Electron-impact ionization of the hydrogen atom is the prototypical three-body Coulomb breakup problem in quantum mechanics. The combination of subtle correlation effects and the difficult boundary conditions required to describe two electrons in the continuum have made this one of the outstanding challenges of atomic physics. A complete solution of this problem in the form of a ‘reduction to computation’ of all aspects of the physics is given by the application of exterior complex scaling, a modern variant of the mathematical tool of analytic continuation of the electronic coordinates into the complex plane that was used historically to establish the formal analytic properties of the scattering matrix. This review first discusses the essential difficulties of the three-body Coulomb breakup problem in quantum mechanics. It then describes the formal basis of exterior complex scaling of electronic coordinates as well as the details of its numerical implementation using a variety of methods including finite difference, finite elements, discrete variable representations and B-splines. Given these numerical implementations of exterior complex scaling, the scattering wavefunction can be generated with arbitrary accuracy on any finite volume in the space of electronic coordinates, but there remains the fundamental problem of extracting the breakup amplitudes from it. Methods are described for evaluating these amplitudes. The question of the volume-dependent overall phase that appears in the formal theory of ionization is resolved. A summary is presented of accurate results that have been obtained for the case of electron-impact ionization of hydrogen as well as a discussion of applications to the double photoionization of helium.

Since the invention of quantum mechanics, even the simplest example of the collisional breakup of a system of charged particles, e(-) + H --> H(+) + e(-) + e(-) (where e(-) is an electron and H is hydrogen), has resisted solution and is now one of the last unsolved fundamental problems in atomic physics. A complete solution requires calculation of the energies and directions for a final state in which all three particles are moving away from each other. Even with supercomputers, the correct mathematical description of this state has proved difficult to apply. A framework for solving ionization problems in many areas of chemistry and physics is finally provided by a mathematical transformation of the Schrodinger equation that makes the final state tractable, providing the key to a numerical solution of this problem that reveals its full dynamics.

We demonstrate experimentally how the time-dependent phase modulation induced by molecular rotational wave packets can manipulate the phase and spectral content of ultrashort light pulses. Using impulsively excited rotational wave packets in CO2, we increase the bandwidth of a probe pulse by a factor of 9, while inducing a negative chirp. This chirp is removed by propagation through a fused silica window, without the use of a pulse compressor. This is a very general technique for optical phase modulation that can be applied over a broad spectral region from the IR to the UV.

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We show that the ''two-potential'' formalism of conventional distorted-wave rearrangement theory, which is formally valid only for short-range interactions, can be used to evaluate amplitudes for the ionization of atomic hydrogen by electron impact. The triply differential cross sections calculated using this method validate earlier results obtained by extrapolating the quantum mechanical flux. Although it uses the same time-independent wave functions, this method offers significant advantages over flux extrapolation. It is more accurate, can be applied in practical calculations over a broader range of collision energies, and unlike the flux-extrapolation method, can be applied for arbitrary values of energy sharing between the ejected electrons. Since this rearrangement formalism provides the complete scattering amplitude for ionization, it can be used to calculate any differential cross section.

. DISCLAIMER DISCLAIMERPortions of this document may be illegible in electronic image products. Images are produced from the best avaiiabie original document.,-. LBNL-4521O Electron-Impact Ionization of Atomic Hydrogen 2000_-------.---.... . . . . . . . . . . . . . . . 72 6.4.2 SDCSfor 17.6,20,25, and30eV . . . . . . . . . . . . . . . . 73 6.5 Integral Ionization Cross Sections . . . . . . . . . . . . . . . . . . . . 74 7 The Three-Body Electron-Impact In the early 1970's Burke and Mitchell [15, 14] showed that cross sections for the elastic and excitation channels could be calculated at energies above the ionization threshold by including positive-energy pseudostates in the expansion. This work was extended in the 1980's by Oza and Callaway [23,22] (Fl, 72) Electron-Impact Ionization of Atomic Hydrogen List of Tables Nature of Ionization Electron-impact ionization is the process in which a target atom or molecule is ionized by a collision with an electron. Scattering theory calculations have progressed to the point of being able to accurately treat non-breakup processes for an electron scattering from relatively complicated target molecules. However, ionization represents a fundamentally different class of problems characterized by a final state in which three particles that interact via long-range by rearranging the Schrodinger equation (Equation 2.1).Since lli~ (?'l, 72) represents the scattered part of the wave function at large distances it must be an outgoing wave in rl and r2. Thus, we define W; (71, 72) Exterior Complex Scaling Application to long-range potentials Finite Difference Implementation ECS on a grid Under ECS, the scattered wave @& (z(rI), Z(T2)) is a continuous function but has discontinuous first derivatives along the lines rl or r2 equal to&.There is no problem representing the wave function on a two-dimensional grid in rl and r2, but in order to correctly approximate its derivatives on each grid point we will require that~be one of the grid points. The scattered wave will be calculated directly on to the ECS contour by solving Equation 3.5 on the two-dimensional, complex-scaled grid. Functions whose analytic forms are known, such as the right-hand side of Equation 3.5 and the potentials in the Hamiltonian, are mapped on to the ECS contour by simply evaluating them on the contour z(r) for both rl and r2. The non-analytic two-electron potential $ is scaled in this way by noting that it is piece-wise analytic and scaling the rl < r2 and rl > r2 regions separately. The potential is unchanged on the real part of the grid and, as will be demonstratedlater in this chapter, the potentials beyond~have very little effect on the wave function in the interior region. Finite difference approximations to derivatives Dimension of the problem Properties of the Calculated Wave FunctionsThe figure on the left shows the absolute value of #~P (rl, Tz) ..................................................j .......................... Total Cross Section Differential Cross ...

This paper discusses different routes to gaining insight from closed loop learning control experiments. We focus on the role of the basis in which pulse shapes are encoded and the algorithmic search is performed. We demonstrate that a physically motivated, nonlinear basis change can reduce the dimensionality of the phase space to one or two degrees of freedom. The dependence of the control goal on the most important degrees of freedom can then be mapped out in detail, leading toward a better understanding of the control mechanism. We discuss simulations and experiments in selective molecular fragmentation using shaped ultrafast laser pulses.

The formulation of the time-dependent close-coupling method is extended to allow the calculation of electron-impact triple-differential cross sections for atoms. The fully quantal method is applied to the electronimpact ionization of hydrogen at an incident energy of 54.4 eV for various scattering geometries. The timedependent close-coupling results are found to be in very good agreement with those obtained by a timeindependent exterior complex-scaling method. On the other hand, even though the incident energy is relatively large, significant differences are found between the two nonperturbative cross-section results and those obtained using perturbative distorted-wave methods. Large differences are found for those geometries that require an accurate knowledge of the correlation between two outgoing continuum electrons in the presence of a Coulomb nuclear field.