Asymptotic forms

Employing simplifications arising from the use of asymptotic forms of the electronic basis functions and the zeroth-order kinetic energy operator, we obtain... 

A convenience of electronic basis functions (53) is that they reduce at infinitesimal-amplitude bending to (28) with the same meaning of the angle 9 we may employ these asymptotic forms in the computation of the matrix elements of the kinetic energy operator and in this way avoid the necessity of carrying out calculations of the derivatives of the electronic wave functions with respect to the nuclear coordinates. The electronic part of the Hamiltonian is represented in the basis (53) by... 

L sirif Lhc above asymptotic forms of the two-ceiiLer Lwo-dccLron integrals, the paramelers A. and can he derived. C. ertainly, parameter A. is different for different orbitals even though they reside on the same atom, Dewar used AM to represent the parameter A obtained via AD to represent the parameter A obtained via and AQ to represent the parameter A obtained from Hpp-... 

However, it is known that the direct correlation functions have an exact long-range asymptotic form, arising due to intramolecular correlations in clusters formed via the association mechanism. This asymptotics is not included in the Percus-Yevick approximation. Other common liquid state approximations also do not provide correct asymptotic behavior of Ca ir). 

To determine the position of the tricritical point and the structure of the ordered phases stable below the bifurcation we analyze the asymptotic form of Qeff for e 0. At local minima the functional derivative of Qeff with respect to all the OPs vanishes. From this condition and from (45), (58), (47), and (64) we find that at the metastable states... 

The asymptotic form of Qeff (corresponding to local minima) is calculated up to 0 e ) for e 0. We thus consistently neglect in our bifurcation analysis all the terms which would lead to contributions to Qeff of order O(e ), without further stating this explicitly. 

These asymptotic forms may be useful for conceptual studies, but the real design calculations must be based on the full Ergun equation. Turning to the case of compressible fluids, scaleup using geometric similarity with Sr = Sl = S is generally infeasible. Simply stated, the reactors are just too long and have too much inventory. 

The Valley theorem leads to simple conditions for the optimised orbitals near the nuclei. However these conditions are not sufficient to characterize these orbitals one needs in addition to take the asymptotic form of the equations into account. 

In the asymptotic region, an electron approximately experiences a Z /f potential, where Z is the charge of the molecule-minus-one-electron ( Z = 1 in the case of a neutral molecule) and r the distance between the electron and the center of the charge repartition of the molecule-minus -one-electron. Thus the ip orbital describing the state of that electron must be close to the asymptotic form of the irregular solution of the Schrodinger equation for the hydrogen-like atom with atomic number Z. ... 

A sequence of approximations, using properties of the confluent hypergeometric function, integration by steepest descents, and judicious discard of all but the dominant terms, gives one the asymptotic form... 

The asymptotic behavior of transitions under the influence of mass-transfer resistances in long, deep beds is important. The three basic asymptotic forms are shown in Fig. 16-2. With an unfavorable isotherm, the breadth of the transition becomes proportional to the depth of bed it has passed through. For the linear isotherm, the breadth becomes proportional to the square root of the depth. For the favorable isotherm, the transition approaches a constant breadth called a constant pattern. 

Casimir and Polder also showed that, at very long range (i.e., separations greater than a characteristic distance R of a few hundred angstrom units), the dispersion interaction takes the modified asymptotic form... 

In the preceding paragraphs of this section we have summed the terms arising from the partial expansion of the exponentials occurring in the coefficients of the powers of particle concentrations to obtain a series of multiple infinite sums, the terms of which are convergent. The terms in S(R) are of the same form as those in the Mayer solution theory, apart from replacement of integration by summation and the fact that mu differs from the solution value because of the discreteness of the lattice. The evaluation of wi - is outlined in the next section. It is found that the asymptotic form is... 

The contribution of the cycle diagrams can be found from Eq. (132) using the asymptotic form for the Fourier transforms (Eq. (155)) and is... 

For longer times, the scalar dissipation rate reverts to the asymptotic form given in (5.372). 

The exact formula was derived by Schulz [80] and the asymptotic equation is due to Zimm [81]. Schulz derived the distribution for/chains that are coupled together, where it has no influence whether one has a head to tail coupling of most probable distributions or whether these distributions are coupled onto a star center. Zimm recognized that the asymptotic form can be efficiently used to describe the distributions of fractions. The index o refers to the primary (most probable) chain distribution and p is the extent of reaction for this primary chain. 

In the two classic viscometric deformations of simple shear and extension, the appropriate components of Q have very different behaviour. For small shear strains, the shear stress depends on the component Q which has the linear asymptotic form 47/15. This prefactor is the origin of tne constant v in the tube potential of Sect. 3.For large strains, however, Qxy 7 and therefore predicts strong shear-thinning. Physically this comes from the entanglement loss on re-... 

Eq. (9.40) reduces to the Gaussian plume equation. Note that the asymptotic condition in Eq. (9.42) corresponds to zh > point source at or near the ground can also be examined. We can take the limit of Eq. (9.39) as h -> 0 using the asymptotic form of... 

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