Potential splitting approach for molecular systems (original) (raw)
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Journal of Physics B: Atomic, Molecular and Optical Physics, 2015
The study of scattering processes in few body systems is a difficult problem especially if long range interactions are involved. In order to solve such problems, we develop here a potential-splitting approach for three-body systems. This approach is based on splitting the reaction potential into a finite range core part and a long range tail part. The solution to the Schrödinger equation for the long range tail Hamiltonian is found analytically, and used as an incoming wave in the three body scattering problem. This reformulation of the scattering problem makes it suitable for treatment by the exterior complex scaling technique in the sense that the problem after the complex dilation is reduced to a boundary value problem with zero boundary conditions. We illustrate the method with calculations on the electron scattering off the hydrogen atom and the positive helium ion in the frame of the Temkin-Poet model.
Physical Review A, 2011
In this paper we suggest a new approach for the multichannel Coulomb scattering problem. The Schrödinger equation for the problem is reformulated in the form of a set of inhomogeneous equations with a finite-range driving term. The boundary conditions at infinity for this set of equations have been proven to be purely outgoing waves. The formulation presented here is based on splitting the interaction potential into a finite range core part and a long range tail part. The conventional matching procedure coupled with the integral Lippmann-Schwinger equations technique are used in the formal theoretical basis of this approach. The reformulated scattering problem is suitable for application in the exterior complex scaling technique: the practical advantage is that after the complex scaling the problem is reduced to a boundary problem with zero boundary conditions. The Coulomb wave functions are used only at a single point: if this point is chosen to be at a sufficiently large distance, on using the asymptotic expansion of Coulomb functions, one may completely avoid the Coulomb functions in the calculations. The theoretical results are illustrated with numerical calculations for two models.
Potential splitting approach to e-H and e-He^+ scattering with zero total angular momentum
2016
An approach based on splitting the reaction potential into a finite range part and a long range tail part to describe few-body scattering in the case of a Coulombic interaction is proposed. The solution to the Schr\"odinger equation for the long range tail of the reaction potential is used as an incoming wave. This reformulation of the scattering problem into an inhomogeneous Schr\"odinger equation with asymptotic outgoing waves makes it suitable for solving with the exterior complex scaling technique. The validity of the approach is analyzed from a formal point of view and demonstrated numerically, where the calculations are performed with the finite element method. The method of splitting the potential in this way is illustrated with calculations of the electron scattering on the hydrogen atom and the positive helium ion in energy regions where resonances appear.
Three-potential formalism for the three-body scattering problem with attractive Coulomb interactions
Physical Review A, 2001
A three-body scattering process in the presence of Coulomb interaction can be decomposed formally into a two-body single channel, a two-body multichannel and a genuine three-body scattering. The corresponding integral equations are coupled Lippmann-Schwinger and Faddeev-Merkuriev integral equations. We solve them by applying the Coulomb-Sturmian separable expansion method. We present elastic scattering and reaction cross sections of the e + + H system both below and above the H(n = 2) threshold. We found excellent agreements with previous calculations in most cases.
Scattering theory with the Coulomb potential
Journal of Physics: Conference Series, 2009
Basic features of a new surface-integral formulation of scattering theory are outlined. This formulation is valid for both short-range and Coulombic longe-range interactions. New general definitions for the potential scattering amplitude are given. For the Coulombic potentials the generalized amplitude gives the physical on-shell amplitude without recourse to a renormalization procedure. New post and prior forms for the amplitudes of breakup, direct and rearrangement scattering in a Coulomb three-body system are presented.
Journal of Physics B: Atomic, Molecular and Optical Physics, 1988
The physical origin of the different contributions to the optical potentials of the electron-molecule scattering problem is analysed. A method is proposed for separating the optical potential into a local, long-range and energy-independent term and into a non-local, short-range and energy-dependent one. Such a separation is desired if one treats the electron scattering off the local and off the non-local part of the interaction by different numerical methods as recently proposed by several different authors. The separation scheme is applied to the e-+N, scattering problem and it is shown that the proposed separation significantly improves the quality of the representation of the interaction.
Scattering theory for arbitrary potentials
Physical Review A, 2005
The fundamental quantities of potential scattering theory are generalized to accommodate longrange interactions. New definitions for the scattering amplitude and wave operators valid for arbitrary interactions including potentials with a Coulomb tail are presented. It is shown that for the Coulomb potential the generalized amplitude gives the physical on-shell amplitude without recourse to a renormalization procedure.
Surface-Integral Approach to the Coulomb Few-Body Scattering Problem
19th International Iupap Conference on Few-Body Problems in Physics, 2009
We present main features of a surface-integral approach to the Coulomb few-body scattering problem. This approach is valid for both short-range and Coulombic longe-range interactions. We give new general definitions for the potential scattering amplitude. For the Coulombic potentials the generalized amplitude gives the physical on-shell amplitude without recourse to a renormalization procedure. New post and prior forms for the amplitudes of breakup, direct and rearrangement scattering in a Coulomb three-body system are also presented. The Green's functions and formal solutions of the Schrödinger equation in integral form are not used. Therefore, for the purpose of defining the scattering amplitudes the knowledge of a complicated analytic structure of the Green's function in the complex-energy plane is not required.
The concept of an optical potential as a physical entity that governs the scattering of a single particle by a composite target is an intuitively appealing phenomenological concept that goes back to the early days of nuclear physics. In principle, the scattering of a nonrelativistic, quantum-mechanical particle off an N -particle target is a many-body problem and governed by the (N + 1)-particle Schrödinger equation. In the so-called optical model [1], the elastic scattering problem is alternatively described by an effective single-particle Schrödinger (or Lippmann-Schwinger) equation. All effects of the interaction of the projectile particle with the target are contained in the so-called optical potential. In general, this optical potential has to be a very complicated object: It becomes a nonlocal operator because exchange and rearrangement of target particles have to be considered. An energy dependence has to account for possible excitations of the target and if inelastic scattering is energetically possible, the optical potential is nonhermitian in order to describe the loss of scattering amplitude into the inelastic channels. One major technical advantage of using optical potentials in numerical calculations is that the scattering problem can be separated from the many-body problem and the latter can be treated using bound-state techniques.
Solving the Coulomb scattering problem using the complex-scaling method
EPL (Europhysics Letters), 2009
Based on the work of Nuttall and Cohen [Phys. Rev. 188 (1969) 1542] and Resigno et al. [Phys. Rev. A 55 (1997) 4253] we present a rigorous formalism for solving the scattering problem for long-range interactions without using exact asymptotic boundary conditions. The long-range interaction may contain both Coulomb and short-range potentials. The exterior complex scaling method, applied to a specially constructed inhomogeneous Schrödinger equation, transforms the scattering problem into a boundary problem with zero boundary conditions. The local and integral representations for the scattering amplitudes have been derived. The formalism is illustrated with numerical examples.