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By Per-Olov Lowdin

ISBN-10: 0080582338

ISBN-13: 9780080582337

ISBN-10: 012034808X

ISBN-13: 9780120348084

ADVANCES IN QUANTUM CHEMISTRY VOL 8.

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Three additional classes A’, B , C‘ are found in that manner and their elements are given in Table IX. (b) Irreducible representations of the group K. The irreducible representations of D, x D,‘ are given in Table X. The conjugate representations x2“ and x4” have the same induced representation in the group K and this one is irreducible. It is the representation Tsgiven in Table XI. The situation is the same for the representations x,” and x7”, x6” and xs“. The induced representation by x3“ (or by x6’7is r6 (Or r7)TABLE IX ADDITIONAL CLASSES OF THE GROUP K Classes Operations A’ B’ S, WW“S, W ~ W ’ S TT’S, , WTW‘T’S, W ~ T W ’ ~ T ’ S c‘ TS, WTS, W’TS, 7‘3, WIT’S, W”T’S, W’T, W ’ W S , W‘W‘TS, wZw”s, WS,W’S, W‘S, W”S, TW’T‘S, wrr’s, w’rr’s, WW”T’S, W’TW’T‘ WW‘S, TW’~T’S, WT’S, WW’T‘S, WW”T‘S, d 2 T , W“WTS, W”WZTS, W’T‘S, W zW’T’S, W zW”T‘S Symmetry Groups of Nonrigid Molecules 33 TABLE X IRREDUCIBLE REPRESENTATIONS OF THE GROUP D3 x D3' Classes E Numberofelements 1 w 2 w'w w' 2 T 3 W'T T' T'W T'T 6 3 6 9 4 TABLE XI IRREDUCIBLE REPRESENTATTONS OF THE GROUPK ~ A rl rz r3 r4 r5 r6 r7 rs rg 1 1 1 1 2 4 4 4 4 B C 1 1 1 1 2 -2 -2 1 1 D 1 1 1 1 1 1 -1 -1 0 2 -2 0 0 E 2 -2 -2 1 1 F 1 1 1 1 1 1 -1 -1 0 -1 A -2 0 1 0 0 0 0 0 ' B ' 1 -1 1 -1 0 0 0 2 -2 -1 C 1 1 -1 0 0 0 -1 1 ' -1 -1 1 1 0 0 0 0 0 At this point we must find the irreducible representations of K corresponding to those of ( D 3 x D3') invariant by s.

3 ) The entire irreducible carrier space can be generated by sweeping X over the group algebra. Suppose that the irrep associated with Young diagram D has dimension dD, and that p,", denotes the permutation that transforms the standard tableau T Dinto TsD: TSD= p:TrD. (16) Any set of dDlinearly independent operators of the form XErDwill be an operator basis for the irrep. Young showed that the operators (r arbitrary; s = 1, 2, . ,dD), E,",= p:ErD form such a basis. Basis functions for the irrep associated with diagram D can then be generated from a suitable primitive function by applying the Young operators { E t ; s = 1, 2, .

It is appropriate at this point to make a reconnaissance. As mentioned previously, the diagonal elements of matric bases multiply according to That is, these elements are idempotent and mutually annihilating. According to group algebra theory, this property is required of generating units for the minimal left ideals into which the group algebra decomposes. Such primitive idempotents are also required to have the property (14). Young idempotents are primitive-they satisfy Eq. Also, two Young idempotents from different diagrams annihilate each other from the left and right.

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Advances in Quantum Chemistry by Per-Olov Lowdin


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