Resonance compound state

G.C. Schatz, A. Kuppermann, Role of direct and resonant (compound state) processes and of the interferences in the quantum dynamics of the collinear H+H2 exchange reaction,... 

Figure A3.12.8. Possible absorption spectrum for a molecule which dissociates via isolated compound-state resonances. Eq is the unimolecular threshold. (Adapted from [4].)...

The ability to assign a group of resonance states, as required for mode-specific decomposition, implies that the complete Hamiltonian for these states is well approxmiated by a zero-order Hamiltonian with eigenfunctions [M]. The ( ). are product fiinctions of a zero-order orthogonal basis for the reactant molecule and the quantity m. represents the quantum numbers defining ( ).. The wavefimctions / for the compound state resonances are given by... 

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... 

When the crystallography of compounds related by polymorphism is such that nuclei in the two structures are magnetically nonequivalent, it will follow that the resonances of these nuclei will not be equivalent. Since it is normally not difficult to assign organic functional groups to observed resonances, solid state NMR spectra can be used to deduce the nature of polymorphic variations, especially when the polymorphism is conformational in nature. Such information is extremely valuable at the early states of drug development when solved single crystal structures for each polymorph or solvate species may not yet be available. 

A relaxation process will occur when a compound state of the system with large amplitude of a sparse subsystem component evolves so that the continuum component grows with time. We then say that the dynamic component of this state s wave function decays with time. Familiar examples of such relaxation processes are the a decay of nuclei, the radiative decay of atoms, atomic and molecular autoionization processes, and molecular predissociation. In all these cases a compound state of the physical system decays into a true continuum or into a quasicontinuum, the choice of the description of the dissipative subsystem depending solely on what boundary conditions are applied at large distances from the atom or molecule. The general theory of quantum mechanics leads to the conclusion that there is a set of features common to all compound states of a wide class of systems. For example, the shapes of many resonances are nearly the same, and the rates of decay of many different kinds of metastable states are of the same functional form. 

The initial and final asymptotic states are always expanded in the time independent basis associated with the molecular hamiltonian the scattering matrix is unitary. Note again that the basis contains all possible resonance and compound states. If there is no interaction, the scattering matrix is the unit matrix 1. Formally, one can write this matrix as S= 1+iT where T is an operator describing the non-zero scattering events including chemical reactions. Thus, for a system prepared in the initial state Op, the probability amplitude to get the system in the... 

The resonance illustrated in fig. 4.5 has a width 0.008 91 eV, corresponding to a lifetime of 7.39 x 10 s. This is much longer than the direct lifetime, justifying the concept of a resonance as a compound state with a characteristic lifetime. The physical constants that enable these calculations to be done easily are... 

A discussion of Van der Waals molecules is a natural component of a treatise on resonance phenomena, for a variety of reasons. The most obvious of these is simply the fact that transitions involving both compound-state and shape resonance levels figure prominently in the spectra of these species. Indeed, in many cases the metastable nature of the final states of such transitions is a key feature of the manner in which they are observed (1-3). Moreover, observations of structure due to resonances in scattering cross sections can provide detailed information regarding intermolecular potential energy functions ( ). 

Such doubly excited states are familiar in nuclear physics as compound state, or Feshbach, resonances, and In atomic spectroscopy as autolonizlng, or Auger, states of atoms and molecules (33). Just as In the atomic case (33.34), sequences of states with quantum numbers of the form (n,n-t-m) do not necessarily have shorter lifetimes as a function of Increasing m, in contradistinction to statistical expectations. This follows from the Increase In period of the local bond modes as dissociation Is neared, and the detuning of any near frequency resonances as m increases in a sequence of (n,n+m) states. 

All resonances are necessarily characterized by the llfetlsie of the compound state, and Smith s (39) definition of the lifetime matrix... 

The same authors investigated resonances in reactive collisions (1971). For the particular case of curvature K = 0, they found one at a total energy = 12-3 kcal/mole with time-delay t = 3 x 10 14 sec and another at = 28-7 kcal/mole. Observing that (s) + (n + j)h(o(s) shows a well capable of sustaining bound states of positive energy only for n > 0, one concludes that the first resonance is analogous to a shape resonance, while the second corresponds to a compound-state resonance. A shape resonance is shown in Fig. 5. 

The phase-space model has been extended by Miller (1970) in order to incorporate the effect of closed channels. He made use of a parametrized form of the S matrix previously developed for compound-state resonances in atom-molecule collisions (Micha, 1967). Indicating with Sd the S-matrix for direct scattering, i.e. in the absence of coupling to closed channels, one can write (omitting the index J). 

A study of long-lived states in atom-molecule collisions (Micha, 1973) has made use of theoretical methods developed for Van der Waals complexes, to discuss a number of reactive atom-molecule pairs where formation of long-lived states appears established. It employs the statistical approximation to the S-matrix for resonance processes, and points out the importance of both compound-state and shape resonances in reactions. It suggests that probabilities of departure from the collision region could be determined from... 

Compound-state resonances are important in quantal theories of unimolecular decomposition. They are prepared in low-energy atom (molecule)-molecule collisions when part of the relative kinetic energy of the motion becomes temporarily converted into excitation of the internal (rotational and/or vibrational) degrees of freedom of either partner. When this excitation occurs, the molecular system has insufficient energy in its relative motion to separate. One can also prepare compound-state resonances by using electromagnetic radiation (e.g., a laser) to excite the molecule. Thus, it is proper to view these resonances as the natural extension of the bound vibration-al/rotational eigenstates into the dissociative continuum. 

Different theoretical methods have been used to calculate the complex energies, Eq. (8.1), for compound-state resonances. They can be divided into time-independent and time-dependent methods. A standard quantum mechanical time-independent method is a close-coupling calculation (Stechel et al., 1978) which considers resonant state formation as a result of a collision such as A + BC —> ABC AB + C. Determined... 

Kosloff, 1994) have also been used to find the complex energies for compound-state resonances. A localized wave packet [i.e., a coherent superposition state, Eq. (4.7)], P(O) is initially placed in the bound region of the potential energy surface and propagated in time to give ( l (0) (r)), which is C t) in Eq. (4.16). If I (O) is a superposition of resonant states, it can be considered a zero-order state (see chapter 4) and can be written as... 

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