Wave function 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. 

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... 

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... 

The boundary conditions (4.39), the orthogonality of the vector harmonics, and the form of the expansion of the incident field dictate the form of the expansions for the scattered field and the field inside the sphere the coefficients in these expansions vanish for all m = = 1. Finiteness at the origin requires that we take y (kjr), where kj is the wave number in the sphere, as the appropriate spherical Bessel functions in the generating functions for the vector harmonics inside the sphere. Thus, the expansion of the field (Ej,H,) is... 

The states correspond to wave packet controlled in the far past and in the far future, respectively. Let us see what this means. In the absence of external time-dependent fields, the scattering component of the time-dependent wave function i/r(f) can be expanded in terms of either of the two sets of scattering states for example, those with incoming boundary conditions... 

The KR variational principle determines a wave function with correct boundary conditions at a specified energy, the typical conditions of scattering theory. Energy values are deduced from consistency conditions. 

These radical functions are normalized according to equ. (7.28c).) Knowing these asymptotic forms, it is now possible to derive the asymptotic limits for the ansatz in equ. (7.21) and the boundary condition in equ. (7.20). A comparison of these expressions will then lead to the unknown coefficients b K) in equ. (7.21) and the scattering amplitude in equ. (7.20). If one starts with the plane-wave part... 

Equations (30) and (31) complete the construction of the scattering wave function in the continuum. According to Eq. (30), the expansion coefficients over the basis functions in the interaction region are uniquely determined by the boundary values of the normal derivatives f. On the other hand, in each dissociation channel (3 one has = (this is the matching condition). Thus, the derivatives / can be evaluated, using Eq. (22), from the solutions F in the outer region by exploiting the matrix Y from Eq.(31). ... 

By assuming harmonic forces and periodic boundary conditions, we can obtain a normal mode distribution function of the nuclear displacements at absolute zero temperature (under normal circumstances). The problem is then reduced to a classic system of coupled oscillators. The displacements of the coupled nuclei are the resultants of a series of monochromatic waves (the normal modes). The number of normal vibrational modes is determined by the number of degrees of freedom of the system (i.e. 3N, where N is the number of nuclei). Under these conditions the one-phonon dispersion relation can be evaluated and the DOS is obtained. Hence, the measured scattering intensities of equations (10) and (11) can be reconstructed. 

F is the bulk collision constant, A is a positive dimensionless factor, Vf is the Fermi velocity and R the particle radius. From a classical point of view, this modification is supported by the fact that, when the radius is smaller than the bulk mean free path of the electrons, there is an additional scattering factor at the particle surface. This phenomenon, known as the mean free path effect, is abundantly discussed in [19]. In a quantum approach, the boundary conditions imposed to the electron wave functions lead to the appearance of individual electron-hole excitations (Fandau damping) [21] resulting in the broadening of the SPR band proportional to the inverse of the particle radius as in Eq. (8) [22]. A chemical interface damping mechanism has also been considered, leading to the l/R dependence of F [23]. 

The coefficients Cm are determined from the boundary conditions which arises from the following considerations. A collimated beam of non-interacting particles may be represented by a plane wave (r) = elkz where z is the beam direction. If target molecules are present and some of the primary beam particles interact with them the scattered beam is represented by a radially outgoing wave,/(0) e kr/r where the angular coefficient,/(0), which is also a function of k, is the scattering amplitude. Since the distance of the particle detector from the point of interaction is effectively infinite compared to atomic dimensions the wave functions at the detector must be of the form... 

In the quantum mechanical treatment of stationary scattering theory the wave function pk(r) corresponding to the wave number k2 = l E/h2 must obey the boundary condition... 

The Korringa, Kohn3 Rostoker (KKR) method [1.17,18] employs an expansion inside the MT spheres similar to the cellular and APW methods. In the interstitial region between the spheres, however, the potential must be flat and the wave functions are expanded in phase-shifted spherical waves. The boundary condition can then be expressed as the condition for self-consistent multiple scattering between the muffin-tin spheres, or alternatively as the condition for destructive interference of the tails of these waves in the core region (Sect.2.1). This is the other most widely used computational technique in band theory. 

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