Atom collision problem

Smith, F. T. Diabatic and adiabatic representations for atomic collision problems, Phys.Rev., 179(1969)111-123. 

For the purpose of normal atomic collision problems the state of an electron in the hydrogen atom is given by the solution of the problem of an electron bound in the Coulomb potential. We choose the nucleus to be infinitely massive for simplicity, although there is no difficulty in eliminating the consequent errors of order 10 by transforming to the system in which the centre of mass is the origin and the particle has the reduced mass memnjime -t- mn). 

Kohn-variational (12), Schwinger-variational, (13) R-Matrix (14), and linear algebraic techniques (15,16) have been quite successful in calculating collisional and phH oTo nization cross sections in both resonant and nonresonant processes. These approaches have the advantage of generality at the cost of an explicit treatment of the continuous spectrum of the Hamiltonian and the requisite boundary conditions. In the early molecular applications of these scattering methods, a rather direct approach based on the atomic collision problem was utilized which lacked in efficiency. However in recent years important conceptual and numerical advances in the solution of the molecular continuum equations have been discovered which have made these approaches far more powerful than those of a decade ago... 

We first divide the dynamical variables into two groups those of heavy particle , which are treated in the coordinate representation R, and those of internal degrees of freedom, which are represented by state labels a, /3,. The former are to be treated in the classical approximation. In his original work, Pechukas considered atomic collision problems. Here we consider a general electron-nucleus coupled dynamics and set the former as the nuclear degrees of freedom and the latter as the electronic degrees of freedom. 

J. A. Belling, The evaluation of slowly converging radial integrals occurring in atomic collision problems, J. Phys. B, 1(1), 136 (1968). 

In many kinds of atomic and molecular collision problem the wavefunction has many oscillations because the... 

Equations of the type (11.2) occur in mathematical physics in the discussion of boundary value problems in potential theory, and in the theory of atomic collisions (see examples 13, 14 below). 

Recently Schulz et aland Fischer et al have had some difficulty in applying the CDW-EIS theory successfully for fully differential cross sections in fast ion-atom collisions at large perturbations. These ionization cross sections are expected to be sensitive to the quality of the target wave function and therefore accurate wave functions are needed to calculate these cross sections. Thus one purpose of this paper is to address this problem theoretically by re-examining the CDW-EIS model and the assumptions on which it is based. We will explore this by employing different potentials to represent the interaction between the ionized electron, projectile ion and residual target ion. For other recent work carried out on fully differential cross sections see and references therein. This discussion is presented in section 4. 

In order to appreciate the size of the basis sets required for fully converged calculations, consider the interaction of the simplest radical, a molecule in a electronic state, with He. The helium atom, being structureless, does not contribute any angular momentum states to the coupled channel basis. If the molecule is treated as a rigid rotor and the hyperfine structure of the molecule is ignored, the uncoupled basis for the collision problem is comprised of the direct products NMf ) SMg) lnii), where N = is the quantum number... 

C.W. McCurdy, F. Martin, Implementation of exterior complex scaling in B-splines to solve atomic and molecular collision problems, J. Phys. B At. Mol. Opt. Phys. 37 (2004) 917. 

It seems, therefore, with the current renewal of theoretical interest in atomic and molecular collision problems, reactive scattering, and predissociation phenomena, that it is worthwhile to examine the VB theory as a useful model that is capable of yielding accurate potential energy surfaces. 

The colinear collision problem of atom A colliding with a molecule BC was first attempted quantum mechanically by Zener [14,15] and then by Jackson and Mott [28] for the purpose of investigating thermal accommodation coefficients for atoms impinging on solid surfaces. An exponential repulsion was utilized, along with the harmonic-oscillator approximation. The distorted-wave (DW) method was employed to obtain a 1 — 0 transition probability of the form... 

In the atomic collision, the v value has been determined extrinsically by the initial relative velocity of the two atoms. In the present problem of electron transfer between donor and acceptor, v takes various values, arising from the thermal fluctuations of atoms along the reaction coordinate. On a trajectory Q t) along the reaction coordinate in the semiclassical picture, the value of v is given by the time derivative Q(t) since Q was defined as the energy difference between the diabatic potentials for the reactant and the product state (Eq. 40). Since fluctuations in Q(t) are statistically independent of those in Q(t), the average value of v is the same at any value of Q(t). 

According to the general expression (1), the energy loss in ion-atom collision can be found if the density of current induced in atomic shell, j(r, t), is known. In general case, this problem can be solved only using some approximate model. In this section, we present the local response approach where the description of current density is based on the results for a uniform electron gas. 

Collisions with atomic clusters represent a relatively new branch of collision physics as compared to the well established fields of ion-atom collisions [1] and ion-solid interaction [2]. The study of cluster collisions is of particular interest and importance because it offers the possibility, to tackle bridge-building questions (like the transition from individual excitations in the elementary ion-atom collision to the macroscopic stopping power in solids) as well as fundamental problems (like phase transitions in finite systems). 

Interactions involving short laser pulses have occasionally been compared to collisions, which also depend impulsively on time. Collision problems require at the outset a rather good description of the correlated atom, without which realistic predictions cannot be made. This analogy suggests that a maximum atomic physics option is required. 

This assumption is acceptable for the near-resonant case (18, 19, 46, 47). Together with this assumption we assume that the direction of P-state points to the other atom throughout collision (the rotating atom approximation). The rotating atom approximation has been employed and discussed by Bates et al. (6,7,8) in some collision problems. In our excitation transfer problem this approximation would also be available. 

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