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Ionic and paramagnetic energy levels algebra
JOURNAL
OF
MOLECULAR
SPECTROSC’OPY
IONIC AND
26, III-130
(1968)
PARAMAGNETIC
ENERGY
LEVELS
ALGEBRA MAURICE
KIBLER
Section de Recherches de M&unique Ondulatoire Facult& des Sciences de Lyon, Villeurbanne,
AppliquCe France
The algebra of coupling and recoupling coefficients relative to a (finite) subgroup G of the special unitary group in two dimensions SU(2) is developed in this paper. Following the work of Schanfeld on the cubic field, and the work of Flato on the trigonal and tetragonal fields, the J symbols are defined from the ClebschGordan coefficients (j,j,n~,n~~ 1jlj2jmz3) expressed in the jr, scheme where I’7 stands for the y-row of the irreducible representation I? of G. The definition of the irreducible tensorial sets under the group G, leads to the Wigner-Eckart theorem. Properties of the J coefficients and of the related f are obtained from the properties of the 3 - j Wigner’s symbols for SU(2). Utilizing the technique developed by Racah, the recoupling coefficients Jv and X are calculated as function off. Numerical values of the f’s for the cubic and tetragonal groups (electronic configurations d2, f’) are given in the appendix. I. INTRODUCTION
Crystalline field and elect,ronic spin resonance problems call upon the operators @.cry (potential energy of the crystalline field) and +,,,, (potential energy of interaction between the external magnetic field and the L and S angular moment,a) . Traditionally, two ways are used in the determination of eigenstates of The Wigner-Eckart theorem is applied either @.cryand %,ag . (a) to the group Xl’(Z), or (b) to a finite group. (a) Taking into account the interelectronic interaction e”/lij and the spinorbit coupling E(ri) s,.Zi , the electronic state funct’ions laSLJM) for the free at(Jm (Russell-Saunders coupling) provide a basis of the irreducible representation D., of SU(2). The energies acry and %,nagare expressed as linear combinations of t,he components TF’ of the t,ensors Tck’ irreducible under SU(2). Consequently the repeated application of the Wigner-Eckart t.heorem leads t,o the evaluation of matrices for @cIYand %,ag in t’he SL JJ, quantization. The method, we have just recalled, is sometimes useful for the weak crystalline field case. But it does not take advantage of the fact that the invariance group of &,, is a subgroup of the three-dimensional rot,ation group R(3) or is isomor111
112
KIBI,lX
phic
to such a subgroup ( whether the total angular momentum is an integer (11 half an odd int,eger; it is only the spectral group which labels the spin that, chwges 1. (b) On the other hand, in the strong field model (1-J), the irreducible wpwgroup G are introduced. Let, us suppos;c~ that, scnt:ltic)ns of the point qmmetr~ @,) > ?,‘I.,, > (,t( ~~ij~,.z,, +‘1 , @l,,:,,) where @h is a crystalline potentkl of high qmmetr>~, cubic for example, and @1 is a ptJklltid of low symmetry, let II* s:r? tckxgonnl or trigcJlld. The monuelectronic crystalline stBates arc coupled int 1I :1 configul:l~t,ion. Taking into account thch interelectronic repulsion, this CI~IIfiguration is split into terms. These terms xrc (Jbtairled according to the l’auli’s rsclusion principle arid calI UptJrl the: irreducible reprcsentationti of G. 3 ~OIYYI\-VI wch of the operators considered is cogredient \vit’h :L given basis function +I, I’., i for t hc irreducible represontntion I’ of the s;?mmetry group G of the JXLIXI~:L~Iletic ion. Then, matrices for a?,,- and $,,:,, c:tn be cvnluated t#hrough the gen(~~ized Wigncr-Eelart theorem. The inconvcnicnce of this procedure is that it dew Ilot IIse the electrostatic and spill-orbit energy mnbrices already c~:Jcul:ltc~d in thtl ,ST,JJ; representution ( for :I det,ailed description, see Ref. ;j ). ~‘(JllWc]UeIlt~y the simplcl, way of solving the problem of iOIliC cncrgy Icvc~l:: ir; to work iI the WA field scheme but, \vith explicit usage of the symmt~tr~ conditions, i.e., using t)he st:Lt’es laSLJI’, 1 rrl:lted to the states iaSL./.l/ I through the unitary tr~tilsfonli:ttioll ( J:ll ~JT’, ). l~ollowing t.hcse principles, Ychijnfeld (.$) calcul:~tcd the cubic field m:~l ric*tls iclcctwnic configurat~ions tl’, r/“) in ./r, re~nw~Ilt:~ti~Jll. I~hLto (s-6) nchieve(i th(h ccmputat ioI1 of t,hc trigonul :111dtetr:lg(JIl~~~ field matrices for t,he same c.ontig~~.;~~ tioils. The nwthod dcvclopcd in these two prrviou:: \~orks may bcl :~pplicd furt1kt.r tI 1 .~omc new. operators distinct of @C,.Y . l~ollowillg the method of Racuh (Y--.9) it swmc>d int.erest.irqg to build the nlgt4xx of the coupling and recoupling coeffi(*i(~~lt~ c?i~Jres.vc~ irl the rep1 e%Wt:~ti(Jll .Ir, . This point of vie\\- 11;~ xlread~- hectr rls;cl(l for the I-, TV, S coefficients introduced in the strong field scheme (10). .I grwt number of formulas introduced in the following n-ill provide u~c~ful means of control for the computntion of the aCVj-and @,,,:,xm:tt,rices OII thth i./r.., ) hasis. l:llrthermore, it is worth \\.hile looking c:trefull\- :it the tr:UlSfcJrnl:~1 jot, 1:tw front the formul:ts writ.tcn ill the ,131 scheme to t.he formulxs it1 th(L ./I’, scheme. ‘The formal substitution of I’, to thca magnetic quantum tllun tw /I/ [\\,hich is related to the introduction of the crystallographic clunntum tlumb(~r of Hell\\-ege (II)] does not reflect exactly this correspondence principle. ?Lhu m;lttclr of fact, ;ip:Lrt from the trivial problem of the dimensions of t,he co1lsidcrt~(l il.rcducibk rcpresc~r~tatioris, cBomplicated phase manipulations enter illto t II<>.I I‘., Ir~JJrc,sellt:Lt,ioIl. 111 :L forthcomirlg cul:ttioIl
paper
the machinery
of iorlic and paramagnetic
)I = 2, 3 I in a cubic cryst:tlline
eIlergy
developed levels
will
of a11 ion
ficltl \vith Inn- s?mmctry
he applied
to thts c:$
(cl”, .I’” c~r~fig~~r:~ti~,l~~
distorsicJI1.
IONIC ENERGY II.COUPLING
113
LEVELS
COEFFICIENTS
I. Clebsch-Gordan Coeficients and S-j Symbols in the jI’, Scheme Prom the angular momentum theory (Id), it is well known that the transformation from the uncoupled representation jl”jz”j,zj,, to the coupled representation jl”jz~~“j,, is achieved through the unitary matrix (j, jsrnlm 1j&j;ma) . This transformation reduces the Iironecker product Dj, @ Dj, of the irreducible representations Dj, and Dj, of SU(2). Expressed in terms of basis functions /jr,) this leads us to Iaj,j,jZ,,,)
=
C Iqljzhrlrzuz) . rlr2,ylr2
(jlj2rlulh2
I jlh~3r3,,),
(11
where the matrix element (jrjJ’1 rl r 2yt 1j, j2j3r3y3), independent of the additional quantum numbers (Y, is related to the Clebsch-Gordan coefficients (j,j,m,rn, 1 j&?j,rn,) by the formula
Wrlylr2y2 l.h.iti3W =
m,gm, Wlrl Ijm)(j2r2y2
1j2m2)
. (ij2mm2 Ijlj2jn)
(2) (j3m8(j3rsy3)
The last matrix reduces the Kronecker product rl @ r2 of the irreducible representations rl and r2 into a sum of irreducible representations I’3 . The Clebsch-Gordan coefficient (jljzrlvlrzrz / j,j&I’,,,) is zero if one of the following situations occurs (i) jr , j, , j, do not satisfy the triangular condition; (ii) the r zirreducible representations of G are not contained in Dji (i = 1,2,3) ; or (iii) the triple direct product r1 $3~rz @ r3* does not contain the identity. With an adapted choice of the phases for the state functions [jr,), the ClebschGordan coefficients in the jl?, scheme can always be taken real. Notations. In the relation ( 1) , c r zyi means summation over all the irreducible representations ri contained in Dj, , followed by the summation over all the rows yi of each irreducible representation I’< . If ri intervene several times in Djj , we will use the notations I’i’, I?’ . . . When necessary for typographical convenience, r7 will be written p and p0 will denote the single row of the identity representation of G. With this convention, 6(p7, pj) = 6(I’, , I’j)6(yi, -yj). The asterisk will denote the complex conjugate and we will use t,he abbreviations Ij] = 2j + 1, [r] = dimension of the irreducible representation I’ and a = b(i)c means that a runs from b to c, inclusive, in increments of i. Finally, note that S(jr , j2 , j,) = 1 if j, , j2 , j, satisfy the triangular condition and S(j, , j, , j,) = 0 otherwise. The highest degree of symmetry is obtained through introduction of the f coefficient defined by
JCIBLIS11
114 with
expressicw can hr where (..,,r ) is ‘d metric tensor of Wigner (13). h symmetrical found for .i from (3) using (3, (41, the unitary property of the tratlsfomutioll m:ttrix (,j)fl / -jr’, ) and the definitJioxl of t,he 3 - .j symbols of Wigner ( ‘“j i”“s I l?12m;<
The j( ~~~~~~ ) coefficient, is zero if (i ) or (ii) are sat,isfied or if the triple tliwctB product, r, @ I?? @ rn does not, contain the idqtity. Tntroducing R pseudo contittcted product. i,,: t 1, it is possible to avoid thcx phase difictllties \t-hich appear in t.he jr, scheme. The signification of (,$I ) is clmr: t,he clunntit;\,
cc)
J, lj,)lj,‘)
w’
PP
=
c (,,;:tL, >ljnl!Ijnh'),
mm ’
is inv:wi:tnt under t#he [j]-dimcl~sio~~sl sinylect~ic group Sr,( 2j + 1 ) ues of [j] (11’the Ij]-dimensicmal rot.:ition group) K( 2j + 1 ) fol odd cp (&j / jp) is t~rmsforrned under rotations cont~~gredierltly i.e., I“rom EC{. (4 ) md the properties of met)ric tensors (,A, I ), it is easy the s~~mhols (,Lt) obey the following relations
for even valv~dues of lj], to the I,jp’ ). to show that.
I ti )
IONIC
ENERGY
LEVELS
115
each component I$: has the same transformation law under rotation as the basis function cp(kr,)l’ Following RI1 we obtain the Wigner-Eckart theorem
This formula is a generalization of the equation for a tT:kg)component (see Ref. 6, p. 30%). The relation between the f symbol as defined in (6) and the j symbol is
equality which can be reversed using (6). The preceding equalit’y can be rewritten in the following forms
= [jJ””
(-_)““(j, hi1 Pl lj,
kP2/JCL).
Thus the conditions for j t,o be different from zero are t,he same as t,he ones for the corresponding Clebsch-Gordan coefficient in the jr, scheme. The main difference between the j symbol and the 17 symbol recently introduced by Tang Au-chin et al. (14) arises in the utilization of a Clebsch-Gordan coefficient of t,he point symmetry group. However, there is also a discrepancy in phase and dimensions of the irreducible representations considered. The advantage of the f symbol is apparent in the simplification introduced in Eq. (8). Furthermore the numerical evaluation of the j coefficients does not require a previous knowledge of the Clebsch-Gordan coefficients ( l?ll?2y1y2I rlr2r3y3) and avoids the uneasy manipulation of phase factors of the type (- ) I‘. III.
Algebraic
Properties
of the f and f-Matrix
Elements
a. Symmetry. The symmetry properties of the f arise from the properties of the 3 - j symbols of Wigner for SG(T),. Inspection of formula (5) shows that an even permutation of columns in j (&i$i) does not change its value,, whereas an odd permutation does introduce a phase factor given by (-)31+92f,f3. For example, we have
p (i;i!J
= f(;I,.+$j;,),
f (.7;itJ
= (_)i’tJ2+i~j(~;~~3).
Symbol f has lower symmetry than the j symbol. One of its symmetries is easily expressible: permutation of t’he last two columns of f(&$~) give rise to a phase change of value (-) 31+j2+J3,
116
KIBLER
6. Orthonormality Relations. The Clebsch-Gordan coefficients (J&L~P~ 1jlj&cca) are evidently elements of a unitary matrix of dimension bJj2]. Thus FG2 (jti?j3Pclpl5,h/w2) jgy k&P2
(j&w2
(5.&&37
I.m3Pc(3~cG.h~~ IjtiZPlG)
= G,‘,
IMP,‘,
= G4’7 d~h’,
PdQl , j2 , id, Pd.
The former relat’ions can be rewritten in terms off symbols. We get
and the dual relat’ion
Relations (9) and (10) remain true after subst’itution off in place off: in this case t’heq’ will lead to the following formulas which are useful for the evaluation of :I J$ mnkix
Finally notice that for [I’] = [r’] = 1, we have
r. 2lliscellane~us Formulas. (A) From (9) we get the particular case
After summation on p3 both sides of the preceding equalities it, is obvious that If/ 5 1 and Ifi 5 1. Thus from (S) and (11) we obtain the following useful suni rulr
CB ) Utilizing orthogonality properties (6 ), (91, and ( lo), further relations involving~ and (,,:I) symbols can be derived from (7 i . For example after multiplication of ( 7) by (p,:;?,,,J, summation of both sides over pR’ and utilization of (6 ), w have
IONIC
ENERGY
117
LEVELS
With a similar method, we get
and
(C)
When j, = 0 the f($~~:~) coefficient takes the simpler form
4 ) =[jll-““s(jl , jlj20 Pl PZ PO
j&(111, Pd.
(12)
Consequently f( E,i,i,) = 1. (D) Using (12) in relation (9) written in terms of f we obtain
p”(I;{,) = [jl%j’,
0M.A
PO>
This sum rule [to be compared with relation (10.21) of Ref. 91 expresses the center of gravity theorem in the jr-, scheme and is specially useful in the verification of the diagonal elements of the field matrices. The f(E;:) coefficients for the svmmetric group (4 !} and the tetragona1 group are compiled in Se&on IV (J, j’ = 0(1)6, k = 2(2)6). III. I)
RECOUPLING
COEFFICIENTS
TV Symbols
a. IlejPnition. At the begimung (RII), the W coefficients (and the @ deduced from the previous one after symmetrization, see Ref. 9) were defined from the V symbols. l,at,er in RIII, the W coefficients have been related to the transformation lalv between two distinct coupling schemes of three angular momenta. In the following, we will use the former presentation which amounts, in last analysis, to t.he construction of convenient rotat,ional invariants. Thus from the highly symmetrical f coefficients we will build the new coefficients Tvf defined by
( )
[j,]-‘s(j,‘, .Mp3”, P3’Wf ;;;;;
11s
BIBLEIt
t-‘utt)ing j,’ = j, , ps” = pa’ and summing
Utilization
both sides of ( 13) on pa’ MT get
of (7 ) in ( 14) leads to t.he simpler
form
(;;;;)*.7’(p;,) J‘ (y;) j (;:;!J).
..T Rewriting 6(
these expressions
in terms of Clehsch-Gordan
coefficients we get
( )
1:3:,i3 j)‘. j, jgcL3’, p3)( -)j1+j3+j4+j6([jJ[j,J)"'~~,. 6
Obviously, this presentation can also be related to the method used in RIII: the 1v.f are relat,ed to the elements of t,he unitary matrix of transformation bet.sveen two different coupling schemes of three angular momenta. In the problem
= C Ijw llj212) ljd41 (.j&ruw2 I j&jw2) g,I”2Pl?PJ
(j12j31112~3
WC might’ as well couple j, and j, and then add their resultant case
The unitary
transformation
which connects
these t,wo states is
/ .itij:~jp).
jfj to jr and in this
IONIC
ENERGY
LEVELS
On the other hand using the unit’ary property (j?n j jr,), (16) leads t,o
of the transformation
119 matrix
Consequently TV, is identical with the T’ coefficient (which is the 6 - j symbol of Wigner also) with well-known properties ($?,I&‘, 15). b. Afiscellaneous Fotwaulas. Relations involving Iv and f coefficients (with the same total number off on both sides) can be proved from the definition of mf and the orthogonality properties (6)) (9)) and (10). Thus, multiplying (13) by [j3’Jj;( p~“~;j;~l) and th en summing on j,’ and P:” we get’ after utilization of (IO) and reordering of the arguments
and to
Multiplying ( 17) by f( $$i’) *, summing both sides on pz , p4 , ~(6and using (9) we get the well-known orthogonality relation for the 6 - j symbols
II. X Symbols The same procedure we have just used in order to define 6 - j symbols as function off can be applied again to t,he recoupling coefficients 3(n - 1) - j of an arbitrary number n of angular momenta. However n-e will limit ourselves to the case n = 4. Let us define the Xf symbol by
120
that leads to t,he remarkably
KIBLER
symmetric expression
This coefficient can be related to the transformation law between two different coupling schemes of four angular momenta. For t,he sake of brevity, we will not discuss that point. A simple development of (19) show-s that Sf = S. Consequently the defined XI coefficient is a 9 - j symbol of Wigner with well-known properties (see for esample Refs. 9, 13, 15). Relat’ions involving j and .X symbols can be derived from (IS) using orthogonality relations of thef. For example we obtain easily
.j
(;i::;hJ
(;;_i;;hJ
and the relation analogous to the relation of de-Shalit. ( 16) is
Appendix rl den,otes the identity representation of the considered point, group G. Since the f( r$$,) coefficient’s are independent of ?, they will be writ.ten .I’(:;. ;, ) . These coefficients with values J, J’ = O(l)4 and J, J’ = ) z( l)l,?,$ have I~erl calculated by Schiinfeld (J), for the (4 !} group and by Flato (5-G) in t,he I:< !} and tetragonal cases. In Table IV of Ref. (6’) we can find the values of t,hr ex-
IONIC
ENERGY
121
LEVELS
pression (70) ““f( &? ,) with J, J’ = 0( 1)4, for each irreducible represent.ation r of t’he tetragonal group. The necessary values of the f’s for the computation of the cubic and tetragonal field matrices have been calculated by the author for t,hef’ configuration. In order to avoid human errors, the coefficientsf( fi:,) with J, J’ = 0( 1)6 for cubic and tetragonal symmetries have been recalculated on the Univac 110s computer with the program set up by Dr. Y. Bordarier. For convenience, we list here the coefficients of interest for the computation of cubic (potential of degree 4 and 6) and tetragonal (potential of degree 2, 4, and 6) crystal field matrices a:,,~, = go4 + (5/14)1’2(g44 @La
= yo2,
&.tra = go6 +
&a
+ ~4~),
= yo4 -
(1/14)“2(yt4
@:ut, = go6 -
(7/10P2(y44
+
(7/2>“‘($4
+ y46),
y44),
+ y4Y
where ypkis the spherical harmonic of degree k. The tetragonal group is a subgroup of (4 !}. Therefore, we will classify the f coefficients relative to the cubic case according to the irreducible representations of t#hetetragonal group. The cubic f’s are derived in a straightforward manner using the following reductibility law Tetragonal representations
Cubic representations r1
--+
r2 r3 r4
-+
r5
+
---)
+
rl(rl) r2( rd r,(c) + r,(c) r,(b) + r4(r2) r3w5) + r5(r4)
Bethe’s notation is used for the irreducible representations of the cubic group, whereas the notations of Ref. (6) are used for the tetragonal group (Bethe’s notations are recalled in bracket). The (JM 1 JI’,) matrix elements necessary for the computation of thef values listed are from Refs. (5, 6) for J = 0( 1)4 and in the cases J = 5, 6 we used the method given in Ref. (8) in order to obtain the results of the Table A. Tabulation of f( Frrrkl)values is given with restriction J < J’ since for the case considered, we obtain easily the following symmetry relation
Explanation of Tables. The squares of thef’s these values, is the indication of the sign of values obtained are rational fractions, we will the denominator. Coefficients that are zero for
are listed here. In front of each of the corresponding f. Since all the write the numerator first and then physical reasons (6( J, J’, k) = 0)
Representations and corresponding J-value
6
-1
5
-1
4
1
1
d/s
- VW
V%
3
2
l
1
0
-1
-2
-6
Normalization factor
IONIC TABLE
123
LEVELS
1
J 0
J'
2
2 41
2
ENERGY
2
FIFI-VALUES 1 5
J 411
J' 5
FIFI-VALUES 7 330
61 611
-3 -7
5
6
a58 715
35
411
1
30
-1
42
411 5
-2
286
2
411
41
41
0
5
61
-4
143
41
411
4
99
5
611
4
1001
41
5
1
66
61
61
0
41
61 611
611 611
-2 -72
61
0
41
611
10
429
411
411
16
3465
TABLE J
2 J'
FI FI-VALUES
J
J'
FIFI-VALUES
2 2
2 3
2
35 14
61 611
5
-1
4 4
390 a58
2
4 3
1 0
42
3 3
4
-4
3 4
5 4
4
715 5005
5
5
-6
715
5
61
-4
455
105
5
611
-4
1001
3 -16
110 3465
61
61
61
611
0 -2
91
5
-7
330
611
611
72
5005
TABLE J
3 J'
FIFI-VALUES
J
J,
Fi FI-VALUES
1
1
1'
30
41
6111
-1
1
2
411 411
4
572 3465
31
10 28
411
1
-1 -1
51
1
aa
1
311 2
3 -1
140 70
411
511
-7
411 411
5111 61
7 14
440 1320 2145
2 2
31
1
56
2 2
311 41
-3 -1
280 24
411
611
0
411
6111
-7
2
411
-1
168
51
51
5
31
31
0
51
511
0
31 31
311
1
28
51
5111
-7
41
-1
60
51
61
3 0
572 658 206 143
31 31
411
1
105
51
611
51
-1
616
6111
0
31
511
-9
440
51 511
31
5111
-3
311 311
311
1
440 105
41
0
311
411
1
311 311
51 511
5 0
311 41
5111 41
1 -7
aa 495
41
411
-1
220
41
51
0
41
511
1
41
5111
1
110 1320
41
61 611
1 3
17160 104
511 511 511 511 5111 5111 5111 5111 61 61 61 611 611 6111
511 5111 61 611 6111 5111 61 611 6111 61 611 6111 611 6111 6111
-3 -9 -27 3 a 3 -16 0 -27 5 0 27 -11 3 -1
J 4
J' 511
-2
55
4 51
6 51
1 -10
143 429
41 TABLE
2002 2002 4004 910 la20 10010
4
J 1
J' 1
1
3
3 3
3 4
3
51 511 4
3 4 4
za 462
1430 1430 20020 364 1001 1430 5005
51
FIFI-VALUES 15 -2 35 -3 105 -4
51
511
0
231
51
6
0
511 495
SII 6
511 6 6
6 -32 2
0 -10 0 28 0
F1 FI-VALUES
715 1001 5005
1‘24
KIBLER
TABLE J
5 J'
2
2
FIFI-VALUES 35 2
J 4
J' 61
F\F -56
3 4
-1
14 42
4 5
611 5
0 -6
715
3
3
0
5
61
64
5005
3 3
4 5
-4 3
105 110
5 61
611 61
0 -10
1001
4 4
4
-16
3465
5
-7
330
61 611
611 61 I-
0 22
435
TABLE J
6 J'
FIF
J
J'
FIFI-VALUES
0
41
1
9
41 411
611 411
0 2
3861
30
411
5
4
143
61 611
0 a
1287
-7
a58
2 2
1
I-VALUES
I-VALUES 2145
0
411
2 2
2 41
1 0
2
4
99
2
411 5
411 411
-1
66
5
5
2
61
0
2
611
4lY
5 5
61 611
0 1
429
dd61
61
bl
-147
9724
61 611
611 611
0 361
29172
41
41
10 Yt(
41
411
0
41 41
5 61
0
TABLE
-40
1287
7
IF I-VALUES
J
J'
F
2
2
1
2
3
2
4
4
99
2
5 61 611
-1 0
66
2 2
10
429
5
3
-2
33
3
4
3
5
0
3
61
-a
3
611
0
TABLE
30
Lid
J
J'
4
4
2
3861
4
5
4
143
4 4
61 611
0 a
1287
5
5
-7
a58
5 5
61 611
0 1
429
61
61
11
2652
61
611
0
611
611
361
FjF(-VALUES
29172
a
J
J'
FIF
1 1
31 311
0 1
I-VALUES
J
J'
FIFI-VALUES
41
41
49
17
41
411
0
L7
41
51
-5
3861
41
511
-7
429
0
1122
1
41
1
1 1
411 51
0 55
1188
41
5111
1
511
1
132
41
61
0
1 2
5111 2
0 -2
41
611
0
135
41
6111
1
429
2 2
31
-1
L-l
411
411
-13
594
311
0
411
51
0
2
41
0
L
411
1
2 L
51
0
511 5111
0 -Li
2 L
411
511
0
411 411
5111 61
1 5
1287 2574
411 411
611
-1
234
297
6111
0
99
61
-13
L576
51
51
7
a58
611
-5
31L
51
311
5
a58
0 -1
51
5111
0
31
6111 31
51
61
0
31
311
0
31 31
41 411
0 5
51 51
611 6111
0 35
3432
511
511
-7
a5a
31
51
0
511
5111
0
31
511
0
511
61
0
;
5v4
198
IONIC TABLE 31
a (CONTINUED) Sill 35
3a61
31
61
529
54054
JI
611
-5
546
31 311 311 311 311 311 311 311 311 311
6111
0
3iI 41 411 51 511 5111 61 611 6111
1 -7 0 5 -7 0 0 0 -25
TABLE J
9 J'
1
3
1
4
1
51
1 35
FtF 1
66
594 429 429
ENERGY
LEVELS
511 511 5111 5:II 5111 5111 61 61 61 611 611 6111
611 6111 5111 61 611 6111 61 611 6111 611 6111 6111
0 -1 14 -1445 -1 0 -3481 5 0 33 0 -64
I-VALUES 27
3536 7293
27 11tla
J'
Fi
51 511 6 51 511 6 511 6 6
-5 -7 7 7 5 35 -7 -1 -64
Fi F I-VALUtS -13 594
1
132
3 4
1 -7
66 594
3
51
5
429
3
511 6
-7 -25
429
3 4
4
49
1722
TABLE
10 J'
F IF I-VALUES
J
J'
-2
4 4 4 4 5 5 5 61 61 611
4 5 61 611 5 61 611 61 611 611
3003
135 27 99 297 2376 312 594 1YB 3861 54054 546
F I-VALUES
J
511
2 3 4 5 61 611 3
1050192 10608
4 4 4 51 51 51 511 511 6
1
2 2 2 2 2 i 3 3 3 2 3
3432 3861 61776 624
3003
3 3
i
125
3861 429 429 a58 858 3432 a58 3432 7293
1
1287
5 -1
2574
5 61 611
-1 1 -8 -13 -5 -1 5 35 529 -5
TABLE J
11 J'
F t F I-VALUES
J
J'
F1F
0
41
0
0
411
611 41 I 5 61 611 5 61 611 61 611 611
a -640 0 128 -56 5 3 11 0 -135 5
41
61 611 41 411 5 61
1 -1 4 16 -7 -3 -7 0 2 -4 0
41 411 411 411 411 5 5 5 61 61 611
TABLE J
12 J'
FIF
2 2 2 2 2 2 3 3
2 3 4 5 61 611 3 4
I-VALUES 42
J 4
J' 4
FIF 640
105
4
5
0
3465 330
4 4
61 611
0 56
6435
-7 7 0
390 a58
5
5
-5
a58
5 5
61 611
-25 -11
1092 5460
-2
77
61
61
0
4
2
2
i. 2
41
2
i ‘? 41 41 41
411 5
1 4 -16 7
9 42 99 3465 330 286 a58 3i)bl 143
14 -1445
234 3861 61776
-1 -3481
624
5
10608 3536
33
1050192
t- -VALUES 1287 27027 6435 6435 858 2860 5460 9724 51051
I-VALUES 27027
TAtiLk 3
5
-4
3 3
61 611
0 8
TABLE
13 J'
J 1
12
IcUNTINU~U) 429
b1
b1 1
121
18564
611
611
-5
51051
FIFI-VALUES LB 1
J
J' 41
FIFI-VALUES
41
35
30888
5
756
41
411
-15
1144
429
1
31 311
1
41
- -7
540
41
51
7
15444
1 i
411 51
-1 L5
60 4752
41 41
511 5111
-5 5
1716 57L
1 I
511
-7
~640
41
61
1
286
-7
LLO
41
611
1
130
i
5111 2
i
lBY
41
6111
5
1716
i
31
1
3780
411
411
125
216216
L
311 41
-1 -1
28 220
411
51
1
429
L
411
511
0
2
411
11
1260
411
5111
35
5148
2 i
51 511
1 7
d&
i i
5111
-7
411 41 I 411
61 611 6111
-7 7 21
10296 4680 2860
bi
L
611
-77 7
51 51
51 311
5 -7
3432 3432
L
6111
-7
5 IL
21
5111
7
1144
31 31
31
-35
Lj
21
01
-5
572
311
-1
616
51
611
31
41
-1
6111
-49
13728
31
-1
88 5544
51
411
511
511
-5
3432
31
51
-10
1001
511
5111
-5
1144
31 >I
511 5111
1 -1
143 1>444
511 511
61 bI1
-11 81
1456 7280
31
61
123
3CJH8b
511
6111
-5
96096
31
bll
1
jli
.I
4111
1
27L
5111 5111
2111 01
-10 --i
3861 17LY7L8
311 311
511 41
5 4Y
1648
5111
611
7
li480
11880
311
411
-ibY
ziL40
2111 61
6111 61
8 59405
5005 iY405376
311 511
51 511
25 4Y
1LOlL d5kO
61 6i
611 0111
-7 52Y
42432 544544
311 311
5111
-1
~860
61
-2
143
611 611
611 6111
165 405
99008 4Y504
311
611
U
6111
6111
-80
51051
311 TABLE
6111 14 J'
5
J'
440 ZY70 56160 iL‘+b
I6
0
1716
3
FI FI-VALiltS -5 1dY
J 4
21
FIFI-7
I
4
7
135
4
511
5
429
I
51
--L5
1188
511
7
660
t, 51
-5 -5
42Y
i
4 51
3 3
3 4
-5 -4Y
461 LY70
51 51
511 6
7 49
858 3432 858
J i
,VALUtS 3861
858
3
51
-25
3003
3
511
-49
2145
511 511
511 6
5 5
3
6
-5
429
6
6
320
4
4
-35
7722
TABLE d L
15 J' i
FIFI-VALUES -8 1bY
J 4
J' 4
FIFI-VALUtS -125 54054
L
3
-1
Y45
24024 51051
4
5
-35
1287
i
4
-11
315
5
14
1485
4 4
61 611
7 -7
2574
i i
6I
77
14040
5
1170 3861
2
611
-7
312
5
432432
IONIC TABLE 3 3
15 3 4
ENERGY
LET-ELS
(CONTINUED) 35 1
594 Ad%6
5 61 61 bI1
611 61 611 611
-7 -59405 7 -162
3120 7351344 lob08 L457L
2
5
1
3861
5
61
-1L5
71LL
3
6II
-1
78
TAULE .I
16 JI
FIFI-VALUES
J
J'
FIFI-VALUtS
u cl
61 611
1 0
41 411
611 411
0 128
6435
2
41
411
5
3
2860
L L
411 5
411 411
61 61 I'
0 -135
97c4
2
61
5
5
-24
12155
2 41
611 41
41
411
41 41
5 5 61 61 bI1
61 bI1 61 611 b11
0 -35 8 0 -72
FIFI-VALUtS
-3 -4
13
186 143
-2 -40
715 ii%7
5 61
0 -14/
YlL4
TAbLE J
17 J'
F iFI-VALUtS
J
J'
i
4
-3
186
2
5
-4
143
L
61
L
--L
713
3
611 3
-24
1001
j
4
3
5
j
61
0 0 1
52
011 4
4 4 4 5 5 5 61 61 bI1
5 61 611 5 61 61 I 61 611 611
-135 -24 0 -35 88 0 -72
11%
6435 J
41
J' 51
FIFI-VALUES 49 228%
4; 41 41
ii1 5111 61
-7 0 0
41 41
611 6111
0 35
411 411 411 411 411 411 411 51 51 51 51 51 51 511 511 511 511 511 5111 5111 5111 5111 61 61
411 51 511 2111 61 611 6111 51 511 5111 61 bI1 6111 511 5111 61 611 6111 5111 61 611 6111 61 611
-5 0 0 -5 -1 135 0 -10 -21 0 0 0 81 -24 0 0 0 35 30 -7 35 0 -Y61 -245
3 4 TABLE
J
i
1% J@ 51
FlFl _j
I
511
35
1
5111
0
1
61
0
1
611
0
1
6111
-1
i
41
u
-VALUES
572 1716
>Y 4iY
i
411
8
L
51
0
L
511
0
i L
5111 61
1 -25
143 1144
2
611
1
520
L 31
6111 31
0 27
2002
31 31
311 41
0 0
31 31 31
411 51 5Ii
7 0 0
85%
31 51
5111 61 6;l 6111
-3 4Y 5 0
L%6 Yl5i
311
-25
bOO6
311
41
5
286
311
411
0
311 311 311
51 511 5111
-7 -15 0
3; 31 311
832
3432 1144
3
2431 46189 461%9
2860
0
9714 12155 243 1 4199 461%Y
11440
1'1448 2574
1716 466752 14144 7293 4862
19448 12155
lY448 2431 35896
3536 369512 33592
1”S
KIBLEI’, 18
TABLE.
~CONTINLJELJI
311
41
0
311
611
0
311
6111
-9
61 611
6111 611
0 11
1144
611
6111
0
lL870
6111
6111
2
FIFI-VALUES
335Y2
41
41
1
41
411
0
TABLE J
19 J'
FjF(-VALUES
J
J'
1
51
-3
572
4
51
49
2288
1 I
511 6
35 -1
1716 IY
4 4
511 6
-7 35
11440 lY448
6006 Lb6
51 51
51 511
-10 -21
7293 4862 19448
46189
3
3
-25
3
4
5
3
51
-7
6
81
511
-15
343/! 1144
51
3
311
511
-L4
12155
3
6
-Y
1144
511
6
35
19448
4
4
1
lLM70
6
6
2
46189
FIFI-VALUES
TABLE
20
J
J'
FIFI-VALUES
J
J’
2
4
8
429
4
5
-5
1716
i 2
5
145 1144
4
61
-I
466152
61
I -25
4
611
135
14144
2
611
1
520
5
5
30
2431
3
J 4
L7
LOO2
5
bl
-7
38896
7
858
5
611
35
3536
286
61
61
-Y61
36Y512
3 3
5
-3
3
61
44
Yl52
bl
bi1
-L45
jd5Yi
611
611
11
33592
F(FI-VALUES
3
bIi
5
b3d
4
4
-5
2574
TABLE J
21 J'
FIFI-VALUES
J
J'
cl
61
0'
41
611
361
29172
lJ
611
1
13
411
411
-56
6435
2
41
10
4L9
411
5
11
5460
L
411
-7
ti58
411
-135
Y7L4
4
1001
61 611
5
51051
5 61
-168 35
12155 L431
2
5
2
61
-i
715
‘
611
-7L
5005
411 5 5
u 8 1
5
611
0
41
41 411 5
1287 429
61 61
61 611
0 -72
46109
41
61
G
611
611
-224
46189
TABLE J
22 J'
F IFI-VALUES
FIFI-VALUES
2 i
4 5
7 -4
i
61
L 3
611 3
3 3 3
J
J'
l35n 1001
4
5
-11
5460
4
61
-121
lb564 51051
91
4
611
-5
;L 0
5005
5
5
16b
12155
5
61
-1
221
4
a
429
5
611
0
5 61
-24 0
1001
61 61
61 611
0 0
611
611
LZ4
3
611
5
4004
4
4
56
6435
TABLE J
23 J'
FIFI-VALUES
J
J'
FlF(-
1
51
~1
ZL88
41
51
7
9152
1 1 i
511 5111
5 5
6864 5li
41 4I
511 5111
4Y i8Y
45760 34320
bI bII
5 YY
LYlL
41
61
lll6Y
I
ZYli
41
611
15
Ab67008 56576
1
6111
-1
1OYL
41
--~45
717YL
z
41
8
4L3
41
-343
51480
CIlI
I
41
I
4618Y
VALUES
IONIC
TABLE
23 411 51
15
22B8
2
511
21
2288
2 2
5111 6I
25 11
2912
2
611
1
311 41 411 51
31
511
31 31 31 31 311 311 311 311 jI1 311 311 311 311 41 41 J 1 I 1 3 3 3 3 3 4
5111 61 611 6111 311 41 411 51 411 5111 61 611 6111 41 411 24 J’ 51 SlI 6 3 4 51 511 6 4
TABLE J
25 J'
2
4
2
5
;
TABLE
3
61 6II 3
3
4
5
5
3
61
3
611 4
4
129
LEVELS
(CONTINUED)
2 2
2 31 31 31 31 31
ENERGY
0
-5
-361 -3 5 -a1 27 529 5 -375 25 5 5 4Y -15 5 105 -363 -9
-7 9
4004
14560 20020 1144 1144 24024 1144 4576 32032 8008 256256 23296 32032 3432 8008 1144 1372ir 32032 8008 36608 23296
32032 51480 5720
411 411 411 411 411 411 51 51 51 51 51 51 511 511 511 511 511 5111 5111 5111 5111 61 61 61 611 611 6111
51 -25 511 77 5111 3 61 -10443 bII 1115 6111 3610 51 35 511 -3 5111 -25 61 lY5 363 611 81 6111 42 511 -567 5111 33 61 15 611 -245 6111 -21 5111 -17 61 611 -15 6111 -119 61 -35 611 -1785 6111 -77 611 61 I I 525 -7 6111
FlFi-VALUES
J
-21
572
-5
1716
1
273
-25
850
-5 -49
2002 3432
15
8008
Y 7
8008 12870
4 4 4 51 51 51 511 511 6
J’ 31 511 6 51 511 6 511 6 6
J 4 4 4 5 5 5 61 61 611
J’ 5 61 611 5 61 611 61 611 611
FI FI-VALUES
-25 -11 -1 -3 3 -27 -529 -5 343
1001 728 3640 286 286 2002 64064 5824 12870
9152 4160 80080 4356352 39603L 10850&H 14586 19448 77792 83716 99008 544544 12155 388Y60 14144 14144 777YL 24310 Y152 14144 4862 86Y44 134368 173888 134368 268736 92378
FIFI-VALutS -7
LL88
-49
11440
245
1944b
-70
7293
3
4862
-al
136136
-i6a
12155
245 14
19448 46189
FtFI-VALUES -3
20020
10443
1089088
-135 42
9Yooa 12155
17
2288
-5
3536
119
21736
35
33592
77
33592
are omitted and accidental zeros are denoted by 0. Each table gives (&:,) for a fixed k and a fixed irreducible representation r of the tetragonal group. Consequently the f’s are indexed only by J and J’. However, if the representation I? appears several times in D, then we use the notations JI, JII . . * and for exam1rr) is denoted 2 31. ple fG& Numerical example: in the table 13 we read
HIBLEI’L
1:50
I. J’( ,$;yf, ) (for &,,,,
potenti:tl):
Table
1: I’ = I’1 ; Table
2: r =
I’? ; Table
3:
I’ = I’:< ; Table 4: r = r,, ; Table 5: I’ = r5 . 11. j( ,::!;.:I ) (for du,, potential) : Table G: r = 1’1 ; Table
7: I? = r2 ; Table S: ; Table 9: r = I’.$ ; Table 10: I’ = IT5 . Table 11: r = I’1; Table 12: r = I’? ; TaIII . ,I’( r+‘,>:I, ) (for @.tutr;lpotential): ble 13: r = rn ; Tdde 14: r = rd ; T:iblc 15: I’ = rs. Ii’. j’( ,::!,::I) fw (&, pot~ent~ial) : Tshle 16: r’ = 1’1 ; T:lble 17: I? = rz ; Txhlc 1s: 1’ = I‘:{ ; Table 19 : I’ = I’d ; T:lble “0: I’ = r5 . \‘.. ./‘I ,.;:‘,.‘~II) (for @tCt,.iLpotential i : Table 21 : I’ = r1; Table 22: r = 1‘: ; TnNC28:r = r:+; T~~hl~ 21: I‘ = I’, ; T:lhle 25: I’ = r5. I’ =
I’;,
The aut her wishes to thank Prof. &I. Plato for his continuous assistarlce awl dt-voted guidalbcc it1 the preparation of this work. IIe also wish(,s to Olank Prof. II. Chabbal for the kind hospitality esteuded to him in the “Laburatoirc Aimi! Cotton” and for fillancial sup port. Thallks are also due to I)r. 1.. Bordarier for utilixatioll of his computer program and IO t)r. ht. (‘aitla.rd for ilrleresliilg dixllssiuns.
/. Y. T\s.\i+E .\Nu S. Suu.\ruo, J. i’hjjs. Sot. Japan 9, 753 (1954). J. I’hp. SW. Japan 13, 394 (1958). .? Y. T.\s.\HE .\su II. KAMIRTJXA, 3. J. S. (~RIFFJTII, “The Theory of Transition ;LIetal-Tolls.” Cambridge Llniv. Press, LUIII.IOII and New York, 19ci-l. 4. (;. SclIONFELx) (M.Sc. Thesis, Cniversity of Jerllsalem, 1958). -5. M. I’I.\w (RI.&-. Thesis, 1:niversity of Jerusalem, 1959). i;, M. E‘L\TO, J.dlOl. SpeCfr!/. 17, 300 (1965). 7. (;. I?.x.\H, Z’hys. Rev. 62, -438 (1942). 8. G. I:.w.\H, Phys. I2ev. 63, X7 (19-U). 9. 11. F.lso .\ND c;. kWH, “Irredrlribte Tellscrrial Sets.” Academic Press, New York. 1959. I/I. J. 5. (GRIFFITH ” The Irrrdrlciblr Tensor Method for Molecular Symmetry (;rr,llps.” Prt’llt ive-Ilat;, Jj;ltglewc)od Cliff’s, 191i2. If. Ii. Il. IlEl,I#\\.EGt:, .IW. Pk!/Sik 4,95 (1918j. I,“. 12. I-. ('ONDON .INI) (i. It. SH~BRT~.EY, “Ttle Theory of Atomic, Spectra.” (‘amhridge Iilliv. Press, Lolldot~ atld New York, 1963. IS. 11;. I’. WIGKEH, ” 011 the Xlatrices which lHI.\N-~I:R, l’ \A’ (~o-SEN., (btr ZIEIS, INI) T\I SHI~-SH.\N. hi. Sirhim (I’ckimJ) 16, (5) (19tiGJ. 1.5. A. It. t';I)MOSI)S. “ Anglllar &lomerlllun itI Quantum Mechanics.” 2nd rd. Pritlcetan I-niv. PEWS. Prillcetoll, New Jorst:y, 19W. /(i. A. I)E~SII\LI'L., I'hqa, I
OF
MOLECULAR
SPECTROSC’OPY
IONIC AND
26, III-130
(1968)
PARAMAGNETIC
ENERGY
LEVELS
ALGEBRA MAURICE
KIBLER
Section de Recherches de M&unique Ondulatoire Facult& des Sciences de Lyon, Villeurbanne,
AppliquCe France
The algebra of coupling and recoupling coefficients relative to a (finite) subgroup G of the special unitary group in two dimensions SU(2) is developed in this paper. Following the work of Schanfeld on the cubic field, and the work of Flato on the trigonal and tetragonal fields, the J symbols are defined from the ClebschGordan coefficients (j,j,n~,n~~ 1jlj2jmz3) expressed in the jr, scheme where I’7 stands for the y-row of the irreducible representation I? of G. The definition of the irreducible tensorial sets under the group G, leads to the Wigner-Eckart theorem. Properties of the J coefficients and of the related f are obtained from the properties of the 3 - j Wigner’s symbols for SU(2). Utilizing the technique developed by Racah, the recoupling coefficients Jv and X are calculated as function off. Numerical values of the f’s for the cubic and tetragonal groups (electronic configurations d2, f’) are given in the appendix. I. INTRODUCTION
Crystalline field and elect,ronic spin resonance problems call upon the operators @.cry (potential energy of the crystalline field) and +,,,, (potential energy of interaction between the external magnetic field and the L and S angular moment,a) . Traditionally, two ways are used in the determination of eigenstates of The Wigner-Eckart theorem is applied either @.cryand %,ag . (a) to the group Xl’(Z), or (b) to a finite group. (a) Taking into account the interelectronic interaction e”/lij and the spinorbit coupling E(ri) s,.Zi , the electronic state funct’ions laSLJM) for the free at(Jm (Russell-Saunders coupling) provide a basis of the irreducible representation D., of SU(2). The energies acry and %,nagare expressed as linear combinations of t,he components TF’ of the t,ensors Tck’ irreducible under SU(2). Consequently the repeated application of the Wigner-Eckart t.heorem leads t,o the evaluation of matrices for @cIYand %,ag in t’he SL JJ, quantization. The method, we have just recalled, is sometimes useful for the weak crystalline field case. But it does not take advantage of the fact that the invariance group of &,, is a subgroup of the three-dimensional rot,ation group R(3) or is isomor111
112
KIBI,lX
phic
to such a subgroup ( whether the total angular momentum is an integer (11 half an odd int,eger; it is only the spectral group which labels the spin that, chwges 1. (b) On the other hand, in the strong field model (1-J), the irreducible wpwgroup G are introduced. Let, us suppos;c~ that, scnt:ltic)ns of the point qmmetr~ @,) > ?,‘I.,, > (,t( ~~ij~,.z,, +‘1 , @l,,:,,) where @h is a crystalline potentkl of high qmmetr>~, cubic for example, and @1 is a ptJklltid of low symmetry, let II* s:r? tckxgonnl or trigcJlld. The monuelectronic crystalline stBates arc coupled int 1I :1 configul:l~t,ion. Taking into account thch interelectronic repulsion, this CI~IIfiguration is split into terms. These terms xrc (Jbtairled according to the l’auli’s rsclusion principle arid calI UptJrl the: irreducible reprcsentationti of G. 3 ~OIYYI\-VI wch of the operators considered is cogredient \vit’h :L given basis function +I, I’., i for t hc irreducible represontntion I’ of the s;?mmetry group G of the JXLIXI~:L~Iletic ion. Then, matrices for a?,,- and $,,:,, c:tn be cvnluated t#hrough the gen(~~ized Wigncr-Eelart theorem. The inconvcnicnce of this procedure is that it dew Ilot IIse the electrostatic and spill-orbit energy mnbrices already c~:Jcul:ltc~d in thtl ,ST,JJ; representution ( for :I det,ailed description, see Ref. ;j ). ~‘(JllWc]UeIlt~y the simplcl, way of solving the problem of iOIliC cncrgy Icvc~l:: ir; to work iI the WA field scheme but, \vith explicit usage of the symmt~tr~ conditions, i.e., using t)he st:Lt’es laSLJI’, 1 rrl:lted to the states iaSL./.l/ I through the unitary tr~tilsfonli:ttioll ( J:ll ~JT’, ). l~ollowing t.hcse principles, Ychijnfeld (.$) calcul:~tcd the cubic field m:~l ric*tls iclcctwnic configurat~ions tl’, r/“) in ./r, re~nw~Ilt:~ti~Jll. I~hLto (s-6) nchieve(i th(h ccmputat ioI1 of t,hc trigonul :111dtetr:lg(JIl~~~ field matrices for t,he same c.ontig~~.;~~ tioils. The nwthod dcvclopcd in these two prrviou:: \~orks may bcl :~pplicd furt1kt.r tI 1 .~omc new. operators distinct of @C,.Y . l~ollowillg the method of Racuh (Y--.9) it swmc>d int.erest.irqg to build the nlgt4xx of the coupling and recoupling coeffi(*i(~~lt~ c?i~Jres.vc~ irl the rep1 e%Wt:~ti(Jll .Ir, . This point of vie\\- 11;~ xlread~- hectr rls;cl(l for the I-, TV, S coefficients introduced in the strong field scheme (10). .I grwt number of formulas introduced in the following n-ill provide u~c~ful means of control for the computntion of the aCVj-and @,,,:,xm:tt,rices OII thth i./r.., ) hasis. l:llrthermore, it is worth \\.hile looking c:trefull\- :it the tr:UlSfcJrnl:~1 jot, 1:tw front the formul:ts writ.tcn ill the ,131 scheme to t.he formulxs it1 th(L ./I’, scheme. ‘The formal substitution of I’, to thca magnetic quantum tllun tw /I/ [\\,hich is related to the introduction of the crystallographic clunntum tlumb(~r of Hell\\-ege (II)] does not reflect exactly this correspondence principle. ?Lhu m;lttclr of fact, ;ip:Lrt from the trivial problem of the dimensions of t,he co1lsidcrt~(l il.rcducibk rcpresc~r~tatioris, cBomplicated phase manipulations enter illto t II<>.I I‘., Ir~JJrc,sellt:Lt,ioIl. 111 :L forthcomirlg cul:ttioIl
paper
the machinery
of iorlic and paramagnetic
)I = 2, 3 I in a cubic cryst:tlline
eIlergy
developed levels
will
of a11 ion
ficltl \vith Inn- s?mmctry
he applied
to thts c:$
(cl”, .I’” c~r~fig~~r:~ti~,l~~
distorsicJI1.
IONIC ENERGY II.COUPLING
113
LEVELS
COEFFICIENTS
I. Clebsch-Gordan Coeficients and S-j Symbols in the jI’, Scheme Prom the angular momentum theory (Id), it is well known that the transformation from the uncoupled representation jl”jz”j,zj,, to the coupled representation jl”jz~~“j,, is achieved through the unitary matrix (j, jsrnlm 1j&j;ma) . This transformation reduces the Iironecker product Dj, @ Dj, of the irreducible representations Dj, and Dj, of SU(2). Expressed in terms of basis functions /jr,) this leads us to Iaj,j,jZ,,,)
=
C Iqljzhrlrzuz) . rlr2,ylr2
(jlj2rlulh2
I jlh~3r3,,),
(11
where the matrix element (jrjJ’1 rl r 2yt 1j, j2j3r3y3), independent of the additional quantum numbers (Y, is related to the Clebsch-Gordan coefficients (j,j,m,rn, 1 j&?j,rn,) by the formula
Wrlylr2y2 l.h.iti3W =
m,gm, Wlrl Ijm)(j2r2y2
1j2m2)
. (ij2mm2 Ijlj2jn)
(2) (j3m8(j3rsy3)
The last matrix reduces the Kronecker product rl @ r2 of the irreducible representations rl and r2 into a sum of irreducible representations I’3 . The Clebsch-Gordan coefficient (jljzrlvlrzrz / j,j&I’,,,) is zero if one of the following situations occurs (i) jr , j, , j, do not satisfy the triangular condition; (ii) the r zirreducible representations of G are not contained in Dji (i = 1,2,3) ; or (iii) the triple direct product r1 $3~rz @ r3* does not contain the identity. With an adapted choice of the phases for the state functions [jr,), the ClebschGordan coefficients in the jl?, scheme can always be taken real. Notations. In the relation ( 1) , c r zyi means summation over all the irreducible representations ri contained in Dj, , followed by the summation over all the rows yi of each irreducible representation I’< . If ri intervene several times in Djj , we will use the notations I’i’, I?’ . . . When necessary for typographical convenience, r7 will be written p and p0 will denote the single row of the identity representation of G. With this convention, 6(p7, pj) = 6(I’, , I’j)6(yi, -yj). The asterisk will denote the complex conjugate and we will use t,he abbreviations Ij] = 2j + 1, [r] = dimension of the irreducible representation I’ and a = b(i)c means that a runs from b to c, inclusive, in increments of i. Finally, note that S(jr , j2 , j,) = 1 if j, , j2 , j, satisfy the triangular condition and S(j, , j, , j,) = 0 otherwise. The highest degree of symmetry is obtained through introduction of the f coefficient defined by
JCIBLIS11
114 with
expressicw can hr where (..,,r ) is ‘d metric tensor of Wigner (13). h symmetrical found for .i from (3) using (3, (41, the unitary property of the tratlsfomutioll m:ttrix (,j)fl / -jr’, ) and the definitJioxl of t,he 3 - .j symbols of Wigner ( ‘“j i”“s I l?12m;<
The j( ~~~~~~ ) coefficient, is zero if (i ) or (ii) are sat,isfied or if the triple tliwctB product, r, @ I?? @ rn does not, contain the idqtity. Tntroducing R pseudo contittcted product. i,,: t 1, it is possible to avoid thcx phase difictllties \t-hich appear in t.he jr, scheme. The signification of (,$I ) is clmr: t,he clunntit;\,
cc)
J, lj,)lj,‘)
w’
PP
=
c (,,;:tL, >ljnl!Ijnh'),
mm ’
is inv:wi:tnt under t#he [j]-dimcl~sio~~sl sinylect~ic group Sr,( 2j + 1 ) ues of [j] (11’the Ij]-dimensicmal rot.:ition group) K( 2j + 1 ) fol odd cp (&j / jp) is t~rmsforrned under rotations cont~~gredierltly i.e., I“rom EC{. (4 ) md the properties of met)ric tensors (,A, I ), it is easy the s~~mhols (,Lt) obey the following relations
for even valv~dues of lj], to the I,jp’ ). to show that.
I ti )
IONIC
ENERGY
LEVELS
115
each component I$: has the same transformation law under rotation as the basis function cp(kr,)l’ Following RI1 we obtain the Wigner-Eckart theorem
This formula is a generalization of the equation for a tT:kg)component (see Ref. 6, p. 30%). The relation between the f symbol as defined in (6) and the j symbol is
equality which can be reversed using (6). The preceding equalit’y can be rewritten in the following forms
= [jJ””
(-_)““(j, hi1 Pl lj,
kP2/JCL).
Thus the conditions for j t,o be different from zero are t,he same as t,he ones for the corresponding Clebsch-Gordan coefficient in the jr, scheme. The main difference between the j symbol and the 17 symbol recently introduced by Tang Au-chin et al. (14) arises in the utilization of a Clebsch-Gordan coefficient of t,he point symmetry group. However, there is also a discrepancy in phase and dimensions of the irreducible representations considered. The advantage of the f symbol is apparent in the simplification introduced in Eq. (8). Furthermore the numerical evaluation of the j coefficients does not require a previous knowledge of the Clebsch-Gordan coefficients ( l?ll?2y1y2I rlr2r3y3) and avoids the uneasy manipulation of phase factors of the type (- ) I‘. III.
Algebraic
Properties
of the f and f-Matrix
Elements
a. Symmetry. The symmetry properties of the f arise from the properties of the 3 - j symbols of Wigner for SG(T),. Inspection of formula (5) shows that an even permutation of columns in j (&i$i) does not change its value,, whereas an odd permutation does introduce a phase factor given by (-)31+92f,f3. For example, we have
p (i;i!J
= f(;I,.+$j;,),
f (.7;itJ
= (_)i’tJ2+i~j(~;~~3).
Symbol f has lower symmetry than the j symbol. One of its symmetries is easily expressible: permutation of t’he last two columns of f(&$~) give rise to a phase change of value (-) 31+j2+J3,
116
KIBLER
6. Orthonormality Relations. The Clebsch-Gordan coefficients (J&L~P~ 1jlj&cca) are evidently elements of a unitary matrix of dimension bJj2]. Thus FG2 (jti?j3Pclpl5,h/w2) jgy k&P2
(j&w2
(5.&&37
I.m3Pc(3~cG.h~~ IjtiZPlG)
= G,‘,
IMP,‘,
= G4’7 d~h’,
PdQl , j2 , id, Pd.
The former relat’ions can be rewritten in terms off symbols. We get
and the dual relat’ion
Relations (9) and (10) remain true after subst’itution off in place off: in this case t’heq’ will lead to the following formulas which are useful for the evaluation of :I J$ mnkix
Finally notice that for [I’] = [r’] = 1, we have
r. 2lliscellane~us Formulas. (A) From (9) we get the particular case
After summation on p3 both sides of the preceding equalities it, is obvious that If/ 5 1 and Ifi 5 1. Thus from (S) and (11) we obtain the following useful suni rulr
CB ) Utilizing orthogonality properties (6 ), (91, and ( lo), further relations involving~ and (,,:I) symbols can be derived from (7 i . For example after multiplication of ( 7) by (p,:;?,,,J, summation of both sides over pR’ and utilization of (6 ), w have
IONIC
ENERGY
117
LEVELS
With a similar method, we get
and
(C)
When j, = 0 the f($~~:~) coefficient takes the simpler form
4 ) =[jll-““s(jl , jlj20 Pl PZ PO
j&(111, Pd.
(12)
Consequently f( E,i,i,) = 1. (D) Using (12) in relation (9) written in terms of f we obtain
p”(I;{,) = [jl%j’,
0M.A
PO>
This sum rule [to be compared with relation (10.21) of Ref. 91 expresses the center of gravity theorem in the jr-, scheme and is specially useful in the verification of the diagonal elements of the field matrices. The f(E;:) coefficients for the svmmetric group (4 !} and the tetragona1 group are compiled in Se&on IV (J, j’ = 0(1)6, k = 2(2)6). III. I)
RECOUPLING
COEFFICIENTS
TV Symbols
a. IlejPnition. At the begimung (RII), the W coefficients (and the @ deduced from the previous one after symmetrization, see Ref. 9) were defined from the V symbols. l,at,er in RIII, the W coefficients have been related to the transformation lalv between two distinct coupling schemes of three angular momenta. In the following, we will use the former presentation which amounts, in last analysis, to t.he construction of convenient rotat,ional invariants. Thus from the highly symmetrical f coefficients we will build the new coefficients Tvf defined by
( )
[j,]-‘s(j,‘, .Mp3”, P3’Wf ;;;;;
11s
BIBLEIt
t-‘utt)ing j,’ = j, , ps” = pa’ and summing
Utilization
both sides of ( 13) on pa’ MT get
of (7 ) in ( 14) leads to t.he simpler
form
(;;;;)*.7’(p;,) J‘ (y;) j (;:;!J).
..T Rewriting 6(
these expressions
in terms of Clehsch-Gordan
coefficients we get
( )
1:3:,i3 j)‘. j, jgcL3’, p3)( -)j1+j3+j4+j6([jJ[j,J)"'~~,. 6
Obviously, this presentation can also be related to the method used in RIII: the 1v.f are relat,ed to the elements of t,he unitary matrix of transformation bet.sveen two different coupling schemes of three angular momenta. In the problem
= C Ijw llj212) ljd41 (.j&ruw2 I j&jw2) g,I”2Pl?PJ
(j12j31112~3
WC might’ as well couple j, and j, and then add their resultant case
The unitary
transformation
which connects
these t,wo states is
/ .itij:~jp).
jfj to jr and in this
IONIC
ENERGY
LEVELS
On the other hand using the unit’ary property (j?n j jr,), (16) leads t,o
of the transformation
119 matrix
Consequently TV, is identical with the T’ coefficient (which is the 6 - j symbol of Wigner also) with well-known properties ($?,I&‘, 15). b. Afiscellaneous Fotwaulas. Relations involving Iv and f coefficients (with the same total number off on both sides) can be proved from the definition of mf and the orthogonality properties (6)) (9)) and (10). Thus, multiplying (13) by [j3’Jj;( p~“~;j;~l) and th en summing on j,’ and P:” we get’ after utilization of (IO) and reordering of the arguments
and to
Multiplying ( 17) by f( $$i’) *, summing both sides on pz , p4 , ~(6and using (9) we get the well-known orthogonality relation for the 6 - j symbols
II. X Symbols The same procedure we have just used in order to define 6 - j symbols as function off can be applied again to t,he recoupling coefficients 3(n - 1) - j of an arbitrary number n of angular momenta. However n-e will limit ourselves to the case n = 4. Let us define the Xf symbol by
120
that leads to t,he remarkably
KIBLER
symmetric expression
This coefficient can be related to the transformation law between two different coupling schemes of four angular momenta. For t,he sake of brevity, we will not discuss that point. A simple development of (19) show-s that Sf = S. Consequently the defined XI coefficient is a 9 - j symbol of Wigner with well-known properties (see for esample Refs. 9, 13, 15). Relat’ions involving j and .X symbols can be derived from (IS) using orthogonality relations of thef. For example we obtain easily
.j
(;i::;hJ
(;;_i;;hJ
and the relation analogous to the relation of de-Shalit. ( 16) is
Appendix rl den,otes the identity representation of the considered point, group G. Since the f( r$$,) coefficient’s are independent of ?, they will be writ.ten .I’(:;. ;, ) . These coefficients with values J, J’ = O(l)4 and J, J’ = ) z( l)l,?,$ have I~erl calculated by Schiinfeld (J), for the (4 !} group and by Flato (5-G) in t,he I:< !} and tetragonal cases. In Table IV of Ref. (6’) we can find the values of t,hr ex-
IONIC
ENERGY
121
LEVELS
pression (70) ““f( &? ,) with J, J’ = 0( 1)4, for each irreducible represent.ation r of t’he tetragonal group. The necessary values of the f’s for the computation of the cubic and tetragonal field matrices have been calculated by the author for t,hef’ configuration. In order to avoid human errors, the coefficientsf( fi:,) with J, J’ = 0( 1)6 for cubic and tetragonal symmetries have been recalculated on the Univac 110s computer with the program set up by Dr. Y. Bordarier. For convenience, we list here the coefficients of interest for the computation of cubic (potential of degree 4 and 6) and tetragonal (potential of degree 2, 4, and 6) crystal field matrices a:,,~, = go4 + (5/14)1’2(g44 @La
= yo2,
&.tra = go6 +
&a
+ ~4~),
= yo4 -
(1/14)“2(yt4
@:ut, = go6 -
(7/10P2(y44
+
(7/2>“‘($4
+ y46),
y44),
+ y4Y
where ypkis the spherical harmonic of degree k. The tetragonal group is a subgroup of (4 !}. Therefore, we will classify the f coefficients relative to the cubic case according to the irreducible representations of t#hetetragonal group. The cubic f’s are derived in a straightforward manner using the following reductibility law Tetragonal representations
Cubic representations r1
--+
r2 r3 r4
-+
r5
+
---)
+
rl(rl) r2( rd r,(c) + r,(c) r,(b) + r4(r2) r3w5) + r5(r4)
Bethe’s notation is used for the irreducible representations of the cubic group, whereas the notations of Ref. (6) are used for the tetragonal group (Bethe’s notations are recalled in bracket). The (JM 1 JI’,) matrix elements necessary for the computation of thef values listed are from Refs. (5, 6) for J = 0( 1)4 and in the cases J = 5, 6 we used the method given in Ref. (8) in order to obtain the results of the Table A. Tabulation of f( Frrrkl)values is given with restriction J < J’ since for the case considered, we obtain easily the following symmetry relation
Explanation of Tables. The squares of thef’s these values, is the indication of the sign of values obtained are rational fractions, we will the denominator. Coefficients that are zero for
are listed here. In front of each of the corresponding f. Since all the write the numerator first and then physical reasons (6( J, J’, k) = 0)
Representations and corresponding J-value
6
-1
5
-1
4
1
1
d/s
- VW
V%
3
2
l
1
0
-1
-2
-6
Normalization factor
IONIC TABLE
123
LEVELS
1
J 0
J'
2
2 41
2
ENERGY
2
FIFI-VALUES 1 5
J 411
J' 5
FIFI-VALUES 7 330
61 611
-3 -7
5
6
a58 715
35
411
1
30
-1
42
411 5
-2
286
2
411
41
41
0
5
61
-4
143
41
411
4
99
5
611
4
1001
41
5
1
66
61
61
0
41
61 611
611 611
-2 -72
61
0
41
611
10
429
411
411
16
3465
TABLE J
2 J'
FI FI-VALUES
J
J'
FIFI-VALUES
2 2
2 3
2
35 14
61 611
5
-1
4 4
390 a58
2
4 3
1 0
42
3 3
4
-4
3 4
5 4
4
715 5005
5
5
-6
715
5
61
-4
455
105
5
611
-4
1001
3 -16
110 3465
61
61
61
611
0 -2
91
5
-7
330
611
611
72
5005
TABLE J
3 J'
FIFI-VALUES
J
J,
Fi FI-VALUES
1
1
1'
30
41
6111
-1
1
2
411 411
4
572 3465
31
10 28
411
1
-1 -1
51
1
aa
1
311 2
3 -1
140 70
411
511
-7
411 411
5111 61
7 14
440 1320 2145
2 2
31
1
56
2 2
311 41
-3 -1
280 24
411
611
0
411
6111
-7
2
411
-1
168
51
51
5
31
31
0
51
511
0
31 31
311
1
28
51
5111
-7
41
-1
60
51
61
3 0
572 658 206 143
31 31
411
1
105
51
611
51
-1
616
6111
0
31
511
-9
440
51 511
31
5111
-3
311 311
311
1
440 105
41
0
311
411
1
311 311
51 511
5 0
311 41
5111 41
1 -7
aa 495
41
411
-1
220
41
51
0
41
511
1
41
5111
1
110 1320
41
61 611
1 3
17160 104
511 511 511 511 5111 5111 5111 5111 61 61 61 611 611 6111
511 5111 61 611 6111 5111 61 611 6111 61 611 6111 611 6111 6111
-3 -9 -27 3 a 3 -16 0 -27 5 0 27 -11 3 -1
J 4
J' 511
-2
55
4 51
6 51
1 -10
143 429
41 TABLE
2002 2002 4004 910 la20 10010
4
J 1
J' 1
1
3
3 3
3 4
3
51 511 4
3 4 4
za 462
1430 1430 20020 364 1001 1430 5005
51
FIFI-VALUES 15 -2 35 -3 105 -4
51
511
0
231
51
6
0
511 495
SII 6
511 6 6
6 -32 2
0 -10 0 28 0
F1 FI-VALUES
715 1001 5005
1‘24
KIBLER
TABLE J
5 J'
2
2
FIFI-VALUES 35 2
J 4
J' 61
F\F -56
3 4
-1
14 42
4 5
611 5
0 -6
715
3
3
0
5
61
64
5005
3 3
4 5
-4 3
105 110
5 61
611 61
0 -10
1001
4 4
4
-16
3465
5
-7
330
61 611
611 61 I-
0 22
435
TABLE J
6 J'
FIF
J
J'
FIFI-VALUES
0
41
1
9
41 411
611 411
0 2
3861
30
411
5
4
143
61 611
0 a
1287
-7
a58
2 2
1
I-VALUES
I-VALUES 2145
0
411
2 2
2 41
1 0
2
4
99
2
411 5
411 411
-1
66
5
5
2
61
0
2
611
4lY
5 5
61 611
0 1
429
dd61
61
bl
-147
9724
61 611
611 611
0 361
29172
41
41
10 Yt(
41
411
0
41 41
5 61
0
TABLE
-40
1287
7
IF I-VALUES
J
J'
F
2
2
1
2
3
2
4
4
99
2
5 61 611
-1 0
66
2 2
10
429
5
3
-2
33
3
4
3
5
0
3
61
-a
3
611
0
TABLE
30
Lid
J
J'
4
4
2
3861
4
5
4
143
4 4
61 611
0 a
1287
5
5
-7
a58
5 5
61 611
0 1
429
61
61
11
2652
61
611
0
611
611
361
FjF(-VALUES
29172
a
J
J'
FIF
1 1
31 311
0 1
I-VALUES
J
J'
FIFI-VALUES
41
41
49
17
41
411
0
L7
41
51
-5
3861
41
511
-7
429
0
1122
1
41
1
1 1
411 51
0 55
1188
41
5111
1
511
1
132
41
61
0
1 2
5111 2
0 -2
41
611
0
135
41
6111
1
429
2 2
31
-1
L-l
411
411
-13
594
311
0
411
51
0
2
41
0
L
411
1
2 L
51
0
511 5111
0 -Li
2 L
411
511
0
411 411
5111 61
1 5
1287 2574
411 411
611
-1
234
297
6111
0
99
61
-13
L576
51
51
7
a58
611
-5
31L
51
311
5
a58
0 -1
51
5111
0
31
6111 31
51
61
0
31
311
0
31 31
41 411
0 5
51 51
611 6111
0 35
3432
511
511
-7
a5a
31
51
0
511
5111
0
31
511
0
511
61
0
;
5v4
198
IONIC TABLE 31
a (CONTINUED) Sill 35
3a61
31
61
529
54054
JI
611
-5
546
31 311 311 311 311 311 311 311 311 311
6111
0
3iI 41 411 51 511 5111 61 611 6111
1 -7 0 5 -7 0 0 0 -25
TABLE J
9 J'
1
3
1
4
1
51
1 35
FtF 1
66
594 429 429
ENERGY
LEVELS
511 511 5111 5:II 5111 5111 61 61 61 611 611 6111
611 6111 5111 61 611 6111 61 611 6111 611 6111 6111
0 -1 14 -1445 -1 0 -3481 5 0 33 0 -64
I-VALUES 27
3536 7293
27 11tla
J'
Fi
51 511 6 51 511 6 511 6 6
-5 -7 7 7 5 35 -7 -1 -64
Fi F I-VALUtS -13 594
1
132
3 4
1 -7
66 594
3
51
5
429
3
511 6
-7 -25
429
3 4
4
49
1722
TABLE
10 J'
F IF I-VALUES
J
J'
-2
4 4 4 4 5 5 5 61 61 611
4 5 61 611 5 61 611 61 611 611
3003
135 27 99 297 2376 312 594 1YB 3861 54054 546
F I-VALUES
J
511
2 3 4 5 61 611 3
1050192 10608
4 4 4 51 51 51 511 511 6
1
2 2 2 2 2 i 3 3 3 2 3
3432 3861 61776 624
3003
3 3
i
125
3861 429 429 a58 858 3432 a58 3432 7293
1
1287
5 -1
2574
5 61 611
-1 1 -8 -13 -5 -1 5 35 529 -5
TABLE J
11 J'
F t F I-VALUES
J
J'
F1F
0
41
0
0
411
611 41 I 5 61 611 5 61 611 61 611 611
a -640 0 128 -56 5 3 11 0 -135 5
41
61 611 41 411 5 61
1 -1 4 16 -7 -3 -7 0 2 -4 0
41 411 411 411 411 5 5 5 61 61 611
TABLE J
12 J'
FIF
2 2 2 2 2 2 3 3
2 3 4 5 61 611 3 4
I-VALUES 42
J 4
J' 4
FIF 640
105
4
5
0
3465 330
4 4
61 611
0 56
6435
-7 7 0
390 a58
5
5
-5
a58
5 5
61 611
-25 -11
1092 5460
-2
77
61
61
0
4
2
2
i. 2
41
2
i ‘? 41 41 41
411 5
1 4 -16 7
9 42 99 3465 330 286 a58 3i)bl 143
14 -1445
234 3861 61776
-1 -3481
624
5
10608 3536
33
1050192
t- -VALUES 1287 27027 6435 6435 858 2860 5460 9724 51051
I-VALUES 27027
TAtiLk 3
5
-4
3 3
61 611
0 8
TABLE
13 J'
J 1
12
IcUNTINU~U) 429
b1
b1 1
121
18564
611
611
-5
51051
FIFI-VALUES LB 1
J
J' 41
FIFI-VALUES
41
35
30888
5
756
41
411
-15
1144
429
1
31 311
1
41
- -7
540
41
51
7
15444
1 i
411 51
-1 L5
60 4752
41 41
511 5111
-5 5
1716 57L
1 I
511
-7
~640
41
61
1
286
-7
LLO
41
611
1
130
i
5111 2
i
lBY
41
6111
5
1716
i
31
1
3780
411
411
125
216216
L
311 41
-1 -1
28 220
411
51
1
429
L
411
511
0
2
411
11
1260
411
5111
35
5148
2 i
51 511
1 7
d&
i i
5111
-7
411 41 I 411
61 611 6111
-7 7 21
10296 4680 2860
bi
L
611
-77 7
51 51
51 311
5 -7
3432 3432
L
6111
-7
5 IL
21
5111
7
1144
31 31
31
-35
Lj
21
01
-5
572
311
-1
616
51
611
31
41
-1
6111
-49
13728
31
-1
88 5544
51
411
511
511
-5
3432
31
51
-10
1001
511
5111
-5
1144
31 >I
511 5111
1 -1
143 1>444
511 511
61 bI1
-11 81
1456 7280
31
61
123
3CJH8b
511
6111
-5
96096
31
bll
1
jli
.I
4111
1
27L
5111 5111
2111 01
-10 --i
3861 17LY7L8
311 311
511 41
5 4Y
1648
5111
611
7
li480
11880
311
411
-ibY
ziL40
2111 61
6111 61
8 59405
5005 iY405376
311 511
51 511
25 4Y
1LOlL d5kO
61 6i
611 0111
-7 52Y
42432 544544
311 311
5111
-1
~860
61
-2
143
611 611
611 6111
165 405
99008 4Y504
311
611
U
6111
6111
-80
51051
311 TABLE
6111 14 J'
5
J'
440 ZY70 56160 iL‘+b
I6
0
1716
3
FI FI-VALiltS -5 1dY
J 4
21
FIFI-7
I
4
7
135
4
511
5
429
I
51
--L5
1188
511
7
660
t, 51
-5 -5
42Y
i
4 51
3 3
3 4
-5 -4Y
461 LY70
51 51
511 6
7 49
858 3432 858
J i
,VALUtS 3861
858
3
51
-25
3003
3
511
-49
2145
511 511
511 6
5 5
3
6
-5
429
6
6
320
4
4
-35
7722
TABLE d L
15 J' i
FIFI-VALUES -8 1bY
J 4
J' 4
FIFI-VALUtS -125 54054
L
3
-1
Y45
24024 51051
4
5
-35
1287
i
4
-11
315
5
14
1485
4 4
61 611
7 -7
2574
i i
6I
77
14040
5
1170 3861
2
611
-7
312
5
432432
IONIC TABLE 3 3
15 3 4
ENERGY
LET-ELS
(CONTINUED) 35 1
594 Ad%6
5 61 61 bI1
611 61 611 611
-7 -59405 7 -162
3120 7351344 lob08 L457L
2
5
1
3861
5
61
-1L5
71LL
3
6II
-1
78
TAULE .I
16 JI
FIFI-VALUES
J
J'
FIFI-VALUtS
u cl
61 611
1 0
41 411
611 411
0 128
6435
2
41
411
5
3
2860
L L
411 5
411 411
61 61 I'
0 -135
97c4
2
61
5
5
-24
12155
2 41
611 41
41
411
41 41
5 5 61 61 bI1
61 bI1 61 611 b11
0 -35 8 0 -72
FIFI-VALUtS
-3 -4
13
186 143
-2 -40
715 ii%7
5 61
0 -14/
YlL4
TAbLE J
17 J'
F iFI-VALUtS
J
J'
i
4
-3
186
2
5
-4
143
L
61
L
--L
713
3
611 3
-24
1001
j
4
3
5
j
61
0 0 1
52
011 4
4 4 4 5 5 5 61 61 bI1
5 61 611 5 61 61 I 61 611 611
-135 -24 0 -35 88 0 -72
11%
6435 J
41
J' 51
FIFI-VALUES 49 228%
4; 41 41
ii1 5111 61
-7 0 0
41 41
611 6111
0 35
411 411 411 411 411 411 411 51 51 51 51 51 51 511 511 511 511 511 5111 5111 5111 5111 61 61
411 51 511 2111 61 611 6111 51 511 5111 61 bI1 6111 511 5111 61 611 6111 5111 61 611 6111 61 611
-5 0 0 -5 -1 135 0 -10 -21 0 0 0 81 -24 0 0 0 35 30 -7 35 0 -Y61 -245
3 4 TABLE
J
i
1% J@ 51
FlFl _j
I
511
35
1
5111
0
1
61
0
1
611
0
1
6111
-1
i
41
u
-VALUES
572 1716
>Y 4iY
i
411
8
L
51
0
L
511
0
i L
5111 61
1 -25
143 1144
2
611
1
520
L 31
6111 31
0 27
2002
31 31
311 41
0 0
31 31 31
411 51 5Ii
7 0 0
85%
31 51
5111 61 6;l 6111
-3 4Y 5 0
L%6 Yl5i
311
-25
bOO6
311
41
5
286
311
411
0
311 311 311
51 511 5111
-7 -15 0
3; 31 311
832
3432 1144
3
2431 46189 461%9
2860
0
9714 12155 243 1 4199 461%Y
11440
1'1448 2574
1716 466752 14144 7293 4862
19448 12155
lY448 2431 35896
3536 369512 33592
1”S
KIBLEI’, 18
TABLE.
~CONTINLJELJI
311
41
0
311
611
0
311
6111
-9
61 611
6111 611
0 11
1144
611
6111
0
lL870
6111
6111
2
FIFI-VALUES
335Y2
41
41
1
41
411
0
TABLE J
19 J'
FjF(-VALUES
J
J'
1
51
-3
572
4
51
49
2288
1 I
511 6
35 -1
1716 IY
4 4
511 6
-7 35
11440 lY448
6006 Lb6
51 51
51 511
-10 -21
7293 4862 19448
46189
3
3
-25
3
4
5
3
51
-7
6
81
511
-15
343/! 1144
51
3
311
511
-L4
12155
3
6
-Y
1144
511
6
35
19448
4
4
1
lLM70
6
6
2
46189
FIFI-VALUES
TABLE
20
J
J'
FIFI-VALUES
J
J’
2
4
8
429
4
5
-5
1716
i 2
5
145 1144
4
61
-I
466152
61
I -25
4
611
135
14144
2
611
1
520
5
5
30
2431
3
J 4
L7
LOO2
5
bl
-7
38896
7
858
5
611
35
3536
286
61
61
-Y61
36Y512
3 3
5
-3
3
61
44
Yl52
bl
bi1
-L45
jd5Yi
611
611
11
33592
F(FI-VALUES
3
bIi
5
b3d
4
4
-5
2574
TABLE J
21 J'
FIFI-VALUES
J
J'
cl
61
0'
41
611
361
29172
lJ
611
1
13
411
411
-56
6435
2
41
10
4L9
411
5
11
5460
L
411
-7
ti58
411
-135
Y7L4
4
1001
61 611
5
51051
5 61
-168 35
12155 L431
2
5
2
61
-i
715
‘
611
-7L
5005
411 5 5
u 8 1
5
611
0
41
41 411 5
1287 429
61 61
61 611
0 -72
46109
41
61
G
611
611
-224
46189
TABLE J
22 J'
F IFI-VALUES
FIFI-VALUES
2 i
4 5
7 -4
i
61
L 3
611 3
3 3 3
J
J'
l35n 1001
4
5
-11
5460
4
61
-121
lb564 51051
91
4
611
-5
;L 0
5005
5
5
16b
12155
5
61
-1
221
4
a
429
5
611
0
5 61
-24 0
1001
61 61
61 611
0 0
611
611
LZ4
3
611
5
4004
4
4
56
6435
TABLE J
23 J'
FIFI-VALUES
J
J'
FlF(-
1
51
~1
ZL88
41
51
7
9152
1 1 i
511 5111
5 5
6864 5li
41 4I
511 5111
4Y i8Y
45760 34320
bI bII
5 YY
LYlL
41
61
lll6Y
I
ZYli
41
611
15
Ab67008 56576
1
6111
-1
1OYL
41
--~45
717YL
z
41
8
4L3
41
-343
51480
CIlI
I
41
I
4618Y
VALUES
IONIC
TABLE
23 411 51
15
22B8
2
511
21
2288
2 2
5111 6I
25 11
2912
2
611
1
311 41 411 51
31
511
31 31 31 31 311 311 311 311 jI1 311 311 311 311 41 41 J 1 I 1 3 3 3 3 3 4
5111 61 611 6111 311 41 411 51 411 5111 61 611 6111 41 411 24 J’ 51 SlI 6 3 4 51 511 6 4
TABLE J
25 J'
2
4
2
5
;
TABLE
3
61 6II 3
3
4
5
5
3
61
3
611 4
4
129
LEVELS
(CONTINUED)
2 2
2 31 31 31 31 31
ENERGY
0
-5
-361 -3 5 -a1 27 529 5 -375 25 5 5 4Y -15 5 105 -363 -9
-7 9
4004
14560 20020 1144 1144 24024 1144 4576 32032 8008 256256 23296 32032 3432 8008 1144 1372ir 32032 8008 36608 23296
32032 51480 5720
411 411 411 411 411 411 51 51 51 51 51 51 511 511 511 511 511 5111 5111 5111 5111 61 61 61 611 611 6111
51 -25 511 77 5111 3 61 -10443 bII 1115 6111 3610 51 35 511 -3 5111 -25 61 lY5 363 611 81 6111 42 511 -567 5111 33 61 15 611 -245 6111 -21 5111 -17 61 611 -15 6111 -119 61 -35 611 -1785 6111 -77 611 61 I I 525 -7 6111
FlFi-VALUES
J
-21
572
-5
1716
1
273
-25
850
-5 -49
2002 3432
15
8008
Y 7
8008 12870
4 4 4 51 51 51 511 511 6
J’ 31 511 6 51 511 6 511 6 6
J 4 4 4 5 5 5 61 61 611
J’ 5 61 611 5 61 611 61 611 611
FI FI-VALUES
-25 -11 -1 -3 3 -27 -529 -5 343
1001 728 3640 286 286 2002 64064 5824 12870
9152 4160 80080 4356352 39603L 10850&H 14586 19448 77792 83716 99008 544544 12155 388Y60 14144 14144 777YL 24310 Y152 14144 4862 86Y44 134368 173888 134368 268736 92378
FIFI-VALutS -7
LL88
-49
11440
245
1944b
-70
7293
3
4862
-al
136136
-i6a
12155
245 14
19448 46189
FtFI-VALUES -3
20020
10443
1089088
-135 42
9Yooa 12155
17
2288
-5
3536
119
21736
35
33592
77
33592
are omitted and accidental zeros are denoted by 0. Each table gives (&:,) for a fixed k and a fixed irreducible representation r of the tetragonal group. Consequently the f’s are indexed only by J and J’. However, if the representation I? appears several times in D, then we use the notations JI, JII . . * and for exam1rr) is denoted 2 31. ple fG& Numerical example: in the table 13 we read
HIBLEI’L
1:50
I. J’( ,$;yf, ) (for &,,,,
potenti:tl):
Table
1: I’ = I’1 ; Table
2: r =
I’? ; Table
3:
I’ = I’:< ; Table 4: r = r,, ; Table 5: I’ = r5 . 11. j( ,::!;.:I ) (for du,, potential) : Table G: r = 1’1 ; Table
7: I? = r2 ; Table S: ; Table 9: r = I’.$ ; Table 10: I’ = IT5 . Table 11: r = I’1; Table 12: r = I’? ; TaIII . ,I’( r+‘,>:I, ) (for @.tutr;lpotential): ble 13: r = rn ; Tdde 14: r = rd ; T:iblc 15: I’ = rs. Ii’. j’( ,::!,::I) fw (&, pot~ent~ial) : Tshle 16: r’ = 1’1 ; T:lble 17: I? = rz ; Txhlc 1s: 1’ = I‘:{ ; Table 19 : I’ = I’d ; T:lble “0: I’ = r5 . \‘.. ./‘I ,.;:‘,.‘~II) (for @tCt,.iLpotential i : Table 21 : I’ = r1; Table 22: r = 1‘: ; TnNC28:r = r:+; T~~hl~ 21: I‘ = I’, ; T:lhle 25: I’ = r5. I’ =
I’;,
The aut her wishes to thank Prof. &I. Plato for his continuous assistarlce awl dt-voted guidalbcc it1 the preparation of this work. IIe also wish(,s to Olank Prof. II. Chabbal for the kind hospitality esteuded to him in the “Laburatoirc Aimi! Cotton” and for fillancial sup port. Thallks are also due to I)r. 1.. Bordarier for utilixatioll of his computer program and IO t)r. ht. (‘aitla.rd for ilrleresliilg dixllssiuns.
/. Y. T\s.\i+E .\Nu S. Suu.\ruo, J. i’hjjs. Sot. Japan 9, 753 (1954). J. I’hp. SW. Japan 13, 394 (1958). .? Y. T.\s.\HE .\su II. KAMIRTJXA, 3. J. S. (~RIFFJTII, “The Theory of Transition ;LIetal-Tolls.” Cambridge Llniv. Press, LUIII.IOII and New York, 19ci-l. 4. (;. SclIONFELx) (M.Sc. Thesis, Cniversity of Jerllsalem, 1958). -5. M. I’I.\w (RI.&-. Thesis, 1:niversity of Jerusalem, 1959). i;, M. E‘L\TO, J.dlOl. SpeCfr!/. 17, 300 (1965). 7. (;. I?.x.\H, Z’hys. Rev. 62, -438 (1942). 8. G. I:.w.\H, Phys. I2ev. 63, X7 (19-U). 9. 11. F.lso .\ND c;. kWH, “Irredrlribte Tellscrrial Sets.” Academic Press, New York. 1959. I/I. J. 5. (GRIFFITH ” The Irrrdrlciblr Tensor Method for Molecular Symmetry (;rr,llps.” Prt’llt ive-Ilat;, Jj;ltglewc)od Cliff’s, 191i2. If. Ii. Il. IlEl,I#\\.EGt:, .IW. Pk!/Sik 4,95 (1918j. I,“. 12. I-. ('ONDON .INI) (i. It. SH~BRT~.EY, “Ttle Theory of Atomic, Spectra.” (‘amhridge Iilliv. Press, Lolldot~ atld New York, 1963. IS. 11;. I’. WIGKEH, ” 011 the Xlatrices which l