By Boca, Roman (Auth.)
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Extra resources for A Handbook of Magnetochemical Formulae
In a usual description, the transformation from the basis set of the atomic terms (ATs) into the basis set of the CFMs is written as jG0 : Γ0 γbi 5 X CiACFM; jAAT jR3 : ln vLSML MS i ð1:97Þ j The combination coefficients Cij form the matrix U1. Second, we need to define a matrix representation between the AT functions for each symmetry operation that depends upon three Euler angles  L S AT D~ ðRαβγ ÞIJ 5 hIjR^ML M0 ðαβγÞ 3 R^MS M 0 ðαβγÞjJi L ð1:98Þ S Such a matrix adopts a block-diagonal form: each block is a direct product of the rotational matrices for spatial and spin variables.
For instance, a standard tableau of the partition  1 3 Γi = 2 4 has the dimension dimðΓλ Þ 5 4! 53 4U 1U2U1 ð1:110Þ An analytical formula says dimðΓλ Þ 5 N! h2 ! . hr ! i,j ð1:111Þ with hi 5 r 1 λðiÞ 2 i ð1:112Þ where λðiÞ is the length of row i, and r is the number of rows in the Young diagram. The Young tableaux are useful for labelling IRs of various groups: a. The N-box Young diagrams label all IRs of the symmetric group SN. b. The standard tableaux of k-box Young diagrams with no more than n rows label the IRs of GLn, in particular they label the IRs of Un.
5. A basis for the IR can be constructed by a process of symmetrisation with respect to the rows and antisymmetrisation with respect to columns. Notice that the symmetrisation can be performed by summing up all the permutations, whereas the antisymmetrisation can be written in the form of a Slater determinant Sdetfa1 b2 . g that naturally involves sign factors (21)p; p À parity of the permutation (a number of elementary transpositions). Þ1=2 Ω rc N! X ½λ Γrc ðPÞU P^ ð1:113Þ P^ ½λ ^ and dλ is ðPÞ is the matrix element of an IR Γ[λ] for the permutation P; where Γrc the dimension of the IR.