State bound

Molecular adsorbates usually cover a substrate with a single layer, after which the surface becomes passive with respect to fiirther adsorption. The actual saturation coverage varies from system to system, and is often detenumed by the strength of the repulsive interactions between neighbouring adsorbates. Some molecules will remain intact upon adsorption, while others will adsorb dissociatively. This is often a frinction of the surface temperature and composition. There are also often multiple adsorption states, in which the stronger, more tightly bound states fill first, and the more weakly bound states fill last. The factors that control adsorbate behaviour depend on the complex interactions between adsorbates and the substrate, and between the adsorbates themselves. 

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... 

In a time-dependent picture, resonances can be viewed as localized wavepackets composed of a superposition of continuum wavefimctions, which qualitatively resemble bound states for a period of time. The unimolecular reactant in a resonance state moves within the potential energy well for a considerable period of time, leaving it only when a fairly long time interval r has elapsed r may be called the lifetime of the almost stationary resonance state. 

Marcus R A 1973 Semiclassical theory for collisions involving complexes (compound state resonances) and for bound state systems Faraday Discuss. Chem. Soc. 55 34—44... 

Wang D and Bowman J M 1995 Complex L calculations of bound states and resonances of HCO and DCO Cham. Phys. Latt. 235 277-85... 

H1] Qiu Y and Bai Z 1998 Vibration-rotation-tunneling dynamics of (HF)2 and (HCI)2 from fulldimensional quantum bound state calculations Advances in Moiecuiar Vibrations and Coiiision Dynamics, Voi. i-ii Moiecuiar dusters ed J Bowman and Z Bai (JAI Press) pp 183-204... 

For a local potential V(r) which supports bound states of angular momentum i and energy < 0, the phase shift linij Q (Ic)) tends in the lunit of zero collision energy to n. When the well becomes deep enough so as to introduce an additional bound level = 0 at zero energy, then linij ... 

In the internal region r> a, the electron-atom complex behaves almost as a bound state so that a configuration... 

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]. 

Figure B3.4.10. Schematic figure of a ID double-well potential surface. The reaction probabilities exliibit peaks whenever the collision energy matches the energy of the resonances, which are here the quasi-bound states in the well (with their energy indicated). Note that the peaks become wider for the higher energy resonances—the high-energy resonance here is less bound and Teaks more toward the asymptote than do the low-energy ones.

B3.4.7.2 NUMERICALLY EXTRACTING BOUND STATES AND RESONANCE FUNCTIONS... 

An alternative is to use iterative methods. The simplest iterative teclniique for calculating bound state or resonances is to pick a random initial wavefimction vi/q(a ) and propagate it forward in time, producing a wavepacket ... 

Balint-Kurti G G, Dixon R N and Marston C C 1990 The Fourier grid Hamiltonian method for bound state eigenvalues and eigenfunctions J. Chem. See. Faraday Trans. 86 1741... 

Zhang D H and Zhang J Z H 1995 Quantum calculations of reaction probabilities for HO + CO and bound states of HOCO J. Chem. Phys. 103 6512... 

Neuhauser D 1990 Bound state eigenfunctions from wave packets—time -> energy resolution J. Chem. Phys. 932611... 

Beck M H and Meyer H D 1998 Extracting accurate bound-state spectra from approximate wave packet propagation using the filter-diagonalization method J. Chem. Phys. 109 3730... 

Gutzwiller M C 1967 Phase-integral approximation in momentum space and the bound states of an atom J Math. Phys. 8 1979... 

Miller S M, Clary D C, Kliesoh A and Werner H J 1994 Rotationally inelastio and bound-state dynamios of H2-OH... 

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). 

For bound state systems, eigenfunctions of the nuclear Hamiltonian can be found by diagonalization of the Hamiltonian matiix in Eq. (11). These functions are the possible nuclear states of the system, that is, the vibrational states. If these states are used as a basis set, the wave function after excitation is a superposition of these vibrational states, with expansion coefficients given by the Frank-Condon overlaps. In this picture, the dynamics in Figure 4 can be described by the time evolution of these expansion coefficients, a simple phase factor. The periodic motion in coordinate space is thus related to a discrete spectrum in energy space. 

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