Atoms three-bodied systems

The D-dimensional scaling properties of the three-body Coulomb problem in atomic and molecular physics have been discussed in part 1 of this volume. Here we examine some new features of the adiabatic molecular problem illustrating the importance of motion near the saddle point of the two-centre potential. Properties of resonant states of the atomic three-body system are shown to be a direct consequence of... 

Heibst, E. In Atomic, Molecular, Optical Physics Handbook Drake, G., Ed. AIP Press New Yoik, 1996, p 429 Adams, N. G. In Atomic, Molecular, Optical Physics Handbook Drake, G., Ed. AIP Press New Yoik, 1996, p 441. For three-body systems, a slightly more complex temperature dependence is observed. For saturated systems, more complex treatments are needed —see Gilbert, R. G. Smith, S. C. Theory erf Unimolecular and Recombination Reactions Blackwell Oxford, 1990. 

Spectral moments may be computed from expressions such as Eqs. 5.15 or 5.16. Furthermore, the theory of virial expansions of the spectral moments has shown that we may consider two- and three-body systems, without regard to the actual number of atoms contained in a sample if gas densities are not too high. Near the low-density limit, if mixtures of non-polar gases well above the liquefaction point are considered, a nearly pure binary spectrum may be expected (except near zero frequencies, where the intercollisional process generates a relatively sharp absorption dip due to many-body interactions.) In this subsection, we will sketch the computations necessary for the actual evaluation of the binary moments of low order, especially Eqs. 5.19 and 5.25, along with some higher moments. 

Induced dipole autocorrelation functions of three-body systems have not yet been computed from first principles. Such work involves the solution of Schrodinger s equation of three interacting atoms. However, classical and semi-classical methods, especially molecular dynamics calculations, exist which offer some insight into three-body dynamics and interactions. Very useful expressions exist for the three-body spectral moments, with the lowest-order Wigner-Kirkwood quantum corrections which were discussed above. 

C.D. Lin, Hyperspherical coordinate approach to atomic and other Coulombic three-body systems, Phys. Rep. 257 (1995) 1. 

The p and He2+ are thus regarded as two atomic centers in a diatomic molecule. Because of the dual character as an exotic atom and an exotic molecule Antiprotonic Helium is often called antiprotonic helium atom-molecule, or for short, atomcule. Since the Is electron motion, coupled to a large-(n, l) p orbital, is faster by a factor of 40 than the p motion, the three-body system pHe+ is solved by using the Born-Oppenheimer approximation, as fully discussed by Shimamura [6]. 

We will see in Section V that the last result plays an important role in the classification of three-body system to atom-like or to molecule-like systems. 

Figure 17. The critical parameter >.r as a function of K in the range 0 < K < 1. The different three-body systems are shown along the transition line that separates the stable phase from the unstable one. Note the minimum value of Km — 0.35 for K > Km we have molecule-like systems, while for K < Km the systems behave like atoms. The inset in this figure shows the two branches in the (E — /. j-plane as K varies in the interval [0, 1],...

The theoretical formulation of atomic excitation and ionization processes is conveniently discussed by introducing the quantum-mechanical Hamilton operator. For a three-body system the Hamiltonian reads... 

Consider a diatomic, AB, interacting with a surface, S. The basic idea is to utilize valence bond theory for the atom-surface interactions, AB and BS> along with AB to construct AB,S For each atom of the diatomic, we associate a single electron. Since association of one electron with each body in a three-body system allows only one bond, and since the solid can bind both atoms simultaneously, two valence electrons are associated with the solid. Physically, this reflects the ability of the infinite solid to donate and receive many electrons. The use of two electrons for the solid body and two for the diatomic leads to a four-body LEPS potential (Eyring et al. 1944) that is convenient mathematically, but contains nonphysical bonds between the two electrons in the solid. These are eliminated, based upon the rule that each electron can only interact with an electron on a different body, yielding the modified four-body LEPS form. One may also view this as an empirical parametrized form with a few parameters that have well-controlled effects on the global PES. 

Introduction - Why study atomic and molecular three-body systems ... 

Our current direction is to study dynamical processes in atomic and molecular two- and three-body systems. We use a technique which formally is based on the mathematical theory of dilation analytic functions. Numerically these results axe realized though a fully three-dimensional finite element method applied to a total angular-momentum representation. We here show how generalizations of our previously published two-body methods to three-body systems are possible without formal approximations. 

The antiprotonic helium system was used as a model when developing our nonzero angular momentum 3D finite element method. This is an example of a system for which the wave function cannot exactly be decomposed into an angular and a radial part. Besides the helium like atoms it is the experimentally most accurately known three-body system. 

There have now been several applications reported for fiilly exponentially correlated four-body wavefunctions [7-9], also limited to bases without pre-exponential r,j. While it was found that pre-exponential are relatively unimportant for three-body systems, they can be expected to contribute in a major way to the efficiency of expansions for three-electron systems such as the Li atom and its isoelectronic ions, as is obvious from the fact that the zero-order description of the ground states of such systems has electron configuration s 2s. 

The three-body parameter ro determines the phase of the wavefunction at small distances and, in principle, depends on 1. The wavefunction 10.56 has infinitely many nodes, which means that in the zero-range approximation there are infinitely many trimer states. This is one of the properties of three-body systems with resonant interactions discovered by Efimov [54], We see that the fall to the center is possible in many angular momentum channels, provided the mass ratio is sufficiently large. However, for practical purposes and for simplicity, it is sufficient to consider the case where the Efimov effect occurs only for the angular momentum channel with the lowest possible / for a given symmetry. This implies that when the heavy atoms are fermions and one has odd I, in order to confine ourselves to / = 1 we should have the mass ratio in the range 14 < M/m < 76. For bosonic heavy atoms where / is even, we set I = 0 and consider M/m < 39 to avoid the Efimov effect for / > 2. In both cases we need a single three-body parameter tq. 

The Born-Oppenheimer approach for the three-body system of one hght and two heavy atoms was discussed in Fonseca, A.C., Redish, E.F., and Shanley, P.E., Efimov effect in a solvable model, Nucl. Phys. A, 320, 273,1979. 

Before introducing the classical path approximation in reactive systems it is necessary to switch to a coordinate system which does not discriminate between the reaction channels. This condition is fulfilled in the so-called hyperspherical coordinates in which the atom-atom distances for the three-body system are expressed in terms of the hyperradius p and the two hyperangles 6, by [193 ... 

Having two nuclei and one electron, H is a three-body system like the helium atom. As was the case for helium, the Schrodinger equation for Hj cannot be exactly solved by analytical integration, but again we can find tractable and accurate approximations. The coordinates we will use for diatomics are drawn in Fig. 5.1. The two nuclei are labeled A and B and lie on the z axis. The position of the electron is given in spherical coordinates, with the origin at either nucleus A or nucleus B. 

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