Bound states scattering

The energies of the selective adsorption resonances are very sensitive to the details of the physisorption potential. Accurate measurement allied to computation of bound state energies can be used to obtain a very accurate quantitative fonn for the physisorption potential, as has been demonstrated for helium atom scattering. For molecules, we have... 

Gordon R G 1969 Constructing wave functions for bound states and scattering J. Chem. Phys. 51 14-25... 

Section B3.4.3. Section B3.4.4. Sections B3.4.5 describe methods of solving the Sclirodinger equation for scattering events. Sections B3.4.6 and Sections B3.4.7 proceed to discuss photo-dissociation and bound states. 

Alternately, absorbing potentials can also be applied to convert scattering to a bound-state-like problem. One method is to write the Sclirodinger wavefiinction as a sum of two temis where... 

The wavepacket is propagated until a time where it is all scattered and is away from the interaction region. This time is short (typically 10-100 fs) for a direct reaction. Flowever, for some types of systems, e.g. for reactions with wells, the system can be trapped in resonances which are quasi-bound states (see section B3.4.7). There are eflScient ways to handle time-dependent scattering even with resonances, by propagating for a short time and then extracting the resonances and adding their contribution [69]. 

Mead and Truhlar [10] broke new ground by showing how geometric phase effects can be systematically accommodated in scattering as well as bound state problems. The assumptions are that the adiabatic Hamiltonian is real and that there is a single isolated degeneracy hence the eigenstates n(q-, Q) of Eq. (83) may be taken in the form... 

Before progressing, it is useful to review the dynamics of typical molecular systems. We consider three types scattering (chemical reaction), photodissociation, and bound-state photoabsorption (no reaction). 

The method of superposition of configurations is essentially based on the assumption that the basic orbitals form a complete set. The most popular basis used so far in the literature is certainly formed by the hydrogen-like functions, which set contains a discrete and a continuous part. The discrete subset corresponds physically to the bound states of an electron around a proton, whereas the continuous part corresponds to a free electron scattered by a proton, or classically to the elliptic and hyperbolic orbits, respectively, in a central-field problem. 

This chapter has focused on reactive systems, in which the nuclear wave function satisfies scattering boundary conditions, applied at the asymptotic limits of reagent and product channels. It turns out that these boundary conditions are what make it possible to unwind the nuclear wave function from around the Cl, and that it is impossible to unwind a bound-state wave function. 

In Section IV we quantify the relation of the information-rich phase of the scattering wavefunction to the observable 8(E) of Eq. (5). Here we proceed by connecting the two-pathway method with several other phase-sensitive experiments. Consider first excitation from g) into an electronically excited bound state with a sufficiently broad pulse to span two levels, Ea and ),... 

The first term of Eq. (34) describes a direct transition from the bound state to the scattering projection of the structured continuum. The second term describes a resonance-mediated transition. 

Next, we discuss the J = 0 calculations of bound and pseudobound vibrational states reported elsewhere [12] for Li3 in its first-excited electronic doublet state. A total of 1944 (1675), 1787 (1732), and 2349 (2387) vibrational states of A, Ai, and E symmetries have been computed without (with) consideration of the GP effect up to the Li2(63 X)u) +Li dissociation threshold of 0.0422 eV. Figure 9 shows the energy levels that have been calculated without consideration of the GP effect up to the dissociation threshold of the lower surface, 1.0560eV, in a total of 41, 16, and 51 levels of A], A2, and E symmetries. Note that they are genuine bound states. On the other hand, the cone states above the dissociation energy of the lower surface are embedded in a continuum, and hence appear as resonances in scattering experiments or long-lived complexes in unimolecular decay experiments. They are therefore pseudobound states or resonance states if the full two-state nonadiabatic problem is considered. The lowest levels of A, A2, and E symmetries lie at —1.4282,... 

In this chapter, we also discussed several schemes that allow for the computation of scalar observables without explicit construction and storage of the eigenvectors. This is important not only numerically for minimizing the core memory requirement but also conceptually because such a strategy is reminiscent of the experimental measurement, which almost never measures the wave function explicitly. Both the Lanczos and the Chebyshev recursion-based methods for this purpose have been developed and applied to both bound-state and scattering problems by various groups. 

The values are taken from [7], where it is shown, that the correlated density contains the contribution of bound states as well the contribution of scattering states. Above the so called Mott density, where the bound states begin to disappear, according to the Levinson theorem, the continuous behavior of the correlated density is produced by the scattering states. 

Within our exploratory calculation we will use a simplified description of the contribution of correlated states, considering only the bound state with an effective shift, which reproduces the correlated density. This shift is taken as a quadratic function in the densities, where the linear term is calculated from perturbation theory and the quadratic term is fitted to reproduce the results for the composition as found by the full microscopic calculation including the contribution of scattering states. 

The semiclassical mapping approach outlined above, as well as the equivalent formulation that is obtained by requantizing the classical electron-analog model of Meyer and Miller [112], has been successfully applied to various examples of nonadiabatic dynamics including bound-state dynamics of several spin-boson-type electron-transfer models with up to three vibrational modes [99, 100], a series of scattering-type test problems [112, 118, 120], a model for laser-driven... 

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