A Handbook of Magnetochemical Formulae by Boca, Roman (Auth.)

By Boca, Roman (Auth.)

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1 Coupling Coefficients for the Rotational Group Vector coupling coefficients are the numbers that properly combine elementary angular momenta into a compound angular momentum. Important note: one should deal with the coupling coefficients and related symbols consistently from a single source since different phase choice and eventually different constant factor conventions exist in literature. 1 Addition of Angular Momenta The addition of the angular momentum follows the principal physical law: the magnitude as well as each component is conserved.

For instance, for N 5 3 identical particles the permutation group S3 has six elements, grouped in three classes; there are three IRs: Γ1(13), Γ2(13), Γ3(23). Individual permutations could be grouped into cycles. A cycle (a1,. a2!. a1 and it can be written in several alternative ways. For instance, the permutation  1 2 2 3 3 1 4 5 4 6 6 5 7 8 8 7  5 ð123Þð4Þð56Þð78Þ ð1:103Þ can be grouped into cycles (123) of length 3, (4) of length 1, (56) of length 2 and (78) of length 2; these commute with each other.

The 6j-symbols can be evaluated through an expansion via associated 3j-symbols & j1 j2 j3 j4 j5 j6 ' 5 XXXXXX m1  3 3 m2 m3 j1 m1  j4 2m4 m4 j2 m5 j3 ð21Þ j4 1j5 1j6 1m4 1m5 1m6 m6  j1 j5 j6  m2 m3 m1 m5 2m6   j4 j2 j6 j5 j3 m1 m6 m4 2m5 ð2:14Þ m6 The three magnetic quantum numbers in each 3j-symbol must sum to zero (instead this vanishes) and consequently only two of the six summation indices are independent. 4. 4 The 9j-Symbols The coupling of four angular momenta can again be performed by different routes using a different set of intermediate spins, say jA; JMi 5 j j1 j2 j12 j3 j4 j34JMi and jB;JMi 5 jj1 j3 j13 j2 j4 j24JMi.

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