Theorem replacement

As a consequence of the Wigner-Eckart theorem the replacement theorem holds true a matrix element of any irreducible tensor operator can be expressed with the help of the matrix elements formed of the angular momenta... 

Another immediate corollary of the Wigner-Eckart theorem is the replacement theorem, which allows one to write the matrix elements of one spherical tensor operator,... 

The effective operator is the product of one operator on three-dimensional spatial coordinates and another which acts on nuclear spin space. This distinction can be brought out even more clearly by making use of the operator replacement theorem in section 5.5.3 to give... 

This is called the first replacement theorem. The set of graphs with z,-circles becomes the set of graphs with pj-circles and no articulation circles. 

The passage from the spherical harmonic functions and/or their combinations to the equivalent operators is based on the Wigner-Eckart theorem and consequently on the replacement theorem. The reduction of a matrix element to the reduced matrix element and a Clebsch-Gordan coefficient... 

With the help of the replacement theorem (Section 1.4) these matrix elements can be replaced by those of the total angular momentum operator... 

In the case of a strong exchange limit ( JAB is large) the classification of the field-dependent molecular states as slightly perturbed zero-field states 5, Ms) is fully justified. Consequently the replacement theorem holds true for S = S and the irreducible tensor operators containing the operator variable V can be substituted for those containing 5 as a variable. The spin Hamiltonian is somewhat different for each manifold of the spin 5, hence... 

The combination coefficients can be expressed with the use of the replacement theorem as follows... 

The combination coefficients depend upon the spin numbers (S, S2, S3, S,2, S). They can be evaluated with the help of the replacement theorem, hence... 

As a consequence of (A3), the replacement theorem is valid if two operators, say V and W, are of the same synunetry (i.e. they are of the same type with respect to an angular momentum), then both are reduced by the same Clebsh-Gordan coefficient C... 

This is clearly only possible in the case of matrix elements between functions with Sa = Sb = S, because otherwise the matrix elements of the pure spin operators all vanish and no proportionality can be established. The validity of the replacement theorem (A3.29), which is essentially another statement of the present results, is confined to similar situations. 

This result, in which the coupling coeffidents no longer appear explicitly, is often called the replacement theorem it is of limited value (although sometimes useful in the applications in Chapter 11). For further discussion of (A3.27), particularly for the point groups, the reader is referred elsewhere (e.g. McWeeny, 1963 Griffith, 1961, 1962). 

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