lcosahedral Matrix Representations as a Function of Eulerian Angles Emllio Martinez-Tones University of Castllla-La Mancha, 13071 Ciudad Real, Spain Juan J& L6pez-Gmw6lez, Manuel Femandez-G6mez University College "Santo Reino" of Jaen. University of Granada, 23071 Jabn, Spain Antonlo Cardenete-Espinosa University of Granada, 18071 Granada, Spain One of the mhst illustrative exercises for teaching molecular symmetry point-group theory consists of obtaining symmetry-adapted functions by means of the projection-operators technique ( I ) . To obtain complete series of such functions in low-symmetry groups, it is usually enough to know the characters of the irreducible representations of the mouDs in question. In hiaher svmmetrv groups, however, it maybecome necessary toknowthe expii& matrix representations, the obtaining of which in itaelf constitutes a useful and valuable exercise. The finite, higher symmetry molecular point group is the isocahedral group, I,,. Typical examples of molecular species (X = with this symmetryare the closoborateanionsB~zX~z~H , D, C1, Br, I) (2). Figure 1 shows an icosahedral spatial arrangement with the numeration along its vertices. This mouo has 120 svmmetrv operations and is the onlv one wit6 irr;ducible representaiions of an order greater th& three. i.e.. four and five, which makes it particularlv interesting. ~ev&heless, thisgroup has beenbaid relatively little attention in the literature, so that its explicit matrix representations have only recently been published (3). In this paper we propose, as a useful exercise for advanced students of representation theory, the obtaining of the representative matrices of the I ichsahedral rotation group by means of the descent symmetry from the K group of rotations of the sphere. This procedure has a general character for everv subprou~ - . of K. while the corres~ondine - Eulerian angles of the different symmetry operations are necessary in each case. Once the matrices of I have been obtained, those corresponding to Ih are arrived at by merely taking the equivalence Ih = I * i into account.
Figwe 1. Iwaahadral spatial mngement
-
Matrlx Representailon of the KGroup Surface spherical harmonies, PI(& @), are widely employed functions in quantum mechanics, their general explicit form being given hy the expression (4):
where
are the associated Legendre polynomials, 1 = 0,1,2,. .. and m = 0, f1, f2, . .. f1. In Table 1the spherical harmonics have been tabulated up to 1 = 3. These functions are the bases of the irreducible representations, Dl, of the K group, so one may write (5)
where R is the rotation operator and Dlmsm(R)are the corresponding transformation coefficients. These coefficients may be written in explicit form as a function of the Eulerian angles, (or, 8, y ) , which define the rotation by the expression (6) where
where the summation over k goes from the larger of the numhers 0 and ( m - m') to the smaller of the numbers ( 1 m') and ( 1 m) (7).
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Table 3. Eulerlen Anglea for lcosahedral Symmetry Tranrtorrnationra
The transformation 3 may be expressed in matrix form as follows:
1
D ~ , .R DI-,,, (R) ...DIO,l-l(R)....D1-I,I-l(R)
Therefore, the matrix of the c~efficientsD~~+,,(R) of eq 6 is a squared matrix of the order 21 1,which is therefore the dimension of the D' representation, which we denote in a . the matrices of the simplified manner as D1(a, 8 , ~ )Thus, different irreducible representations of the K group would be Doh, 8, y), DYa, 8, y), D2(u,8, y), . .. , respectively.
+
Malrlx Representatloas of the IGrwp The representative matrices of the I group, just as any other subgroup, are obtained by descent in symmetry from those of the K group. The reduction scheme of the first four D' representations of K into the irreducible representations of the I group are shown in Table 2. The representations A, T,G , andH have the dimensions 1, 3.4. and 5. res~ectivelv. 'ihe matrix elements of the DO,Dl, D2, and D"epresentations are obtained bv makine I = 0.1.2. . . . and 3. resnectivelv in eqs 4 and 5. For reasons of ;pace the explicit forks of tgese elements are not given in this paper, but are available upon request. These elements satisfy the following relations (7):
Table 2. Reductlm ot the First Four D,Representallms Into the lneduclble Repmaentatlonsof I
SVmmetry Operation
E C'dl. 2) C'&. 11) C7d7. l o ) C's(8, 9) C'd5.8) cfs(4,12) C'd1.2) O d 3 , 11) 0d7, 10) O d e , 9) Os(5.8) Os(4. 12) @ d l . 2) @ 4 . 11) &(7. 10) CBS(6.9) @&8) br(4,12) &(I, 2) br(3.11) 6 6 7 . 10) 046.9) bd5.8) G d 4 . 12) C'dl. 3) C'dl. 4) O d l . 5) cldi. 8) C'dl. 7 )
8
a
0
-e -E
2 2r
a
+ 2r r +a r
6 6
+r r +c r
-a +6 2x - e r -a u u
+ +
r a r a r c r +e 2r r r -c 2r a 2r 4
-
r 2%
+
r 6
-c c r
r
a w -a r- e
symmeby Operation
y
0 -15 r 9rl5 r 5~13 2r15 n 2r15 5~13 9rl5 r15 5x13 T 8~15 r 8x15 d 3 u 5~13 U I ~ ,
7r/5 r 8rI5 r 8rl5 3r15 r r13 r 5sl3 r 7nl5 4r15 r 4x15 3r/5 0 2r15 s 2 ~ 1 3u 4x13 8r15
0 C'2(3. 4) Od3.7) C12(3,8) c4a3.9) C'd4. 5 ) r C'i4.9) a c7a4. 10) e Cq2(5.6 ) r C'r(8. 7 ) a c1d7.8) r G3(1,3 , 4 ) r & ( I , 3.7) a b,(i.4.5) a e,(l.5,s) a G,(l. 6.7) 6 Gs(4,5.10) 6 bs(5,6,11) e bs(5. 10, 11) 6 Gd6.7.12) a G3(8. 11.12) 8 C 1 ~ (3.4) l. r C1*(l,3.7) r C ' S ( ~4.5) . r C'd1.5.6) c F d l . 6.7) o ~ ' ~ (5,4 10) . c Ct3(5.6, 11) a ~ ' ~ (10, 5 .11) a cT3(6.7.12) r C's((6, 11.12)
-c -6 a -E
+
-
-
u
8
r - < 2-15
c as15 r - c a 4~13 -a r a 2~13 a c 4r15 r e x r 4rl5 r O r 0 n u 0 r e 6~15 r c BUI~ T- c r c d 5 n a c 9rl5 r 3 ~ 1 2 3r12 o 2~ c 3 ~ 1 5 s r/2 3~12 e 2 r a 4x13 a 2 r 6 7r15 r r 0 3rl2 -12 r a 4 ~ 1 3u - a r12 r12 o r rl5 6 e e r15 3r12 3r12 0 2 r c 8r15 r r r12 3 r / 2 0 d 2 r a 2r13 2 r 6 7x15 n e o 3r12 r12 T a 2~13 a r rl2 rl2
-
-
-
-
-
-
+
-
+
-
-
=arcam [WJsl"2=arccoa [is- i ) ~ J s l " ~ , a ~ a = a r c s i n ( i =arcmr ~~&~ binems "goldsn n m w .
(W&l,
*
Thus, in order to obtain the matrices of the irreducible representations of the I group, it is necessary to know the Eulerian angles that correspond to the rotations of this group, for which we have followed Rose's agreement (8) by choosing three equivalent binary axes of the icosahedron (3, 9) as a system of coordinate axes. This system can be seen in Figures 2a and 2b. The values of the Eulerian angles corresponding to the icosahedral transformations, denoted as in (3),are given in Table 3. According to Table 2: For the TI andH symmetry species the representativematrices are obtained by directly substitutingthe values of the Ederian angles shown in Table 3 into D1(a,8, y) and D2(n,8, y), respectively. For the T* and G speeies it is necessary to find a similarity transformation (10) to reduce the D3representation of K down to the Tz and G representations of I. The bases of functions of the irreducible representations of the I group are ordinarily expressed as homogeneous polynomials of the Cartesian coordinates, W x , y , z ) such as those in Table 4 (3,9). The representative matrices, U'(u, 8, y ) , of these bases are related to the D1(u,8, y) in the form:
.
where S' is the matrix which transforms the spherical harmonica Y"'r(9, 6 )inta the homogeneous polynomials, 'Z"j(x,y, z ) , i.e., -I
X'\(X,Y,2 ) =
S~,,Y'"'l(O,9)
(9)
... . ,"k,
Figure 2. (a)Upper and (b) frontal pmiections of a regular icoakdron showing th$adopted cwrdlnates axes.
where S'j,,* are the elements of the S' matrix. This matrix is squared to the order 21 1and has the general form
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The explicit form of the matrices SLwith 1 = 1,2, and 3 are shown in Table 5. In accordance with Table 2 and m 8 the renresentative. matrices of the T Land H symmetry'~peciesare obtained from the matrices D1(a,p, 7) and D2(n,p',~),respectively, while those of the Tz and G species are obtained from the 0 3 matrix, i.e., Table 4. Bads Fundlorn ol the Irreducible Rapreunlatlonsol & Normallred to 4 d 2 1 + 1 over the UnU Sphere
The UrW = T I ,H, T2, and G) matrices may be written as follows:
up=
..... ......
For reasons of space the Uri;matrix element. as a function of the Eulerian angles are not explicitly shown here but are available upon request. If the values of the Eulerian aneles of Table 3 are substituted into them, the matrices givenubyeq 12 reproduce those of ref 3. Bearing in mind that Ih = I * C;,the matrices that correspond to the symmetry operations of the Ih gmup not contained in I, and those of the latter group are interrelated in the following way: The improper rotations Slo and Ss correspond to the proper rotations Cs and C3,respectively, in the farm: ~,ol.S,7.9(i ') - c 8 4 1 2 ( . . 81 6 " ' ~ 3 1 ) S2s(i,j, k)
-C,Z,I(i,j,k )
The reflectionsin the planes of symmetry correspond to the CI axes in the form:
Table 5. S ' Matrlwr and Their lnvene8, [q-' -1
0 4 2 0
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1
-I -I
Upon establishing the above correspondences it is neces-
Literature Cltad 1. Cottan, F. A. Chemical Appliealioim of Group Theory. 2nd d.:Wilsy-bteraeiana: New York, 1971:Chapten610. 2 Lei- L. A,; B"!mIov, 8. 8.; Kwbalrwe, A. P.: Ksganaki. M. M.: G& Yu. k Kuzoctmu,N. T.; %ova, I. A. Sprtmchim.A 1982,WA. 1041-1056.
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3. ~em~nd&maa. M.; u p a - ~ d -J. .I.; h-. J. F. A". mim. 1m5.81.311-
1964: pp 103 and 110. 7. Bradlay, C. J.; Craeknell, A. P. The Mathernotied Theory of Symmetry h Solids. Repmliomtatbn Theory for Point Gmups a d Spat. Gmupa; Clarendon: Oxford, 1972: p 53. 8. Rose, M. E. E k m n f q Theory of A w l o r Momentum; Wiley: New York. 1951: Chapter 4. 9. Boyle, L. L.; 0 ~ 0 . 2Ibt. . J. QuantumChem. 1978.7. % 3 4 ~ . 10. Kettle. S.F. A. Symmfrymd Stnxture; Wiley: Chicheater, 1986: p 267.
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Number 9
September 1989
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