Inelastic time dependent

The basic assumption here is the existence over the inelastic scattering region of a connnon classical trajectory R(t) for the relative motion under an appropriately averaged central potential y[R(t)]. The interaction V r, / (t)] between A and B may then be considered as time-dependent. The system wavefiinction therefore satisfies... 

The preferable theoretical tools for the description of dynamical processes in systems of a few atoms are certainly quantum mechanical calculations. There is a large arsenal of powerful, well established methods for quantum mechanical computations of processes such as photoexcitation, photodissociation, inelastic scattering and reactive collisions for systems having, in the present state-of-the-art, up to three or four atoms, typically. " Both time-dependent and time-independent numerically exact algorithms are available for many of the processes, so in cases where potential surfaces of good accuracy are available, excellent quantitative agreement with experiment is generally obtained. In addition to the full quantum-mechanical methods, sophisticated semiclassical approximations have been developed that for many cases are essentially of near-quantitative accuracy and certainly at a level sufficient for the interpretation of most experiments.These methods also are com-... 

It is important to differentiate between brittie and plastic deformations within materials. With brittie materials, the behavior is predominantiy elastic until the yield point is reached, at which breakage occurs. When fracture occurs as a result of a time-dependent strain, the material behaves in an inelastic manner. Most materials tend to be inelastic. Figure 1 shows a typical stress—strain diagram. The section A—B is the elastic region where the material obeys Hooke s law, and the slope of the line is Young s modulus. C is the yield point, where plastic deformation begins. The difference in strain between the yield point C and the ultimate yield point D gives a measure of the brittieness of the material, ie, the less difference in strain, the more brittie the material. 

Bar-Joseph I, Gurvitz SA (1991) Time-dependent approach to resonant tunneling and inelastic scattering. Phys Rev B 44 3332... 

We seek to nnderstand the response of a material to an applied stress. In Chapter 4, we saw how a flnid responds to a shearing stress through the application of Newton s Law of Viscosity [Eq. (4.3)]. In this chapter, we examine other types of stresses, snch as tensile and compressive, and describe the response of solids (primarily) to these stresses. That response usually takes on one of several forms elastic, inelastic, viscoelastic, plastic (ductile), fracture, or time-dependent creep. We will see that Newton s Law will be useful in describing some of these responses and that the concepts of stress (applied force per unit area) and strain (change in dimensions) are universal to these topics. 

The quantum theory of molecular collisions in external fields described in this chapter is based on the solutions of the time-independent Schrodinger equation. The scattering formalism considered here can be used to calculate the collision properties of molecules in the presence of static electric or magnetic fields as well as in nonresonant AC fields. In the latter case, the time-dependent problem can be reduced to the time-independent one by means of the Floquet theory, discussed in the previous section. We will consider elastic or inelastic but chemically nonreac-tive collisions of molecules in an external field. The extension of the formalism to reactive scattering problems for molecules in external fields has been described in Ref. [12]. 

It has already been pointed out that the power spectrum of this function at zero frequency determines the translational diffusion coefficient, D. The full-time dependence of this function can be obtained indirectly from inelastic slow neutron experiments.57 Unfortunately, these experiments are not yet precise enough to say anything quantitatively about this function. /(t) s memory function, K t), is defined by... 

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. 

The time evolution in Eq. (7.75) is described by the time-dependent Schrodinger equation, provided the molecule is isolated from the rest of the universe. In practice, there are always perturbations from the environment, say due to inelastic collisions. The coherent sum in Eq. (7.75) will then relax to the incoherent sum of Eq. (7.74), that is, the off-diagonal interference terms will vanish and cn 2 — pn corresponding to the Boltzmann distribution. As mentioned earlier, the relaxation time depends on the pressure. In order to take advantage of coherent dynamics it is, of course, crucial that relaxation is avoided within the duration of the relevant chemical dynamics. 

Utilization of both ion and neutral beams for such studies has been reported. Toennies [150] has performed measurements on the inelastic collision cross section for transitions between specified rotational states using a molecular beam apparatus. T1F molecules in the state (J, M) were separated out of a beam traversing an electrostatic four-pole field by virtue of the second-order Stark effect, and were directed into a noble-gas-filled scattering chamber. Molecules which were scattered by less than were then collected in a second four-pole field, and were analyzed for their final rotational state. The beam originated in an effusive oven source and was chopped to obtain a velocity resolution Avjv of about 7 %. The velocity change due to the inelastic encounters was about 0.3 %. Transition probabilities were calculated using time-dependent perturbation theory and the straight-line trajectory approximation. The interaction potential was taken to be purely attractive ... 

It is supposed that a collision can be characterized by the Hamiltonian 2 written as a sum of the Hamiltonian of noninteracting partners J 0(r) and the interaction f (r, R) that vanishes at large interatomic distance R (r represents a set of electron coordinates). Given the classical trajectory R = R(/), the problem of inelastic scattering is equivalent to the solution of the time-dependent Schrddinger equation ... 

Viscoelasticity was introduced in Section 11.5. A polymer example may be useful by way of reeapitulalion. Imagine a polymer melt or solution confined in the aperture between two parallel plates to which it adheres. One plate is rotated at a constant rate, while the other is held stationary. Figure 11-3la shows the time dependence of the shear stress after the rotation has been stopped, r decays immediately to zero for an inelastic fluid but the decrease in stress is much more gradual if the material is viscoelastic. In some cases, the residual stresses may... 

A few examples may demonstrate this. The approximation has been used to treat the photodissociation processes of NOHCl [33] and NO2 [34] which include three internal modes. Five dimensional time dependent calculations were performed on the photodissociation of CH3I [35, 36]. The state-to-state chemistry has been investigated for the reactive scattering of H-1-H2 (v=0,l) —> H2 (v=0,l) -1- H collinear system [37]. The MCTDSCF approach has also been applied to surface chemistry, in particular H2 dissociation on a transition metal surface [38] the photodissociation of CH3I on MgO surface [39, 40] and to inelastic molecule-surface scattering [41, 42]. Recently, the MCTDSCF method has been used to investigate multimode effects in the absorption spectrum of pyrazine taking into account 14 vibrational modes [43]. 

M. Ehara, H.-D. Mayer and L.S. Cederbaum, Multiconfiguration time-dependent Hartree (MCTDH) study on rotational and difractive inelastic molecule-surface scattering, J. Chem. Phys., 105 (1996) 8865. 

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