The angular basis

As discussed in Section 6.3, the angular part of the atomic basis functions is universal - that is, the same for all kinds of basis functions. The purpose of the present section is to consider this angular part in more detail, preparing us for the discussion of radial functions in Sections 6.5 and 6.6. 

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For the prototype level (n,Jl,v, j, J)=(0,0,1,0,0T"of H2-, D2-and HD-Ar, the results of close coupling calculations for the partial and total predissociation widths are listed in Table V. Tests showed that the angular basis sets used to obtain these results were fully converged (25) they included diatom rotation states up to j 8 for v l and up to j" 10 for v -O. The absence of odd-j" dissociation products for H2- and D2-Ar merely reflects the fact that their potentials have no odd-j anisotropy terms. 

Convergence Considerations and Approximate Methods. Another consi-deration which bears on the reliability of results such as those presented above is the question whether or not the calculations are really fully converged. For the prototype (n,Jl,v, j, J)=(0,0,1,0,0) level of H2-Ar, this question is examined in Tables VI and VII, which show how the partial and total widths depend on the size of the angular basis set used for the v l and v" 0 levels, respectively. The calculations of Table VI used a fully-converged open... 

The angular basis functions have a well defined parity block functions with different K are mixed through the off-diagonal Coriolis interaction. 

An alternative approach to solving the coupled equations is to use a basis set expansion for the R coordinate as well as for the angular variables. The angular basis sets used in such calculations are generally the same as in coupled channel calculations. This approach was pioneered by Le Roy and Van Kranendonk, who used numerical basis sets for the radial (R) functions. Such basis sets are adequate for the rare gas-H2 systems, but converge very poorly for more strongly anisotropic systems. An alternative basis set, based on Morse-oscillator-like functions, has been used extensively by Tennyson and coworker... 

Since the angular basis of Yim serves just for projecting to /-subspaces and since for any kind of atomic-like radial calculations (including the Environment-Reflecting Pseudopotentials and Projector Augmented Waves) the wavefunctions and all the quantities can be considered real, we can reach considerable simpliflcation by using real spherical harmonics i instead of Yim so that all the vectors for rank-n update also remain real ... 

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. 

To incorporate the angular dependence of a basis function into Gaussian orbitals, either spherical haimonics or integer powers of the Cartesian coordinates have to be included. We shall discuss the latter case, in which a primitive basis function takes the form... 

These new wave functions are eigenfunctions of the z component of the angular momentum iij = —with eigenvalues = +2,0, —2 in units of h. Thus, Eqs. (D.l 1)-(D.13) represent states in which the vibrational angular momentum of the nuclei about the molecular axis has a definite value. When beating the vibrations as harmonic, there is no reason to prefer them to any other linear combinations that can be obtained from the original basis functions in... 

There are r vo basie r pes of misalignment, angular and parallel. Within eaeh of the.se basie types of misalignment there are eombinations of both. The.se are the most eommon eombinations ... 

The orbitals d. and dy can be expressed in terms of the complex forms di and whose angular parts are given by the spherical harmonics and respectively. The matrix of orbital angular momentum about the z axis in the complex basis is... 

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