[structure and bonding] structure and bonding volume 12 || phase-fixed 3-Γ symbols and coupling...

Phase-fixed 3- f" Symbols and Coupling Coefficients for the Point Groups

S. E. Harnung and C. E. Schi l ler

D e p a r t m e n t I, ( Inorganic Chemis t ry) , The H. C. Ors ted In s t i t u t e , Un ive r s i t y of Copenhagen , U n i v e r s i t e t s p a r k e n 5, DK-2100 C o p e n hagen O, D e n m a r k

Table of Contents

1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 a) The 3-/" Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 b) The I m p o r t a n c e of S t anda rd i z ing t h e Bas is Func t i ons . Conse-

quences of S c h u r ' s L e m m a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 c) The L i g a n d Field as an I m p e t u s to S y m m e t r y S t anda rd i za t i on . 205

I. The 3-l Symbols of R 3, Genera ted b y Use of Spherical Harmonics , and Those of R s l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

2. Solid H a r m o n i c Bases a n d The i r S y m m e t r y Proper t ies . . . . . . . . . . . 206

3. Bas is F u n c t i o n Opera to r E q u i v a l e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . 211

4. The T h r e e - D i m e n s i o n a l R o t a t i o n Group R 8 . . . . . . . . . . . . . . . . . . . . . 213 a) E v e n 3-l S y m b o l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 b) Odd 3-l Symbol s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

5. The Th ree -Dimens iona l R o t a t i o n - I n v e r s i o n Group R 3 ~ . . . . . . . . . . . 218

II. The Sub-Groups of R 3 a n d R s i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

6. S u r v e y of t h e Crys ta l lographic Po in t Groups . . . . . . . . . . . . . . . . . . . 219 a) In t roduc t ion . C o m m u t a t i v e Groups . . . . . . . . . . . . . . . . . . . . . . . . . 219 b) N o n - C o m m u t a t i v e Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

7. Non-Cubic F in i te Po in t G r o u p s as S u b - G r oups of t he Holohedr ic Inf ini te Group Doon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

I I I . The 3 - / ' Symbols for t h e Sub-Groups of Rs a n d R s i . . . . . . . . . . . . . . . . . 232

8. The Non-Cubic P o i n t Groups. R a c a h ' s L e m m a . . . . . . . . . . . . . . . . . . 232 a) The Group D ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 b) The S u b - G r o u p s of Doo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 c) Dooh a n d I t s Sub -Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

9. T h e 3 - / ' Symbo l s for t h e Cubic P o i n t Groups . . . . . . . . . . . . . . . . . . . 239 a) T h e Oc tahed ra l R o t a t i o n Group, O . . . . . . . . . . . . . . . . . . . . . . . . . 239 b) T h e Oc tahed ra l Group Oh a n d I t s S u b - G r oups . . . . . . . . . . . . . . . 240

10. Conclus ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

A p p e n d i x I. T h e E v a l u a t i o n of 3-l S y m b o l s . . . . . . . . . . . . . . . . . . . . . . . . 241 A p p e n d i x 2. R a c a h ' s L e m m a Appl ied to R s t - Coot . . . . . . . . . . . . . . . . . . 243 A p p e n d i x 3. Tab les 15--22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

201

s. E. Harnung and C. E. Sch~.ffer

1. Introduction

a) The 3-Y Symbol

The most general problem in quantum mechanics is the evaluation of integrals of the form

where ~) is some linear operator, T1 and T3 are functions characterizing the states with labels 1 and 3, and where ~ represents all the variables involved in the problem.

The simplest situation, as far as symmetry is concerned, arises when represents a constant of motion, in which case the operator is invariant

under the symmetry group of the system. However, ~) may be any tensorial property and have any symmetry, so that selection rules and polarization properties are among the problems of the form of Eq. (1).

It is always possible to choose state functions as well as tensor operators to transform irreducibly under the symmetry group of the system. So Eq. (1) may be written in the form

<al r l Ia 'l 3/"3 (2)

where the F 's refer to the irreducible representations and the 7's to their components. The a's specify additional quantum numbers and other properties of the functions which are necessary for the full characteriza- tion of the states.

Often basis functions are chosen which are bases for irreducible representations of the three-dimensional rotation-inversion group Rs,, even though the physical system has a sub-group symmetry. In this case the tensorial methods exibit their particular potency because the tensor o p e r a t o r s - also those representing constants of motion -- can be expanded into components of irreducible representations of Rs~.

In the present paper we shall be concerned with simply reducible groups (1). For these, the direct product of two irreducible representations contains no irreducible representation more than once. This implies that the direct product of three irreducible representations never contains the totally symmetric representation more than once, and has the consequence that the [Yl, /"2, /"3t matrix elements 1) embodied in

1) We use the symbol [/"1,/"2 . . . . ] for the product of the dimensions of the irreducible representations ['1, F2, . . .

202

Phase-fixed 3-/' Symbols and Coupling Coefficients for the Point Groups

Eq. (2), when the components are varied in all possible ways, can be factorized according to the Wigner-Eckart theorem,

.Ic(21 3 n y3> = nlic 2tI 3 n > (n nr l , /3)

which involves only one non-symmetry-determined parameter. This parameter, <~1 F1[[~)/'21[~3 P3:>, which has the dimension of

the operator, is generally called a "reduced matrix element". The term "reduced" means "having got rid of the three variables referring to the components of the irreducible representations" and has no numerical sense. In fact, the reduced matrix element has an absolute value which is always bigger than or equal to that of the matrix element itself. Accord- ing to Eq. (4) the square of the reduced matrix element is equal to the sum of the squares of all the IF1, F2,/'3] matrix elements comprised in Eq. (3).

The symbol r l / ' 2 that expresses the matrix element of Eq. (3) 71 72 73

as a function of the components 71, 72, and 73 is a 3-F symbol which is analogous to Wigner's 3-j symbol. The 3-I" symbols are dimensionless numbers and with the components in standard form their absolute values, but not necessarily their phases, are determined solely by the properties of the symmetry group of the system. For the 3-F symbols the normalization condition

(/1171 /'~'~. ral~73/= o(-F'I Pz/'3) (4)

is valid. Here the sum runs over all the [/~1, /"2, /'3] components and the symbol O(F1 /"~ /"3) is one or zero, depending on whether or not the direct product of Pz, P2, and/"~ contains the unit representation.

The 3-/" symbols are invariant under an even permutation of the columns in the symbol and are multiplied by (--1)rz+r2 +rs under an odd permutation. Thus their behaviour under odd permutations classifies them as symmetrical (even 3-/' symbols, Sect. 4a) or antisymmetrical (odd 3-F symbols, Sect. 4b). If a 3-/" symbol belongs to the even or odd class, the exponent/"1 +/"2 + / ' 3 is an even or odd number, respectively. This classification, which will be discussed on p. 210 and treated in detail in Chapt. I l l , is associated with the relationship

(/~1 F2 F3)

from which six coupling coefficients can be produced out of one 3-/' symbol.

203

S. E. Harnung and C. E. Schaffer

b) The Importance of Standardizing the Basis Functions.

Consequences of Schur's Lemma

When the direct product of two irreducible matrix representations of a group is reducible, it can be reduced to a direct sum of irreducible representations by an equivalent transformation with a constant matrix, i.e. the same matrix for all the matrix representatives of the symmetry operators of the group (2). We shall assume the irreducible representn- tions in unitary form; then the constant matrix can be chosen as the real orthogonal matrix whose elements are the coupling coefficients occuring in Eq. (5). The orthogonality properties can be expressed as

which, when combined with Eq. (5), leads to Eq. (4). An important theorem which derives from Schur's lemma is the

fundamental theorem for irreducible representations (3). This can be formulated as the Wigner-Eckart theorem, Eq. (3). In order to obtain this theorem in the present formulation, each irreducible representation must be chosen in identically the same form each time it occurs, rather than in an equivalent form. Therefore the irreducible representations are conveniently generated in standard form by applying the operators of the symmetry group to a properly chosen set of standard bases for the irreducible representations.

I t can be shown also from Schur's lemma that a 3 - r symbol can be defined by means of the coupling coefficients as in Eq. (5). Therefore the reduction of the irreducible product of the two sets la1 rl} and luz P z ) , each forming a basis for an irreducible representation, can be written in terms of the 3-P symbol,

Although the absolute values of the 3 - r symbols are fully determined by the standard bases and the properties of the group, their phases are not. As has been pointed out by J. S . Griflith, who has prev-

Phase-fixed 3-/" Symbols and Coupling Coefficients for the Point Groups

iously introduced 3-/" symbols for the point groups (4), there is an independent choice of phase for each ordered trio o f / " s in the symbol. In the present paper the spherical harmonics will be used as standard bases for the real representations of the point groups. The proposed method, by focusing on the analytical properties of these functions and keeping the group theoretical apparatus in the background, defines such symmetry quantities as the 3-/' symbols, not only with respect to their absolute values, but also with respect to their signs.

A lemma by Racah (5), a consequence of Schur's lemma, provides a relationship between the coupling coefficients of a group and those of a sub-group. This relationship is here illustrated and standardized by introducing differential operator equivalents for the real standard bases of the three-dimensional rotation group R,.

c) The Ligand Field as an Impetus to Symmetry Standardization

The authors have long been concerned with the application of the kind of ligand field models in which the concept of semi-empirical parameters is important. Whether the electrostatic model (EM) or the Angular Overlap Model (AOM) of the ligand field is used, the energies of the elec- trons of the partly filled shell are expressed as linear combinations of semi- empirical parameters whose coefficients are proportional to certain completely specified integrals over the angular part of hydrogen atom orbitals. The ligand field models generally assume the validity of the Laplace equation. This has the consequence that the potential can be expanded into spherical harmonics, each term being expressed as a product of three factors, a potential parameter ~ , a radial factor r ~, and an angular factor ~ representing a surface harmonic, normalized to [4 a/(2 l + 1)] [see Eqs. (15)]. A general term in the matrix elements of the ligand field may then be written

= 2~) t* S~'I~: (~:~'l'¢~:)tlY~*,dT < ~ l l l t l ,, ~ 3 1 3 / 3 > (8)

where the T's represent central ion functions, which from now on will be assumed to be real. The validity of the Laplace equation also allows the functions to be expressed as one factor depending on the coordinate r only, and another factor depending only on the angular coordinates 0 and ?. This is the reason why a function in Eq. (8) may be characterized by the quantum number l associating it with a particular irreducible representation of Rs, and by the t specifying the particular component of this representation.

By use of the factorization, Eq. (8) may be written

l,

205

S. E. Harnung and C. E. Sch~iffer

in striking analogy with Eq. (3). The factors in parenthesis on the right- hand side of Eq. (9) are the formal expressions of the semi-empirical ligand field parameters. The last factor, in which two hydrogen atom angular functions and a functionally analogous expression from the ligand field potential are integrated over the angular coordinates, may be further factorized according to Eq. (3). I t is customary to include in the radial parameters certain constant factors (i. e. factors, independent of tz, t2, and ta) from the angular integral, so as to obtain simple numbers for the coefficients to the radial parameters. Consequently the chemical literature offers many different choices of semi-empirical parameters, varying with respect both to sign and size, but representing the same observed quantity. This irrational state of affairs has prompted us previously (6) and now to go into the problem of standardization.

A more important aspect in this connection arises from the resemblance between Eqs. (3) and (9). Even 3-F symbols, where the F's refer to the irreducible representations of the group R3, can be derived on the basis of this resemblance, as discussed in section 4a.

I. The 3-I Symbols of Ra, Generated by Use of Spherical Harmonics, and Those of R,,

2. Solid Harmonic Bases and Their Symmetry Properties

Here we use as basis functions the spherical harmonics on a real solid harmonic form. They are eigenfunctions of the orbital angular momentum operator,

I 2 = t ~ + I ~ + I 2 (10a) Z X Y

- ~ + y ~ + z ~ + x ~ + y ~ + z N

= s + s2 - - r~ V 2 (10c)

where r 2 ---- x 2 + y2 + zg.,and V2 is the Laplacian in Cartesian coordina-

tes. The operator s ----- x ~ + y ~y + z ~ is totally symmetric in the

three-dimensional rotation group Rs as well as in the three-dimensional rotation-inversion group Rs,. Homogeneous polynomials fl of the degree 1 are eigenfunctions of the operator s with the eigenvalue l so that the two first terms of 12 acting on f l give

(s + s2)/~ _- z(z + z).5 (11)

206


When f~ is a solid harmonic hi, it is subject to the condition

so one has

172 hl = 0 (12)

12 h~ = t(l + 1)hz (13)

and hi belongs to an irreducible tensorial set 1~ l} of Rs. Since the parity of hl is well defined, ht also belongs to an irreducible tensorial set of Ra,.

We shall define sets of real solid harmonics which generally are not eigenfunctions of the operator

x 0 (14)

These real solid spherical harmonics may be given explicitly by one of the alternative expressions, normalized to 4 ~/(2 l +1) over the surface of the unit sphere (/1, sec. 3d and 3e),

~ = r~ (~]~

= [V ~ (1 - o(Lo)) + o(Lo)] [(l + ~)!]1/2 [ ( / _ ~)t]-~/2 2-~(2!)-~

× [fl_a __ (l--~.)2 × (l--~.--I)(22+2) Zl-a-2 (x2 + y 2 ) (15a)

+ (l--~)(l--~--1)(l--2--2) (l--2--3) zl_a_4 (x 2 +y2)2 _ ] 2 × 4 × ( 2 2 + 2 ) ( 2 2 + 4 ) " " " 2

× ra sinaO f(2~o)

or

~/ , = [V2(1 -- (~(~, 0)) ~- 0(~, 0)] [(l ~- ,~)! (l -- )0!] -1/2 [(2 l)!] (l!)-1 2-t

[Zl_ ~ __ (l--~.) (l--~t--l) Zl_~_2 r2 x [ 2 x (2t -- 1) (155)

+ (l--,~)2X4×(l--,~--(2l_1)1)(l--,~--2)(2l_3)(l--,~--3) Zl_~_4 r4 . . . . ]

× r a sinaO 9(2¢?)

Here ~ stands for either s (for sine) or c (for cosine), but for a particular application of a given formula 9 refers to a definite choice. In the two cases the factors # sina0 f(2~) of Eqs. (15) are

r~ sina0 sin(2~) ---- [(~)xa-ly -- (~)x~-3y3 + • • .] (16a)

207

S. E. Harnung and C. E. SchAffer

and

r~ sinX0 cos(~0) = [(~0)x x -- (~)xX-2y ~" + • • .] (16b)

where the condition 0 < ), K l is valid for the integer ~. The Z-axis is the unique axis and the spherical polar coordinates are chosen in the usual way,

Z : r COS 0

x = r sin 0 cos 9 (17) y = r sin 0 sin 9

The functions . ~ are given for 0 < l < 9 in appendix 3, Table 15.

Each function . ~ is fully characterized in the group Rs by the three quantum numbers l, it, and C, where I characterizes the irreducible representation and it~ its component.

The quantum number l which is common to the 2 1 + 1 functions of the set .~t is accordingly determining for the trace z[R(~)] of the (2 l + 1)-dimensional matrix representative of Rz that corresponds to a rotation by the angle a around an arbitrary axis.

1

z[R(a)] - -1 -F ~ 2 c o s (ita) : ( s i n (21+ 1 )~ ) / s in2 (18)

At the same time, l by its even- or oddness defines the parity g(erade) or u(ngerade) of the set .~ under the inversion operator [ of Rs~.

The quantum number it is for it > 0 common to the two functions of the set .~ which forms a basis for the two-dimensional irreducible representations A and Ex of the groups Coov and Dooh, respectively. The trace of the 2 × 2 matrix representative corresponding to the rotation by a around the infinity axis, called the Z-axis, contains it as a parameter,

z[RZ(~)] = 2 c o s ( ~ ) (19)

The two functions~) ~ and tb~ are not eigenfunctions of the operator lz although they are eigenfunctions of

O 0 0 2 0 2 0 ~ = _ _ _ x 2 _ y 2 _ _ ( 2 0 ) Iz x ~-x + Y-O--y + 2 Xy ox oy ~ Ox2

z) For the three-dimensional rotation group R a one obtains the highest symmetry of the irreducible matrix representations (7) by taking the X-functions in the order sine before cosine. For the two-dimensional dihedral group Doo it is more natural (vide p. 230) to chose the opposite order of the X-functions,

2 0 8

Phase-fixed 3-F Symbols and Coupling Coefficients for the Point Groups

with the eigenvalues 23. They may be characterized by either of the relations

or

RZ(,¢) -D~e *Dae Lsin ;tc¢ cos Ace

(22)

where RZ(~) is the operator which rotates the contour of the functions by the angle ,¢ around the Z-axis, so that (7)

RZ(a) = exp (-- i a tz) (23)

The solid harmonics -52~ or the corresponding surface harmonics

¢2~ = r- ' . ~ (24)

are taken as the real standard basis functions for the irreducible representations of the group Rs. As mentioned above, the functions are also basis functions for the group Rs,, but in this case they only span half the irreducible representations

sg, p~, dg, A . . . . (2s)

This does not complicate our issue since Rs~ is the direct product group of Rs with the inversion group S~, which is commutative and of the order two, and therefore trivial to account for (see Sect. 5).

The functions ~g~, summarized in Eq. (25), generate the real standard irreducible representations of Ra uniquely, and in order to obtain the standard matrices of Rs, we make the following definitions: A matrix representative associated with a proper rotation R, is identically the same in the two groups for corresponding gerade and ungerade representations. A representative associated with an improper rotation S of Ra,, corresponding to a proper one, R, through

S ---- RI----IR (26)

is identically the same as that of R for gerade irreducible representations and identically the same as that of R, except for a sign change of all matrix elements, for ungerade representations. By these definitions it is

209

s. E. Harnung and C. E. Sch~ffer

immed ia t e ly obvious t h a t the real i r reducible tensor ia l sets 3) which genera te the i r reducible represen ta t ions

su, pg, d,,, fg . . . . (27)

not appear ing in Eq. (25), are comple te ly defined, except for a common sign wi th in each l set. I t will be shown below (p. 218), however, t ha t in fact only one choice of sign, common to all the l sets, is a rb i t l a ry .

I t is va luable to consider the direct p roduc t of the two sets ]61 l} and ]a2 l} which form the bases for the same i r reducible represen ta t ion of R3. One obta ins in the direct sum symmet r i ca l and skew-symmet r ica l i r reducible tensor ia l sets

lal l} X [62 l} ----- ]aS} -~- laP} + laD} + ]aF} + laG} + . . . +

] a L ( = 2 l)} (28)

where the skew terms are under l ined. I t appears t h a t the skew te rms are all associa ted wi th o d d / - v a l u e s and on this basis the basic rule for the choice of s t a n d a r d basis funct ions for any of the sub-groups of R s is suggested.

Basic rule: S t a n d a r d basis funct ions for i r reducible represen ta t ions t ha t occur in an t i symmet r i zed (skew) di rec t p roduc ts are chosen out of the spher ical harmonics wi th l odd and the remain ing ones out of those wi th l even. This rule is immed ia t e ly appl icable to R s and wi thou t diff icul ty to all non-abel ian poin t groups except the icosahedral group which, however, is not a s imply reducible group4).

We shall an t ic ipa te some consequences of this choice of basis funct ions for the i r reducible represen ta t ions of the poin t groups.

When a 3-/" symbol contains an odd F (i. e. a F chosen from an odd /-value) an even number of t imes, this 3 - F symbol will, a p a r t f rom a (by defini t ion posit ive) cons tant , be represented b y an in tegra l over a p roduc t

3) The basis sets of Eq. (27) are the axial tensorial sets. I t should be noted that the concept ungerade has a significance in an absolute sense as discussed in connection with Eq. (26), but this is not true of the concept axial. Axial means having the same transformation properties as a given (polar) standard set of functions under proper rotations and the same, except for a sign change, under improper rotations.

4) Here a given trio of irreducible representations will sometimes give rise to two independent sets of coupling coefficients. The analysis along the lines of the present paper requires for the four and five-dimensional irreducible representations of the icosahedron in each case two independent sets of basis functions. For the four-dimensional representation, one basis function out of the f set and one out of the g set, and for the five-dimensional representation one out of the d set and one out of the g set form the simplest choice.

210


of three spherical harmonics. So the sign of the symbol is independent of the order of the F 's and we have an even 3-F symbol.

When a 3-F symbol contains an odd F an odd number of times the corresponding integral vanishes because of its ungerade character. So in this case one of the F 's must be represented by a corresponding axial operator (and the others still by their spherical harmonic standard basis functions) in order that the 3-F symbol can be represented by an integral. The 3-F symbol defined in this way will then automatically change sign under odd permutations of its columns, so we have an odd 3-F symbol.

3. Basis Function Operator Equivalents

0 The three differential operators _~' ~-~y, and ~ and the three functions

x, y, and z transform under proper and improper rotations by identically the same matrices as do the Cartesian unit base vectors i, j, and k. The products of the differential operators and linear combinations of them also have the same transformation properties as the corresponding Cartesian unit base tensors or their linear combinations. The same holds true for the functions x, y, and z. For example,

and

~2 1 ~2 1 ~2 3 ~2 1 - - - - V 2 (29) Oz 2 2 c3x 2 2 Oy 2 2 ~z 2 2

1 1 2 .~2 = z 2 _ 2 x 2 _ _ ~ Y (30)

transform as the unit base tensor

(2 k k - - i i - - jj)/V ~ (31)

It is useful to define in general differential operators from the solid harmonics ~]~ by performing in Eqs. (15a) or (15b) the substitutions

0 a 0 x * -~-;~ ; y * -g-~y ; z . a----;

r 2 - ~ F2 - - + + - - Ox 2 0 - ~ ~z2

(32)

where the last expression is a consequence of the first three. These differential operators will be denoted ~)]~; they have the fol-

lowing property (8)

211

S. E. H a r n u n g and C. E, Schi~ffer

(2or

=. 1 X 3 X 5 X ' ' ' X ( 2 l - - 1) O()./t') #(fl;') (33)

where 't9 and 2'9' both refer to the standards of Eq. (15), using the same coordinate axes of quantization. This simple result obtains because the operator ~]~ operates on a solid harmonic of the same degree, l. It is advantageous to define an operator, ~R]~,

so that

~1~¢ -- 2~ll ( -~ ~L, (34a)

l l 9la¢ .~a¢ = 1 (34b)

The operator ~ will be called a unification operator. On the basis of Eq. (34) we further define projection operators va¢"a) by

0 ~ = .~¢ 9]ate, (35a)

o(i) (3Sb)

It is noted that the projection operator is an element of a reducible tensorial set and that its action as a projection operator is limited to homogeneous polynomials and spherical harmonics of the degree l.

To illustrate the properties of these operators it is observed that any homogeneous polynomial f~ of the degree 1 can be written

f~ = h~ + r2f~_2 (36)

where the spherical harmonic h~ is a linear combination of the elements of the set ~ ,

hz = Xk(~tf) . ~ (37) ),¢

Now, the projection operator ~(~) ~,x~ extracts (projects) the term k(2f)52~ out of f~, and analogously the unification operator ~ yields the coefficient k(,lf) to 5~¢ [no!realized to 4 ~/(2 l + 1)] when operating on ft. The analytical form of the operator ~3~ may be obtained by using either Eq. (15a) or Eq. (15b), giving the two forms which appear in Eq, (29). The latter form is particularly useful when the operators ~R~ and O~ are to be used on polynomials which are spherical harmonics

212


If 1-2 = 0 in Eq. (36)]. In this case all the terms containing V 2 or powers of: V 2 will give vanishing contributions.

It is sometimes necessary to obtain the content of a harmonic 5~ of the degree l in a homogeneous polynomial fz+2n of the degree l + 2 n. In this case one has to annihilate the terms of higher degree than l by operating On the polynomial f~+2n with the totally symmetric operator V 2". This can be done because

V2(rqh~) = q[q + (2 l + 1)Jrq-2h2 (3s)

Further applications of Eq. (38) will be shown elsewhere (9). Fol our purpose the important fact is that the coefficients to ht on the right hand side of Eq. (38) are positive.

4. The Three-Dimensional Rotation Group R,

a) Even 3-l Symbols

We now have the necessary preiequisites to obtain the 3-1" symbols for the group R,. In this group they will, however, be termed 3-l symbols. We consider an integral over three basis functions of Eq; (15) and use the Wigner-Eckart theorem:

1 f It~l,~itdS k(lll2la)(lll2la) (39) 47t ~ t l ~-dtt ~.dta = t 1 t 2 t~

Here t is an abbreviation for Xf and we shall make the convention that such an integlation is performed over the surface of the unit sphere. i.e. with the condition r 2 = x2 + y~ + z2 = 1. Thus we have

fdS = 1 (40)

Firstly, it is noted that the integral of Eq. (39) is independent of the order of the functions. Sec0ndly~ the integral can only be non-vanishing if the degree, ll + 12 + la, of the integrand is even. We define the constant k(l11213) to be positive and independent of the order of the functions and thereby ensure that the 3-l symbolS are unchanged under even as well as odd permutations of their columns. 3-1 symbols with these properties will be called even 3-l symbols.

In Eq. (39) one can replace the functions ~3~ or ~3~ or both by their differential operator equivalents of ~) type or 9~ type. This will alter the magnitude of the constant k(l112 lz), which depends on the choice of

213

S. E. H a r n u n g and C. E. Schgffer

operators, but not its sign (10). Therefore, using Eq. (3) and its subse- quent discussion, the constant can in each case be obtained as the positive square root of the sum of the squares of the matrix elements, the sum ranging over tl, t2, and t3.

An important example of this, is the case of Eq. (39), where the first function is the s-function, 9 ° = 1, and where the second function is replaced by ~i:

1 f j t, ~.~e, dS = ~t(lz la) (~(t~. t3) (41)

0 1~ la = k'(0 ls 18) \ ~ t2 t3)

The Kronecker deltas arise for the following reasons. For 12 <18 the integrand is necessarily a solid harmonic and its integral is accordingly vanishing. For 12 = ls and t~ 4= ta, ~1l~: annihilates [Eq. (33)] ~ : , and for 12 = 13 and t2 = t3, ~ : is unified. For 12 > la the integrand vanishes. By squaring Eq. (41) and taking the sum of the (2 12 + 1) non-vanishing terms, application of Eq. (4) yields k '2 = 2 12 -}- 1, so that k' = + (2 12 + 1) t and as a consequence

s Z2la) : (0 12/a) = (212 + 1)-' ~(12 la)8(tz ta) (7 12 t 3 \ a t2 t3

(42)

The next example shows the procedure carried out in different ways with the values ll = 1, 12 = 1, and 18 = 2 fixed (Appendix 3, Table 15). It is convenient, however, to use the orthogonality conditions, Eq. (6), for the coupling coefficients together with the relation, Eq. (5), which connects them with the 3-l symbols.

Firstly, the two harmonics ~[~ and ~ : of Eq. (39) are replaced by their differential operator equivalents. Then with t8 = a the only non- vanishing integrals are

4n ~ ( z ~ - - ½ x 2 - { y 2 ) d S = 2

i r a o -~x -'~'x (zZ - - { x Z --½y~) dS = - - I

i f o o - ½ - d S = - 1

(43)

The corresponding coupling coefficients <ptpt[da> are obtained simply by multiplication of the numbers on the right-hand side by the normalization factor [22 + ( _ 1) 2 + ( _ 1) 21 -~ = 1/V6and the 3-l symbols by the normalization factor 1/]/U-~ = 1/y~6.

214

Phase-fixed 3-1" Symbols and Coupling Coefficients for the Point Groups

Secondly, only the first function in Eq. (39) is replaced by its differential operator equivalent. The product 612 6~z is a homogeneous polynomial of the third degree which could be decomposed into a solid harmonic off-type, and one of p-type multiplied by r~. Since only the p-type part will give a non-vanishing contribution to the integral of Eq. (39), we replace the first function by ~31~ V z so that V 2 annihilates the f-term and, according to Eq. (38), leaves the p-type term multiplied by some positive constant. Analogously with Eq. (43) one obtains

f ' l 0 1 t (3

and corresponding expressions for the remaining integrals. This leads to the same result as before, except that this time the normalization constants are 1/(2V6 ) and 1/(2V30), respectively.

Thirdly, only the second harmonic of Eq. (39) is replaced by its differential operator equivalent giving

--4z~ z~-z [z z _ l x 2 _ ½ y 2 ] d S = 2 / 3 (45)

and analogous expressions. All 3-l symbols of ppd-type are given explicitly in Appendix 3, Table 16.

Finally, it is noted that the results obtained, by means of Eq. (7) can be used to derive that irreducible product of 61 and ~31, which is of d-type, i.e. of the degree 2,

2 0 1 0 1 0

1 o 1 O [61 X ~)l ]~s :~--#y-~z--k~-~z~y

1 O 1 a [61 x ~)l]~c=~-~z~x+~x-~z (46)

1 O 1 O [51 × )I]L =F# Uy +

1 0 1 0 [61 x ~l]$c=~x-#-#x--~--#y~y

These operators are determined with sign and can be used to find the 2 1 3-1 symbols of the type (~1 t2 t3)- In the case where 1 = 1, results equi-

valent to those of Eq. (43) will be generated. It should be noted that when the operators of Eq. (46) are introduced instead of the function 6~ of

215

S. E. Harnung and C. E. Sch~ffer

Eq. (39), they have an effective degree of zero; i.e. the degree of the integrand becomes ll + ls. However, it is not required that ll = 13 in order to have non-vanishing matrix elements of the operatois, Eq. (46). This is in contradistinction to the operators of Eq. (47), which, since they are proportional to the angular momentum operator, Eq. (14), are diagonal in l.

b) Odd 3-l Symbols

In the previous sub-section it was seen that the principle of associating with a basis function its differential operator equivalent, defined also with respect to sign, has many facets. However, even though the standard transformation properties of the basis functions of R~ and R3, are fully defined, 3-l symbols with ll + 12 + 13 equal to an odd number cannot be handled in the same direct way. As mentioned at the- end of Sect. 2, one must in this case replace one of the spherical harmonic basis functions, which themselves form polar tensorial sets, by a corresponding axial tensoz. This is done here by considering first 3-l symbols of the type

I 1 l • • • l (tz t2 t3) in which case the axial tensor Rt2, which corresponds to the 1 • polar .~ t2, is skew-symmetric.

A convenient choice of a skew tensorial set of operators consists of the components of the angular momentum operator i [Eq. (14)], which transform according to the irreducible representation pg of Rs,. When these are divided by V~2, one obtains a set of real operators whose elements are the irreducible products of the degree one (effective degree zero) of the sets ~1 and 31,

1 O 1 0 [ ~1 x ~1] ~ = _ ._~ X .~y + ._v.~ y __ff_;.

~)l]'e~t = 1 a 1 0 [.$1 x - y + z

1 0 1 0 [51 x - z -g-; + - f f x 0-7

(47)

In Eq. (47) we have made a definite phase choice which is actually the opposite of that occurring in the vector product of xi + yj + zk

o a 0 and i + ~ j -t- ~ k. This particular choice has the consequence

that reduced matrices derived by means of Wigner's 3-j symbols are identical to those derived in the present scheme (10). As was mentioned in connection with Eq. (27) and as will appear in this sub-section, this choice of phase determines the phases of all odd 3-l symbols.

216

Phase-fixed 3-1" Symbols and Coupling Coefficients for t h e Point Groups

By means of the operators of Eq. (47) we shall now derive odd 3-l symbols by a procedure similar to that of Sect. 4a.

Consider first the integral

.~L[~51 × ~ ~o~.,~odS = ~ y - - f f ~ + - f f y ~ , d S

1 ( p p p ) - sV~- - k (4s) ~ S O" ~ C

Again the constant k is taken as the positive square root of the sum of the squares of all integrals of this kind [Eq. (4)]. There are six non-

6 vanishing integrals of the same absolute value, and one obtains k s = I--if' k = + l/V3and

( p p p ) + i ~ s ~ c - - V ~- (49)

This procedure can be carried out quite genelally for 3-l symbols of

(' ') rom o mow , the type tlt~ta "

1 ]" z z d i j Sac Iz ~a, S - 2 l + 1 - - a n d ~ ~ , Iz -~c dS -- 21+~

2 = 1, 2 . . . . . l (50)

are the only non-vanishing integrals that can occur with the operator Iz. Therefore the sum of the squares of the matrix elements of the operator [8 1 × ~1101 is

l l

2 2 ( 2 / + 1 ) ~ - ( 2 / + 1 ) ~ A-1 A- I

1 l(l+ 1) (2t+ 1) Z(t+ 1) : (2/+1) 2 X 6 -- 6(2/+1~

(51)

or, in other words

I 2

6(2/+1) t~ t,

=k 2 l pl 2 k 2 x t l a ta 3

(s2)

217

S. E. Harnung and C. E. Sch/iffer

where in the last step the Eqs. (5) and (6a) have been used. Thus for the constant k one has

/z ( l+ 1)

generally applicable for these 3-l symbols, which can be found from the relation

f ~ [~ ~11 , ] / / ( /+1) { , , l ~ (53) 1

_~1._ 1 X ]t~St3dS : F2(2 l+ 1) \tlt2ta] 4re

[d p d~ As an example of this, we calculate \~snc nc] '

1 'V3-xy ( 1 --~--~ -~- 1 a ) V3zxdS= -1 V2~-3 ( 2 1 2 )

( d : c d ) --1 = (54)

All the 3-1 symbols of dpd type are given in Appendix 3, Table 17. By means of the 3-l symbols evaluated through Eq. (53) one can

now calculate any odd 3-I symbol. To illustrate this, it is noted that if ll + li -}- 13 is odd, then the integral whose integrand is of the degree ll + 12 + la -- 1 can be calculated by the methods of Sect. 4a. Therefore, in the integrand of Eq. (39), we replace the middle function 5~, which is a component of a polar tensorial set, by that of its operator equivalents in the rotation group Rs, which is a component of an axial tensorial set and at the same time has an effective degree of 12 -- 1. This is

× (55)

where the irreducible product is expanded by means of the 3-I symbols of Eq. (53). Then Eq. (38) may be used to reduce the degree of [515 ×

i I, l, ]*, St, down to the degree ll of the first factor in the integrand, and finally the projection operator Oqt~ ) can be used, if desired.

The important thing to note is that, since the constant of Eq. (38} is positive, the phase convention made in Eq. (47) together with the deci- sion to choose the constant k in Eq. (39) positive, determine the phases of all 33 symbols.

5. The Three-Dimensional Rotation-Inversion Group Rs~

It was mentioned in connection with Eq. (25) that the spherical harmonics which span the irreducible representations of the group Rs span only half

218


the irreducible representations of the group R3,. Consequently it is not possible to define the 3-l symbols of R3, on the same basis as was used in Sect. 4.

However, we shall make use of the fact that the group R3, is the direct product group of the group Rs with the inversion group $2. For a 3-/" symbol of a direct product group the relation (4)

(/Pl /"1' /"9/"2" /"2/"3'~ = (/"11P2/"2 / [/"l'/"2'/"2"~ ?i 71' 72 79' ?3 73'1 ~'I 79 '~3/ ~ ?I' 72' 73'1

(56)

is valid. Here the unprimed and primed symbols refer to the irreducible representations of the constituent groups.

The 3-F symbols of the group $2 are most simply defined by

/"1 I'2 1"3 / = ~(F1 F2 /'8) Yl 79 731

(57)

where the F's take on the values Ag or Au, so that the 3-/ ' symbol vanishes whenever Au occurs an odd number of times. We conclude that the 3-l symbols of the group R,, are equal to the corresponding ones of the group Rs whenever they do not vanish for parity reasons.

The relations Eqs. (56) and (57) will be used similarly in Sect. 8c to generate the 3-/ 'symbols of those point groups which can be described as direct product groups of a rotation group and either the inversion group $2 or the reflexion group C/h.

II. The Sub-Groups of Ra and Rn~

6. Survey of the Crystallographic Point Groups

a) Introduction. Commutative Groups

The symmetry displayed by chemical entities such as molecules or ions is known as point symmetry* because the symmetry elements intersect in a specific point. To put this in another way: all symmetry operations which bring such an enti ty into a configuration equivalent to the original one leave one point unchanged. The assemblage of symmetry operators characterizing the geometry of an enti ty constitutes its point group.

*) For the groups Cn and Cnv the intersection takes place in a line, also for n = oo. For the group C2n -----C8 it takes place in a plane.

219

S. E. Harnung and C. E. Schiiffer

The point groups were first studied by crystallographers, who used them to characterize the macroscopic properties of crystals. An ideal crystal must belong to one of the thirty-two crystallographic point groups. (This limited number arises from the fact that only one-, two-, three-, four-, and six-fold axes can occur in crystals because of space- filling requirements.) These point groups are also the point groups of most importance for chemistry; however, a few others may have to be added.

Before giving a general survey of point groups, we study three simple ones, each of the order two: the inversion group $2 (or C,), the reflexion group Clh (or Cs), whose only symmetry element is a mirror plane, and the group C2, which has one two-fold axis. Although they are isomorphic, they represent quite different physical systems. Their character table is displayed in Table 1.

Table 1. Characters for the rotation group C2 and the isomorphic groups Se and Clh, which are hemihedrics of the hoiohedric C2n. $2 is the holohedric triclinic group

$2 : Ci

Clh ~ C s

C2

E I = C ~ ah =S2

E ~ =1 C z =S2 C z

E C z =1 ~h =S~ ah

Ae A' A 1 1

Au A" B 1 -- 1

The operators I ---- $2 and en = IC z commute with all other symmetry operators. Therefore the direct product group of a group G Of the order g with the inversion group gives a new group of the order 2g such that each of the irreducible representations of G occurs twice with the further labels g and u. Similarly, if a group does not contain the operator IC z, then the direct product of this group with C1n gives rise to a new group with twice the number of elements and of irreducible representations. In this case the irreducible representations are labeled with a prime when symmetrical with respect to IC z and with a double prime when antisymmetrical.

Irreducible representations of dimension one are generally termed A if they are symmetrical with respect to the main axis operation and are termed B if antisymmetrical.

Using Table 1, one can form the direct product group of any pair of the three groups mentioned. As an example we take the direct product

220


Table 2. Characters for the holohedric rnonoclinic group C2~

Csn- C2 X S~ E C~ S2 ah =S2C z

Ag 1 1 1 1

B o 1 --1 1 --1

Au 1 1 --1 --1

Bu 1 --1 --1 1

group C2n of the groups C2 and St, Table 2. I t follows from the remarks in the previous paragraph that the four irreducible representations are labelled with A and B, and with g and u. I t is seen that the character table for the group C~ occurs four times, three times multiplied by + 1 and the fourth time multiplied by --1. The same group C2~ could equally well have been obtained as the direct product group of the groups C~ and C1~. However, when the same group can be generated as the direct product group from either the group S~ or the group Clh, the inversion group is the perferred one by convention, as far as labeling of the resulting irreducible representations are concerned. Further, it is a useful convention to state the operations of the group with the rotations first and then the corresponding improper rotations in the same order,

One can obtain a group D~ which is isomorphic with the group C2h by using the generators C z and C~. Obviously these generators imply that the non-trivial operation C x is in the group. The character table for the group D~ is given in Table 3 and it is seen that with the irreducible representations of C2h taken in the order Ag, Au, Bg, Bu the two character tables are identical. In the group D2 none of the three axes is a priori the principal one and the symbols B1, Ba, and Bs are used for antisymmetrical about two axes. This implies symmetrical about one axis, the Z, Y, and X axes, respectively.

Table 3. Characters for the dihedral orthorhombi¢ rotation group D2 and the hemihedral orthorhombic group C~v

c~ E C~ %-zx -o2"-s-~ "Y %Yz =S2C x E

A1 A 1 1 1 1

A2 B1 1 1 --1 --1

B1 B2 1 --1 1 --1

Bz B3 1 --1 --1 1

221

S. E. Harnung and C. E. SchAffer

It is a general feature that a cyclic group (Cn) of the order n with the Z-axis as symmetry axis becomes dihedral (Dn) by addition of the generator C~, and thereby embraces twice as many elements.

The direct product group of the groups Dz and $2 gives the group Dzh with eight elements and with the irreducible representations of D2 further characterized by g and u (Table 4). Besides the groups D2 and

Table 4. Characters for the holohedric orthorhombi¢ group D~

c,

Ag 1 1 1 1 1 1 1 1

B ig 1 1 - -1 - -1 1 1 - -1 - -1

B2g 1 - -1 1 - -1 1 - -1 1 - -1

B30 1 --1 --1 1 1 --1 ~ 1 1

Au 1 1 1 1 --1 --1 --1 --1

B lu 1 1 --1 --1 --1 --1 1 1

B2u 1 - -1 1 - -1 - - 1 1 - - 1 1

B3u 1 - -1 - -1 1 - -1 1 1 - -1

$2 also the previously mentioned groups C2, C1h, and C2h are sub-groups of D~h. Further the group C2v (Table 3), consisting of the first proper rotation and the last two improper ones, is a sub-group of D2~ which is isomorphic with D2.

The point groups considered till now correspond to the three crystal systems which are optically biaxial. These are the triclinic system, the monoclinic system, and the orthorhombic system. As illustrated in Table 9, the groups $2, C2h, and D2h may be called the holohedric triclinic, monoclinic, and orthorhombic groups. The other groups within each crystal system (Table 9) are sub-groups of the holohedric ones, and when they are isomorphic with the pure rotation group which makes up the first half of the elements of the holohedric group, they are called hemihedric groups.

The point groups discussed in this section are commutative (Abelian) groups. For these groups the number of irreducible representations equals the number of elements and all the irreducible representations have the dimension one. It is generally not the case, however, that the irreducible representations are real. Thus for n even and greater than two, all but two of the irreducible representations of the groups C• and Sn fall in pairs whose characters generally are complex, and conjugate to one another. A similar statement holds true of all but one of the irreducible re -

2 2 2

Phase-fixed 3-1" Symbols and Coupling Coetficients for the Point Groups

presentations in the case of Cn groups with n odd and greater than one. For these groups we shall employ a real basis which connects such pairs into pseudo two-dimensional representations, which we shall call the most reduced representations with a real basis. This is somewhat artificial from a group theoretical point of view but has compensating advantages (Sect. 8). We shall mention that with a real basis the cyclic groups Cn (n > 2) and the alternating groups Sn (n even and > 2) are not simply reducible groups.

b) Non-Commutative Groups

The holohedric point groups that correspond to the optically uniaxial crystal systems are non-commutative groups. In such groups some of the symmetry operators fall in the same class, and at least one of the irreducible representations will have its dimension greater than one. We shall discuss these groups in the following general manner. From the pure rotation group with the highest number of elements the direct product group with the inversion group is formed, giving the corresponding holohedric group. Next the invariant sub-groups of the holohedric groups will be considered.

The lowest order possible for a non-commutative group is six. This follows from the basic group theoretic proposition, that the number of elements equals the sum of the squares of the dimensions of the irreducible representations. The pure rotation group Ds which is of the order six and has three irreducible representations, is obtained by combining the generators C z and C~. This group is characterized in Table 5 by its standard irreducible representations together with the basis functions which have been used to generate these representations through relations analogous to Eq. (22). In this way it is always possible to generate the matrix representatives whenever standard basis functions have been chosen.

In order to discuss the rest of the crystallographic point groups, one further has to consider the dihedral rotation groups D~ and D6, and the cubic rotation group O. Their character tables, standard basis functions, and a useful choice of group generators are displayed in Tables 6, 7, and 8. In this way the material required for symmetry considerations is directly available.

In the same way as the holohedric orthorhombic group D2h was obtained as the direct product group of the rotation group D2 with the inversion group Sz, the direct product groups of Ds, D4, Ds, and 0 with $2 are formed. As illustrated in Table 9, the groups Dsa, D4h, D6h, and Oh may be called the holohedric trigonal (or rhombohedral), tetragonal, hexagonal, and cubic groups, respectively. The corresponding crystal

223


2 2 4

0

o .

u

0

A ° ~

I

U

u

t ~ o ~

U

i

" 7 + +

J ÷ I I

'~J~ _ 1 . ° t ÷

- I ~ ~ I ~ J I

I I

! !

I

I J

1

I 1 I J

'~1 ¢~ - I ~ ' i I

I I I

I !

I I

Phase-fixed 3-T' Symbols and Coupling Coefficients for the Poin t Groups

Table 6. Characters of the dihedral tetragonal rotation group D4. The basis functions for standard irreducible representations are also given together with their matrix representatives of the standard generators of the group

~, E 2 c~ c~ 2 c'~ 2c

. ~ A1 1 1 1 1 1

.~ma A 2 1 1 1 --1 --1

~ e B1 1 - -1 1 1 - -1

~ s B2 1 - -1 1 - -1 1

~ e , - ~ , E 2 o - 2 o o

Generators C~ C~

Standard basis functions

Table 7. Characters of the dihedral hexagonal rotation group D e . The basis functions for standard irreducible representations are also given, together with their matrix representatives of the standard generators of the group

~ E 2c] 2 c ~ c] ~c; 3c;

~ Ax 1 1 1 1 1 1

. ~ A2 1 1 1 1 -- 1 -- 1

. . ~ s B2 1 - - 1 1 - - 1 - - 1 1

tMe ,~ , r:l 2 1 - 1 - 2 o o

• ~ a e , - ~ s E2 2 - - 1 - - 1 2 0 0

Generators C z C~

-~-

2 2 5

S. E. H a r n u n g a n d C. E. Sch~ffer

Table 8. Characters of the octahedral rotation group O. The basis functions for standard irreducible representations are also given, together with their matrix representatives of two generators of the group

O E 8Ca 3C2 6C4 6C~

~*a A1 1 1 1 1 1

~ 6 s A 2 1 1 1 - - 1 - - 1

d d ~a, ~oc E 2 -- 1 2 0 0

~, ~L, $L T1 a o - -1 1 - - 1

~nc, ~6s T 2 3 0 - - 1 - - 1 1

Genera to r s C x:cz C z

[ [: :] S t a n d a r d basis func t ions 1 2

E [5~, 9£] Y~

2 . ~ j -

TI [Sv, 5v, ~vc] 0 0

0 --1

5~.] 1 o - 1 o

0 1 0 0

i] i]

systems are given in Table 9, and in Table 10 they are characterized by their symmetry operators collected into classes.

The invariant sub-groups of the non-cubic holohedric groups fall into two equally large collections: the first one contains the non-commutative groups and the second one the Abelian or commutative ones. For example, the holohedric group D4h contains the non-commutative, isomorphic, and invariant sub-groups D4, C4v, and D2a, and the Abelian invariant sub-groups C4h, C4, and $4. For these latter groups, the operations which belong to the same class of the holohedric group will now fall into classes of their own.

It may be noted that the holohedric group D3a is an invariant sub- group of the holohedric group D6h. This fact has led some workers to eliminate the trigonal system and classify all the crystallographic groups

226

Phase-fixed 3-_P Symbols and Coupling Coefficients for the Point Groups

Table 9.

1. Triclinic crystal system: the holohedric group $2, and its $2 sub-group (no specific direction is defined) C1

2. Monoclinic crystal system: the holohedric group Cgh, and C2h those sub-groups which do not belong to class 1 (one C~, C 8 specific direction is defined)

3. Orthorhombic crystal system: the holohedric group D~h, D2h and those sub-groups which do not belong to classes 1 and D2, C2v 2 (three specific directions are defined)

4. Trigonal crystal system: the holohedric group D3a, and D3a those sub-groups which do not belong to classes 1 and D3, C3v 2 Ss, Cs

5. Tetragonal crystal system: the holohedric group D4h , and D4h those sub-groups which do not belong to classes 1, 2, D4, Car, Dza and 3 C4~, C4, $4

6. Hexagonal crystal system: the holohedric group D6h, and D6h those sub-groups which do not belong to classes 1, 2, 3, D6, Csv, D~h and 4 Cab, Ca, Cab

7. Cubic crystal system: the holohedric group Oh, and those Oh sub-groups which do not belong to classes l, 2, 3, 4, O, Ta and 5 Th, T

In this table all the crystallographic point groups have been collected together. They can be briefly characterized as all the sub-groups of Oh plus the sub-groups of Dab of class 6 of this table.

which are sub-groups of the g roup D6h as belonging to the hexagona l c rys ta l sys tem except those belonging to the c rys ta l sys tems of classes 1, 2, and 3 of Tab le 9.

F o r the cubic groups Th and T the opera tors of the class 8 C3 of Oh

fall in to two classes, one corresponding to a ro ta t ion of + - ~ and ano the r

2zt one corresponding to a ro t a t ion of - - - -

3 '

7. Non-Cubic F in i t e Po in t Groups as Sub-Groups of the Holohedr ic

Inf ini te Group Dooh

Before the holohedr ic g roup Dooh is genera ted as the d i rec t p roduc t g roup of the infini te d ihedra l g roup Doo wi th the invers ion group $2, the former group and i ts sub-groups will be s tud ied in some detai l .

The d ihedra l group Doo which m a y be genera ted b y add ing the operator C~ to the two-d imens iona l ro t a t ion group Coo is charac te r ized in

227

tO

Tab

le 1

0. R

evie

w o

f sy

mm

etry

cla

sses

of

the

holo

hedr

i¢ c

ryst

allo

grap

hic

poin

t gr

oups

and

the

ir s

ub-g

roup

s

Ds

x $

2=

D

za

E

2C3

3C~

S2

2Se

3aa

Dz

0

0 0

Csv

0

0 0

o~

03 x

sz:

s6

0 0

O

0 C

3 0

0 m

P

D4

X S

z :

D4

X C

zt, =

D4h

E

2C

4 C

~.

2C~

2C,~

S2

2S

4 at

, 20

"v

2 aa

D4

0 0

0 O

0

¢4v

0 0

0 0

0 D

za

0 0

0 0

0 C

4 x S

z:

C4 x C

z~:

C4n

0

0

0

0

0

0

c4

0 0

0 s4

0 O

0

De

X S

e=D

a X

Czn

=D

sn

E

2Cs

2C8

C~

3C~

3C2

$2

2S3

2S6

at,

Sat,

3aa

De

0 0

0 0

0 0

Ca,

, 0

0 0

0 0

0 D

ab

0 O

0

0 0

0 te

n

0 0

0 O

0

0 0

0 c3

~ 0

0 0

0 ca

0

0 0

0

.D8

X C

lh

C6

X S

2 =

Ca

X C

lh =

C

a X

Clh

:

0 x

Sz=

O

h E

8C3

3C=

6C4

6C~

S2

8S6

3a~

6S4

6~

o 0

0 0

0 0

T~

0 0

0 0

0 T

x S

z=

T

n 0

0 0

0 0

0 T

0 0

O


TaMe 1 I. Characters of the infinite dihedral rotation group Dao. The basis functions for standard irreducible representations are also given, together with their matrix representatives of the standard generators of the group

Doo E 2 R(~o) C2

~ A1 1 1 1

~5~ A2 1 1 --1

[~a¢, S as] Eli/-/] 2 2 cos 9 0

[~ae, ~as] E2[zJ] 2 2 cos 2 ~0 0 g e ] [-~,c, ~ , . E3[¢] 2 2 cos 3 q0 0

[~(~-l)c, ~(~21-I),] l= 2i E(2i-I) 2 2 cos [(21 -- I)9 ] 0

[~(Z2De, ~(~)s] l = 2A E (24) 2 2 cos [(2t)q0] 0

Generators Rz(io) C~

Standard basis functions

_:] Table 11. The group D~o has all the finite axial pure rotation groups, i.e. the cyclic groups Cn as well as the dihedral groups Dn as its sub-groups. The structures of these groups depend on whether n is an even or an odd integer.

When n is odd, the Cn groups have one one-dimensional irreducible representation A, and the D,, groups have two such representations, A1, the totally symmetric representation, and A 2 which is antisymmetrical under C~. The number of two-dimensional irreducible representations (or, for Cn, the number of most reduced representations with a real basis) is

When n is even, the Cn groups have two one-dimensional irreducible representations A and B. The representation B shows the

transformation properties of a spherical harmonic with ;t =~- , and it is

noted that both the cosine and the sine function belonging to this ;t- value transform in this way. For even n, the Dn groups exhibit four one-

229


dimensional irreducible representations A1, A2, B1, and B2, arising from A and B of Cn, each of which splits into two representations, one symmetrical and the other antisymmetrical with respect to C~. The number of two-dimensional irreducible representations (the most reduced

ones for C•)isin this case ( 2 - 1).

The antisymmetrized direct product of the two-dimensional representations with themselves is A for Cn and A~ for Dn, independent of whether n is even or odd. The basic rule (p. 210) for the choice of standard basis functions for the irreducible representations naturally proposes the function .~a ~ as a standard function for this representation. It is noted that Cn in this way requires two standard basis functions for A, namely further the function . ~ representing the symmetrized direct product of the most reduced representations with themselves. This is due to the fact that the group Cn with a real basis is not a simply reducible group since in

Ea × E~ = A [symm] + A (skew) + Ei~ (ss)

A occurs twice. One A is symmetric and the other A skew. These are different in the group Dn (which is simply reducible}, namely A1 and A2, respectively.

The basic rule for the choice of standard basis functions requires that the basis functions for all the two-dimensional irreducible representations have 1 even. The simplest choice for D~o therefore is that given in table 11. This choice is also valid for Coo with real bases, and generally for the groups Dn and Cn, provided that the sequence El, E2 . . . . in the table is truncated from below according to the number of irreducible representations (or most reduced representations for Cn) of the group in question. This choice of l even makes it natural to choose the operator C~ as one of the standard generators for all the dihedral groups, since by this choice ~ c is always symmetrical and ~ s always antisymmetrical under the C~ operation. If C x had been chosen, the behaviour of ~ c functions (l even) under the standard twofold operator would depend on their 4 values. Thus, for (l -- 4) even, the set [~c, ~ s ] transforms under C x with the same matrix as under C~, whereas for (l -- 4) odd the matrix representatives differ by a factor of -- 1.

With the standardization mentioned in the previous paragraph, the standard basis functions for the B1 and B2 representations of the Dn

n

z and .~z2s , respectively, where l = -~- when y groups with n even are ~2 c

$¢ n .

is even and l = -~- + 1 when -~- is odd and greater than one.

230

Phase-fixed 3-]" Symbols and Coupling Coefficients for the Point Groups

The set of standard basis functions for the two-dimensional irreducible representations of Doo and Dn commonly used in the literature, is the set [cos(~9), sin(,~9)] whose elements are not simultaneously basis functions for R3. It may be noted that if the standard generator C x is chosen for Doo, this set of functions spans the Ex representations with identically the same matrices as do the present choice of standard bases with C~ as the standard generator. This has the important consequence that the conventional symbols for the irreducible representations of the groups Doo and Dn agree with those of the present work. For the group D6 (Table 7), for example, E1 has the set .~an, E2 the set .~a, B1 the function ~$c, and B 2 the function ~ 8 as generating functions. The last two functions

n $¢ mentioned have ,~ = ~ = 3 and l = ~- + 1 -- 4.

The holohedric group Doon is obtained as the direct product group of the groups Doo and $2. Its sub-groups are all the non-cubic point groups, including also the groups which do not belong to the crystal systems.

In the classification of the point groups we shall for our purpose be particularly interested in their relationship with the dihedral groups Dn for which the 3-F symbols can be uniquely defined. The groups Cn will always be described in terms of Dn symbols, whereby A [symm] and A (skew) of Eq. (58) become distinguishable. For isomorphic groups the same 3-/" symbols can be chosen, and by establishing a one-to-one corre- spondence between the components of the irreducible representations of the group Dn and those of the isomorphic ones, the relation between these groups and R3, becomes unique. Further 3-/' symbols of direct product groups can be evaluated using Eq. (56). For these reasons, we note the following relations and isomorphisms (indicated by the symbol oo)

Dna = Dn × Clh (n odd) Dn1, = Dn × $2 (neven) D n a = D n × $2 (nodd) Dna oo D2n (n even) Cnv oo Dn (n odd and even)

We add the foUowing 5)

Cna=Cn × Clh (nodd) Cna=Cn × $2 (n even) Cn oo Sn (n even) Cnh O0 S2n (n odd)

which, however, are of less importance.

(59)

(60)

5) I t m a y be noted tha t the group Sn (n odd) = Cnh is of the order 2n. Therefore the group symbol Sn is only used for n even, in which case the order of the group is n, Similarly the relation Coo × $2 = Coo × Clh = Soo is valid.

231


III. 3-/" Symbols for the Sub-Groups of Ra and Rs,

8. The Non-Cubic Point Groups. Racah's Lemma

a. The Group Doo

Once the choice of basis functions for the irreducible representations of Doo has been made, the 3-F symbols can be calculated as integrals analogously to the 3-I symbols (Sect. 4). Therefore, they may also be calculated from the corresponding 3-l symbols of Rs using a positive constant of renormalization. The scheme for direct multiplication of the irreducible representations of Doo is given in Table 12.

Table 12.

Doo A1 A~ Ex Ez

A1 AI A2 Ex Ex,

A~ A2 A1 Ex Ex,

Ex Ex E~ AI+A2+ Et2x} E{x,-x) +E(x,+a)

E~, E~, E~, E{~,-a) + E(x'+a) AI+A2+E(~,)

Scheme of direct multiplication of the irreducible representations of Doo, where $' > $. From this table the multiplication schemes for the other point groups can be derived using the relationships discussed in Sect. 7. The antisymmetrized direct product (E~) is always A2 and this is true also of all the dihedral sub-groups of Doo

An alternative way of generating the 3-F symbols of Doo is to consider the tlansformation properties of the standard functions under the symmetry operator Coo, and in this way Table 13 has been obtained.

The method of constructing the 3-F symbols by means of the standard basis functions implies that another choice of basis functions generally yields a set of 3-F symbols where some phases are changed. To discuss this in greater detail, we shall consider a lemma by Racah (5). This lemma describes a relationship between the coupling coefficients of a group and those of a sub-group and shows the value of the choice of basis functions for irreducible representations as specified by the basic rule, p. 210.

The irreducible representations of a group are generally reducible as representations of a sub-group. In Racah's lemma this statement is considered by introducing the concept of a group which is reduced with respect to a sub-group. By this is meant that the irreducible representa-

232


tions of the group are in a form where the representatives of the elements of the sub-group appear in block-diagonal form, each block being chosen so as to be standard for the sub-group. The lemma states that the coupling coefficients for the irreducible representations of the group in this form are proportional to the coupling coefficients for the corresponding irreducible representations of the sub-group. One has, for example, taking R, as the group,

</1 Cl F1 yl 12 C2 F2 y2113 ca F3 78>

where the labels c~ may be introduced if more than one irreducible sub- group representation of the type F, arises from the irreducible representation l, of the gaoup. The constants of proportionality, which are independent of the components y,, form orthogonal matrices,

I, e a

= 0(cl F1 , Cl F1) ~(c2/'2, c~/'~) (62a)

el .PI e , / ' t = ~( ls , 13) (~(c3, c3) (62b)

These orthogonal matrices, which we shall call Racah lemma matrices, are characterized by two irreducible representations of the group and one irreducible representation of the sub-group [underlined in Eqs. (61), (63), and (64)]. The number of times this sub-group representation occurs in the direct product of the two irreducible representations of the group is equal to the order of the Racah lemma matrix.

As an example, we consider the direct product of thep and d representations of Rs and the resultant ~ representation of Doo. Each of the terms in the direct sum, /~ X d = /~ + d + f contributes z~ once, so that the Racah lemma matrix is three by three. By use of the Tables 11, 16, 17, and 18 and Eq. (5) one obtains

p d f En d En p E~

4 l

2 1 1 A1 +

+ Vs

(63)

233

S. E. Harnung and C. E. SchS.ffer

and with the Rs basis functions taken in the opposite order the matrix changes to

d p f Ea d E,~ p E~,

__4 _' y ~ .~. + ~ o + ~ +

2 1 1 AI E~, + - -

1/~o V2 V~o 1 1 1/~-

(64)

where the first row and the second column have had sign changes because of the odd character of the 3-/" symbols involving the trios (En A2 E~) and (d p d), respectively.

To illustrate the Racah lemma matrices in more detail, the middle element of Eq. (64) is calculated. According to Eq. (61) it is expressed by

<d A1 a l p E~ ~l[d E~ a ' l> < d A l a l p E~ z~2[d E~ z2> ~Alal En~21E~2~ (6s)

It is noted that among the Rz bases used in Eq. (65) the function ~a and the set (.~ac, .~as) transform in the standard way according to the A1 and E1 = E~ representations of D~ (Table 11), whereas the set (5ge, ~8 ) is not standard under C~. The function 5gs is symmetrical under Cz Y and the function .~c antisymmetrical. However, in order to transform by the standard matrices, i.e. by those of [.~ae, .~as], one of the p-functions will have to change its sign. We take the set [---~s, .~c] as the standard choice so that the set (.~, --~5~s, ~Sge) = (z, --y, x) forms a right-handed coordinate system. By this choice, the two last columns of the Racah lemma matrix are completely specified. One has, returning to the middle element,

and < d A l a i p E n ~ l l d E ~ l > = - - < d a p~sldz~c> (66)

<dA1 al p Eaz~2ld Eaz~> = .<de pzl c[dz~ s> (67)

which may be evaluated from Table 17 and Eq. (S),

< d A1 a l p Euz~I [d E u ~ I > = < d A1 a l p Enzi2 ]d E u ~ 2 >

1 1 = -V~0 x 1/s = - g-~ (6s)

234


The coupling coefficients for Doo are found from Table 13 and Eq. (5),

1 <AI al E~t~llE#~I> = <A1 al E~z21E~z2> = ~-~ ×

x V = 1

Combination of Eqs. (68) and (69) according to Eq. (65) then gives the matrix element desired.

It is noted that the adaptation of 5~ functions to Doo can always be done analogously to the case of d and/5 functions. For I even the set [-~c, 5~s] and for l odd the set [-- 5~s, 5~e] transform in the standard way under Doo. By means of this convention, the first column of the Racah lemma matrices of Eqs. (63) and (64) is completely specified.

In order to apply Racah's 1emma to the group R3 reduced with respect to Coo (most reduced representations with a real basis), one simply uses the classification of irreducible representations according to Doo as explained in Table 13. As the two sigma terms in A × A cannot be distinguished in the group Coo, the Racah lemma matrices whose elements are characterized by (lA l_A I l'X__), will appear in block- diagonal form, since the 2: can be either 2:1 or X2 (Table 13).

Table 13. Non-vanishing 3-I ~ symbols for Coo. based upon the most reduced representations with a real basis

(.~V'l 271 .~r'l) = (~v'2 Z1 Z 2 ) = 1 0"1 0"1 0"1 0"2 0"1 0"2

( A 271al A ' ) = ( A 271al A s ' ) = (1/[/2, d(A ,A ' )

(A 272a~ A') = - - (A 27~ A') = (1/V~)$(a, A') (odd)

(AA'A+A')c = (A A'A+A')s s

(AA'A+cA") (As A'A+A' ) = c s (1/2)

The group Coo with a real basis is not a simply reducible group, because A X A = 27+27+ (2A) where (2A) stands for the representation with 2 = 2A. The 3-Fsymbols containing 27 fall into an even class called (F 271 F) and an odd class called (F 2J2 F) with reference to the group Doo. All other 3- / ' symbols are evcn. When in Doo the Y-axis is used to standardize the components (sub-indices 1 and 2 for symmetrical and antisymmetrical under C~) and when all standard basis Junctions are chosen with even/-values, all cosine functions are symmetrical and all sine functions antisymmetrical under C2 v and the table is valid for Doo by the substitution, 271 a l -~ A1 al, • ~v'2 a2 ~ A2 a2, Ac ~ E,t 21, and As ~ Ea 22.

235

S. E. H a r n u n g and C. E. Schi~ffer

b) The Sub-Groups of Doo

As mentioned in Sect. 7, the structure of the groups Dn and Cn is different for n odd and n even. The groups Ds and D4 are used to exemplify this.

The group D, has the irreducible representations A1, A2, and E (Table 5), where E is the conventional abbreviation for El. Racah's lemma is fulfilled foi these representations when they are associated with the first three irreducible representations of Doo (Table 11, compare Table 5) and the 3-P symbols can be directly read out of Table 13, except for the EEE symbols, which require additional considerations. The basis functions for E2 of Doo span the E representation of D~ in the standard manner when taken as [ ~ a _ .~0a,]. Hence the 3-P symbols follow from Table 17 (using a positive renormalization constant) or from Table 13. From Table 13 one obtains

. . . . . . . ~ (70) e2 el e2 el e2 e2 el el el e2 es el

where the first two symbols and the last one are the same, as they should be, since this 3-F symbol is even.

The irreducible representations of D4 are Aa, A2, B1, B2, and E (Table 6) where again E means El. As discussed in Sect. 7, the standard basis functions are .~ , ~ , -~0c, .~oas, and [~ac, .~nas], respectively. The 3-F symbols follow from Table 13 using a positive constant of renormalization for those involving the ~-functions. One has

(B. A1Bq (BeA1Bs bl al bl ] = b2 a l be ] ~--- 1

B2 As Bl~ = __ [B1 As = 1 (odd) b2 as bl] \ bl as b2

eselbe elesbe] = elelba] esezbl]- - ~

(71)

In order to obtain general expressions for the 3-Y symbols of the Dn groups it is noted (Table 11) that for E~ the standard representative of the Ca operator is

cos ~ -- sin - -

ksin 2n~ 2~z| n COS n /

(72)

236


where the standard order of the basis functions is [&~e, ~ s ] and I even. For 2 = n, this is a unit matrix and the functions span the A1 and A2

n representations. For 2 = ~ (n even), the matrix is minus one times

the unit matrix and the functions span 131 and B2. The En-a representative of the Cn operation is the same as that of Ea except for a sign change on the non-diagonal elements in Eq. (72). Therefore, in order to fulfil the conditions for Racah's lemma, one of the basis functions of En-a has to change sign so that the set transforms with the standard Ea matrices. We choose, as above with Ds, the standard [~n-a)e, --~(~-x)s] for the representations En-a. For the Dn groups the Ea

representations all have 2 < ~ so that when the direct product Ea X Ea'

n contains Ea+a' with ,t + ,1' > ~ we have, for example, using Table 13,

(Z~. Ez. E,~+x.I = __ (Ea E t" En_~._a. ) 1

e2 el e2 1 keg el e2 - - 2 (73)

Table 14. Non-vanishing 3-F symbols for the Dn group. For n odd, all those containing the B representations vanish

( ) (a.) A1 = = 1 A1 A1 A2 A1 a l a l a l a2 a l a2

) = ( - . ) E~, = (1/V~) 8(,L ;¢) Ea A1 Ez A1 \ el al el e2 al e2

B1 = = 1 \ b l al bl b2 al b2

(B2 A2BI~ =_(BI A2 B2~ = 1 (odd) b2 a2 bl ] bl a2 b2 ]

el e2 / \ e l e2 e2 / el el / - - \ e 2 e2 el /

-(~,~,.~.-,-,.1=~,., .. , -(~'~"~--'-"/=( ~ ' ~ ' ~ - - ' - ' ~ - - - ( ~ ' ~ ' ~ - - ' - ' ~ = ( , . , . . .. ,, , ~, ~1 ~, , - ,~., , ,, ( ( ) o,)

= = B ~ = - = 1 / V ~ Ex Eln-x B2] Ea E½n-a B2~ (Ea E,n-a (E~ E½n-a e2 el b2] el e2 bg.] \ e l el bl \e2 e2 bl

(odd)

This table can be used for all the other finite non-cubic point groups as well, as discussed

insect . 8. Similar tables have been given by Griffith (4), but his signs are in some cases differ-

ent from ours.

237

S. E. Harnung and C. E. Sch~.ffer

where the minus sign arises because it is minus one times the sine component of Ea+~.,, which gives the standard form of En-a-x,. Generally,

whenever one has the sine component of Ea+~, and 2 + 2' > ~, the 3-/"

symbol changes sign relative to Table 13. When 2 + 2 ' = n/2 (n even), the cosine component transforms as

B1 and the sine component as Be, and the four last 3-/" symbols of Table 13 fall into two groups; both are even and keep the sign of Table 13. The generalized results for Dn groups are collected in Table 14.

As has been mentioned (Sect. 7 and Table 13), the real basis choice makes it possible to classify a cyclic sub-group Cn by its corresponding dihedral group Dn and to use its 3-F symbols. This can be done in all cases where the degeneracies within the Cn groups are not lifted, as is the case with electric fields, for example, in contrast to magnetic fields, which will lift the degeneracies and thereby most conveniently require a complex basis for the description.

c) D~n and Its Sub-Groups

The 3-/" symbols of Dooh may be obtained through the relation Dooh = Doo X $2 in an exactly analogous way to that used for Rs~ -= Rs x $2 in Sect. 5. For the groups Dnh similar relations apply [Eq. (59)] and the same is true of the groups Dna, so that the 3-F symbols for the holohedric groups can be obtained from those of the corresponding pure rotation groups.

For the group Cwv the isomorphism Doo co C~v makes it possible to use the 3-1" symbols of the group Doo for the group C~v. However, in order to give such a choice a practical meaning, one has to associate the irreducible representations and components of the two groups in a unique way. This is done by the following relations

A1 al -~--+ ~'+ a+; Ag. a2 .,-o. ~'- a-

E~ 21 ~ A 2+; Ea 22 -*--,- A 2- (74)

Here, the components, which in Doo are labeled 1 and 2 for symmetrical and antisymmetrical under the C~ operation, are in C~v labeled + and -- for symmetrical and antisymmetrical under the operation ozx = IC~.

In order to use Racah's lemma to the group Rs~ reduced with respect to C~ov, it is noted that the standard basis functions of the irreducible representations in Eq. (25) [which are the functions of Eq. (15)] are symmetry-adapted to Co~v in such a way that sigma and cosine functions

238


are symmetrical under a zx, and sine functions antisymmetrical. The basis functions of the irreducible representations in Eq. (27) have under the proper rotations Rz(~) the same behaviour as the corresponding ones of Eq. (25) but transform with the opposite signs under the improper rotation, a zx. In analogy with previous choices6 t (p. 234 and p. 236), we make the choice of standard basis functions [-- 2R~, 2R~] for those irreducible representations of Rs, whose bases are not represented by spherical harmonics, i.e. the axial tensorial sets which have l even, parity p ungerade or l odd, parity p gerade. Thereby the Racah ]emma matrices for R,~ reduced with respect to Coov are fully defined.

For the groups Cnv, the isomorphism Dn oo Cnv makes an asso- ciation with the corresponding Dn groups possible in the same way as for the infinite groups. However, here the conventional labeling of the irreducible representations is a different one in that the subindices 1 and 2 in the irreducible representations of Cnv mean symmetrical and antisymmetrical with respect to a zx. So, for the groups Dn and Cnv, one associates those irreducible representations and components which have the same set of labels, and one has only to remember the different meanings of the sub-indices in the two groups.

For the group Dna (with n even) the isomorphism Dna oo D2n may be used to define the 3-1" symbols. Again the irreducible representations and components that have the same set of labels will be associated. One has to remember that A and B in D2~ refer to the symmetry operator C(~n) and in Dna to the operator S(2n). In this case, however, the sub-

Y indices 1 and 2 refer to the C2 operator m both groups. Finally, in the commutative point groups Cn~ and Sn one can apply

the 3-/' symbols of Dnh and Dn, using the relations of Eq. (60).

9. The 3-1" Symbols for the Cubic Point Groups

a) The Octahedral Rotation Group, 0

In the three-dimensional rotation group Rs every other irreducible representation occurs in the antisymmetrized direct products [Eq. (28)], whereas in the dihedral group Do only one particular irreducible representation occurs in these direct products [Table 12]. For the octahedral rotation group O, one has an intermediate situation.

6) We note t ha t whenever in this paper the sets [~lxc, ~ , ] or [ ~ c , ~;~,] cannot be taken to t ransform standard, we have always, independent of the s tandard order within a particular set, achieved the s tandard behaviour by allotting a minus sign to the sine component.

239

S. E. Harnung and C. E. SchAfler

In the group 0 the irreducible representations A2 and T1 occur in the antisymmetrized direct products only, while A1, E, and T2 occur in the symmetrized direct products only. According to the basic rule (p. 210), the spherical harmonic basis functions are chosen so that odd /-values are associated with A2 and T1 and even/-values with A1, E, and T2. Using for the group 0 always the lowest possible/-value, the sets of standard basis functions 7) are as stated in Table 8.

With this choice of standard basis functions the 3-/" symbols of the group 0 are completely defined from the 3-l symbols, provided that a positive constant of renormalization is used for each trio of irreducible representations of 0. The trivial 3-/" symbols are the even symbols,

( A1 A1 A1 ~ A2 A1 = 1 al a l a l / a2 a l a2

(75)

The non-trivial ones may be classified as follows. 7. Those containing the E or the T2 representation twice. They may be found by renormah- zation of the 3-1 symbols of Table 17 and are given in the Tables 19 and 21.2. The 3-/" symbols that contain T1 twice. They may be found from Table 16 and are given in Table 20. 3. The 3-/" symbols of the form (T, r T,, v,v,/. They can be found from Table 18 and are given in Table 22.

The 3-/" symbols defined in this way have the same character of even- and oddness under permutations of their columns as have their parent 3-l symbols. Thus, when A2 and T1 occur an odd number of times, one has an odd 3-/' symbol. By use of the permutation properties it appears that the 3-/" symbols that involve the trios (T2T1T1) and (T2T2T1) in Table 22 also occur in Tables 20 and 21, respectively.

b) The Octahedral Group On and Its Sub-Groups

The 3-/" symbols of the group 0 can be taken as 3-/" symbols also for the isomorphic group Ta, and these symbols may again be used for the tetra- hedral rotation group T when a real basis is sufficient.

The 3-/" symbols of the group of the octahedron On can be found through the relation Oh = 0 × $2 using Eq. (56) andthese 3-Fsymbols may again be applied to Th, provided that the most reduced representations with a real basis are sufficient for the description of this group.

7) This choice is the same as t h a t of Griff i th (d), to whom we are greatly indebted for the idea of the basic rule. Apparently, he himself did not realize its scope, since he did not apply it to associate the groups Ra and O and did not use i t for the D . groups at all.

240


10. Conclusions

A standard choice of real basis functions for the three-dimensional rotation group Rs and its sub-groups has been used to generate phase- fixed even 3-l symbols for Rs and even 3-/" symbols for the sub-groups, in each case together with the corresponding coupling coefficients.

The odd 3-l and 3-F symbols for the pure rotation groups have similarly been defined through the introduction of axial operators involving partial differentiation. When three given functions (or one operator and two functions) are used to define a 3-/' symbol (or a 3-/symbol) for more than one group, the sign is fixed by the functions and independent of the group.

For the holohedric groups which arise as the direct product of the pure rotation groups and the inversion group $2, the 3-F symbols (or for Rs, = Rs × $2 the 3-I symbols) have been defined on the basis of those for the rotation groups themselves.

For the hemihedric groups the 3-F symbols have been chosen to be equal to those of their isomorphic pure rotation groups.

Through these simple conventions, plus a few other cases of choice of basis functions transforming standard, the relation between the coupling coefficients of the group Rs, and those of its sub-groups has been completely specified.

As a consequence of a lemma by Racah, it has been necessary to distinguish between standard basis sets which are used to generate with sign the 3-F symbols of the rotation groups, and basis sets transforming standard. The latter sets would in general have generated the 3-F symbols with different phases.

Appendix 1

The Evaluation of 34 Symbols.

The even 3-I symbols can be found directly from Eq. (39) by integration, provided that the constant, k(lll21a) is known. As it will be shown in the following paper (10), Eq. (39) may be written

4 ~ J a, Jt, ~,Jt, ~3~, \a g \tlt~}3

where ll + 12 + 13 = L is even, and

241

S. E. Harnung and C. E. Schgffer

The integrand may be found by multiplication of the spherical harmonics, Eq. (15a) or Table 15, in which they are listed up to l = 9. The integration is conveniently carlied out by use of the expression

if x2py2qz 2r dS = (2p)!(2q) !(2r)!(p+q+r)! p !q!rl (2 p + 2q + 2r + l) [

(78)

which we have derived by the trigonometric substitution, Eq. (17), and the integrals

f: cosma sinna do~ - - m + n sinn~ d~

._if: - - ~n+n cosm~ sinn-2~ dx

(79)

f2gcosmo¢ s i n n a da m - 1 f2= - - cosm-2~ sinna da

j o m + n o

- - ~ + 1 c°smasinn-2e de

(80)

In the special case of an even 3-l symbol which has l l - - - -13- 12 one can use Eq. (39) in the form

4~j~.at, t ,~ t d S = k ' ( l l l z l 3 ) qt2t3]' 11-----t3--12 (81)

where the constant k' is given [(70), Eq. (48)] by

213 + 1) ½ k'(lll2/3) = (2ll + 1) -½ \ 212 (82)

The odd 3-l symbols can be evaluated as outlined in Sect. 4b by an operator equivalent method which ultimately brings the calculation into the form of Eq. (76). However, although this method has considerable theoretical interest, it is laborious and not feasible for high /-values. We refer to the following paper (10) for a general and useful way to calculate any 3-l symbol in terms of a 3-j symbol.

Generally, any 3-l symbol can be expressed as the integral

if (83)

242

Phase-fixed 3-/~ Symbols and Coupling Coefficients for the Point Groups

where the reduced matrix element [see Eq. (3) and comments on Eq. (4)] is given by

(84)

Eq. (84) is a general form of the set lIk of unit tensorial operators, whose reduced matrices are

< ]] u ][ la > = a(l, a(l, k z.) (ss)

and the operators ~ , of which there is an infinite number, may be found from Ref. (lO), Eq. (33), and operator equivalents for ~ may be found from Ref. (lO), Eq. (57). In Tables 16--22 we have represented a few 3-l symbols of Ra and 34 ' symbols of 0 in terms of the general unit tensorial operators ~ and 2~v r, respectively.

A p p e n d i x 2

Racah's Lemma Applied to the Group Rs, Reduced with Respect to the Group Coov.

In Sect. 8a Racah's lemma was illustrated for the group R3 reduced with respect to D~o. It is valuable also to consider the group Ra~ reduced with respect to Coov, since this will exemplify the standardizations made in section 8c.

We shall use as an example one of the coupling coefficients of Rs which corresponds to the middle element of Eq. (64),

1 < d,~ #:~c ] d:~s > = -- 1/~ (86)

By use of Eqs. (56) and (57) one can generate for Ra, the four coupling coefficients

< dg,~ pgnc ] dgzs > = dg,~ p,,:~c [ d, ,m > 1

= < dua pg~c I du:~s > = < du~ pu:w [ dg~s > = - - (87)

Each of these yields one Racah lemma constant after Ra, has been reduced with respect to Coov. Since the basis set of the irreducible representation pg is an axial tensorial set R~g (p. 210), and tile basis set

243


of dg is a polar tensorial set ~eg, the first of the coefficients of Eq. (87) becomes

1 < dg Z'+a + pg/- /e- [ dg H e - > --

V~

(1t( t = ( ~ z+ p, I / I d, _n / < z + ~ + / z = - I n = - > = - 7 ~ + 1 (ss/

Here, the coupling coefficient of Coov follows from Eq. (74), Table 13, and Eq. (5), and the value ~- 1 implies that the Racah lemma constant

1 is -- ]/-y. Similarly the remaining three coupling coefficients of Eq. (87)

become

1 < da I:+a+ P u / / e + [ & He+ > = + ~ff

1 < du 2:-~- p g / / e - [ &/ - /e+ > = +

1 < d~ 2:-~- p ~ / / e + [ d g / / e - > = -- V-T

when reduced to Coov.

Appendix 3 (Tables 15--22)

244

Tab

le

15.

Soli

d H

arm

onic

s ~

, no

rmal

ized

to

4~/(

2l +

1)

over

the

Uni

t Sp

here

. W

hen

norm

aliz

ed t

o un

ity

they

are

the

usu

al

real

hyd

roge

n an

gula

r fu

ncti

ons

g tm

l~O

l =

1

or'

o C~

0

~ =

z

l ~

2

o~

C~ g g

t'O

,t

a.

carl

$~

= (1

/2)

(2z~

-.3-

y2)

25L

= 0/

~2)

12 *

y)

co

~0

o.

O

Tab

le

15 (

cont

inue

d)

l =

3

~ =

(1/2

)(2z

~-3

x 2-

3y

2)z

~6

=

(V

6/4)

( 4z

2 -x

2-y2

)y

.~c

= (V

6/4)

(4

z2-x

2-

y2)x

~5~

= (l

/~/2

) ~

(2 .

y)

~c

=

(V15

/2)z

( x2

- y2

)

~,

= (V

~/4

) (3

x2y

_ ya

)

oq

9~

f~

l =

4

-~ga

=

(1/8

) (S

z 4 +

3x

4 -

~ 3y

4 -

- 24

z2x

2 --

24y

2z 2

+

6x2y

2)

~L

=

(Vi~

/4)

(4z2

-

3~,~

-

3y~

)~y

5% =

(V

~/4)

(G

~2

_ ~2

_

y2)

(2 xy

) ~

c :

(V5/

4) (

6z 2

-x2-

y2)(

x2-y

2)

~ :

(Vfo

/4)~

(3~

y -

y~)

~%c

: (y

~/4

) z(

x3 -

3x

y2)

-~g~

s =

(V~-

/8)

(4xS

Y -

- 4x

Y 3

)

-~c

= (V

~/8

) ( x

4 --

6x2

y 2

+

y4)

l =

5

.~ha

=

(1/8

) (8

z 5

-- 4

0za

x ~

-- 4

0zZ

y 2

-~

15

zx 4

-~-

15

zy 4

q-

30

zyS

x 2)

~s

: (V

~-/

8) (

8z4

-~-

X4

-~ y

4 +

2x~

y2

__

12Z

~X

2 __

12z

~y2

) y

~h

e

=

(V~

/8)

(8Z

4 _~

X4

-~_

y4 _

1_ 2

x2y2

__

12

Z2X

2 __

12z

2y~)

X

~s

= (V

1-05

/4)

( 2z2

-x

2

-y2

) z

(2x

y)

~n

c =

(V~

/4)

(2z9

~ _

x2

_y

2)

z(x

2-y

~)

.~nc

s =

(Vff

~-/1

6)

(8z

2 _

x2

_ y2

) (3

x2y

_ yS

)

$~

= (V

~/1

6)

(8~

-

~ -

y~)

(~a

- 3x

28)

~5~

= (3

V~-

/8)

~ (4

;,~y

-

a~y~

)

~c

= (3

1/5s

-/8)

~ (

~4 _

6

~y

~

+ 24

)

.~,~

=

(3 V

]T/1

6) (

S~

ay

_ 10

~y8

+

y~)

~hnc

=

(3V

~/1

6)

(x 5

- lO

x3y

2 q-

5xy

4)

=

6

oq

cy"

o o fro C~

o

.~

=

~,

=

.~,

=

.~$o

=

(1/1

6)

(16z

6 -

- 1

20

z4x

~ -

- 12

0z4y

2 q

- 90

z2;¢

4 -

-[- 9

0z2

y 4

if-

18

0z2

x2

y 2

--

5x

6 -

- 5

y 6

--

15a~

4y 2

--

15

~2

y 4

)

(V2]

-/8)

(S

z 5

- 2

0zS

x2

_

20

zay

2

--}-

5gx

4 -q

- 5z

y4 q

- lO

zx2

y 2)

y

(y~

-/8

) (8

z 5

- 2

0za

x 2

-

20

z3y

2 q

- 5

zx 4

-1-

5zy

4 -

q- l

Ozx

2y

2)

x

(V~

32

) (1

6z4

_ 16

z2x2

_

16z~

y2 -

{- ~

4 -J

r- y

4 ~

2

zZy

2)

(2xy

)

(V2-

-~/3

2)

(16z

4 _

16

z2x2

_

16z2

y2 +

x

4 +

y4

+

2x

2y

2)

(X 2

__ y

~)

(V21

0-/1

6)

(8z

2 -

3x

2 -

3

y ~

:) z

(3x2

y --

y3

)

(V2i

-0/1

6)

(8z

2 -

3x

2 -

3

y 2

) z

(x 3

-- 3

xy

2)

o7

o

oo

Tab

le

15

(co

nti

nu

ed)

~,

= (3

V7-

/16)

(10

z 2

- x

2 -

y2)

(4.v

Sy

_ 4~

:y8)

-~c

: (3

V7-

/16)

(lO

z 2

- x

2 -

y2)

(.,.4

_

6x

Sy

2

_~_ y

4)

3~

s =

(3 V

154/

16)

z (5

x4

y

- lO

xZ

y 3

-[.

- ys)

~c

: (3

V1-

54/1

6) z

(x

5 -

lOz3

y 8

-~

5x

y 4

)

S3~,~

=

(V~

/32)

(6

xSy

- 2

0~

3y

8

-b 6

xy

5)

l =

7

oo

oq

f~

~q

3~a ~

: 1/

16 (

16z

7 --

16

8zS

x 2

--

16

8z5

y 2

@

21

0z3

x 4

-~-

21

0z3

y 4

-}-

42

0z3

x2y

~ --

3

5zx

6 -

- 3

5zy

6 -

- 1

05

zxa

y 2

--

10

5zx

2y

4)

~nks

=

(V7/

32)

(64z

6 -

2

40

z4x

2

_ 2

40

z4y

2

-~

120z

2x4

-~

120z

2y4

-~-

24

0z2

xSy2

_

5x

6

_ 5

y6

-

15

x4

y2

_

15

x2

y4

) y

~c

= (V

7/32

) ( 6

4z6

- 2

40

zax

2

- 2

40

z4y

2 -

~ 12

0z2x

4 -~

- 12

0z~

y 4

-~ 2

40

z2x

2y

2 -

5

x6

-

5y

6 -

1

5x

4y

2 -

1

5x

2y

4)

x

-~s

= (V

~3

2)

( 48z

4 -

80

z2x

2

- 8

0z2

y 2

-~-

15x

4 -~

- 15

y 4

-~ 3

0x

2y

2)

z (2

~y)

-~c

= (V

42-/

32)

(48z

4 -

8

0z

zz2

-

80

zay

2 -}

- 15

x 4

-1-

15 y

4 _~

30x

ZyZ

) z

(x 2

- y~

)

~s

= (V

~-/

32)

(SO

z 4 _

6

0z2

x2

_ 6

0z2

y2

~_ 3

x4

q_

3y4

.~_

6x2y

~)

(3x2

y __

y3)

-~c

= (~

21/3

2)

(80z

4 -

- 6

0z~

x 2

--

60

z2y

2 "q

- 3

x 4

q-

3y

4 -

b 6

x2y

~)

(x 3

--

3

xY s

)

-~s

= (~

231/

16)

(lO

z 2

--3

x

2-3

y

2) z

(4x

3x

-4x

y

s)

~c

: (V

2-31

/16)

(lO

z 2

-3x

~

-3y

2)

z(x

4-6

x2

y

s-y4

)

• ~8

=

(V2-

31/3

2)

(12z

2 -

x2

- y2

) (S

x4

y

_ lO

x2

y3

q_

yS)

.~nk

c =

(V~

32

) (1

2z 2

_

x 2

_ y2

) (x

5 _

lo

x3

22

+

5xy4

)

~8

:

(V6-

0~/3

2)

z(6

xSy

- 2

0xa

y 3

+

6xy

5)

.~Po

= +5

++

=

l =

8 (l/6

-~/3

2)

z(x6

--

lS

x4

y2

27

15

x2

y4

_

y6)

(V42

9-/3

2) (

7x

6y

_

35

x4

y3

27

2!x

2y

5

_ yT

)

(V42

~/3

2)

(x 7

-- 2

1xT

y2 2

7 3

5x

3y

4 -

- 7x

y6)

¢,D

.~

= 1/

128

(128

z s

--

1792

z6x

2 --

179

2z6y

2 2

7 3

36

0z4

x 4

27

33

60

z4y

4 2

7 6

72

0z4

x2

y 2

--

l12

0z2

x 6

--

l12

0z2

y 6

--

33

60

z2x4

y2

--

33

60

z2x2

y4

27

35

x 8

27

35y

8 27

14

0x

6y

2 2

7 1

40

x2

y 6

27

21

0x

4y

4)

~/s

=

(3/3

2)

(64z

6 -

- 3

36

zax

2 -

- 3

36

z4y

2 2

7 28

0zZ

x 4

27 2

80

z2y

4 2

7 5

60

zZx2

y 2

--

35

x 6

--

35

y 6

--

10

5x

4y

2 -

- 1

05

x2

y 4

) zy

~/~c

=

(3/3

2)

(64z

6 -

- 3

36

z4x

2 -

- 3

36

z4y

2 -

~- 2

80

z2x

4 2

7 2

80

z2y

4 2

7 5

60

z2x2

y 2

--

35

x 6

--

35

y 6

--

10

5x

4y

2 -

- 1

05

x2

y 4

) zx

~t

8 =

(3V

7076

4)

(32z

6 -

80

z4x

2

-80

z4y

2

27

30

z2x

42

7 3

0z2

y 4

27

60

z2x

2y

2

- x

6-y

6

- 3

x4y

2 -

3x

2y

4)

(2

xy)

~$e

:

(3 V

~6

4)

(32z

6 -

8

0za

x 2

-

80

z4y

2 2

v 3

0z2

x 4

27

30

z2y

4 2

7 6

0z2

x2y

2 -

x 6

- y6

_

3x4

y2

_ 3

x2

y4

) (x

2 _

y2)

.~,

: (V

l155

/32)

(1

6z 4

-

20

z2x

2 -

20

z2y

2 2

7 3

x 4

27

3y

4 2

7 6

x2y

2) z

(3

x2y

- y3

)

~e

= (V

l155

/32)

( 1

6z4

-20

z2x

2

-20

z2y

2 27

3

x 4

27

3y

42

7 6

x2

y 2

) z(

x a

-

3xy

2)

-~s

= (3

V77

/64)

(4

0z 4

-

24

z2x

2 -

2

4z2

y 2

27

x 4

27 y

4 2

7 2

x2

y2

) (4

xsy

_ 4

xy3

)

~e

~--

(3V

7776

4)

(40z

4 -

2

4z2

x 2

-24

z2y

22

7x

42

7y4

27

2

x2

y2

) (x

4 _

6x2

y2

_y4

)

.~,

= (3

V10

01-/

32)

(4z

2 -

x 2

- y2

) z

(5x4

y -

lOx2

y 3

27 y

s)

~ =

(3 1

/1~6

~/32

) (4

z2

_ .8

_

y2)

z (.

5 _

10.~

y2

+ sx

y4)

~,+

=

(V8~

8/64

) (1

4z2

_ x2

_

y2)

(6x5

y _

20

x3

y3

27

6xyS

)

-~c

= (V

858/

64)

( 14z

2 -

x 2

- y2

) (x

6 _

15x4

y2 2

7 1

5x

2y

4

_ y6

)

.~s

= (3

V77

5/32

) z

(7x

6y

_

35

x4

ya

27 2

1x

2y

5 -

yT

)

~c

= (3

V77

5/32

) z(

x 7

-2

1x

5y

2

27

35

x~

y a

-7zy

6)

~/j~

=

(3 V

715/

128)

(8

x7

y

- 5

6x

ay 3

27

56

x3

y 5

--

8x

y 7

)

.~t~

c :

(3 V

7~5/

128)

(x

a -

- 2

8x6

y 2

27 7

0x

ay 4

--

2

8x

2y

6 2

7 yS

)

oo

¢T

o C)

o C~

~° g o

Tab

le

15 (

con

tin

ued

) ~o

/ ~-

~ 9

sS,~

=

~=

1/12

8 (1

28

z 9

--

23

04

zTx

2 -

- 2

30

4zT

y2

-~

- 6

04

8zs

x4

-[

- 6

04

8zS

y4

~

- 1

20

96

z5x

2y

2

--

33

60

z3x

6

_ 3

36

0z3

y6

--

10

08

0z3

x4

Y 2

--

10

08

0z3

x2

y 4

-}-

31

5zx

s -b

31

5zy

8 ~

- 1

26

0zx

6y

2 -

[-

12

60

x2

y 6

-[-

18

90

zx4

y 4

)

(3 V

5/12

8)

(12

8z

s -

89

6zO

x 2

-

89

6z6

y 2

~-

l12

Oza

x4

-~-

11

2O

z4y

4 -

[- 2

24

0z4

x2

y 2

-

28

Oz2

x 6

-

28

Oz2

y 6

- 8

40

zUx

4y

2 -

8

40

z2x

2y

4 -

b 7

x s

-]-

7y

s -

k 2

8x

6y

2 q

- 2

8x

2y

6 -

[- 4

2x

ay 4

) y

(3 V

5/12

8)

(12

8z

s -

89

6z6

x 2

-

89

6z6

y 2

q-

ll2

Oz4

x 4

q-

11

2O

z4y

4 ~

- 2

24

Oz4

x2

y 2

-

28

Oz2

x 6

-

28

Oz2

y 6

- 84

Oz2

x4,1

2 -

84

Oz2

x2

y 4

~-

7x

8 -

~- 7

y s

-k

28

xS

y 2

-b

28

x2

y 6

-k

42

x4

y 4

) x

(3 V

17o/

64)

32

z 6

- ll

2z4

x

u -

ll2

z4y

2 q

- 7

oz2

x 4

q-

7o

z2y

4 ~

- 1

4o

z2x

2y

2

_ 7

x6

_

7y

6

_ 2

1x

ay2

_

21

x2

y4

) z

(2x

y)

(3 V

~/6

4)

32

z 6

- l1

2z4

x 2

-

l12

z4y

2 -

]- 7

Oz2

x 4

-}-

7o

z2y

4 -

~

14

Oz2

x2

y2

_

7x

6

_ 7

y6

_

21

x4

y2

_

21

x2

y4

) z

(x 2

- yU

)

(V2

~0

-/1

28

) 6

4z

6 _

12

oz4

x2

_

12

oza

y2

q_

36

z2x

4 ~

_ 3

6z2

y4

~

72

z2x

2y

2

_ x

6

_ y

6

_ 3

x4

y~

_

3x

2y

4)

(3x

2y

_

y~)

(V2-

3~-/

128)

6

4z6

-

12

oz4

x2

-

12O

z4y

2 -[

- 3

6z2

x4

q-

36

z2y

4 -

b 7

2z2

x2

y 2

-

x6

_

y6

_

3x

4y

2

_ 3

x2

y4

) (x

3 _

3x

y2

)

(3 ]

/~/6

4)

8z

4 -

8z2

x ~

-

8z2

y 2

~

x 4

-k y

4 q

_ 2

x2

y2

) z

(4x

3y

-

4x

y3

)

(3 [

/~/6

4)

8z

4 -

8z2

x 2

-

8z2

y 2

-[-

x 4

-~

y4

_[_

2x

2y

2)

z (x

4 -

6x

2y

2 -

y4

)

(3 [

/~/1

28

) 5

6z

4 -

28z2

;¢ 2

-

28

z2y

~ -

}- x

4 q

- y

4 _

[_ 2

~2

y2

) (5

x4

_

lOx

2y

3

_~ y

S)

(3 V

~/1

28

) 5

6z

4 -

28

z2x

~ -

2

8z2

y 2

~-

x 4

+ y

4 _

[_ 2

x2

y~

) (x

5 _

lOx

ay2

-k

5x

y 4

)

(V42

90/6

4)

'14

z 2

- 3

x 2

-

3y

2)

z (6

xay

-

2o

xsy

3 ~

6

xy

5)

(V4

2~

/64

) '1

4z

2 -

3x

2 -

3

y 2

) z

(x 6

-

15

x4

y ~

Jr

15

x2

y 4

-

y6

)

(3 [

/14

~/2

56

) (1

6z

2 -

x 2

- y

2)

(Tx

6y

-

35

x4

y ~

~

21

x2

y ~

-

yS

)

(3 V

14

30

/25

6)

(16

z 2

-- x

2 -

- y2

) (x

5 _

21

~5

y2

~_

35

x3

y4

_

7x

y6

)

(3 V

~]1

28

)

z (S

xT

y

--

56

xS

y 3

q-

56

xay

5 -

- 8

xy

7)

(3 V

~/1

2s)

z

(z s

- 2

8x

6y

2 -

~ 7

0x

4y

4 -

2

8x

~y

6 -

k yS

)

(V~

/25

6)

(gx

Sy

_

84

x6

y3

q_

12

6x

ay5

_

36

x~

y7

~_

y9)

(V~

3]0

/25

6)

(x 9

--

36

x7

y2

-~

- 1

26

x5

y4

--

84

x3

y6

-{

- 9

xy

8)

o~


( p l p ) represented as matrix elements of the general unit Table 16. The 3-1 symbols _ t2 t3 \ /

tensorial operator ~z [Eqs. (83) and (84)]. In the table, the entries concerned with l = s must be divided by V ~, those with l = p divided by V ~, and those with I = d by V ~ . i t is noted that except for l = s the trace of the matrix is zero. This is known as the barieenter rule or the center of gravity rule and is a consequence of Eqs. (6a) and (5),

72 73/ \71 72 73/ = [F3]-I 6(F3 F~) 6(y3 73) 6(F1 F2 F3). 'FAT,

t/~IF2 ~ ) = 1/V[F1],and Here, i f F~ = s, then F1 = F2, ?~ = 72, and \71 ?2

_ra = 1/[- ~ d(ra, s) ~(73, G). ?i 71 ?3

7~

The matrix is symmetrical/or even 3-I symbols. Further, the square sum rules implied from Eqs. (4), (5), and (6) may be noted

~ + ~3v + ~a p a p~s p~e

P~

p ~ s

pz~c

1 2 s da

--1 p~c

1 pres

V S 1 d~rs s

d~c

p ~ e d~s

- 1

da d6c

pa dds

--1 p~rs

1 pa

1 S

V ~ d~e

dds

--1 da

V~ ddc

251

bo

bo

d l

d)

mat

rix

elem

ents

th

e u

nit

ten

sori

al

?St.

Fo

r 1

an

d d

the

ent

ries

T

able

17

. T

he 3

-l s

ymbo

ls

tl t

~ ta

~e

pres

ente

d as

of

ge

nera

l op

erat

or

s, P

, of

/

the

tabl

e m

ust

be d

ivid

ed

by

V ~

, V

so,

and

V7o

, re

spec

tive

ly.

For

l =

f on

ly

the

elem

ents

o

f fB

los

are

give

n,"

they

m

ust

be

div

ided

by

V

~.

The

ele

men

ts o

f ~

e ar

e om

itte

d

ctq

fB 8

+

fB2 °

+

fB a

+

~le

s I

da

d

~s

d:,r

c d

os

do

e

da

dn

s

dne

des

d(Se

1 2

s d

a

-V~

'

p~

c

d~

s

V S

1 p

~c

d~

s

1 I

-V3

s

da

d

6c

p~

s d

~c

x V

~s

pa

d¢

5

--2

d

os

-1

Vs

p~

s d

~c

V~

i p

~s

d~

c

--2

d

~s

--2

1

d~

c f~

s

--1

~3

pa

d~

s

pn

s d

ne

p~rc

d

vrs

s d

a

ddc

-1

Va

pn

c d

ns

V~

p~

s d

zc

1 p~

rc

1 s --2

Pa

Vg

dT~s

--2

d

a

--2

--

1

ddc

fds

--1

-V ~

p

~c

d~

s

-1

1/~

p~

s d~

ro

2 Pa

1 --

2

s d

a

u~

o ~a

Phase-fixed 3-_P Symbols and Coupling Coefficients for the Point Groups

II

.~.

z 2 ~.~

s ~

ICe~ tn V,D

7 ~ ,~,~ , ~ ,~,,

253

S, E. Harnung and C. E. Sch~ffer

E I" E l of the octahedral Table 19. The 3-Fsymbols \ y l y l y a ]

rotation group 0 represented as matrix elements of the general set of unit tensorial operators 53P, F = A 1, A 2, and E

53A1 +53A2 + ~ E E0 E8

E0

Ee

1/V2 1/2 A1 E0

IlV~ -1/2 A2 Ee

-1/V2 --I/2 A2 Ee

1/V~ - 1 / 2 A1 E0

T 1 1"~ of the octahedral rotation group 0 represented as Table 20. The 3-F symbols \?1 72 ?a

matrix elements of the general set of unit tensorial operators 531, F = A 1, E, T1, and T2

!itA1 +~v.._{_ 53Tx +!3Tu T1 z Tly T1 x

T1 z

T l y

T1 x

A1 E0

-1/V6 l/V6 TlX T25

x/Vg 1/Vg T1 y T2

~/V~ 1/V~ T1 x T2

llV~ -UV~ -1/z A1 EO Es

--1/V6 I/V6 T 1 z T2 $

- i]V6 l/V6 T1 y T2 r~

1/V~ ~/V~ T1 z T 2 $

1/V~ -IlV~ +1/2 A 1 E0 E~

Table 21. The 3-I' symbols [T, ~2I" T \_.2] of the oetahedralrotation groupOrepresentedas \71 Y~ yz/

matrix elements of the general set of unit tensorial operators 53 1 , F = A 1, E, T1, and T2

~AI +53E +53T1 +53T2 T2 $ T2 ~ TS

T2~

T~ ~/

T2

1/V5 1/V~-U2 A1 E0 Ee

-1/V6 1/V6 T1 z T2 $

1/V6 l/V6 TI y T~

1]V-6 1/V6 T I z T2

1/V~ 1/V~ i/2 A 1 E0 E~

-1/V6 1/V6 T1 x T~

- l /V6 l/V6 T l y T2*/

l/V-6 l[V-6 TIX T25

1/Vg -x/Vg AI EO

254

Phase-fixed 3- / ' Symbols and Coupling Coefficients for the Point Groups

{ ~'~ v ~rq of the octahedral rotation group 0 represente~ as Table 22. The 3-V symbols \Yl Y2 Ya/

matrix elements of the general set of unit tensorial operators 73 r, F = A 2, E, T1, and T2

~A 2 _]_~E jl_ ~T1 _~_ ~T2 T1 y T1 x

T2~

T2 ~/

T2 ~"

TlZ

l /V6 -1 /V -~ l/V6 T l y T27[ T l z

1/V~ 1/V6 1/V~ T1 x T2 ~ A2

1 / V 3 - 1 / V 3 l/V6 A 2 E ,~ T 1 x

1/V-6 T~

- 1 / 2 l/V12 EO Es

- l / V 6 T2

l/V3 1/2 1 /V~ A2 EO Ee

1/V~ -1/V~ T1 z T2

l/V6 l/V6 T1 y T2

References

1. Wigner, E. P.: Am. J. Math. 68, 57 (1941).

2. - -Group Theory. New York-London: Academic Press 1959.

3. L6wdin, P. O. : Rev. Mod. Phys. 39, 259 (1967).

4. Griffith, J. S. : The Irreducible Tensor Method for Molecular Symmetry Groups. New Jersey: Prentice-Hall, Inc. 1962.

5. Ravah, G.: Phys. Rev. 76, 1352 (1949).

6. Sehtiffer, C. E.: Proc. Roy. Soc. (London) A 297, 96 (1967).

7. -- Struct. Bonding 5, 68 (1968).

8. -- Intern. J. Quantum Chem. 5, 379 (1971).

9. -- Struct. Bonding 14, inprint.

10. Harnung, S. E., Schiiffer, C. E.: Struct. Bonding 12, 257 (1972).

11. Svhtiffer, C. E. : Pure Appl. Chem. 24, 361 (1970).

255

[structure and bonding] structure and bonding volume 12 || phase-fixed 3-Γ symbols and coupling...

Documents