Recursion coupling scheme

We have seen that the coupling of more angular momenta results in an appearance of the 3/-symbols (or Clebsch-Gordan coefficients) which bear all the fundamental information about the coupling route. 

For the coupling of three angular momenta we have the basis set lxl2hmxm2mi) = 

The second transformation matrix U(123) is an expansion of the matrix of the previous generation U(l2) and Clebsch-Gordan coefficients for a new set of 

The initial coupling matrix becomes a unit matrix of the order (2/( + 1) 

For example, the coupling of three spins 5] = s2 = s3 = 1/2 results in the transformation of the column vectors 

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Based on a perturbation expansion using the KS Hamiltonian [26,27], recently a new systematic scheme for the derivation of orbital-dependent Ec has been proposed [12]. While this representation is exact in principle, an explicit evaluation requires the solution of a highly nonlinear equation, coupling Exc and the corresponding x>xc [19]. For a rigorous treatment of this Exc one thus has to resort to an expansion in powers of e, which allows to establish a recursive procedure for the evaluation of Exc and the accompanying Vxc-... 

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