .1224i523 .50869245 -.16681938
-.01055860 -,16857112 -.55209205
s .62915620 -.55901699 .67700320 .73598007 .43301270 -.90138752 .15728822 .80201569 945927933 .66536331 .21840635 -,82679728 .49738902 -.493446o4 .24853u84 .77754329 . O O O 0 0 8 ~ U .64864926 ,31816159 .63494689 -.71999430 .278~489i .52454407 - . 3 Z 5 6 2 9 9 2 .38b92100 .0 0 0 0 0 0 ~0 .75912017 .26518649 .OOOOOOO0 , 7 1 9 6 7 9 6 9 ,23456023 .61870199 .319~2370 -.62543930 .2053333~ ,O00OO000 .53909081 -.233447i6 .496G4914 .OOOOOOdO .26352167 ,7~492582 .72638498 ,8Q880000 .18141272 ,60885633 .i0263144 .O000COOO .34725003 -.55401582 .29390486 ,00000000 .54796963 -,17872898 .OOOUO80D . 5 5 4 4 4 0 g 5 .25649218 ,73362353 .O00OOOO0 ,O0000OUO .717D5311 .0800000U .14543202 .68237325 .168532s 00000008 36816849 -.499464~I ,35201683 ooO00OOO 55386384 -.14255974 ,DOODODOO OOOOOOOO .58638190 000000o0 .24802451 .72436677
contains only spherical harmonics with m = 2rood 4. All coefficients with m = 0 mod 4 have to be zero, if the corresponding function YEs is the second component of the E representation and is not mixed with an A 1 function. This is a good check of the numerical calculation. For all orders this numbers were zero to all eight places. The functions belonging to A 2 are then orthogonal to the E 1 functions just as the E 2 functions are orthogonal to the A t functions. All numerical calculations were done on the CDC computer of the ETH using the Algol programm library.
4. Conclusion In addition to the coefficients of the expansion in spherical harmonics there were also calculated the functions belonging to At, A 2 and E as linear combinations of powers of x/r, y/r and z/r, which is useful in some applications since this form represents the symmetry of the functions best. The spherical harmonics which are needed in applications are found using the recursion relations ql+lo
]/2/+3 /+1
(]/2]/2]/2]/2]/2]/2]~lcosO.qlo ~ q l -'~
o)
Vol. 23, 1972
Cubic Harmonics as Linear Combinations of Spherical Harmonics
and
2'+3 ql+lm+l=
~]/(/-.)(/-.i)
(l+m+l)(l+m+2) ~[/
2~--1
317
_~
q,.)
q / - lrn+l
with 1 q00=]/~,
qlo:
]/~ cos 0 ]//~
-1/~ and q i a = 2 F / ~ .
The spherical harmonics are then Ylm=(sign m)mqll,~l sinlml0
e imCsignm
References [1] [2] [3] [4] [5] [6] [-7] [-8]
S.L. ALTMANNand C.J. BRADLEY, Rev. rood. Phys. 37, 33 (1965). S.L. ALTMANNand A. P. CRACKNELL, Rev. mod. Phys. 37, 19 (1965). S.L.ALTMANN, Proc. Cambr. Phil. Soc. 53, 343 (1957). S.L. ALTMANNand C.J. BRADLEY, Phil. Trans. 255, 193 and 199 (1962). J. MUGGLI, Phys. kond. Mat. 12, 237 (1971). F.M. MULLER and M.G. PRIESTLEY, Phys. Rev. 148, 638 (1966). H. PUFF, phys. sat. sol. 41, 11 (1970). A.R. EDMONDS, Angular Momentum in Quantum Mechanics (Princeton Univ. Press, 1957).
The cubic harmonics belonging to different irreducible representations expressed as linear combinations of real spherical harmonics Yz~,(/even: ycz,,- ~ 1]/~ (Y//m+ Yt-,,), load: yl,y =l ]/_i _, (yzm_ , ) s
for even m).
1 2 2 1 The two components of the E-representation are defined as El ~ ~ - _ (x - y ), E 2 ~ -
g'J
(2z 2 _ x 2_y2).
Abstract Orthonormal functions belonging to different irreducible representations of the cubic point group Oh are calculated by a simple numerical procedure. The coefficients of the expansion in spherical harmonics are tabulated for A1, A 2 and E up to l = 30.
Zusammenfassung Es wird ein einfaches numerisches Verfahren beschrieben zur Berechnung eines Orthonormalsystems yon Funktionen, die wie die versehiedenen irreduziblen Darstellungen der kubischen Gruppe 0 n transformieren. Die Koeffizienten der Entwicklung nach Kugelfunktionen sind bis l= 30 fiir A1, A z und E tabelliert. (Received: January 7, 1972)
ZAMP23/21
