Continuum states

As was emphasised above, continuum states are included in the description of the complete excitation channel. The part lying below threshold (bound states) is called closed, while the continuum states are the open 

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Rule A. The transition rate (probability per unit tune) for a transition from state O of a quanPim system to a number p( ) of continuum states 4) by an external perturbation V is... 

A partial acknowledgment of the influence of higher discrete and continuum states, not included within the wavefunction expansion, is to add, to the tmncated set of basis states, functions of the fomi T p(r)<6p(r) where dip is not an eigenfiinction of the internal Flamiltonian but is chosen so as to represent some appropriate average of bound and continuum states. These pseudostates can provide fiill polarization distortion to die target by incident electrons and allows flux to be transferred from the the open channels included in the tmncated set. 

Substitution of Eq. (18) into the partial wave expansion formula [Eq. (A.3)] gives for the continuum state... 

The frequency dependence is taken into accoimt through a mixed time-dependent method which introduces a dipole-moment factor (i.e. a polynomial of first degree in the electronic coordinates ) in a SCF-CI (Self Consistent Field with Configuration Interaction) method (3). The dipolar factor, ensuring the gauge invariance, partly simulates the molecular basis set effects and the influence of the continuum states. A part of these effects is explicitly taken into account in an extrapolation procedure which permits to circumvent the sequels of the truncation of the infinite sum-over- states. 

References 29-33 introduce the notion of coherence spectroscopy in the context of two-pathway excitation coherent control. Within the energy domain, two-pathway approach to coherent control [25, 34—36], a material system is simultaneously subjected to two laser fields of equal energy and controllable relative phase, to produce a degenerate continuum state in which the relative phase of the laser fields is imprinted. The probability of the continuum state to evolve into a given product, labeled S, is readily shown (vide infra) to vary sinusoidally with the relative phase of the two laser fields < ),... 

Considering again the case of a structureless continuum, we have that 8j3 arises from excitation of a superposition of continuum states, hence from coupling within PHmP [69]. The simplest model of this class of problems, depicted schematically in Fig. 5b, is that of dissociation of a diatomic molecule subject to two coupled electronic dissociative potential energy curves. Here the channel phase can be expressed as... 

M. Shapiro and P. Brumer, Quantum control of bound and continuum state dynamics, Phys. Rep. 425, 195 (2006). 

The validity of our analytical model for the multielectronic targets lies in the description of the bound and continuum states of the target atom. The analytical... 

Figs. 10 and 11 for helium. This feature is closely analogous to the one observed for ve = vp for electron capture to the continuum mechanism or to electron loss to the continuum which is often referred to as the cusp. The electrons with v, vr 0 in the low-lying continuum states of the target ion also form a sharp peak known as the target cusp. ... 

However, in contrast to XPS, electrons are promoted into bound or continuum states and still provide partial screening to the excited atom, so that these absorption energies are less sensitive to relaxation effects. Shifts in absorption energies are thus often more easily detected than those in XPS BEs. Moreover, XANES is a bulk-sensitive technique, and avoids the ambiguities associated with XPS in attributing trends to a bulk vs. surface phenomenon. 

XAS data comprises both absorption edge structure and extended x-ray absorption fine structure (EXAFS). The application of XAS to systems of chemical interest has been well reviewed (4 5). Briefly, the structure superimposed on the x-ray absorption edge results from the excitation of core-electrons into high-lying vacant orbitals (, ] ) and into continuum states (8 9). The shape and intensity of the edge structure can frequently be used to determine information about the symmetry of the absorbing site. For example, the ls+3d transition in first-row transition metals is dipole forbidden in a centrosymmetric environment. In a non-centrosymmetric environment the admixture of 3d and 4p orbitals can give intensity to this transition. This has been observed, for example, in a study of the iron-sulfur protein rubredoxin, where the iron is tetrahedrally coordinated to four sulfur atoms (6). 

Matsumoto, A. (1988), Generalized Matrix Elements in Discrete and Continuum States for the Morse Potential, J. Phys. B 21, 2863. 

Wulfman, C. E., and Levine, R. D. (1983), A Unified Algebraic Approach to Bound and Continuum States of Anharmonic Potentials, Chem. Phys. Lett. 97, 361. 

As noted earlier, for each particle i, there is a discrete spectrum of positive energy bound states and positive and negative energy continuum states. Let us consider a product wave function of the form = i//i(l)i//2(2), a normalizable stationary bound-state eigenfunction of... 

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