Relations between various coupling schemes

Usually, the first way is utilized in practice. This is due to the well developed mathematical technique necessary, by the presence of the expressions for both the matrix elements of the energy operator and of the electronic transitions in various coupling schemes. However, the second method is much more universal and easier to apply, provided that there are known corresponding transformation matrices. Now we shall briefly describe this method. 

Two full orthonormal bases of wave functions xpj) and q j), describing the system under consideration, are related by a unitary transformation (moreover, if the corresponding transformation matrices are chosen to be real, then the transformation itself will be orthogonal) 

Here the corresponding matrices satisfy the following condition  

Let us recall that the transformation matrix D1 is the Hermitian conjugate to matrix D and obtained from D by its transposition (D) and complex conjugation ( ). The unitarity of the matrix implies that 

Formulas, similar to (12.1), (12.2), may be written for the wave functions of intermediate coupling (ic), too. Namely, 

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The relations between the various interactions are given by the first multiple inequality of section 3.2.3, thus leading to the conclusion that in the 4f 6p configurations there exists a f,j coupling, and the most appropriate coupling scheme for characterizing the levels of these configurations is ... 

Models related to spin-forbidden reactions are discussed in this chapter. Coupling between two surfaces of different spin and symmetry is given by various levels of approximation for spin-orbit operators from the reduction of relativistic quantum mechanics. Well-established methods such as the Breit-Pauli Hamiltonian exist, but new relativistic methods such as the Douglas-Kroll Hamiltonian and other new transformation schemes are also being investigated and implemented today. 

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