Bound-free dipole matrix elements

In analogy to (3.4) the bound nuclear wavefunction in the electronic ground state, multiplied by the transition dipole function, may be expanded according to 

Inserting Equations (3.4) and (3.11) into (3.1) yields for the bound-free matrix elements 

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To accomplish this two things are required the replacement of the free-particle with the Coulomb Green s function and the calculation of the bound-free dipole matrix element. From the latter quantity, the angular distribution of the photoelectrons... 

The matrix elements By(n,m) and Dy(n,E) are calculated numerically without difficulty since they contain at least one bound orbital. On the contrary, the free-free dipole matrix elements Cy(E,E ), whose role is crucial in the process of sfrong field ionization and of ATI, contain an on-shell singularity. Fortunately, this singularity turns out to be integrable, meaning that the integrals containing Cy(E,E ) in Eq. (27) have a finite value. Let us see how. 

There are a number of interesting cases in which full control is still attainable. For example, when only one n state at energy E for a given q exists. In particular when the number of product channels is only two, (as in diatomic molecules), the Schwarz equality (Eq. 13a) holds and full control can be attained. In the polyatomic case, the Schwarz equality holds whenever the bound-free transition-dipole matrix elements, are facorizeable. 

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