Cross Sections in Electron-Nuclear Dynamics

Quantum Theory Project, University of Florida, P.O. Box 118435, Gainesville, FL 32611-8435, USA 

Scattering theory has been a subject of interest from the beginning of quantum theory as a way to probe interactions between atoms and molecules. The many-body character of the interaction has proven a very difficult problem to deal with. For a proper description of the interaction, any model should incorporate dynamical effects such as electron transfer, rotations and vibrations, nuclear displacement, bond breaking and bond making (chemical reactions), photon emission and absorption, and ionization. 

In this chapter we describe a procedure to obtain reliable cross sections within the END theory [1] by incorporating a well-known semiclassical treatment by Schiff [2]. This topic requires the combination of classical. 

In Section 2 we briefly outline the salient features of END theory. In Section 3, we discuss the treatment of the END trajectories and their connection to the deflection function and differential cross section, as well as the derivation of the Schiff approximation. In Section 4, we present some simple applications and results of our approach. Finally, Section 5 contains our conclusions. 

END theory treats the collisional system in a Cartesian laboratory frame and each level of approximation [3] is defined by a choice of system wave function characterized by a set of time-dependent parameters and choice of basis set. Minimal END employs a system state vector of the form [3] 

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Calculation of Cross Sections in Electron-Nuclear Dynamics... 

This level of theory outhned above is implemented in the ENDyne code [18]. The explicit time dependence of the electronic and nuclear dynamics permits illustrative animated representations of trajectories and of the evolution of molecular properties. These animations reveal reaction mechanisms and details of dynamics otherwise difficult to discern, making the approach particularly suitable for the study of the subtleties of contributions to the stopping cross section. 

The integrated total absorption cross section, divided by Ephoton> is independent of the nuclear wavefunction in the excited electronic state and therefore it is independent of the nuclear dynamics in the upper state. 

The problem of cross-section calculation for various inelastic collisions is mathematically equivalent to the solution of a set (in principle, infinite) of coupled wave equations for nuclear motion [1]. Machine calculations have been done recently to obtain information about nonadiabatic coupling in some representative processes. Although very successful, these calculations do not make it easy to interpret particular transitions in terms of a particular interaction. It is here that the relatively simple models of nonadiabatic coupling still play an important part in the detailed interpretation of a mechanism, thus contributing to our understanding of the dynamic interaction between electrons and nuclei in a collision complex. 

From a dynamical (and/or spectroscopic) perspective, we may ask ourselves how to describe and predict the vibronic structures which are superimposed on many low resolution Abs. Cross Sections. These vibronic structures are deeply linked to the time evolution of the wavepacket, after the initial excitation, over typical times of a few hundreds of femtoseconds as discussed by Grebenshchikov et al. [31]. In ID, for a diafomic molecule, fhe fime evolufion is rafher simple when only one upper electronic state is involved. In contrast, for friafomic molecules fhe 3D character of the PESs makes the wavepacket dynamics intrinsically complex. So, for most of the polyatomic molecules, the quantitative interpretation of fhe vibronic structures superimposed to the absorption cross section envelope remains a hard task for two main reasons first because it requires high accuracy PESs in a wide range of nuclear coordinates and, second, it is not easy to follow fhe ND N = 3 for triafomic molecules) wavepackef over several hundred femtoseconds,... 

In the previous section we noticed that all the scattering information can be obtained from the transition operator T which satisfies Lippmann-Schwinger equation (7). In order to obtain cross sections for vibrationally inelastic processes we project the equation (7) onto electronic space spanned by plane waves and a two-dimensional space for nuclear dynamics defined by a vibrational ground state and its first excited state /j. The resulting equation follows ... 

INS spectroscopy avoids this problematic feedback effect, in the sense that it measures nuclear motion directly. This provides a simple relation between the amplitude of the motion, the cross section of the nuclei and the vibrational frequency to determine the intensity and shape of the spectral lines. The electron density is, of course, used to determine the potential energy surface that controls the nuclear motion but it is not involved in the calculation of the spectra. The combined use of INS spectroscopy, ab initio calculations and ACLIMAX can provide a good test of the molecular model, assessing the quality of structure and dynamics. 

The integration of this set of coupled first-order differential equation can be done in a number of ways. Care must be taken since there are basically rather two different time scales involved, i.e. that of the nuclear dynamics and that of the normally considerably faster electron dynamics. It should be observed that this END takes place in a Cartesian laboratory reference frame, which means that the overall translation as well as overall rotation of the molecular system is included. This offers no complications since the equations of motion satisfy basic conservation laws and, thus, total momentum and angular momentum are conserved. At any time in the evolution of the molecular system can the overall translation be isolated and eliminated if so should be deemed necessary. This level of theory [16,19] is implemented in the program system ENDyne [20], and has been applied to atomic and molecular reactive collisions. Calculations of cross sections, differential as well as integral, yield results in excellent agreement with the best experiments. 

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