Order description

One of the most widely used - and successful representations of the constitution, the topology, of a molecule is the HOSE code (Hierarchical Ordered description of the Substructure Environment) [9]. It is an atom-centered code taking into account... 

With nanosecond time resolution, sensitive, accurate detectors, studies of these release waves have proven to be particularly revealing. First-order descriptions of release properties were obtained with rudimentary instrumentation from the earliest studies [65A01] it has required the most sophisticated modern instrumentation to provide the necessary detail to obtain a clear picture of the events. Characteristically different profiles are encountered in the strong-shock, elastic, and elastic-plastic regimes. 

Release waves for the elastic-plastic regime are dominated by the strength effect and the viscoplastic deformations. Here again, quantitative study of the release waves requires the best of measurement capability. The work of Asay et al. on release of aluminum as well as reloading, shown in Fig. 2.11, demonstrates the power of the technique. Early work by Curran [63D03] shows that limited time-resolution detectors can give a first-order description of the existence of elastic-plastic behavior on release. 

It has been a persistent characteristic of shock-compression science that the first-order picture of the processes yields readily to solution whereas second-order descriptions fail to confirm material models. For example, the high-pressure, pressure-volume relations and equation-of-state data yield pressure values close to that expected at a given volume compression. Mechanical yielding behavior is observed to follow behaviors that can be modeled on concepts developed to describe solids under less severe loadings. Phase transformations are observed to occur at pressures reasonably close to those obtained in static compression. 

In spite of these representative first-order descriptions, experiments, theory, and material models do not typically agree to second order. Compressibility (derivatives of pressure with volume) shows complex behaviors that do not generally agree with data obtained from other loadings. Mechanical yielding and strength behavior at pressure show complexities that are not... 

The study clearly shows that the observed electrical signals are electrochemical in origin, and the first-order description of the process is consistent with that expected from atmospheric pressure behaviors. Nevertheless, the complications introduced by the shock compression do not permit definitive conclusions on values of electrochemical potentials without considerable additional work. 

TYPICAL WIRE-ROPE CONSTRUCTIONS WITH CORRECT ORDERING DESCRIPTIONS... 

The next update of the null hypothesis would incorporate a zero-order description of bonding, in terms of a prior prejudice of standard chemical groups. The MaxEnt map then will tell us about the subtle differences induced in formally equivalent chemical bonds by conjugation, stacking, and other intra- and intermolecular interactions. To achieve this degree of accuracy, the refinement of structural parameters... 

A simple — and generally accepted — approach of a kinetics for the hydrolysis is a 1-order description (Henze et al 1995) ... 

The following compounds from the databank match the 1. order description ... 

The Landau-Zener expression is calculated in a time-dependent semiclassical manner from the diabatic surfaces (those depicted in Fig. 1) exactly because these surfaces, which describe the failure to react, are the appropriate zeroth order description for the long-range electron transfer case. As can be seen, in the very weak coupling limit (small A) the k l factor and hence the electron transfer rate constant become proportional to the absolute square of A ... 

Type of Values Linearly-Ordered Descriptive Phrases Values ... 

Moller-Plesset perturbation theory (MPPT) aims to recover the correlation error incurred in Ilartree- Fock theory for the ground state whose zero-order description is ,. The Moller-Plesset zero-order Hamiltonian is the sum of Fock operators, and the zero-order wave functions are determinantal wave functions constructed from HF MOs. Thus the zero-order energies are simply the appropriate sums of MO energies. The perturbation is defined as the difference between the sum of Fock operators and the exact Hamiltonian ... 

The corrected rate constants provide a basic description of hopping between states, but it is necessary to determine self-diffusivities from these to enable comparison with experimental measurements. In the case of potential minima within a zeolite pore, the lattice of sorption sites is often anisotropic. The probability of a molecule residing in a certain site is dependent on the type of site, and the rate constants, k,n may be different for each ij pair. A Monte Carlo algorithm, based on a first-order description of the hopping process, is usually used to determine the diffusivities. 

R. D. Levine To answer the question of Prof. Lorquet, let me say that the peaks in the ZEKE spectra correspond to the different energy states of the ion. From the beginning one was able to resolve vibrational states, and nowadays individual rotational states of polyatomics have also been resolved. The ZEKE spectrum is obtained by a (weak) electrical-field-induced ionization of a high Rydberg electron moving about the ion. The very structure of the spectrum appears to me to point to the appropriate zero-order description of the states before ionization as definite rovibrational states of the ionic core, each of which has its own Rydberg series. Such a zero-order description is inverse to the one we use at far lower energies where each electronic state has its own set of distinct rovibrational states, known as the Bom-Oppenheimer limit. 

In both cases what the basis set provides is only a zero-order description because in both cases one knows that the system can exit to the continuum. 

The theoretical results described here give only a zeroth-order description of the electronic structures of iron bearing clay minerals. These results correlate well, however, with the experimentally determined optical spectra and photochemical reactivities of these minerals. Still, we would like to go beyond the simple approach presented here and perform molecular orbital calculations (using the Xo-Scattered wave or Discrete Variational method) which address the electronic structures of much larger clusters. Clusters which accomodate several unit cells of the crystal would be of great interest since the results would be a very close approximation to the full band structure of the crystal. The results of such calculations may allow us to address several major problems ... 

An issue that has been explored is how the relative distribution of charge and mass affect the viscosity of an ionic liquid. Kobrak and Sandalow [183] pointed out that ionic dynamics are sensitive to the distance between the centers of charge and mass. Where these centers are separated, ionic rotation is coupled to Coulomb interactions with neighboring ions where the centers of charge and mass are the same, rotational motion is, in the lowest order description, decoupled from an applied electric field. This is significant, because the Kerr effect experiments and simulation studies noted in Section III. A imply a separation of time scales for ionic libration (fast) and translation (slow) in ILs. Ions in which charge and mass centers are displaced can respond rapidly to an applied electric field via libration. Time-dependent electric fields are generated by the motion of ions in the liquid... 

Both CIS and TDHF have the correct size dependence and can be applied to large molecules and solids (we will shortly substantiate what is meant by the correct size dependence ) [42-51], It is this property and their relatively low computer cost that render these methods unique significance in the subject area of this book despite their obvious weaknesses as quantitative excited-state theories. They can usually provide an adequate zeroth-order description of excitons in solids [50], Adapting the TDHF or CIS equations (or any methods with correct size dependence, for that matter) to infinitely extended, periodic insulators is rather straightforward. First, we recognize that a canonical HF orbital of a periodic system is characterized by a quantum number k (wave vector), which is proportional to the electron s linear momentum kh. In a one-dimensional extended system, the orbital is... 

While CIS provides adequate zeroth-order descriptions of excited states, it does not account for the effects of electron correlation and hence does not have quantitative accuracy. An inexpensive and size-correct method to incorporate those effects for large systems is desired. Low-order perturbation corrections to CIS are important in this context along with TDDFT considered in the next section and the so-called GW method in solid state physics. 

Indeed for the Neel-based theory to work best it is better to have a bipartite system (i.e., a system with two sets of sites all of either set having solely only members from the other set as neighbors). Of course, when there is a question about the adequacy of the zero-order description questions about the (practical) convergence of the perturbation series arises. But for favorable systems these [38] or closely related [41] expansions can now be made through high orders to obtain very accurate results. 

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