Asymptotic Hamiltonian

The total Hamiltonian of the collision system can be most generally written as the sum of three terms the kinetic energy of the relative motion, the interaction potential between the colliding particles, and the asymptotic Hamiltonian describing the colliding particles at infinite separation. We make the following approximations ... 

The interactions with electromagnetic fields are included in the asymptotic Hamiltonian and are assumed to be independent of the relative separation of the colliding particles. This is a good approximation because at short range the interaction potential between the colliding particles is always much stronger than the field-induced interactions. 

In order to solve the coupled equations (8.34), it is necessary to introduce the eigenstates of the asymptotic Hamiltonian,... 

Most generally, the asymptotic Hamiltonians of the separated particles can be written as [22]... 

This is an analog of propagators known in perturbation expansions in many-body theories such as the M0ller-Plesset theory. Hamiltonian H0 represents asymptotic Hamiltonian, e.g. without the interaction of the molecule with the scattering electron. 

Let us examine the processes described in the reactant channel (1) corresponding to the collision pair R1+R2. Elastic scattering effects are hidden in Hn. To see this, one uses with the asymptotic Hamiltonian (HR1+HR2) to represent the 1-state. The Hamiltonian H = Ku + Hm (T) can be written in terms of the asymptotic system Hm = HR1 + HR2 + WR1R2. The term WR1R2 is the intermolecular interaction operator measured from the local stationary frames whose origins are... 

This classification is useful for molecules which do not have stereoisomers. This point is important in chemistry. The asymptotic hamiltonians of normal molecules are invariant to parity. For stereoisomers, the molecule assumes under inversion a configuration in space which cannot be made to coincide with the original configuration by rotation. For these type of molecules, we will talk of a symmetry-broken molecular hamiltonian. These right and left hand modifications exist as real molecules that can interconvert into each other via transition structures having appropriate symmetry. From the present standpoint, there exists different electronic wave functions for the R- and L-molecules. Thus, each subset cannot be used to expand wave functions of the other. 

The reference scattering state can be chosen as a distorted wave (with any level of distortion) or as a free wave, as long as it is regular at the origin and is an eigenstate of the asymptotic Hamiltonian, Ho — lirnH oo H Groenenboom et al... 

A convenience of electronic basis functions (53) is that they reduce at infinitesimal-amplitude bending to (28) with the same meaning of the angle 9 we may employ these asymptotic forms in the computation of the matrix elements of the kinetic energy operator and in this way avoid the necessity of carrying out calculations of the derivatives of the electronic wave functions with respect to the nuclear coordinates. The electronic part of the Hamiltonian is represented in the basis (53) by... 

Asymptotic Condition.—In Section 11.1, we exhibited the equivalence of the formulation of quantum electrodynamics in the Coulomb and Lorentz gauges in so far as observable quantities were concerned (t.e., scattering amplitudes). We also noted that both of these formulations, when based on a hamiltonian not containing mass renormalization counter terms, suffered from the difficulty that the... 

In the present section we shall make this difficulfy apparent in a somewhat different way by showing that it is not possible to satisfy the asymptotic condition when the theory is formulated in terms of an unsubtracted hamiltonian of the form jltAll(x) — JS0JV. We shall work in the Lorentz gauge, where the relativistic invariance of the theory is more obvious. 

In the present implementation, the unperturbed functions are not subject to any orthogonality constraint nor are required to diagonalize any model hamiltonian. This freedom yields a faster convergence of the variational expansion with the basis size and allows to obtain the phaseshift of the basis states without the analysis of their asymptotic behaviour. 

It should be stressed that in this first group of works (1956-1970) there is nowhere any deviation from the Hamiltonian laws of dynamics. This feature was taken as a preliminary postulate of our work. Irreversibility appeared as an asymptotic property of the evolution of certain classes of systems. The term asymptotic refers to the large size of the system, as well as to the long time scale of observation. 

Fig. 2. Global bifurcation for equations (21)—(23). (a) Hamiltonian reference system, (b) Stable separ-atrix loop arising from the effect of the dissipative" perturbation I2. (c) Asymptotically stable...

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... 

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