Wave functions Continuum

The inclusion of the iodine rotations resulted in an effective centrifugal barrier in the potential that pushed the continuum wave function out of the bound-state region, thereby reducing the overlap integral and decreasing the VP rate. 

In addition to bound-state wave functions and energies, there are continuum wave functions and energies all positive energies are allowed. 

Figure 15. Three-dimensional plots of a regular (a) and an irregular (b) continuum wave function in the dissociation of HNO. The energies are in the same range.

The kinematics of the situation for the case of optical limit type (e,2e) experiments are illustrated in Fig. 2b (Fig. 1 b of Hamnett et al.23), which shows the direction of the ejected electron j as a function of the two polar angles x and y. The angle between j and the vector K is denoted by i//. Provided the forward scattering kinematics are such that K is small and we may approximate /(K) by /0m(0), then, as is well known, regardless of the detailed form of the continuum wave function, provided that k I is orthogonal to Pq 35... 

The other source of a channel phase is the complex continuum wave function at the final energy E. At first it would appear from Eq. (15) that the phase of ESk) should cancel in the cross term. This conclusion is valid if the product continuum is not coupled either to some another continuum (i.e., if it is elastic) or to a resonance at energy E. If the continuum is coupled to some other continuum (i.e., if it is inelastic), the product scattering wave function can be expanded as a linear combination of continuum functions,... 

Fliflet, A.W. and McKoy, V. (1978). Discrete basis set method for electron-molecule continuum wave functions, Phys. Rev. A 18,2107-2114. 

Nestmann, B.M. and Peyerimhoff, S.D. (1990). Optimized Gaussian basis sets for representation of continuum wave functions, J. Phys. B 22, L773-L777. 

Continuum wave functions are spatially extended and are not normalizable in the usual spatial sense. Instead an energy normalization is chosen.186... 

Figure 23 Predissociation of the v = 2 vibrational level of the bound electronic state by a vibrational continuum wave function of the dissociative electronic state after radiative excitation (arrows) from the electronic ground state Po- The circle around the potential curve crossing point indicates an area of large overlap between the vibrational wave functions.

In practical applications, the continuum is often approximated by a discrete spectrum. To this end, one conveniently introduces a potential wall at long internuclear separations and solves for the artifically bound states.171,172 Alternatively, basis set expansion techniques can be employed.195,196 In either case, the density of states depends on external conditions, that is, the size of the box or the number of basis functions. This dependence on external conditions has to be accounted for by the energy normalization. Instead of employing a single continuum wave function with proper energy E in Eq. [240], one samples over the discrete levels with energy E -... 

Because the bound and continuum wave functions usually belong to different electronic manifolds, they are orthogonal to one another, and the only term that contributes to 4n(E i) derives fromVP/, the excited part of the wave packet. It follows from the boundary conditions on E, n ) [Eq. (2.57)], that the t —> oo limit of Eq. -(2.73) can be written as ... 

Eq. (8.31) to fonn Sx3 and E, n, a ) to form Sa2. The resultant Ss3Sa2 has the form of Eq. (8.32) with tif1> and associated with the symmetric and antisymmetric continuum wave functions, respectively. Consider now the effect of summing over m. Standard formulas [273, 281] imply that this summation introduces a Si 0, which, in turn, forces k = X via the first and second 3/ symbol in Eq. (8.32). However, it is possible to show that /-matrix elements associated with symmetric continuum eigenfunctions and those associated with antisymmetric continuum eigenfunctions must have k of different parities. Hence summing over m eliminates all contributions to Eq. (8.13) that involve both E n, a ) and E, n, s ). Thus, we find after m summation ... 

Figure 3-1. Schematic representation of atom in dielectric continuum. wave function in vacuo (solid line) and solvated (dashed line) e dielectric constant a cavity radius g(r) radial distribution of solvent molecules...

For the structure at higher energies, the agreement becomes better when the basis set is extended. The molecular orbital approximation has been proved to be efficient even for generation of continuum wave function of free-electron state when the extension of the basis set is sufficient. 

Electron-jump in reactions of alkali atoms is another example of non-adiabatic transitions. Several aspects of this mechanism have been explored in connection with experimental measurements (Herschbach, 1966 Kinsey, 1971). The role of vibrational motion in the electron-jump model has been investigated (Kendall and Grice, 1972) for alkali-dihalide reactions. It was assumed that the transition is sudden, and that reaction probabilities are proportional to the overlap (Franck-Condon) integral between vibrational wavefunctions of the dihalide X2 and vibrational or continuum wave-functions of the negative ion X2. Related calculations have been carried out by Grice and Herschbach (1973). Further developments on the electron-jump mechanism may be expected from analytical extensions of the Landau-Zener-Stueckelberg formula (Nikitin and Ovchinnikova, 1972 Delos and Thorson, 1972), and from computational studies with realistic atom-atom potentials (Evans and Lane, 1973 Redmon and Micha, 1974). 

The present method to obtain the Rydberg basis functions is based on the universal Gaussian basis sets devised by Kaufmann et al. for representing Rydberg and continuum wave functions [45], and the contraction coefficients are obtained in the following way A CASSCF wavefunction for the cation lowest in energy is determined with the uncontracted, extra basis set placed at the center of charge. From the... 

Let us therefore consider the relationship between the outgoing Green s function and the continuum wave function of the problem. The continuum wave function satisfies the Schrbdinger equation... 

Since at x = x = 0, the expansion of the outgoing Green s function is divergent, it is not possible to obtain a purely discrete expansion of r k). As discussed in Ref. [18] for the half-line, the expansion for the reflection amplitude requires at least of two subtraction terms and will not be pursued here. Substitution of Eq. (80) into Eqs. (93) and (94) leads, respectively, to resonance expansions for the continuum wave function along the internal region and the transmission amplitude, namely. 

Expansions in terms of continuum wave functions As is well known, the time evolution of the decaying wave solution given by Eq. (47) may also be calculated by expanding the retarded Green function in terms of the complete set of continuum wave functions of the problem, the so-called physical wave solutions r) to obtain... 

G. Garcia-Galderon, An expansion of continuum wave functions in terms of resonant states, Nucl. Phys. A 261 (1976) 130. 

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