Scattering state many-body

Y. Ohm, Int J. Quantum Chem., Quantum Chem. Symp., 1985, 39-50, from conference Proceedings of the International Symposium on Atomic, Molecular and Sohd-State Theory, Scattering Problems, Many Body Phenomena, and Computational Quantum Chemistry, 18-23 March 1985. 

Liquids are difficult to model because, on the one hand, many-body interactions are complicated on the other hand, liquids lack the symmetry of crystals which makes many-body systems tractable [364, 376, 94]. No rigorous solutions currently exist for the many-body problem of the liquid state. Yet the molecular properties of liquids are important for example, most chemistry involves solutions of one kind or another. Significant advances have recently been made through the use of spectroscopy (i.e., infrared, Raman, neutron scattering, nuclear magnetic resonance, dielectric relaxation, etc.) and associated time correlation functions of molecular properties. 

Thus far we have examined the determination of a field that will control the quantum many-body dynamics of a system when all that is specified is the initial and final states of the system and the constraints imposed by the equations of motion and physical limitations on the field. When posed in this fashion, the calculation of the control field is an inverse problem that has similarities to the determination of the interaction potential from scattering data. Despite the similarities, the mathematical methods used are very different. Because only the end points of the initial-to-final state transforma-... 

As described in the main text of this section, the states of systems which undergo radiationless transitions are basically the same as the resonant scattering states described above. The terminology resonant scattering state is usually reserved for the case where a true continuum is involved. If the density of states in one of the zero-order subsystems is very large, but finite, the system is often said to be in a compound state. We show in the body of this section that the general theory of quantum mechanics leads to the conclusion that there is a set of features common to the compound states (or resonant scattering states) of a wide class of systems. In particular, the shapes of many resonances are very nearly the same, and the rates of decay of many different kinds of metastable states are of the same functional form. It is the ubiquity of these features in many atomic and molecular processes that we emphasize in this review. 

Interactions between neighboring atoms shift and broaden the line. This can be described from a many-body picture as a result of the mean field energy shift, AEe = (Anh2aeyg/ m) rig, where ae,g is the s-wave scattering length of atoms in state e and g, m is the atomic mass, and ng is the density of g-state atoms. From the an atomic standpoint it is equal to a cold collision frequency shift which we... 

Molecular-orbital approaches to edge structures differ for semiconducting and isolating molecular complexes. The latter and transition-metal complexes allow one to minimize solid-state effects and to obtain molecular energy levels at various degrees of approximation (semiempirical, Xa, ab initio). The various MO frameworks, namely, multiple-scattered wave-function calculations (76, 79, 127, 155) and the many-body Hartree-Fock approach (13), describe states very close to threshold (bound levels) and continuum shape resonances. 

We found the contraction of the f-wave functions to an atomic like character. This has an important consequense, because we see the onset of a cooperative effect, the orbital magnetic moment to spin conduction band electrons interaction and other correlations will generate an effective mass for the f-electrons which, in turn, will, by contraction of the f-wave function, enhance the atomic like character and then enhance the full process, hence the process will not be smooth or show linear behaviour the process will tend to present a sudden change of the f-electrons from a band character to an almost atomic like behaviour. This many-body effect is responsible for further scattering between the conduction electrons and the rare earth or actinide f electron states,... 

Methods for calculating collisions of an electron with an atom consist in expressing the many-electron amplitudes in terms of the states of a single electron in a fixed potential. In this chapter we summarise the solutions of the problem of an electron in different local, central potentials. We are interested in bound states and in unbound or scattering states. The one-electron scattering problem will serve as a model for formal scattering theory and for some of the methods used in many-body scattering problems. 

In an actual scattering experiment from condensed matter, we do not measure the cross-section for a process in which the scattering system goes from a specific initial state c 7 to another state 0/, both being unobserved states of the many-body system. Therefore, one takes an appropriate average over all these states [Squires 1996 van Hove 1954], as done in Eq. (6). 

In the framework of many-body perturbation theory, one first defines the scattering matrix. S as a time-ordered exponential in terms of the perturbing Hamiltonian and field operators [471. Then, one considers the matrix elements corresponding to the proccs.s in which the recoiless probe particle carries the system either from an initial state a to a final state af >(, (single excitation) or from an initial state to a final state a/a (I>(f... 

For the purpose of comparison with the measured absorption coefficient, the theoretical spectra are convoluted with a Lorentzian broadening function F(E). This function is the sum of two terms. The first takes account of the core hole width and the second term is the width of the excited band energy, which is a function dependent on the mean free path of the excited electrons, and takes account of the photoelectron inelastic scattering which is energy dependent and varies for each material as shown in Fig. 1. Note that in this theory any broadening effect due to the experimental resolution and many-body effects, such as the influence of the core hole on the band states, are not included. 

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