Angular Matrix Elements

The most straightforward way of evaluating the angular matrix elements of Eqs. (17.12) and (17.13) is to use the method of Edmonds.5 The matrix elements are evaluated as the scalar products of tensor operators operating on the wave-functions of electrons 1 and 2. Using this approach we can write Wdas... 

The two-electron spin-angular matrix element of the Coulomb potential (3.102) is given by... 

This contribution examines current approaches to Coulomb few-body problems mainly from a methodological perspective, in contrast to recent reviews which have focused on the results obtained for benchmark problems. The methods under discussion here employ wavefunctions which explicitly involve all the interparticle coordinates and use functional forms appropriate to nonadiabatic systems in which all the particles are of comparable mass. The use of such wavefunctions for states of arbitrary angular symmetry is reviewed, and the kinetic-energy operator, written in the interparticle coordinates, is presented in a convenient form. Evaluation of the resultant angular matrix elements is discussed in detail. For exponentially correlated wavefunctions, problems of integral evaluation are surveyed, the relatively new analytical procedures are summarized, and relations among matrix elements are presented. The current status of Gaussian-orbital and Hylleraas methods is also reviewed. 

For a coupled spin system, the matrix of the Liouvillian must be calculated in the basis set for the spin system. Usually this is a simple product basis, often called product operators, since the vectors in Liouville space are spm operators. The matrix elements can be calculated in various ways. The Liouvillian is the conmuitator with the Hamiltonian, so matrix elements can be calculated from the commutation rules of spin operators. Alternatively, the angular momentum properties of Liouville space can be used. In either case, the chemical shift temis are easily calculated, but the coupling temis (since they are products of operators) are more complex. In section B2.4.2.7. the Liouville matrix for the single-quantum transitions for an AB spin system is presented. 

TV. Matrix Elements of Angular-Momencum-Adopted Gaussian Functions... 

IV. MATRIX ELEMENTS OF ANGULAR-MOMENTUM-ADOPTED GAUSSIAN FUNCTIONS... 

To form the only non-zero matrix elements of Hrot within the J, M, K> basis, one can use the following properties of the rotation-matrix functions (see, for example, Zare s book on Angular Momentum) ... 

The only difference is that a(0) is now an operator acting in jm) space of angular momentum eigenfunctions. This space consists of an infinite number of states, unlike those discussed above which had only four. This complication may be partly avoided if one takes into account that the scalar product in Eq. (4.55) does not depend on the projection index m. From spherical isotropy of space, Eq. (4.55) may be expressed via reduced matrix elements (/ a(0 /) as follows... 

The operator Tang contains the cross-terms that give rise to the Coriolis coupling that mixes states with different fl (the projection of the total angular momentum quantum number J onto the intermolecular axis). This term contains first derivative operators in y. On application of Eq. (22), these operators change the matrix elements over ring according to... 

Calculation of the angular part of the matrix elements thus remains, which can be performed exactly using tensor algebra techniques based on group theory. Since the calculation of the matrix elements is not straightforward, we provide here some details on it for the interested reader. The treatment follows the procedure described in Ref. [17]. 

---