Transformations Between Hunds Case Basis Sets

4 Transformations Between Hund s Case Basis Sets 

Throughout this book, matrix elements of various perturbation operators are evaluated in the case (a) basis set. To illustrate the irrelevance of the choice of basis, it is instructive to see that it is always possible to express the basis functions of one Hund s case in terms of those of another. Several methods exist whereby the basis functions of any Hund s case may be expanded in terms of those of case (a). 

One especially simple procedure for defining transformations between cases 

Instead of CG coefficients it is often more convenient to use 3-j coefficients. These are related to the CG coefficients as follows  

An extremely useful listing of 3-j coefficients is found in Edmonds (1974, p. 125). Some of the properties of 3-j coefficients are discussed in Section 3.4.5. For the special case of J — N = there is a convenient closed-form expression for the 3-j coefficients, 

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The reason there are so many Hund s cases is that each Hund s case corresponds to an arrangement of terms in H in order of relative importance. For each arrangement there is a different Hund s case. Since each Hund s case is associated with a complete set of commuting angular momentum operators, explicitly defined transformations between any two Hund s case basis sets may be specified independent of the details of a particular molecular example. 

Watson (1999) analyzes two subcases of case (d), which he calls (o+,d) and (b+ d). The good quantum numbers in the former are A+5+S+ J+Q+ls(J — sr)J (2S+ + 1 values of ft+) and in the latter subcase A +S+N+J+ls(J — sr)J (2S+ + 1 values of N+). Consider an example in which the ion-core has 1A symmetry, then the nonzero quantum numbers in both (a+,d) and (b+. d) sets of basis functions are reduced to A+N+lsNJ, which is the set of quantum numbers traditionally used to describe a Hund s case (d) Rydberg state. The larger set of quantum numbers, expressed in the form specified by Watson (1999) rather than the more familiar quantum numbers used here, is needed to uniquely specify the transformations between (core, Rydberg) composite cases when S+ 0. 

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