University of Arkansas Foyetteville, 72701 and S. J. Cyvin Institute of Theoretical Chemistry Technical University of N O ~ W O Y N-7034, Trondheim-NTH, Norway
Complete Reduction of Representations of Infinite Point Groups
I
For many purposes (e.g., vibrational analyses) the complete reduction of a reducible representation of a point group is of great interest, and the formula
is of fundamental importance in group theory (1-6). I n eqn. (1)n(y) gives the number of times the symmetry species (or irreducible representation), y, shall occur in the completely reduced representat,ion; x, are the characters of the reducible representation, I?, for every symmetry element R; xP"are the characters for the given irreducible representation y ; and y is the number of elements (i.e., group order). For most students who are eager to test the above relation (I), it is usually shocking to observe that it does not give meaningful results for molecules belonging to infinite poiut groups; i.e., the linear molecules. I t is a drawback of most of the introductory textbooks dealing wit,h group theory that they do not mention this imperfection and, even worse, they do not outline a procedure for treating the linear molecules. Within this category are the test cases as popular as C02,C2H2, HCN, etc. Since the exact group theoretical treatment (6) of this question appears to be too advanced for most of the general and undergraduate courses, we think it will be useful to present the following elementary approach. We restrict our examples to the application of group theory to normal modes of molecular vibrations (I), and our exposition is as elementary as the useful articles by Meister, et al. (7,s). The present method employs the formula
fmn I n the case of finite groups eqn. (1) is obt:~i~~r!d eqn. (2) by using the ort,hogonality thcorrm. 111 wsc of infinite groups eqn. (1) cannot he applied to 1111t:lin the n(y) from a given set, of xn. If o w considers, hrnvever, that eqn. (2) simply states that the cli:~r:~ctrvx,, of r is equal t o the sum of the characters for R of all thn y which combine to form I?, theu it is easily srrw that eqn. (2) is also valid for t,he infinitegmups. IVitIwut using eqn. (1) one can then solve for the n(y) 11y:11)1)1ying formula (2) to each symmetry rlcmtwt H or thr group. This produces a system of linwr t ~ ~ u : r t i ~ ~ ~ ~ s from which the unlcno~ms,~ ( y ) cut , be cv:~lu:~trd. Formula (2) will usually provide n sufficit-ntnumlw~~ of equations to determiuc all the coeficiruts I I ( ~ if) ,r m i s able to guess which of them do not v:tnirh. Oncr 11 solution has been found, the problem ir solvl~llwr;~usr the obtained set of n(y) coefficients is uniqnv i l l w t ~ y case. I n the following we s l ~ i ~use l l B f ' ~ \ \ - vi1)riitirmtl problems as examples to outline the procetlurc i n detail. We shall first cousider some general t!xprwsions for t l ~ : charact,ers of the reducible rcpresent:~tio~~s for 1lor111:1l vibrations. Using them \ye sl~allcoustrurt for somrr concrete examples the reducible rt~present:~tio~~s \I-Iiirh will then be reduced by the proposed proccdnrr~. Below we give the general expressions ( f t ~ r m u l(:i)-~ :~~ (8)) for the characters of thc reducible n . p r ~ w n I : ~ t i ~ n , I?,ib, for the molecular vibrntiol~sof ; ~ n.N-:ltrnuic l i n ~ w molecule. The given formulas are easily u n ~ l r r 4 o dif one considers that t,he characters of the rrprrscut:rtions of the internal coordinates 1. (bond stretchins) ;111d0 (valence angle bonding) of an N-atomic linear molecule are as given in the Tables 1 arid 2. Thew tnhlrs were obtained in the usual way (1-5).
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295
Table 3.
Chorocters of the Reducible Representations for Vibrations of Linear Molecules XYZ
Table 4.
Characters of the Reducible Representations for Vibrations of Linear Molecules XYz oos
It follows common experience that the irreducible representations occurring in problems of molecular vihrations involve only y = Z+ and 11 in the case of asymmetrical h e a r molecules (point group C,,); and y = &+, II,, 2,+, and II, in the case of symmetrical linear molecules (D,A). That this is so can also he seen from the Tables 1and 2. The reducihle representation for the valence bond stretchings of an N-atomic linear Table 1. Characters for the Representation of the Bond Stretchinas of an N-atomic Linear Molecule
N even N odd
N-1 N-1
N-1 N-1
1 0
N-1 N-1
1 0
1 0
Table 2. Characters for the Representation of the Valence Angle Bendings of an N-atomic Lineor Molecule
N even N odd
E
2CQ
s.
i
2S*
C*
2N-4 2N-4
( 2 N 4 ) cos m (2N-4) cos m
0 0
0 -2
0 2 cos .P
0 0
molecule can he checked to contain N/22,+ and (N/2 - I)&+ (N even, point group Dan); (N/2 - '/2)&+ and (N/2 - '/z)X.+ (N odd, D,A); and (N - 1 ) 2 + (C,,). For the 0 coordinates we obtain similarly (N/2 - l)n, (N/2 - l)II, (N even, Dooh);(N/2 a/,)n, (N/2 - l/,)n. (N odd, D,,,); (N - 2)n Consequently only these just mentioned ir(C,.). reducible representations have to be considered for the reduction of rYib of linear molecules. Example I. Linear XYZ molecule (C,,); e.g., HCN. The characters of the reducible representation for normal vihrations are shown in Tahle 3. They are found as the special case of N = 3 from eqns. (3)-(5). It is sufficient to apply eqn. (2) with R = E and R = v,, which gives the two independent equations
+
+
x(=) = 4 = n ( 2 + ) x'=-~' = 2 = n ( 2 +)
+ 2n(n)
(9)
(10)
For the characters of irreducible representations; see, e.g., reference (1). The eqns. (9) and (10) yield the answer n(Z+) = 2,
n(n)
=
1
(11)
or written in terms of the group-theoretical symbolism r u t b ( x ~ z=) 2 2 +
+ 11
(12)
Alternatively the same answer may he obtained by
296 / Journal of Chemical Education
Table 5.
m
2
i
S*
CB
-2
2 cos @
0
Choracters of the Reproducible Representations for Vibrotions of ~inedrMolecules X;Y~
.
m
k!
i
S*
*+
+
C"
*.
utilizing X(@' = 2 2 cos with arbitrary angles In particular the values of = 0 and @ = 90" give equations identical to eqns. (9) and (10) respectively. Example 2. Linear symmetricalXYz molecule (Dm&); e.g., COz. The characters of the reducihle represent* tion for normal vihrations are shown in Tahle 4. They are found as the special case of N = 3 from eqns. (3)-(8). With R = E, u,,i and Czeqn. (2) gives X ( ~ )=
4 = n(&+)
=
2 = n(&+)
X(aa)
= -2
x ( ~ d=
= n(2,+)
0 = n(&+)
+ 2n(II,) + n(Z,+) + 2n(n,)
(13)
+ %(xu+) + 2n(II,) - n(Z,+) - 2n(lI.)
(15)
- n(&+)
(16)
(14)
The solution is n(2,+) = 1, n(n,) = 0,
n(n,) = 1, n(Z,+) = 1 (17)
or written in group-theoretical symbolism T , I ~ ( X Y S=) 2.+ ?,+ nu
+
+
(18)
Example 3. Linear symmetrical XzYz molecule (Dm&);e.g., CzHz. The characters of the reducible representation in question are shown in Tahle 5. Following the procedure outlined above the reader may easily verify the result ryib(XX2) = 2x0'
+ n9+ 2"' + nu
(I9)
I n conclusion we want to point out that, as far as the reduction of reducihle representations is concerned, the present method is generally applicable, not restricted to the problem of molecular vibrations. Literature Cited (1) WILSON, E. B.. D~crus,J.. A N D Cnosa. P. C.. "Molecular Vibrations," MaGraw-Hill, N e w Yark, 1955. (21 S o m m m o . D. S., "Malecul~r Symmetry," Ven Nostrsnd, London. -~~-~
(3) COTTON. F. A,. "Group Theory: Wiley-Intarscience,New Yark. 1963. (4) J. C.. "Molecular Structure." Edward . . BRLND.J. C. D.. SPEARMAN. Arnold. London, 1960. ( 6 ) Mxrossr, F., "Gruppentheorie," Springer, Berlin,1961. (6)WIGNER, "-4. ,.,en E. P.. 8 * G r ~Theory" ~p (English Ed.). Academic Press, N e w .7"7.
(7) MEISTER,A. G.. CLEVELAND, F. F.. MUBRAY.M. J.. Amcr. J . Phys.,
11,239 (1943).
(8) MEISTEB, A. G . , AND C L E Y E ~ ~F. ~N F.,DAmer. , J. Phys.,
14,13(1946).
