The full solution

In the full quantum mechanical picture, the evolving wavepackets are delocalized functions, representing the probability of finding the nuclei at a particular point in space. This representation is unsuitable for direct dynamics as it is necessary to know the potential surface over a region of space at each point in time. Fortunately, there are approximate formulations based on trajectories in phase space, which will be discussed below. These local representations, so-called as only a portion of the FES is examined at each point in time, have a classical flavor. The delocalized and nonlocal nature of the full solution of the Schtddinger equation should, however, be kept in mind. 

Euler s equation is thus recovered as a direct consequence of momentum conservation, but only via the zeroth-order approximation to the full solution to the Boltzman-equation. 

Chapman-Enskog Expansion As we have seen above, the momentum flux density tensor depends on the one-particle distribution function /g, which is itself a solution of the discrete Boltzman s equation (9.80). As in the continuous case, finding the full solution is in general an intractable problem. Nonetheless, we can still obtain a useful approximation through a perturbative Chapman-Enskog expansion. 

The rate constant combinations and panel designations are those listed in Table 4-2. The exact solution is shown as A3 in A-D, and as exact in E-F, where the full solution to Eq. (4-34) is required. Each panel also displays various approximations improved steady-state, steady-state, and prior-equilibrium. Some of the solutions coincide. 

The solution of the resultant set of differential equations is more complex than the situation involving one rate-determining step, but it is still simpler than the full solution. 

There is compelling evidence that reducing agent oxidation and metal ion reduction are, more often than not, interdependent reactions. Nonetheless, virtually all established mechanisms of the electroless deposition fail to take into account this reaction interdependence. An alternative explanation is that the potentials applied in the partial solution cell studies are different to those measured in the full electroless solution studies. Notwithstanding some differences in the actual potentials at the inner Helmholtz plane in the full solution relative to the partial solutions, it is hard to see how this could be a universal reason for the difference in rates of deposition measured in both types of solution. 

It is clear in both of these studies that the small cavity size (which fails to entirely contain all of the atoms given standard van der Waals radii) causes electrostatic solvation free energies to be seriously overestimated — the difference in the 4-nitroimidazole system seems much too large to be physically reasonable. This overestimation would be still more severe were a correct DO model to have been used (i.e., one which accounted self-consistently for the full solute polarization using eq 30). Nevertheless, the D02 results may be considered qualitatively useful, to the extent that they identify trends in tautomer electrostatic solvation free energies. 

Fig. 10.10 Values of dimensionless period, lo = v/coa, for given amplitude ratio rj and phase shift p for spheres. The continuous lines give the full solution, while the broken lines are for the case where the history component is neglected.

Figure 11.4 shows the velocity-time curves from the full solution for weightless rigid spheroids (y = 0) and for density ratios typical of particles in liquids (y = 2.65) and gases (y = 10 ). Figure 11.5 shows the ratio of the value of t for which 14 = 0.5 to the corresponding value for a sphere. The effect of spheroid... 

If the reaction mechanism contains more than one or at most two steps, the full solution becomes very complicated and we will have to solve for the rates and coverages by numerical methods. Although the full solution contains the steady state behavior as a special case, it is not generally suitable for studies of the steady state as the transients may make the simulation of the steady state a numerical nightmare. 

A fundamental property of the Fourier transform is that of superposition. The usefulness of the Fourier method lies in the fact that one can separate a function into additive components, treat each one separately, and then build up the full result by summing the individual results. It is a beautiful and explicit example of the stepwise refinement of complex problems. In stepwise refinement, one successfully tackles the most difficult tasks and solves problems far beyond the mind s momentary grasp by dividing the problem into its ultimately simple pieces. The full solution is then obtained by reassembling the solved pieces. 

As illustrated, here a single variable (the maximum temperature) is chosen as a characteristic function of the solution. For the premixed twin flame, this is a good choice. However, in other circumstances, like an opposed-flow diffusion flame, the choice of a characteristic scalar is less clear. Vlachos avoids the need for a choice by using a norm of the full-solution vector to characterize the solution in the arc length [415,416], The Nish-... 

In the related work of Kim and Hynes [50], Equations (3.107) and (3.112) have been designated, respectively, by the labels SC (self-consistent or mean field) and BO (where Born-Oppenheimer here refers to timescale separation of solvent and solute electrons). More general timescale analysis has also been reported [50,51], Equation (3.112) is similar in spirit to the so-called direct RF method (DRF) [54-56], The difference between the BO and SC results has been related to electronic fluctuations associated with dispersion interactions [55], Approximate means of separating the full solute electronic densities into an ET-active subspace and the remainder, treated, respectively, at the BO and SC levels, have also been explored [52],... 

Is resonance a real phenomenon The answer is quite definitely no. We cannot say that the molecule has either one or the other structure or even that it oscillates between them. .. Putting it in mathematical terms, there is just one full, complete and proper solution of the Schrodinger wave equation which describes the motion of the electrons. Resonance is merely a way of dissecting this solution or, indeed, since the full solution is too complicated to work out in detail, resonance is one way - and then not the only way - of describing the approximate solution. It is a calculus , if by calculus we mean a method of calculation but it has no physical reality. It has grown up because chemists have become used to the idea of localized electron pair bonds that they are loath to abandon it, and prefer to speak of a superposition of definite structures, each of which contains familiar single or double bonds and can be easily visualizable. [30]... 

To define the oscillator-bath interaction term, we write the full solute-... 

The solution to the fiill Hamiltonian is rendered difficult by the electron-electron repulsion term that depends on ri 2. The full solution can be approximated by initially ignoring this term, solving the remaining simplified Hamiltonian, and then reintroducing the term as a perturbation. 

The product of the functions S and R (equation 18.4) gives the full solutions of the Schrodingcr equation which have been given in Table IV. The functions corresponding to various atomic states, are orthogonal. 

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