Total Angular Momentum Eigenfunction

We consider a nuclear wave function describing collisions of type A + BC(n) AC(n ) + B, where n = vj, k are the vibrational v and rotational j quantum numbers of the reagents (with k the projection of j on the reagent velocity vector of the reagents), and n = v, f, k are similarly defined for the products. The wave function is expanded in the terms of the total angular momentum eigenfunctions t X) [63], and takes the form [57-61]... 

The TD wavefunction satisfying the Schrodinger equation ih d/dt) F(f) = // (/,) can be expanded in a basis set whose elements are the product of the translational basis of R, vibrational wavefunctions for r, r2, and the body-fixed (BF) total angular momentum eigenfunctions as41... 

The parity-adapted total angular momentum eigenfunctions [84] are defined as... 

The term in curly brackets in this equation can be rewritten in terms of total angular momentum eigenfunctions, (R, f), which are obtained by coupling... 

The total angular momentum eigenfunctions, 7(R,f), have parity (—1)- therefore, the summation in Eqs. (A.4) and (A.5) extends over both positive and negative parities. The function is the space-fixed radial scattering... 

In order to transform to the body-fixed representation, we will need to relate the angular functions Wj (R,r) to angular functions defined relative to the body-fixed axes [L., J,K,M,p)QjK ), where J,K,M,p) are the parity-adapted total angular momentum eigenfunctions of Eq. (4.5) and x(0) normalized associated Legendre polynomials of the body-fixed Jacobi angle]. 

The coupled angular basis function YJ in Eq. (88) is the BF total angular momentum eigenfunction which can be written as... 

To find the total-angular-momentum eigenfunctions, one must evaluate the coefficients in (11.33). These are called Clebsch-Gordan or Wigner or vector addition coefficients. For their evaluation, see Merzbacher, Section 16.6. 

Thus each total-angular-momentum eigenfunction jij2JMj) is a Unear combination of those product functions jimi) j2fn2) whose m values satisfy mi + m2 = Mj. 

As discussed in the appendix of ref 23, the total angular momentum eigenfunction in the BF representation can generally be written as ... 

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