Atomic radial matrix elements

Here B[E/(M)X is the reduced nuclear probability, atomic radial matrix elements of electric (magnetic) [E/M] multipolarity A ji f and 7i,f are the angular momenta of the electronic and nuclear states correspondingly. The atomic radial matrix elements P (ft)jv) of electric (magnetic) [E/M] multipolarity A are expressed by means of the integral ... 

In Chapter 3 we considered briefly the photoexcitation of Rydberg atoms, paying particular attention to the continuity of cross sections at the ionization limit. In this chapter we consider optical excitation in more detail. While the general behavior is similar in H and the alkali atoms, there are striking differences in the optical absorption cross sections and in the radiative decay rates. These differences can be traced to the variation in the radial matrix elements produced by nonzero quantum defects. The radiative properties of H are well known, and the radiative properties of alkali atoms can be calculated using quantum defect theory. 

For t n states the same reasoning does not apply. For the t = n - 1 state the only allowed transition is to the n = n — 1, = n — 2 state. In this case the frequency of the transition is not constant but is given by 1 In3. When cubed, this frequency contributes an n 9 scaling to the decay rate. Offsetting this scaling is the fact that the radial matrix element represents the size of the atom in the n and n — 1 states. Since (n — In — 2 r n n — 1) n2the lifetimes of the highest t states scale as n5, i.e. 

Ionization at a given photon energy may proceed in several channels. For example, the dipole selection rule, A l- 1, permits an initial electronic state of angular momentum / to decay into two degenerate ionization channels, the / +1 and / -I channels in which the photoelectrons have angular momenta (/ + 1) h and (/ - 1 )h. Since the parameters a and P contain the radial matrix elements for ionization into the two channels, and since these elements are proportional to the overlap of the electronic wavefunctions for the initial and final states of the ionization process, it follows that a and P are functions of these overlaps. Secondly, since the two photoelectron waves have different phase and nodal structures, they may interfere this interference is also determinative of o and p values. For atomic photoionization and LS coupling, one finds ... 

In this paper we modify and extend this approach in several ways. In particular, we consider the magnetic fine structure effects in the presence of a uniform electric field F for ls2p Pj- excited states of helium. We introduce two separate differential polarizabihties to describe the quadratic part of the electric field splitting and three differential hyperpolarizabilities to describe the terms the order of in the fine-structure splitting of the atomic multiplet s2p Pj. We have developed a calculational approach that allows correct estimation of potential contributions due to continuum spectra to the dipole susceptibilities j3 and 7. In the next section we briefly outline our method. The details of the calculations of the angular and radial matrix elements have been described elsewhere [8,9] and are omitted here for brevity. Atomic units are used throughout. 

Usually the solutions of any version of the Hartree-Fock equations are presented in numerical form, producing the most accurate wave function of the approximation considered. Many details of their solution may be found in [45], However, in many cases, especially for light atoms or ions, it is very common to have analytical radial orbitals, leading then to analytical expressions for matrix elements of physical operators. Unfortunately, as a rule they are slightly less accurate than numerical ones. 

All three forms of the dipole matrix element are equivalent because they can be transformed into each other. However, this equivalence is valid only for exact initial- and final-state wavefunctions. Since the Coulomb interaction between the electrons is responsible for many-body effects (except in the hydrogen atom), and the many-body problem can only be solved approximately, the three different forms of the matrix element will, in general, yield different results. The reason for this can be seen by comparing for the individual matrix elements how the transition operator weights the radial parts R r) and Rf(r) of the single-particle wavefunction differently ... 

Tike all effective one-electron approaches, the mean-field approximation considerably quickens the calculation of spin-orbit coupling matrix elements. Nevertheless, the fact that the construction of the molecular mean-field necessitates the evaluation of two-electron spin-orbit integrals in the complete AO basis represents a serious bottleneck in large applications. An enormous speedup can be achieved if a further approximation is introduced and the molecular mean field is replaced by a sum of atomic mean fields. In this case, only two-electron integrals for basis functions located at the same center have to be evaluated. This idea is based on two observations first, the spin-orbit Hamiltonian exhibits a 1/r3 radial dependence and falls off much faster... 

For an atom the evaluation of the potentials in Eq. (2) is achieved straightforwardly, using trial wavefunctions, by integrating the charge density radially. Once the potentials are known the wavefunctions are obtained by numerical integration of the radial part of the Schrodinger equation, and the process is continued iteratively to self-consistency. In molecules the loss of spherical symmetry makes the procedure much more difficult, and in particular for molecules containing heavy atoms the number of matrix elements which must be evaluated becomes prohibitive. Calculations on the uranyl ion have employed three different approaches to circumvent this difficulty. 

The examples discussed here show that even in the case where the generalized Sturmian method is applied to atoms, a case in which radial orthonormality between different configurations is sometimes lost, simplified formulas for the matrix elements can often be derived. However, even when this is not possible, the... 

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