Numerical calculations coordinates

The numerical calculations have been done on a two-coordinate system with q being a radial coordinate and <() the polar coordinate. We consider a 3 x 3 non-adiabatic (vector) mabix t in which and T4, aie two components. If we assume = 0, takes the following form,... 

The order parameter can be defined in two different ways. It can be either a function of atomic coordinates or just a parameter in the Hamiltonian. Examples of both types of order parameters are given in Sect. 2.8.1 in Chap. 2 and illustrated in Fig. 2.5. This distinction is theoretically important. In the first case, the order parameter is, in effect, a generalized coordinate, the evolution of which can be described by Newton s equations of motion. For example, in an association reaction between two molecules, we may choose as order parameter the distance between the two molecules. Ideally, we often would like to consider a reaction coordinate which measures the progress of a reaction. However, in many cases this coordinate is difficult to define, usually because it cannot be defined analytically and its numerical calculation is time consuming. This reaction coordinate is therefore often approximated by simpler order parameters. 

The critical value (V0/B)CTit is difficult to calculate exactly Anderson originally obtained for z=6 a value near 5, but subsequent calculations (Edwards and Thouless 1972) gave a value near 2. Numerical calculations by Schonhammer and Brenig (1973) confirm this. The value should be roughly proportional to the coordination number (Economou and Cohen 1972). 

For a polyatomic reactant with many degrees of freedom the numerical calculations required to execute the program outlined above can easily achieve a scale that is impossible to handle even with a vectorized parallel processor supercomputer. The simplest approximation that reduces the scale of the numerical calculations is the neglect of some subset of the internal molecular motions, but this approximation usually leads to considerable error. A more sophisticated and intuitively reasonable approximation [72, 73] is to reduce the system dimensionality by placing constraints on the values of the internal molecular coordinates (instead of omitting them from the analysis). 

Unlike the case of the neutral reactants, where analytical solution reveals the auto-model behaviour in coordinated r/ , in our case of charged particles the singular solutions arise on the spatial scale of the order of the recombination radius ro thus preventing us from such a simplified analytical analysis. Therefore, we will compare semi-qualitative arguments for the new law, n(t) oc r5/4, with numerical calculations of our kinetic equations. 

Here t, x are the dimensionless time and coordinate along the propagation of the front 6 = (T — Tc)E/RTq is the dimensionless temperature counted down from the combustion temperature Tc and measured in the characteristic intervals RTq/E E is the activation energy f3 = RTC/E and A is the scale coefficient. In the numerical calculations we took (3 = 0. 

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]. 

When the gap width between two particles becomes very small, numerical calculations involved in both the bispherical coordinate method and the boundary collocation technique are computationally intensive because the number of terms in the series required to be retained to achieve a desired accuracy increases tremendously. To solve this near-contact motion more effectively and accurately, Loewenberg and Davis [43] developed a lubrication solution for the electrophoretic motion of two spherical particles in near contact along their line of centers with the assumption of infinitely thin ion cloud. The axisymmetric motion of the two particles in near contact can be approximated as the pairwise motion of the spheres in point contact plus a deviation stemming from their relative motion caused by the contact force. The lubrication results agree very well with those obtained from the collocation method. It is shown that near contact electrophoretic interparticle... 

For the percolation model, the situation is rather different, where the numerical calculation is easily accessible with the aid of the probability theory combined with a computational calculation. As a matter of course, the same methodology cannot be applied to real branching reactions, since real molecules are not fixed on lattices, and one cannot define the number of configurations that corresponds to the coordination number, q, of the percolation model. 

The coordinate transformation makes all the velocities coincide. The boundary layer approaches the core flow asymptotically and in principle stretches into infinity. The deviation of the velocity wx from that of the core flow is, however, negligibly small at a finite distance from the wall. Therefore the boundary layer thickness can be defined as the distance from the wall at which wx/wx is slightly different from one. As an example, if we choose the value of 0.99 for Wj/uico, the numerical calculation yields that this value will be reached at the point r]+ m 4.910. 

Viscosity is determined by integration of the stress relaxation modulus [Eq. (7.117)] Tj = jG(t)dt. The numerically calculated dependence of viscosity on the number of reptons (for the coordination number z — 6) is... 

In general, before any numerical calculation, and in order to explore poten-tied energy surfaces of higher dimensionality, it emerges the crucial necessity to consider cuts as these, where some coordinate is held constant or is adjusted to minimize the potential energy [41]. 

If the transformation matrix U permutes the coordinates x,-, it is generally necessary to apply relation (4.7) which is easy to program for numerical calculations by computer. 

The discrete variational (DV) method numerically calculates the basis atomic orbitals using the following wave equation for the radial atomic orbital function Rja r) in spherical coordinates... 

Our aim is to develop and test practical, numerically stable means of treating nonsepaxable potentials in which tuimeling occurs in two or more degrees of freedom. The system is particularly suitable for this purpose. Exact numerical calculations [4] are available for comparison over a wide range of R. Also, in spheroidal coordinates the double-minimum potential is separable and timneling occurs in only one coordinate [5] whereas in cylindrical coordinates [6] the potential is nonseparable and tunneling occurs in two coordinates. This offers an opportunity to compare approximation methods for separable and nonseparable versions of the same system. 

To is a vibrational period in the reaction coordinate, or the collision time, n is a small number dependent on the complex structure, D is the dissociation energy D A -B) of the complex, and s is the number of effective degrees of freedom, s 3n — 6. The expression of Eq. (7) is discussed in most kinetics texts, for example in the excellent text of Johnston. Numerical calculations of Dugan et suggest that Tq for nonpolar neutral reactants... 

Each top experiences the same on-site potential 1 3 and the coupling depends only on the phase difference of the two rotors. In contrast to the textbook case of coupled harmonic oscillators, this Hamiltonian cannot be diagonalized by simple transformation into normal coordinates [86], On the other hand, numerical calculation of the eigenstates and eigenfunctions cannot be carried out with standard methods, for it requires very large basis sets with dimension 10, and the analysis of the eigenfunctions would be cumbersome. 

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