Resonance resonant wavefunctions

The time-dependent resonance wavefunction ijftes(x,t) decays in time because of the negative imaginary part of the complex energy. 

Although most resonances in the F+HCl system are clearly of the reagent type or product type, there are some mixed" cases. Because the resonance manifold is dense, degeneracies can occur between zero-order reagent-type or product-type states. Thus, the resonance wavefunctions for these states are linear combinations of entrance and exit channel expressions. 

For transparency, we now use exterior complex scaling [Eq. (9)]. With Z = 2 the lowest resonance of -symmetry is known to be —0.78 a.u., see, for example, Ref. [15]. The bound part is dominated by 2s2 and the outgoing component is lses. Projection of the resonance wavefunction onto Is should then, for large enough r, gives an outgoing Coulomb wave... 

The final rotational state distributions qualitatively reflect the shape of the resonance wavefunctions along the transition line, i.e., along a line which is roughly perpendicular to the minimum energy path. 

Resonance phenomena have been shown to play a significant role in many electron collision and photoionization problems. The long lived character of these quasi-stationary states enables them to influence other dynamic processes such as vibrational excitation, dissociative attachment and dissociative recombination. We have shown it is possible to develop ab initio techniques to calculate the resonant wavefunctions, cross sections and dipole matrix elements required to characterize these processes. Our approach, which is firmly rooted in the R-matrix concept, reduces the scattering problem to a matrix problem. By suitable inversion or diagonalization we extract the required resonance parameters. 

Resonant wavefunctions - asymptotically expanding Non-Integrable eigensolutions. 

In this sense, K is more fundamental than R. For example, if b is chosen as above to make A = 0, then all E coincide with the resonance energies in S. It follows that the xa equal the resonance wavefunctions, at least for r < o. Now consider the dependence of the theory on a, which can normally be omitted since ka is usually small (see, e.g., [370]) and imagine o to be unphysically large. The dependence of R on a is then simple and explicit in this case, instead of (8.12),... 

The classical counterpart of resonances is periodic orbits [91, 95, 96, 97 and 98]. For example, a purely classical study of the H+H2 collinear potential surface reveals that near the transition state for the H+H2 H2+H reaction there are several trajectories (in R and r) that are periodic. These trajectories are not stable but they nevertheless affect strongly the quantum dynamics. A study of the resonances in H+H2 scattering as well as many other triatomic systems (see, e.g., [99]) reveals that the scattering peaks are closely related to the frequencies of the periodic orbits and the resonance wavefunctions are large in the regions of space where the periodic orbits reside. 

Figure B3.4.12. A schematic ID vibrational pre-dissociation potential curve (wide full line) with a superimposed plot of the two bound functions and the resonance function. Note that the resonance wavefunction is associated with a complex wavevector and is slowly increasing at very large values o R. In practice this increase is avoided by using absorbing potentials, complex scaling, or stabilization.

Figure 2.15 The resonance wavefunctions and their corresponding self-overlaps at two different intensities / = 0.0510 W/cm (left panels) and / = 0.394910 W/cm (right panels). Only the real part of each channel wavefunction is displayed.

At the core of the analysis and methods that are discussed in this Chapter is the consistent consideration of the fact that the form of each resonance wavefunction is = fl I o+Xas (Eq. (4.1) of text), if necessary, the extension to multi-dimensional forms is obvious. Depending on the formalism, the coefficient a and the asymptotic part, Xas, are functions of either the energy (real or complex) or the time. The many-body square-integrable, %, represents the localized part of the decaying (unstable) state, i.e., the unstable wavepacket which is assumed to be prepared at f = 0. its energy, Eo, is real and embedded inside the continuous spectrum, it is a minimum of the average value of the corresponding state-specific effective Hamiltonian that keeps all particles bound. 

As far as fhe present discussion on understanding and computing wave-functions and properties of unsfable sfafes of polyelecfronic sysfems is concerned, the crucial point is how to define and compufe and, subsequently, how to determine its correct superposition with the function space that represents the open channels, thereby leading to the formation of resonance in the vicinity of Eq. The practical success of fhis quesf is dependenf on the choice of forms of trial resonance wavefunctions and of corresponding matrices that appear in the formalism. 

The matrix M(z) consists of two parts, in correspondence with the form (1). The solution of Eq. (12) or the diagonalization of M(z) implies the accurate construction and handling of the complex self-energy matrix A(z), which is the same as the complex quantity in the resonance formula in Feshbach s theory [2a, p. 367,2b, p. 304]. For a many-electron system, such a goal is very difficult to achieve rigorously. Therefore, one has to search for a practical computational method which produces the resonance wavefunction and the corresponding complex energy. 

It was in response to criticism of the method proposed in Ref. [92a] (and implemented in Ref. [93]) as regards the practical computation of matrix elements with resonance wavefunctions that the integration technique later called "ECS" was suggested—see Section 4.3 below. Its conceptual background is the form (1). 

This idea is at the root of the ECS technique for the regularization of matrix elements between resonance wavefunctions, which was written explicitly in Ref. [92b] (see Eq. (20) below) and was used in Ref. [95] for various potentials. The name ECS was later given by Simon [96], who proposed the same scheme for handling difficulties with the calculation of molecular resonances in the B-O approximation. 

This type of mode selectivity may be rationalized by the structure of resonance wavefunctions in a rather similar way as pointed out for the Henon-Heiles model [34]. On the experimental side it is a challange to discover this type of hyperspherical versus local mode specifity via spectroscopic investigation of line widths in highly excited ABA" molecules (possible candidates are dihydrides like H O [10,50,31,33] or other systems like SO2 [36]). 

The covalent-ionic resonance wavefunction for the A B electron-pair bond can be expressed according to Eqs. (27) and (28). 

Fig.l Radial part of resonance wavefunction clearly showing bound state charatcter in the strong interaction region and oscillatory, continuum character in the non-interacting region. 

It is also possible using the approach to obtain the distribution of the internal states of the products of the unimolecular dissociation The rigorous, asymptotic (R goes to infinity) form of the resonance wavefunction for a triatomic molecule at the energy Ej, is given by... 

In words, equation (Al.6.89) is saying that the second-order wavefunction is obtained by propagating the initial wavefunction on the ground-state surface until time t", at which time it is excited up to the excited state, upon which it evolves until it is returned to the ground state at time t, where it propagates until time t. NRT stands for non-resonant tenn it is obtained by and cOj -f-> -cOg, and its physical interpretation is... 

If all the resonance states which fomi a microcanonical ensemble have random i, and are thus intrinsically unassignable, a situation arises which is caWtA. statistical state-specific behaviour [95]. Since the wavefunction coefficients of the i / are Gaussian random variables when projected onto (]). basis fiinctions for any zero-order representation [96], the distribution of the state-specific rate constants will be as statistical as possible. If these within the energy interval E E+ AE fomi a conthuious distribution, Levine [97] has argued that the probability of a particular k is given by the Porter-Thomas [98] distribution... 

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