Spatial motion

Quantum mechanical calculations of reactions for three atoms moving on a plane, or even more generally in space, have been for some time the aim of researchers in the field. In this section we consider efforts that are directly linked to use of curvilinear coordinates, or that are an extension of the approaches for collinear motion. Approaches from different angles are left for the other main sections. 

Early attempts to extend collinear calculations have allowed for rotation of the axis on which the three atoms are positioned, so that velocities may point in other directions and impact parameters do not need to be zero. More recently, the first computed results on coplanar motion have appeared. Studies on spatial motion are being developed at present, but equations have only been solved for schematic potentials. 

Collinear models may be extended to include the centrifugal forces that arise from rotation of the three-atom axis. This has been done by Wyatt (1969), who studied H + H2 reactive collisions with a vibrational adiabatic model. The required coordinates are (xAB,xBC), and new ones (0, p) which specify the axial orientation. Transforming (xAB,xBC) to curvilinear coordinates (s, r) he expressed the total wavefunction as 

Another modified collinear model was developed by Connor and Child (1970). They constrained atoms A, B and C to a line, but allowed this to rotate. In this way impact parameters could be different from zero, although velocities remained always in the same plane. The two coordinates (X, x) in centre of mass were replaced by natural ones, (s, p). The wavefunction F was taken as a product of an adiabatic vibrational state (s, p) times a translational function i6), with 0 the axial angle. Then 

The first preliminary calculations for coplanar H + H2 were published by Saxon and Light (1971). This system is favourable for exploratory calculations because of the large rotational spacing of H2. Total energies were varied from 13 to 17-5 kcal/mole and total angular momenta from 0 to 12 h. Calculations were made for an analytical fit to the Shavitt et al. (1968) surface, within a manifold of 19 rotational states (from j = — 9 to 7 = 9) 

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Recapitulating, the SBM theory is based on two fundamental assumptions. The first one is that the electron relaxation (which is a motion in the electron spin space) is uncorrelated with molecular reorientation (which is a spatial motion infiuencing the dipole coupling). The second assumption is that the electron spin system is dominated hy the electronic Zeeman interaction. Other interactions lead to relaxation, which can be described in terms of the longitudinal and transverse relaxation times Tie and T g. This point will be elaborated on later. In this sense, one can call the modified Solomon Bloembergen equations a Zeeman-limit theory. The validity of both the above assumptions is questionable in many cases of practical importance. 

It is now easy to add the terms describing the spatial motion under influence of an applied force F. Of course, (4.10a) is not affected, because the donors are fixed, but (4.10b) becomes... 

In this chapter, dielectric response of only isotropic medium is considered. However, in a local-order scale, such a medium is actually anisotropic. The anisotropy is characterized by a local axially symmetric potential. Spatial motion of a dipole in such a potential can be represented as a superposition of oscillations (librations) in a symmetry-axis plane and of a dipole s precession about this axis. In our theory this anisotropy is revealed as follows. The spectral function presents a linear combination of the transverse (K ) and the longitudinal (K ) spectral functions, which are found, respectively, for the parallel and the transverse orientations of the potential symmetry axis with... 

Employing the additivity approximation, we find dielectric response of a reorienting single dipole (of a water molecule) in an intermolecular potential well. The corresponding complex permittivity jip is found in terms of the hybrid model described in Section IV. The ionic complex permittivity A on is calculated for the above-mentioned types of one-dimensional and spatial motions of the charged particles. The effect of ions is found for low concentrated NaCl and KC1 aqueous solutions in terms of the resulting complex permittivity e p + Ae on. The calculations are made for long (Tjon x) and rather short (xion = x) ionic lifetimes. 

Here multiplier 2 approximately accounts for doubling of integrated absorption due to spatial motion of a dipole, which is more realistic than motion in a plane to which LCs(Z) corresponds. For representation (235), only one (Debye) relaxation region with the relaxation time rD is characteristic. At this stage of molecular modeling it was not clear (a) why the CS potential, which affects motion of a dipole in a separate potential well, is the right model of specific interactions and (b) what is physical picture corresponding to a solid-body-like dipole moment pcs. 

The trajectory of an ion moving in such a potential presents a sequence of rectilinear sections placed between the points of elastic reflections of an ion from the walls of the well. We consider two variants of such a model related to one-dimensional and spatial motion of ion, depicted, respectively, in Figs. 47a and 47b. In the first variant the ion s motion during its lifetime59 presents periodic oscillations on the rectilinear section 2 lc between two reflection points. In the second variant we consider a spherically symmetric potential well, to which a spherical hollow cavity corresponds with the radius lc. 

For the case of spatial motion, Figs. 47b and 47c, it is convenient to start from the following expression for the autocorrelator of the F-projection rE(t) of the radius vector r(t) [147] ... 

Figure 49. Frequency dependencies of the real (a) and imaginary (b) parts of the normalized complex conductivity S. Normalized collision frequency, Y, is 0.1. Spatial motion Solid curves for d=, dashed curves for d = 2. One-dimensional motion Dashed-and-dotted curves for d = 1, dotted curves for d = 2.

First of all, it is difficult to see what extension might be. Several philosophers have appealed to the notion, but without some robust characterization of the sort I shall give, such an appeal can appear to lack much content. Second, the account of extension I shall give provides a perspicuous way of understanding not only Aristode s conception of the spatiality of material objects but also how he might have understood various phenomena associated with spatial motion. Finally, and most importandy, my account of Aristotelian extension will prepare the way for some speculations I make in the final section about the nature of prime matter as distinct from but importandy related to extension. 

First, consider spatial motion. Let us posit a primitive relation of enforming that holds between a form and a particular region of points. And let us suppose that there is in addition to the form and region of points a third entity, namely the composite of form and matter. Under such a supposition, there could be a composite A of form Ft and... 

In condensed phases the spatial motion of reactants takes place by diffusion, which is described by the Smoluchowski equation. To be specific we consider a... 

The force balance or momentum balance equation describes the spatial motions of small particles or droplets in an unsteady, nonuniform flow. The acceleration of the particle mass increased by the added mass results from action of several forces on the particle. Assuming that there are not any nonlinear interactions between the various forces acting on the particle, one can write (42)... 

Computer modelling has provided a wealth of numerical results. Many accurate cross sections are now available for collinear collisions of atoms with diatoms, where motion is constrained to a line at all times. These models are clearly not for immediate application to physical situations, but they are useful in other respects. They provide exact results with which approximations may be tested. This testing may prevent waste of effort while developing approximations for physical (spatial motion) situations. Results on coplanar motion are available and others are likely soon on spatial motion,... 

A large part of the computational work has been influenced by the introduction of curvilinear coordinates, designed to take advantage of the topography of potential surfaces. These coordinates allow for a smooth change from reactant to product conformations and in effect transform the rearrangement problem into the much simpler one of inelastic collisions. The various treatments have employed reaction-path (or natural collision) coordinates less restricted reaction coordinates atom-transfer coordinates, somewhat analogous to those used for electron-transfer and, for planar and spatial motion, bifurcation coordinates. 

To illustrate the general procedure let us consider collinear reactions of type A + BC - AB + C. This means that the three atoms move on a line, e.g. the x-axis, and furthermore that velocities are also along the same line. Indicating relative atomic distances by xAB, xBC and xCA, we introduce centre-of-mass coordinates, which for reactants are x = xBC and X, the distance from A to the centre of mass of BC. Similar coordinates could be defined for products. Because of restrictions in the type of motion it is not possible to simultaneously account for the B + CA rearrangement. To do this one must proceed to planar or spatial motion and, for example, introduce bifurcation coordinates. 

In the following paragraphs we review these and other developments in the last few years, including the more recent treatments that extend to coplanar and spatial motions. 

The natural collision coordinates for spatial motion introduced by Marcus were made the starting point for a development (Wyatt, 1972) of reactive collision equations for AB + C -> A + BC. The treatment may be regarded as an extension of previous work (Curtis and Adler, 1952) for inelastic collisions. The kinetic energy operator was simplified by introducing two approximations appropriate for linear intermediates, and the interaction was chosen of form... 

It is at present too soon to judge the usefulness of curvilinear coordinates for coplanar and spatial motions. From the numerical results and analytical developments published so far, it is clear that generalizations to a plane or space have been possible only through a large increase of effort over col-linear calculations, due to the increased complexity of the equations to be solved. However, this is a usual situation in the preliminary stages of most work and may change in the future. Another open question, that will have to be answered, is the extent to which curvilinear coordinates will be useful in situations where surface hopping , i.e. electronic transitions, are involved. 

What happens with the interaction between the rotational and spin symmetries once the system is characterized as being defined by at least different spinors Wigner and von Neumann [10] combined both types of symmetries with the permutation aspect [11]. They intuitively reached the idea using atomic spectroscopy that the H operator has to be constructed by two terms H, resulting from the spatial motion of the single electron only (and the electromagnetic interaction with the field of the atomic core), and (//2), which has to visualize the electron spin. For simplicity, we can consider the eigenvalue problem of the spinless wave function i r without the second term as... 

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