Bound states scattering theory

The fixed nuclear approximation is extremely useful, not only for the bound states, but also for the treatment of electronic scattering by molecules. For instance, stationary-state scattering theories within the fixed nuclear approximation have been extensively developed for molecular photoionization [18, 250, 336, 516] and electron scattering from polyatomic molecules [210, 418, 419]. These scattering phenomena are quite important in that they are widely foimd in nature as elementary processes and even in industrial applications using plasma processes. These scatterings may be referred to as stationary-state electron dynamics in fixed nuclei approximation. 

The low-temperature chemistry evolved from the macroscopic description of a variety of chemical conversions in the condensed phase to microscopic models, merging with the general trend of present-day rate theory to include quantum effects and to work out a consistent quantal description of chemical reactions. Even though for unbound reactant and product states, i.e., for a gas-phase situation, the use of scattering theory allows one to introduce a formally exact concept of the rate constant as expressed via the flux-flux or related correlation functions, the applicability of this formulation to bound potential energy surfaces still remains an open question. 

Within our exploratory calculation we will use a simplified description of the contribution of correlated states, considering only the bound state with an effective shift, which reproduces the correlated density. This shift is taken as a quadratic function in the densities, where the linear term is calculated from perturbation theory and the quadratic term is fitted to reproduce the results for the composition as found by the full microscopic calculation including the contribution of scattering states. 

The first part of the review deals with aspects of photodissociation theory and the second, with reactive scattering theory. Three appendix sections are devoted to important technical details of photodissociation theory, namely, the detailed form of the parity-adapted body-fixed scattering wavefunction needed to analyze the asymptotic wavefunction in photodissociation theory, the definition of the initial wavepacket in photodissociation theory and its relationship to the initial bound-state wavepacket, and finally the theory of differential state-specific photo-fragmentation cross sections. Many of the details developed in these appendix sections are also relevant to the theory of reactive scattering. 

The two-particle Boltzmann collision term if and the three-particle contribution for k = 0 were considered in Section II. It was possible to express those collision integrals in terms of the two- and three-particle scattering matrices. It is also possible to introduce the T matrix in if for the channels k = 1, 2,3, that is, in those cases where three are asymptotically bound states. Here we use the multichannel scattering theory, as outlined in Refs. 9 and 26. 

In Section 2.1 we derived the expression for the transition rate kfi (2.22) by expanding the time-dependent wavefunction P(t) in terms of orthogonal and complete stationary wavefunctions Fa [see Equation (2.9)]. For bound-free transitions we proceed in the same way with the exception that the expansion functions for the nuclear part of the total wavefunction are continuum rather than bound-state wavefunctions. The definition and construction of the continuum basis belongs to the field of scattering theory (Wu and Ohmura 1962 Taylor 1972). In the following we present a short summary specialized to the linear triatomic molecule. 

Resonances are common and unique features of elastic and inelastic collisions, photodissociation, unimolecular decay, autoionization problems, and related topics. Their general behavior and formal description are rather universal and identical for nuclear, electronic, atomic, or molecular scattering. Truhlar (1984) contains many examples of resonances in various fields of atomic and molecular physics. Resonances are particularly interesting if more than one degree of freedom is involved they reflect the quasi-bound states of the Hamiltonian and reveal a great deal of information about the multi-dimensional PES, the internal energy transfer, and the decay mechanism. A quantitative analysis based on time-dependent perturbation theory follows in the next section. 

This part introduces variational principles relevant to the quantum mechanics of bound stationary states. Chapter 4 covers well-known variational theory that underlies modern computational methodology for electronic states of atoms and molecules. Extension to condensed matter is deferred until Part III, since continuum theory is part of the formal basis of the multiple scattering theory that has been developed for applications in this subfield. Chapter 5 develops the variational theory that underlies independent-electron models, now widely used to transcend the practical limitations of direct variational methods for large systems. This is extended in Chapter 6 to time-dependent variational theory in the context of independent-electron models, including linear-response theory and its relationship to excitation energies. 

In bound-state calculations, the Rayleigh-Ritz or Schrodinger variational principle provides both an upper bound to an exact energy and a stationary property that determines free parameters in the wave function. In scattering theory, the energy is specified in advance. Variational principles are used to determine the wave function but do not generally provide variational bounds. A variational functional is made stationary by choice of variational parameters, but the sign of the residual error is not determined. Because there is no well-defined bounded quantity, there is no simple absolute standard of comparison between different variational trial functions. The present discussion will develop a stationary estimate of the multichannel A -matrix. Because this matrix is real and symmetric for open channels, it provides the most... 

In the case of a scattering resonance, bound-free correlation is modified by a transient bound state of fV+1 electrons. In a finite matrix representation, the projected (fV+l)-electron Hamiltonian H has positive energy eigenvalues, which define possible scattering resonances if they interact sufficiently weakly with the scattering continuum. In resonance theory [270], this transient discrete state is multiplied by an energy-dependent coefficient whose magnitude is determined by that of the channel orbital in the resonant channel. Thus the normalization of the channel orbital establishes the absolute amplitude of the transient discrete state, and arbitrary normalization of the channel orbital cannot lead to an inconsistency. 

To understand how resonances affects the molecular phase, we briefly outline the basics of the theory of scattering resonances. We consider bound states <))s) interacting with a set of continuum states denoted E, n 1), where, as for the full scattering states , n-) [see Eq. (2.66)], the states E, n 1) approach the fine asymptotic solutions at infinite time ... 

In our applications we are mainly interested in the bound states of simple quantum systems. The choice J2, J3 is necessary for a proper description of such states whereas the other choices would be more suitable for the description of the continuum states arising from scattering theory (Wybourne, 1974). Thus, we shall choose a representation space of states kq) on which J2 and J3 are simultaneously diagonal ... 

Physical chemistry of the positron and Ps is unique in itself, since the positron possesses its own quantum mechanics, thermodynamics and kinetics. The positron can be treated by the quantum theory of the electron with two important modifications the sign of the Coulomb force and absence of the Pauli exclusion principle with electrons in many electron systems. The positron can form a bound state or scatter when it interacts with electrons or with molecules. The positron wave function can be calculated more accurately than the electron wave function by taking advantage of simplified, no-exchange interaction with electrons. However, positron wave functions in molecular and atomic systems have not been documented in the literature as electrons have. Most researchers perform calculations at certain levels of approximation for specific purposes. Once the positron wave function is calculated, experimental annihilation parameters can be obtained by incorporating the known electron wave functions. This will be discussed in Chapter 2. 

Methods for calculating collisions of an electron with an atom consist in expressing the many-electron amplitudes in terms of the states of a single electron in a fixed potential. In this chapter we summarise the solutions of the problem of an electron in different local, central potentials. We are interested in bound states and in unbound or scattering states. The one-electron scattering problem will serve as a model for formal scattering theory and for some of the methods used in many-body scattering problems. 

Scattering theory concerns a collision of two bodies, that may change the state of one or both of the bodies. In our application one body (the projectile) is an electron, whose internal state is specified by its spin-projection quantum number v. The other body (the target) is an atom or an atomic ion, whose internal bound state is specified by the principal quantum number n and quantum numbers j, m and / for the total angular momentum, its projection and the parity respectively. We... 

Heller [1. 2. 3] introduced and popularised wavepacket dynamics in the context of the theory of nioleculm iihotodissociation. In a photodissociation process, the molecule starts in a. well defined initial state and ends up in a firiiil scattering state. The initial bound state vibrational-rotatioiuil wavefunction provides a natural initial wavepacket in this case. 

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