Scattering boundary conditions

Until very recently, however, the same could not be said for reactive systems, which we define to be systems in which the nuclear wave function satisfies scattering boundary conditions. It was understood that, as in a bound system, the nuclear wave function of a reactive system must encircle the Cl if nontrivial GP effects are to appear in any observables [6]. Mead showed how to predict such effects in the special case that the encirclement is produced by the requirements of particle-exchange symmetry [14]. However, little was known about the effect of the GP when the encirclement is produced by reaction paths that loop around the CL... 

This chapter has focused on reactive systems, in which the nuclear wave function satisfies scattering boundary conditions, applied at the asymptotic limits of reagent and product channels. It turns out that these boundary conditions are what make it possible to unwind the nuclear wave function from around the Cl, and that it is impossible to unwind a bound-state wave function. 

This coupling potential is smooth everywhere, which allows numerical calculations with high precision. There is no nonadiabatic coupling since the basis functions [0< )( 2C) are independent of p in each sector. The solution I Wf/o, 2C) is connected smoothly, in principle, from sector to sector by a unitary frame transformation from the /th set of channels to the (/ + l)st set [97-99]. The coordinate system is transformed from the hyperspherical to the Jacobi coordinates at some large p, beyond which the conventional close-coupling equations are employed for determining the asymptotic form of the wavefunction appropriate for the scattering boundary condition [100]. 

This equation formally includes the scattering boundary conditions. 

The use of a separable potential of the form of Equation 8 in Equation 6 to obtain solutions of the form of Equation 9 can be shown to be equivalent to using the functions ct (r) in the Schwinger variational principle for collisions (13). At this stage the functions ai(r) can be chosen to be entirely discrete basis functions such as Cartesian Gaussian (lA) or spherical Gaussian (15) functions. We note that with discrete basis functions alone the approximate solution satisfies the scattering boundary condition. Such basis... 

Indicating with ,(u, v, 0) the internal vibration-rotation states and with Yfo) the orbital states, scattering boundary conditions for direct and reactive collisions are, respectively... 

This implies that Vv(r, t) can be used to describe an exponentially decaying probability distribution. The time-independent partial wave solutions above a threshold obey scattering boundary conditions. They are thus proportional to the regular solution tpe(E, r) a origin and tend to infinity like Jost solutions f E,r) ... 

One way to determine the rate constant rigorously is to solve the complete state-to-state reactive scattering Schrodinger equation (with approximate scattering boundary conditions) to obtain the -matrix Snfhnr(E, 7) as a function of total energy E and total angular momentum J (where nr (np) label the reactant (product) quantum states), from... 

V (z) describes a decreasing in time quasi-stationary state. Contrary to the Lippmann-Schwinger equation, which requires scattering boundary conditions, V (z) does require outgoing boundary conditions commensurate with the Gammow-Siegert method. It is inherent in the complex technique and defined in a nonambiguous manner as a continued wavefunction in the second Riemann sheet. 

If the limit is negative, then the channel is closed and the nuclei remain bounded to each other. The form of the solutions of the coupled equations, Eqs. (39 or 40), depends on the imposed boundary conditions. Consider first the case where there is at least one open channel. Scattering boundary conditions consist of imposing, for an open channel n, a combination of incoming (exp [ - ik R]) and outgoing (exp [ - - iknR]) waves, while all closed channel functions are constrained to vanish asymptotically. In the case... 

In the next section (Sec. 2), we will develop the theory of the BCRLM. We discuss the solution of the coupled-channel equations in both natural collision coordinates " and hyperspherical coordinates. " Both coordinate systems are widely used to treat collinear reactive scattering processes. We will discuss the projection " of the hyperspherical equations on coordinate surfaces appropriate for applying scattering boundary conditions and review the definition of integral and differential scattering cross sections in this model. 

The radial strength functions [fg(r) are in general complex. However, it is almost always possible to transform the problem to work with real functions and to express the complex solutions, when needed, in terms of these. For both open and closed channels, the boundary conditions require that l (r) 0 at short range. However, at long range the scattering boundary conditions corresponding to real solutions are... 

Scattering boundary conditions are applied to the solution of the coupled-... 

We can numerically solve for Xk > th continuum state at an energy that corresponds to the eigenvalue of a pseudostate. Recall toat Xk satisfies scattering boundary conditions, and Xn zero for large r, square-integrable and normalised. If however we project Xk O " sis subspace, P)(P, we find for a good choice of basis that... 

The quantum dynamics of these phenomena is described by the time-independent Schrodinger equation with scattering boundary conditions, or alternatively by the operator equations of scattering theory. The Hamiltonian for the system is... 

The variational approach received a major boost also when it was realised [79] that the simplest variational method - the Kohn variational principle, which is essentially the Rayleigh-Ritz variational principle for eigenvalues modified to incorporate scattering boundary conditions - is free of anomalous (i.e., spurious, unphysical) singularities if it is formulated with S-matrix type boundary conditions rather than standing wave boundary conditions as had been typically used previously. It is useful first to state the Kohn variational approach for the general inelastic scattering. Thus the variational expression for the S-matrix is... 

This is precisely the RRVP. As such, the KVP can be seen as the propei generalization of the RRVP for case of scattering boundary conditions. 

The ab initio treatment of electron-molecule collisions requires the solution, or approximate solution, of the Schrodin-ger equation subject to scattering boundary conditions. These conditions are more complicated than those corresponding to the familiar bound states of quantum chemistry in that they involve the very quantities we seek to compute, namely the scattering amplitudes. If we focus only on electronic degrees of freedom, for example for an N-electron target in the initial state To, we can write the asymptotic boundary conditions for the N -f-1 electron wave function, for which we must solve as... 

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