Theoretical descriptions

The experimental data shown in Figs. 15.3 and 15.5 were obtained with a microwave frequency oj/2ji lit where r is the time or duration of the collision and 1/r is the linewidth. In this case the resonances corresponding to the absorption or emission of different numbers of photons are resolved. Here we describe radiative collisions starting from the high frequency regime, oj/2ji 1/r and progressing to the low frequency regime, a 2n 1/r. 

To describe the shifts and intensities of the m-photon assisted collisional resonances with the microwave field Pillet et al. developed a picture based on dressed molecular states,3 and we follow that development here. As in the previous chapter, we break the Hamiltonian into an unperturbed Hamiltonian H(h and a perturbation V. The difference from our previous treatment of resonant collisions is that now H0 describes the isolated, noninteracting, atoms in both static and microwave fields. Each of the two atoms is described by a dressed atomic state, and we construct the dressed molecular state as a direct product of the two atomic states. The dipole-dipole interaction Vis still given by Eq. (14.12), and using it we can calculate the transition probabilities and cross sections for the radiatively assisted collisions. 

We begin by developing the wavefunction for one atom in combined static and microwave fields, both of which are in the same direction. If the nominal n state has an energy Wn((E) in a static field E, we can describe the energy in the vicinity of the static field Es by 

If we now imagine that AE varies in time, but slowly, its only effect is to cause a time variation of the energy of the n state. We assume that the spatial wavefunction is unaffected by AE and that no transitions occur. This approximation is the adiabatic approximation of Autler and Townes.11 Now let us consider the time variation of AE of particular interest to us, Emw cos cot. If the assumptions stated above are valid, we can use the energy of Eq. (15.6) as the unperturbed Hamiltonian H0 in the Schroedinger equation. Explicitly, 

Carrying out the integrations and discarding the phase factors from t0 yields r(0 = V re-f ) e- -- e-(15.9) The exponentials with sinusoidal arguments may be expanded as Bessel functions using12 

Apart from these, one has to either resort to simulation methods, in particular to MD, or follow a top-down approach by augmenting macroscopic hydrod3mamic equations such that they can be used down to the scale at which the molecular structure becomes relevant. 

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The function of this chapter is to review these methods with emphasis on the types of phenomenology involved and information obtained. Many of the effects are complicated, and full theoretical descriptions are still lacking. The wide variety of methods and derivative techniques has resulted in a veritable alphabet soup of acronyms. A short list is given in Table VIII-1 (see pp. 313-318) the lUPAC recommendations for the abbreviations are found in Ref. 1. 

Unlike most words in a glossary of tenns associated with the theoretical description of molecules, the word synnnetry has a meaning in every-day life. Many objects look exactly like their mirror image, and we say that they are syimnetrical or, more precisely, that they have reflection synnnetry. In addition to having reflection synnnetry, a pencil (for example) is such that if we rotate it tln-ough any angle about its long axis it... 

Progress in the theoretical description of reaction rates in solution of course correlates strongly with that in other theoretical disciplines, in particular those which have profited most from the enonnous advances in computing power such as quantum chemistry and equilibrium as well as non-equilibrium statistical mechanics of liquid solutions where Monte Carlo and molecular dynamics simulations in many cases have taken on the traditional role of experunents, as they allow the detailed investigation of the influence of intra- and intemiolecular potential parameters on the microscopic dynamics not accessible to measurements in the laboratory. No attempt, however, will be made here to address these areas in more than a cursory way, and the interested reader is referred to the corresponding chapters of the encyclopedia. 

Lin S H and Villaeys A A 1994 Theoretical description of steady-state sum-frequency generation in molecular absorbates Phys. Rev. A 50 5134-44... 

These theoretical descriptions of the thennal etching reaction between F2 and Si(lOO) have been reviewed in some detail in the context of ah initio methods in surface chemistry [60]. 

The simplest theoretical description of the photon capture cross-section is given by Fermi s Golden Rule... 

The problem of the theoretical description of biopolymer water adsorption isotherms has drawn the attention of researchers for a long time. In the works [19], [20] a rigorous statistical basis for equations describing the isotherms for the case of homogeneous adsorption surfaces and noninteracting adsorption sites of N different types has been suggested. The general equation is ... 

At very low densities It Is quite easy Co give a theoretical description of thermal transpiration, alnce the classical theory of Knudsen screaming 9] can be extended to account for Che Influence of temperature gradients. For Isothermal flow through a straight capillary of circular cross-section, a well known calculation [9] gives the molar flux per unit cross-sectional area, N, In the form... 

The theoretical description of a non-isothermal viscoelastic flow presents a conceptual difficulty. To give a brief explanation of this problem we note that in a non-isothennal flow field the evolution of stresses will be affected by the... 

A theoretical description of hydrogen bonding effects can be made from model of charge-controlled adsorption. It was found that the energy of adsorption of organic molecules ai e determined by the ratios between the effective chai ges of their atoms and atoms in polai solvent molecules ... 

Ion-selective electrodes (ISEs) with ionophore-based membranes allow for quantification of a large number of analytes in various matrixes. Tailoring of the composition of the membranes to comply with the analytical task, requires advanced theory of membrane response. Most of theoretical descriptions include nonrealistic extra-thermodynamic assumptions, in the first place it is assumed that some kind of species strongly predominate in membranes. Ideally, a rigorous theory of ISE response should be based on strict thermodynamics. However, real ISE membranes are too complex. Therefore, known attempts aimed at rigorous thermodynamic description of ISEs proved to be fraritless. 

Because of the complex situation on the surface, satisfactory theoretical description of the ionization process leading to secondary ion formation has not yet been possible. Different ionization mechanisms have been proposed ... 

The shock-compression events are so extreme in intensity and duration, and remote from direct evaluation and from other environments, that experiment plays a crucial role in verifying and grounding the various theoretical descriptions. Indeed, the material models developed and advances in realistic numerical simulation are a direct result of advances in experimental methods. Furthermore, the experimental capabilities available to a particular scientist strongly control the problems pursued and the resulting descriptions of shock-compressed matter. Given the decisive role that experimental methods play, it is essential that careful consideration be given to their characteristics. 

J. Stafiej, A. Ekoka, Z. Borkowska, J. P. Badiali. New theoretical description of electrified interfaces. J Chem Soe, Faraday Trans 92 3677-3682, 1996. 

The Gaussian program contains a hierarchy of procedures corresponding to different approximation methods (commonly referred to as different levels of theory). Theoretical descriptions for each of them may be found in Appendix A. The ones we ll be concerned with most often in this work are listed in the following table ... 

From the molecular point of view, the self-diffusion coefficient is more important than the mutual diffusion coefficient, because the different self-diffusion coefficients give a more detailed description of the single chemical species than the mutual diffusion coefficient, which characterizes the system with only one coefficient. Owing to its cooperative nature, a theoretical description of mutual diffusion is expected to be more complex than one of self-diffusion [5]. Besides that, self-diffusion measurements are determinable in pure ionic liquids, while mutual diffusion measurements require mixtures of liquids. 

Since a considerable amount of review articles on both theoretical frameworks and calculated results have been reported[15-25], the main objective of the present study is placed on the comparisons with experimental results. The organization of the present report is as follows In the next section, for the sake of completeness, a brief theoretical description of the PPM is summarized from the previous articles. In the third section, disorder-LIq transition is focused and visualized atomic (spin) configuration is compared with recent high resolution electron micrograph. In the fourth section, ordering relaxation... 

In summary, a fully relativistic theoretical description of photo emission for magnetic solids has been presented that is based on multiple scattering theory. For the VB-XPS case a very simple expression for the photo current intensity is found that can... 

Miska, S., and P. Skalle, Theoretical description of a new method of optimal hydraulic program design, Society of Petroleum Engineering Journal, August 1981. 

An important hierarchy of parameterization schemes based on successive refinements to a mean-field theoretic description, called local structure theory, has been developed by Gutowitz, et. al. ([guto87a], [guto87b], and [guto88]), and is discussed in detail in chapter five. Below we summarize results of what is essentially a zeroth order local structure theory developed by Langton [lang90]. 

SlaZ81 Slanina, Z. Chemical isomerism and its contemporary theoretical description. Advances in Quantum Chemistry 13 (1981) 89-153. 

The most popular theoretical description of the potentiometric behavior of ion-selective membranes makes use of the three-segmented membrane model introduced by Sollner53), Teorell 30,54), and Meyer and Sievers 31-5S). In this model the two phase boundaries and the interior of the membrane are treated separately. Here, the... 

Theoretical Description of Base-Stabilized Silylene Complexes... 

Quantum Electrodynamics in the Heisenberg Picture.— With the present section we begin our discussion of the quantum theoretical description of the interaction between the negaton-positon field and the radiation field i.e., of quantum electrodynamics proper. 

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