Summation approximant

C.3.b. Two parallel cylinders, pairwise-summation approximation, Hamaker-Lifshitz hybrid, retardation screening neglected C.3.b.l. All separations... 

II.C. van der Waals Attraction. The van der Waals attraction between the hollow shells of the ferritin can be easily computed in the pairwise summation approximation. Let us consider a large sphere B made up of a small sphere b and a spherical shell S. The interaction free energy Fbb between two large spheres can be written (assuming pairwise interactions) as... 

Starting from eqn (28) and applying the Pade summation approximation, it is possible to define a family of correlation functionals, which has the general form ... 

In Fig. 4.41, the results of the singularity method are compared with two approximate solutions—the APPI and pairwise summation approximation. For Ka = 5, there is obviously no deviation between the approximations and the numerical calculation—independent from the aggregate type. In the case of DLCA aggregates, both approximations perform weU even for xa = 1. However, both approaches fail as soon as the double layer thickness is large compared to the primary particle size Ka < 1). The APPI systematically underestimates the interaction energy, whereas the pairwise summation approximation overvalues it. Regarding the deviation from the numerical solution, none of the two approaches can be preferred. However, the APPI requires less computation steps than the pairwise summation approach. 

The summation of pair-wise potentials is a good approximation for molecular dynamics calculations for simple classical many-body problems [27], It has been widely used to simulate hyperthennal energy (>1 eV) atom-surface scattering ... 

The second summation is over all the orbitals of the system. This equation is used in IlyperChem ah imiio calculations to generate contour plots of electrostatic potential, [fwe choose the approximation whereby we n eglect the effects of the diatomic differen tial overlap (NDDO). then the electrostatic potential can be rewritten... 

It is common practice to omit the second summation on the right hand side of (11.118) on the groiands that it is small compared with the contribution of the conductive flux, which appears on the left hand side. However, this may not be so If the reactions are rapid and the thermal conductivity of the pellet material is low. One should, therefore, at least be aware of the approximation involved in the fona of the enthalpy balance most commonly seen in the literature. 

A relatively simple example of a group contribution technique is the method for estimating Hquid and soHd heat capacities (159). This method is a modification of Kopp s rule (160,161) which was originally proposed in 1864. Kopp s rule states that, at room temperature, the heat capacity of a soHd compound is approximately equal to a stoichiometric summation of the heat capacities of its atoms (elements). The Hurst-Harrison modified equation is as follows ... 

Numerical Because of the property of definite integrals another method for obtaining their solution is available which cannot be apphed to indefinite integrals. This involves a numerical approximation based on the previously outlined summation definition ... 

Summation of Separate Contributions to Gas or Flame Emissivity Flame emissivity g -t-, due to joint emission from gas and soot has already been treated. If massive-particle emissivity ., such as from fly ash, coal char, or carbonaceous cenospheres from heavy fuel oil, are present, it is recommended that the total emissivity be approximated by... 

H type columns must be used at a flow rate and pressure drop below maximum values listed in Tables 4.12-4.16. Standard flow rates are also listed in these tables. They are flow rate range recommendable for long-term usage in tetrahydrofuran at 25°C and vary with temperature. H type columns can be operated at a higher flow rate at elevated temperatures. They also vary with solvent depending on the viscosity. They are approximately inversely proportional to the solvent viscosity. The maximum pressure drop listed in the tables is for one column. When some columns are used in series, the total maximum pressure drop is a summation of values of all columns. 

As the number of grid points increases, this approximation becomes better. The reduction in the formal scaling from to comes from the fact that the summations involve GM operations, G being the number of grid points, which typically will be linearly dependent on the number of basis functions M, i.e. GM- M. ... 

The above treatment has made some assumptions, such as harmonic frequencies and sufficiently small energy spacing between the rotational levels. If a more elaborate treatment is required, the summation for the partition functions must be carried out explicitly. Many molecules also have internal rotations with quite small barriers, hi the above they are assumed to be described by simple harmonic vibrations, which may be a poor approximation. Calculating the energy levels for a hindered rotor is somewhat complicated, and is rarely done. If the barrier is very low, the motion may be treated as a free rotor, in which case it contributes a constant factor of RT to the enthalpy and R/2 to the entropy. 

Tlie suffices i and J refer to individual atoms and S and Sj to the species of the atoms involved. The summation over j extends over those neighbors of the atom i for which ry, the separation of atoms i and J, is within the cutoff radii of these potentials. The second term in Equation (la) is the attractive many-body term and both V and are empirically fitted pair potentials. A Justification for the square root form of the many-body function is provided in the framework of a second moment approximation of the density of states to the tight-binding theory incorporating local charge conservation in this framework the potentials represent squares of the hopping integrals (Ackland, et al. 1988). 

Under most circumstances the equations given in Table 10.4 accurately calculate the thermodynamic properties of the ideal gas. The most serious approximations involve the replacement of the summation with an integral [equations (10.94) and (10.95)] in calculating the partition function for the rigid rotator, and the approximation that the rotational and vibrational partition functions for a gas can be represented by those for a rigid rotator and harmonic oscillator. In general, the errors introduced by these approximations are most serious for the diatomic molecule." Fortunately, it is for the diatomic molecule that corrections are most easily calculated. It is also for these molecules that spectroscopic information is often available to make the corrections for anharmonicity and nonrigid rotator effects. We will summarize the relationships... 

Employing this simplification (known as the external field approximation - EFA) in Eq. (4.58) and replacing summation over l with integration over the impact parameters, we obtain... 

X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

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