Theoretical investigation of scattering of an atomic projectile confined in a harmonic surface potential

In this paper, scattering of a projectile atomic gas confined in an atomic harmonic surface is studied using the Lippmann–Schwinger. The nonlocal separable potential of rank one has been assumed between the projectile gas and surface, because this potential is useful to investigate the few-body systems. The analytical solution of Lippmann–Schwinger equation has been derived, and the scattering properties including transition and scattering matrices, phase shift, scattering amplitude and time delay are calculated analytically as a function of incident atomic gas energies.     

A parallel Green's operator for multidimensional quantum scattering calculations

A parallel algorithm for computing multidimensional scattering wave functions is introduced. The inhomogeneous scattering (Lippmann–Schwinger) equation is solved within the discrete variable representation with absorbing boundary conditions, using iterative (Krylov) methods. A parallel Green's operator enables one to distribute the wave function to orthogonal subspaces in which it is processed in parallel. Application to a model problem of electron scattering in a three-dimensional rectangular quantum wire is given. Speedup is demonstrated with an increasing number of processors and with increasing dimensions and/or sampling density. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 167–173, 1998     

Multi-variate Bateman method for two-body scattering without partial-wave decomposition

The use of Bateman method for solving the two-variable version of the two-body Lippmann–Schwinger equation without recourse to partial-wave decomposition is investigated. Bateman method is based on a special kind of interpolation of the momentum representation of the potential on a multi-variate grid. A suitable scheme for the generation of a multi-variate Cartesian grid is described. The method is tested on the Hartree potential for electron-hydrogen scattering in the static no-exchange approximation. Our results show that the Bateman method is capable of producing quite accurate solutions with relatively small number of grid points.     

Path integrals and optical potentials for elastic and inelastic scattering

When relative degrees of freedom are taken to traverse a fixed trajectory R(τ), the properties of Feynman path integrals are used to derive a series expansion of the logarithm of the internal state elastic scattering amplitude in terms of the internal relative coupling potential. Likewise transition amplitudes for inelastic scattering are expressed in terms of an exponential of an infinite series of 2 x 2 matrices. These results represent a reordering of the usual perturbation series of time dependent quantum mechanics to obtain a series expansion in the exponential as well as truncation from the space of all channels to that of only one or two (or more, if necessary). When the relative degrees of freedom are to be treated quantum mechanically, R(τ) is now the integration variable in a Feynman path integral. The above noted exponentiated series then correspond to the exact optical potentials for elastic and inelastic scattering within the Feynman path integral formulation of quantum mechanics.     

Bound and scattering states for a hyperbolic‐type potential in view of a new developed approximation

A new developed approximation is used to obtain the arbitrary l‐wave bound and scattering state solutions of Schrödinger equation for a particle in a hyperbolic‐type potential. For bound state, the energy eigenvalue equation and unnormalized wave functions in terms of Jacobi polynomials are achieved using the Nikiforov–Uvarov (NU) method. Besides, energy eigenvalues are calculated numerically for some states and compared with those given in the literature to check accuracy of our results. For scattering state, the wave function is found in terms of hypergeometric functions. Furthermore, scattering amplitude and phase shifts are achieved using scattering solutions. Also it is shown that the energy eigenvalue equation obtained from analytic property of scattering amplitude is same with one obtained using NU method. © 2015 Wiley Periodicals, Inc.     

Quantum trajectories for resonant scattering

Time‐dependent wave packet resonant scattering for the double square barrier has been studied in terms of Bohm quantum trajectories. The high transmission probability for the wave packet with a resonant energy can be explained by the behavior of the quantum trajectories under the influence of the relatively slow formation of a node within the first barrier. This node splits the trajectories into reflected and transmitted components. During this stage, many particle trajectories pass through the double‐barrier region and contribute to the transmitted part of the wave packet. Due to the transient nature of the nodes, trajectories in the reflected wave packet bunch together between the nodes for a finite period of time so that temporary structure (localization of particles and accompanying increase in the probability density) develops on small length scales. These calculations also show that the particles gain high momentum near the nodal points, and they reach a uniform momentum distribution after transmitting the barrier region. We have found that the presence of a node between the two barriers influences the different lifetimes of the quasi‐bound states. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 206–213, 2001     

Construction of a nonlocal potential from the scattering amplitude in the born approximation

Within a framework of the Born approximation simplified by certain assumptions, a procedure is given which explicitly constructs a nonlocal potential that provides a fit to elastic scattering data. The data are assumed to consist of the scattering amplitude as a function of the momentum transfer over a finite range of energies. The constructed potential is unique only when the kernel of the nonlocal interaction is energy independent, or when its explicit energy dependence is known from information other than scattering data.     

Theory of half-collision cross sections

Standard Lippmann–Schwinger theory does not apply to decay of metastable or unstable states due to failure of the adiabatic hypothesis. In the Fock–Tani representation, discrete unstable states can be described by state vectors orthogonal to the asymptotic states representing their decay fragments, and a decay/formation interaction is exhibited explicitly as a portion of the total interaction Hamiltonian. This allows a straightforward derivation of a generalized Lippmann–Schwinger “half-collision” differential decay cross section without the need of projection operators. It reduces in first order to the product of a resonance line shape by a Golden Rule matrix element squared. In the general case the line shape is non-Lorentzian and the matrix element factor contains final-state interaction contributions. The exact expression is expected to be applicable to a variety of processes such as predissociation, autoionization, and Auger effect. The derivation employs the van Hove resolvent formalism to exhibit the dependence of the cross section on the complex self energy of the decaying state.     

Approximate analytical versus numerical solutions of Schrödinger equation under molecular Hua potential

We solve the D‐dimensional Schrödinger equation under the Hua potential by using a Pekeris‐type approximation and the supersymmetry quantum mechanics. The reliability of the spectrum is checked via a comparison with the finite difference method. This interaction resembles Eckart, Morse, and Manning–Rosen potentials. Some useful quantities are reported via the Hellmann–Feynman Theorem. © 2012 Wiley Periodicals, Inc.     

Self‐energy fields solutions to the generalized reduced Liouville equation in the perturbative approach

A direct approach for the generalized reduced Liouville equation of motion decoupling problem associated with open quantum systems within the superoperator formalism is presented. The procedure is based on inversion of the perturbation series for the energy representation of the density operator so as to obtain one for the proper self‐energy fields that emerge as a consequence of the analytic character of the associated spectral representation. Thus, the perturbation series that arises from the iteration of the energy‐dependent matrix elements hierarchy involved in the statistical operator allows, upon further expansion of the inverse of such series, to get formally exact expressions for the corrections to all orders of the self‐energy fields. The lower order corrections of these fields are discussed in terms of resonant and nonresonant contributions. The present approach provides matrix equations that show the close relation between the environment effects represented by the self‐energy fields and the relaxation kernel that drives the system–reservoir interaction. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 280–290, 2000     

Exact solution of multidimensional hyper‐radial Schrödinger equation for many‐electron quantum systems

In quantum theory, solving Schrödinger equation analytically for larger atomic and molecular systems with cluster of electrons and nuclei persists to be a tortuous challenge. Here, we consider, Schrödinger equation in arbitrary N‐dimensional space corresponding to inverse‐power law potential function originating from a multitude of interactions participating in a many‐electron quantum system for exact solution within the framework of Frobenius method via the formulation of an ansatz to the hyper‐radial wave function. Analytical expressions for energy spectra, and hyper‐radial wave functions in terms of known coefficients of inverse‐power potential function, and wave function parameters have been obtained. A generalized two‐term recurrence relation for power series expansion coefficients has been established. © 2016 Wiley Periodicals, Inc.     

A numerical method for the determination of atom—atom scattering amplitudes from the measured differential cross sections

A numerical method is given for the determination of the scattering amplitude, hence all the phase shifts, from the differential cross section at fixed energy. To obtain the phase of the scattering amplitude, the unitarity equation of scattering theory is solved, using the (experimental) cross section as input information. A straightforward iterative approach diverges for atom—atom input data, thus an appropriately modified method of solution is introduced to overcome this difficulty. The method was applied to two test cases, in both of which calculated atom—atom cross sections were used as simulated input data. Convergence to the correct phase was obtained in both examples, starting in each case with a guess phase function that differed considerably from the true solution. Convergence cannot be obtained, however, from an extremely poor starting phase. This study shows that the scattering amplitude for atomic scattering at thermal energies can be determined systematically, without use of parametrized functions, if a sufficiently accurate experimental cross section is available. A subsequent article describes a quantum mechanical procedure whereby the interaction potential can be constructed from the determined scattering amplitude.     

Effects of complex-valued electronic wave functions on nuclear motions

Moleculer species and colliding groups of atoms are considered for which the electronic wave functions are complex-valued, having arguments that depend parametrically on the nuclear coordinates. The effective Hamiltonian for nuclear motions in the adiabatic approximation that arises in the present case differs from the ordinary Born–Oppeneheimer Hamiltonian, the latter being obtained when restriction to real-valued electronic functions is made. The asymptotic boundary conditions imposed in collision theory lead to in- and out- states [8], and hence to complex-valued wave functions in the coordinate representation. The study of the influence of electron–molecule scattering on nuclear motions therefore necessitates the use of the new effective Hamiltonian, which leads to results differing from those predicted on the basis of the Born–Oppenheimer operator. It is shown that momentum-dependent potentials occurring in the new Hamiltonian might cause “distortions” to the vibrational patterns of some electron–molecule metastable states. Also, these terms can give rise to non-Born–Oppenheimer resonances when motions in an internuclear coordinate become unbounded. We derive expressions for the relevant level widths and line shapes, showing them to be subject to an isotope effect. Even when real-valued electronic functions may be used, the selections of complex-valued functions in their linear span is still optional. Although exact treatments lead to the same results in both real and complex cases we show how the choice of the argument of the electronic function to be non-zero and dependent on nuclear coordinates may be useful for the application of certain approximation schemes. It is demonstrated that for certain systems a suitable choice of the argument assures convergence when the related Lippmann–Schwinger Equation is iterated. It is also shown that in this way an nth order term in the series expansion of the T matrix [8] for moleculer systems can be made negligibly small.     

Applications of automatic differentiation to the time‐dependent Schrödinger equation

A new approach based upon the Taylor series method is proposed for propagating solutions of the time‐dependent Schrödinger equation. Replacing the spatial derivative of the wave function with finite difference formulas, we derive a recursive formula for the evaluation of Taylor coefficients. The automatic differentiation technique is used to recursively calculate the required Taylor coefficients. We also develop an implicit scheme for the recursive evaluation of these coefficients. We then advance the solution in time using a Taylor series expansion. Excellent computational results are obtained when this method is applied to a one‐dimensional reflectionless potential and a two‐dimensional barrier transmission problem. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Quantum trajectories in complex space: one-dimensional stationary scattering problems

One-dimensional time-independent scattering problems are investigated in the framework of the quantum Hamilton-Jacobi formalism. The equation for the local approximate quantum trajectories near the stagnation point of the quantum momentum function is derived, and the first derivative of the quantum momentum function is related to the local structure of quantum trajectories. Exact complex quantum trajectories are determined for two examples by numerically integrating the equations of motion. For the soft potential step, some particles penetrate into the nonclassical region, and then turn back to the reflection region. For the barrier scattering problem, quantum trajectories may spiral into the attractors or from the repellers in the barrier region. Although the classical potentials extended to complex space show different pole structures for each problem, the quantum potentials present the same second-order pole structure in the reflection region. This paper not only analyzes complex quantum trajectories and the total potentials for these examples but also demonstrates general properties and similar structures of the complex quantum trajectories and the quantum potentials for one-dimensional time-independent scattering problems.     

Complex trajectories sans isochrones: quantum barrier scattering with rectilinear constant velocity trajectories

One of the major obstacles in employing complex-valued trajectory methods for quantum barrier scattering calculations is the search for isochrones. In this study, complex-valued derivative propagation method trajectories in the arbitrary Lagrangian-Eulerian frame are employed to solve the complex Hamilton-Jacobi equation for quantum barrier scattering problems employing constant velocity trajectories moving along rectilinear paths whose initial points can be in the complex plane or even along the real axis. It is shown that this effectively removes the need for isochrones for barrier transmission problems. Model problems tested include the Eckart, Gaussian, and metastable quadratic+cubic potentials over a variety of wave packet energies. For comparison, the "exact" solution is computed from the time-dependent Schrodinger equation via pseudospectral methods.     

Electron scattering with H2S and PH3 molecules

Calculated total, differential and momentum transfer cross sections are reported for the vibrationally elastic scattering of electrons from H₂S and PH₃ molecules in the range of energy 0.1–50 eV. The scattering process is approximated by two incoherent scatterings caused, separately, by a central field and a long-range electric dipole interaction. The central field is calculated with a spherical approximate molecular wave function, in which the exchange interaction is treated in two ways: (i) exactly within the accuracy of the molecular wave function; (ii) approximately by a local model potential. The scattering by the central field is calculated with partial wave expansion technique, while the scattering by the electric dipole potential is calculated by using the first Born approximation for a rotating dipole model with experimental values of the dipole moments of H₂S and PH₃. The total cross sections are approximated by the incoherent sum of the cross section due to the central potential and the cross section of 00→10 rotational transition caused by the electric dipole potential. The effects of the polarization interaction are also tested. The contribution of small-angle scattering to the integral cross section is analyzed for these weakly polar molecules with some quantitative comparison.     

Configuration interaction study of X-ray and fast electron scattering factors for light atomic systems

A study of X-ray and fast electron scattering by light atoms and ions has been carried out in the first Born approximation. Coherent and incoherent scattering factors calculated with configuration interaction wave functions are compared with those obtained with Hartree–Fock wave functions. These configuration interaction wave functions involve only L-shell correlation. It is shown that the changes in the coherent scattering factors due to configuration interaction are not negligible and that the electron correlation effects on the incoherent scattering factors are important. Tables of coherent and incoherent scattering factors for light atomic systems are given.     

Equilibrium inversion barrier of NH3 from extrapolated coupled‐cluster pair energies

The basis‐set convergence of singlet and triplet pair energies of coupled‐cluster theory including single and double excitations is accelerated by means of extrapolations based on the distinct convergence behaviors of these pairs. The new extrapolation procedure predicts a nonrelativistic Born–Oppenheimer inversion barrier of 1767±12 cm^?1 for NH₃. An effective one‐dimensional, vibrationally averaged barrier of 2021±20 cm^?1 is obtained when relativistic effects (+20 cm^?1), Born–Oppenheimer diagonal corrections (?10 cm^?1), and zero‐point vibrations (+244 cm^?1) are accounted for. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1306–1314, 2001