Theoretical formulation of the photoprocess

The differential cross section then follows from (see Section 8.1)  

The fine-structure constant a indicates that first-order perturbation theory has been applied the linear dependence on the photon energy Eph is due to the length form of the dipole operator used in equ. (2.1), and the wavenumber k compensates the 1 /k which appears if the absolute squared value of the continuum wavefunction is used (see equ. (7.29)). The summations over the magnetic quantum numbers M, of the photoion and ms of the photoelectron s spin are necessary because no observation is made with respect to these substates. Due to the closed-shell structure of the initial state with f — 0 and M = 0, the averaging over the magnetic quantum numbers M simply yields unity and is omitted. 

The matrix element is understood to be on-the-energy-shelF, i.e., the energy e of the photoelectron has to be calculated according to equ. (1.29a). Due to the different binding energies of electrons ejected from different shells of the atom, it is therefore possible to restrict the calculation of the matrix element to the selected process in the present example to photoionization in the Is shell only. As a consequence, the matrix element factorizes into two contributions, a matrix element for the two electrons in the Is shell where one electron takes part in the photon interaction, and an overlap matrix element for the other electrons which do not take part in the photon interaction (passive electrons). The overlap matrix element is given by 

In the frozen atomic structure approximation, where the same orbitals are used in the initial and final states, this overlap matrix element yields unity. Hence, one obtains for the remaining matrix element 

Renaming the electron numbers shows that the first and fourth terms in the matrix element M i(M1,ms) and also the second and third terms are identical, and they can be combined. As a next step one calculates the action of the photon operator on the single-particle wavefunctions. Omitting for simplicity the wavefunction 

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In this chapter, we are concerned with various theoretical formulations that allow us to treat nonadiabatic quantum dynamics in a classical description. To introduce the main concepts, we first give a brief overview of the existing methods and then discuss their application to ultrafast molecular photoprocesses. 

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