Continuum state wavefunction

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. 

This is referred as BO ansatz. This ansatz is taken as a variational trial function. Terms beyond the leading order in m/M are neglected m is the electronic and M is nuclear mass, respectively). The problem with expansion (4) is that functions /(r, R) contain except bound states also continuum function since it includes the centre of mass (COM) motion. Variation principle does not apply to continuum states. To avoid this problem we can separate COM motion. The remaining Hamiltonian for the relative motion of nuclei and electrons has then bound state solution. But there is a problem, because this separation mixes electronic with nuclear coordinates and also there is a question how to define molecule-fixed coordinate system. This is in detail discussed by Sutcliffe [5]. In the recent paper by Kutzelnigg [8] this problem is also discussed and it is shown how to derive adiabatic corrections using, as he called it, the Bom-Handy ansatz. There are few important steps to arrive at formula for a diabatic corrections. Firstly, one separates off COM motion. Secondly, (very important step) one does not specify the relative coordinates (which are to some extent arbitrary). In this way one arrives at relative Hamiltonian Hrd [8] with trial wavefunction If we make BO ansatz... 

Since Vd(r) is only nonzero near r = 0 the matrix element of Eq. (6.51) reflects the amplitude of the wavefunction of the continuum wave at r 0. Specifically, the squared matrix element is proportional to C, the density of states defined earlier and plotted in Fig. 6.18. From the plots of Fig. 6.18 it is apparent that the ionization rate into a continuum substantially above threshold is energy independent. However, as shown in Fig. 6.18, there is often a peak in the density of continuum states just at the threshold for ionization, substantially increasing the ionization rate for a degenerate blue state of larger This phenomenon has been observed experimentally by Littman et al.32 who observed a local increase in the ionization rate of the Na (12,6,3,2) Stark state where it crosses the 14,0,11,2 state, at a field of 15.6 kV/cm, as shown by Fig. 6.19. In this field the energy of the... 

Direct calculation from equation (29) requires a knowledge of all excited-state wavefunctions. If these are known, equation (29) becomes a sum over all discrete states and an integration over continuum states. For systems of more than two electrons, excited-state wavefunctions are difficult to come by Even less is known about the continuum states, but for the diamagnetic susceptibility their contribution is thought to be of the same order of magnitude as that of the discrete states. For other systems the direct use of equation (29) is clearly a non-starter. 

In the field of photoionization, the Fano formula for the cross section has often been used for resonance fitting. Note, however, that the same resonance can sometimes stand out sharply from the background, but can also fail to manifest themselves clearly in the photoionization cross section, depending upon the initial bound state of the dipole transition [51]. Thus, the cross-section inspection might miss some resonances. The asymptotic quantities of the final continuum-state wavefunction, if available, should be much more convenient in general for the purpose of resonance search and analysis. 

In fig. 3. 1, wavefunctions for bound states and for the continuum state lying near threshold (e —> 0) are shown. In order to compare them, one has to normalise the continuum functions, which are not square-integrable. This difficulty is resolved by requiring... 

One then gets the correct expression for the wavefunction at distances r Re. When a Re, Equation 10.8 correctly describes the wavefunction of weakly bound and continuum states even at distances much smaller than a. 

The final state of the system always corresponds to a dissociative or continuum state. Let us suppose that the complex breaks up to yield fragments in final internal states denoted by "f". We may then write the continuum state wavefunction in the form... 

This is efficiently performed by block inverse iteration [4]. The continuum states so obtained are then normalized, fitting the asymptotic part to the regular and irregular Coulomb wavefunctions. 

Although the abovementioned theories were formulated in the context of either photoionization of atoms or electron scattering with atoms, they lend themselves equally well to photodetachment of negative ions, where an essential question is again to construct the wavefunction for the total continuum state in the presence of transient states embedded in continua. The main results from Mies s work [54] for calculating cross sections are summarized below. 

The wavefunctions for the initial bound state the resonant states , and the neutral thresholds are generated in separate valence RCI calculations. The valence RCI calculation for the resonant state yields energy positions of unperturbed resonant states, i.e., the E s in equation 1.3. Since virtual orbitals are used for the correlation configurations that capture not only the bound orbitals but also a portion of the continuum orbitals, it s important to avoid in cj) the correlation configurations that are equivalent to the continuum state. For example, in Ce [5], 4/ 5d 6s vp and 4/ 5d 6s vf were excluded from the basis set for resonant state 4/ 5d 6p. Otherwise, the variational optimization for 4/ 5d 6s 6p may collapse into the continuum 4/ 5d 6s ep (f) in which it lies. 

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