Angular functions general expression

The energy spectrum of atoms and ions with j j coupling can be found using the relativistic Hamiltonian of iV-electron atoms (2.1)-(2.7). Its irreducible tensorial form is presented in Chapter 19. The relativistic one-electron wave functions are four-component spinors (2.15). They are the eigenfunctions of the total angular momentum operator for the electron and are used to determine one-electron and two-electron matrix elements of relativistic interaction operators. These matrix elements, in the representation of occupation numbers, are the parameters that enter into the expansions of the operators corresponding to physical quantities (see general expressions (13.22) and (13.23)). 

The Is atomic orbital of the hydrogen atom is spherically symmetrical and so the angular function is a constant term = l/(4n) see Table 2.4] chosen to normalize the wave function so that the integral of its square over all space has the value +1, as expressed by the general equation ... 

It is often useful (and in quantum, mechanics necessary) to express the rotational energy as a function of the components of the angular momentum. In the general case this transformation can be carried out as Mows. The inverse of Eq. (18) is given by... 

All the knowable information about a physical system (i.e., energy, angular momentum, etc.) is contained in the wave function of the system. We shall restrict our discussion to one-body systems for the present. (We could easily generalize to many body systems.) The wave function can be expressed in terms of space coordinates and time or momenta and time. In the former notation we write,... 

The exploitation of the community of the transformation properties of irreducible tensors and wave functions gives us the opportunity to deduce new relationships between the quantities considered, to further simplify the operators, already expressed in terms of irreducible tensors, or, in general, to offer a new method of calculating the matrix elements. Indeed, it is possible to show that the action of angular momentum operator Lf on the wave function, considered as irreducible tensor tp, may be represented in the form [86] ... 

As was mentioned in Chapter 2, there exists another method of constructing the theory of many-electron systems in jj coupling, alternative to the one discussed above. It is based on the exploitation of non-relativistic or relativistic wave functions, expressed in terms of generalized spherical functions [28] (see Eqs. (2.15) and (2.18)). Spin-angular parts of all operators may also be expressed in terms of these functions (2.19). The dependence of the spin-angular part of the wave function (2.18) on orbital quantum number is contained only in the form of a phase multiplier, therefore this method allows us to obtain directly optimal expressions for the matrix elements of any operator. The coefficients of their radial integrals will not depend, except phase multipliers, on these quantum numbers. This is the case for both relativistic and non-relativistic approaches in jj coupling. 

Let us also mention that using a number of functional relations between the products of 3n/-coefficients and submatrix elements (/ C(k) / ), the spin-angular parts of matrix elements (26.1) and (26.2) are transformed to a form, whose dependence on orbital quantum numbers (as was also in the case of matrix elements of the energy operator, see Chapters 19 and 20) is contained only in the phase multiplier. In some cases this mathematical procedure is rather complicated. Therefore, the use of the relativistic radial orbitals, expressed in terms of the generalized spherical functions (2.18), is much more efficient. In such a representation this final form of submatrix element of relativistic Ek-radiation operators follows straightforwardly [28]. 

Another option is the exploitation of the so-called generalized spherical functions D m, instead of the usual spherical functions (harmonics) Y K In this approach one can express all operators in terms of these D-functions. It turned out [28] that it is possible to present a new form of relativistic atomic wave function, the angular part of which is of the form ... 

In Eqn. (5), the angular brackets impley averages over the asymmetric rotor wave-function as well as the vibrational wavefunction. Rz is the component of R along the space-fixed z-axis. The final step is to relate the coupling constants in Eqn. (5) to those of the monomer. In general, the expressions depend on the complexity of the monomers and on the dimer rotational state observed. For a large number of cases, a linear type dimer in a K=0 rotational state may be assumed, and Eqn. (5) may be expressed as... 

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