By Gerhard Fieck (auth.)
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Additional resources for Symmetry of Polycentric Systems: The Polycentric Tensor Algebra for Molecules
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Whereas the first coupling mode takes into account the separation of the interaction according to electron pairs, the second mode is more capable for the calculation of the integrals. i) n,Dl~ddq] Since the square brackets are defined without complex conjugationlthe conjugated representations a + and b + show up explicitly. 2) =Ee~~aa~O~cC)~erlr~I(BJ~bb~Dl~d d)~fsJ~dime "Glmltma+Cp r-e}e(f+b+d)~-s n q" Note that#in contrast to the one-centre integrals of scalar operators! the representations e and f may be different.
E. positions being invarlant to no symmetry operation). The normalization is chosen different allowing for continuity if ~ approaches an element of symmetry. 17)~we use the scalar product = with respect to the discrete set R in order to orthonormalize the standard functions. We now compile the properties of the standard functions in the following theorem: For each irreducible representation a of a point group G there is a set of standardized ~unctions ~ s t . , dima. 2) ~ s t . ~ a p ~ g ~ > ~ g ~ s t .
Triangle functions. 16/17) : (Zi IF) : ~-(n i l~ar) (n~ar IF) ga (7. 4i)(~. i2) 1 If such a function depends on the intrinsic parameters of the triangles only, but not on their orientation in space, then it is invariaut on the set ~. In this cdseywe simply write F(A) or FA. i). ~3) ~ZCa~ACB~c~ACB~ .... 3). 8) have four columns of indices. A direet treatment in the sense of section 6 would lead to a parallelism with the group-theoretical 4jm symbols, which couple four irreducible representations [34].