Approximate accuracy

Interestingly, the correction factor is to a good accuracy approximated by the Wigner formula (2.15) and it is practically independent of friction [Hanggi 1986], by contrast with the relatively strong dependence of on r. ... 

An accuracy approximately comparable to the ASTM method is probably feasible. This method may be superior for coal samples in which a substantial fraction of the pyrite is totally surrounded by organic matter. 

Inclusion of spin-orbit effects at zero magnetic field is critical primarily in two arenas, structure determination of light actinides (notably Pu) [48] and certain spectroscopic characterizations. At non-zero field their inclusion is clearly more important. Calculating spin-orbit effects however raises the computational requirements by approximately an order of magnitude because one must use double-group symmetries (complex-valued KS orbitals are already possible in GTOFF, which works with Hermitian matrices rather than real-symmetric ones). Therefore, a rapid, high-accuracy approximate scheme for inclusion of spin-orbit effects has been developed [48]. 

By solving such inequalities it proves possible to find the relative levels of the values Qak with accuracy approximately within 1 kcal. If one of the obtained bond energies of this system, e.g., Qhni is matched with a corresponding value found by the thermochemical method, the zero reading is determined. 

Methods of analyzing this material are time-consuming and, in certain cases, of uncertain accuracy. Approximately 14 hours are needed to complete the analytical procedures currently being used. TG, however, was found to be an effective quality control tool in screening out samples having an assay value below 97.0% CaCr04. 

It must be emphasized that Eq. (6.7) is only an approximate method for calculating the performance of refrigeration cycles. If greater accuracy is required, the refrigeration cycle must be followed using thermodynamic properties of the refrigerant being used. °... 

At each phase of a project cost information is required to enable decisions to be taken. In the conceptual phase these estimates may be very approximate (e.g. + 35% accuracy), reflecting the degree of uncertainty regarding both reservoir development and surface options. As the project becomes better defined the accuracy of estimates should improve. 

Other limitation for the spatial resolution can be found in the detector. A limited number of pixels in the camera array can be a reason for pure resolution in the case of a big field of view. For example, if field of view should be 10 by 10 nun with camera division 512x512 pixels the pixel size will be approximately 20 microns. To improve the relation of the field of view and the spatial resolution a mega-pixel sensor can be used. One more limitation for the spatial resolution is in mechanical movement (rotation) of the object, camera and source. In the case of a mechanical movement all displacements and rotations should be done with accuracy better than the spatial resolution in any tested place of the object. In the case of big-size assemblies and PCB s it is difficult to avoid vibrations, axle play and object non-planarity during testing. 

It is well established that GO approximation leads to aceurate results if both the source and the observation points are not close to the interface. In practice, this means that both points must be distant from the interface of at least one wavelength. This condition is always fulfilled for the source point. For the field-point, the accuracy of the solution is ensured if the above condition is completed, as shown by comparing exact and approximate results [5]. 

The basic requirements for the Mephisto model was satisfactory accuracy, that means prediction of amplitude, position and phase relation between the various signals, and short computation times, typically a few minutes for the simulation of a whole Cscan, compatible with an intensive use. These a priori contradictory characteristics have been contented by means of appropriate approximations based on physical considerations. 

The maximum bubble pressure method is good to a few tenths percent accuracy, does not depend on contact angle (except insofar as to whether the inner or outer radius of the tube is to be used), and requires only an approximate knowledge of the density of the liquid (if twin tubes are used), and the measurements can be made rapidly. The method is also amenable to remote operation and can be used to measure surface tensions of not easily accessible liquids such as molten metals [29]. 

A highly readable account of early efforts to apply the independent-particle approximation to problems of organic chemistry. Although more accurate computational methods have since been developed for treating all of the problems discussed in the text, its discussion of approximate Hartree-Fock (semiempirical) methods and their accuracy is still useful. Moreover, the view supplied about what was understood and what was not understood in physical organic chemistry three decades ago is... 

The osmotic coefficients from the HNC approximation were calculated from the virial and compressibility equations the discrepancy between ([ly and ((ij is a measure of the accuracy of the approximation. The osmotic coefficients calculated via the energy equation in the MS approximation are comparable in accuracy to the HNC approximation for low valence electrolytes. Figure A2.3.15 shows deviations from the Debye-Htickel limiting law for the energy and osmotic coefficient of a 2-2 RPM electrolyte according to several theories. 

By integrating over the hard cores in the SL expansion and collecting tenns it is easily shown this expansion may be viewed as a correction to the MS approximation which still lacks the complete second virial coefficient. Since the MS approximation has a simple analytic fomi within an accuracy comparable to the Pade (SL6(P)) approximation it may be more convenient to consider the union of the MS approximation with Mayer theory. Systematic improvements to the MS approxunation for the free energy were used to detemiine... 

If we consider the optical response of a molecular monolayer of increasing surface density, the fomi of equation B 1.5.43 is justified in the limit of relatively low density where local-field interactions between the adsorbed species may be neglected. It is difficult to produce any rule for the range of validity of this approximation, as it depends strongly on the system under study, as well as on the desired level of accuracy for the measurement. The relevant corrections, which may be viewed as analogous to the Clausius-Mossotti corrections in linear optics, have been the... 

To improve upon die mean-field picture of electronic structure, one must move beyond the singleconfiguration approximation. It is essential to do so to achieve higher accuracy, but it is also important to do so to achieve a conceptually correct view of the chemical electronic structure. Although the picture of configurations in which A electrons occupy A spin orbitals may be familiar and usefiil for systematizing the electronic states of atoms and molecules, these constructs are approximations to the true states of the system. They were introduced when the mean-field approximation was made, and neither orbitals nor configurations can be claimed to describe the proper eigenstates T, . It is thus inconsistent to insist that the carbon atom... 

The accuracy of most TB schemes is rather low, although some implementations may reach the accuracy of more advanced self-consistent LCAO methods (for examples of the latter see [18,19 and 20]). However, the advantages of TB are that it is fast, provides at least approximate electronic properties and can be used for quite large systems (e.g., thousands of atoms), unlike some of the more accurate condensed matter methods. TB results can also be used as input to detennine other properties (e.g., photoemission spectra) for which high accuracy is not essential. 

In the connnonly used atomic sphere approximation (ASA) [79], the density and the potential of the crystal are approximated as spherically synnnetric within overlapping imifiBn-tin spheres. Additionally, all integrals, such as for the Coulomb potential, are perfonned only over the spheres. The limits on the accuracy of the method imposed by the ASA can be overcome with the fiill-potential version of the LMTO (FP-LMTO)... 

The projector augmented-wave (PAW) DFT method was invented by Blochl to generalize both the pseudopotential and the LAPW DFT teclmiques [M]- PAW, however, provides all-electron one-particle wavefiinctions not accessible with the pseudopotential approach. The central idea of the PAW is to express the all-electron quantities in tenns of a pseudo-wavefiinction (easily expanded in plane waves) tenn that describes mterstitial contributions well, and one-centre corrections expanded in tenns of atom-centred fiinctions, that allow for the recovery of the all-electron quantities. The LAPW method is a special case of the PAW method and the pseudopotential fonnalism is obtained by an approximation. Comparisons of the PAW method to other all-electron methods show an accuracy similar to the FLAPW results and an efficiency comparable to plane wave pseudopotential calculations [, ]. PAW is also fonnulated to carry out DFT dynamics, where the forces on nuclei and wavefiinctions are calculated from the PAW wavefiinctions. (Another all-electron DFT molecular dynamics teclmique using a mixed-basis approach is applied in [84].)... 

By substituting these expressions into Eq. (55), one can see after some algebra that ln,g(x, t) can be identified with lnx (t) + P t) shown in Section III.C.4. Moreover, In (f) = 0. It can be verified, numerically or algebraically, that the log-modulus and phase of In X-(t) obey the reciprocal relations (9) and (10). In more realistic cases (i.e., with several Gaussians), Eq. (56-58) do not hold. It still may be due that the analytical properties of the wavepacket remain valid and so do relations (9) and (10). If so, then these can be thought of as providing numerical checks on the accuracy of approximate wavepackets. 

The Hemian-Kluk method has been developed further [153-155], and used in a number of applications [156-159]. Despite the formal accuracy of the approach, it has difficulties, especially if chaotic regions of phase space are present. It also needs many trajectories to converge, and the initial integration is time consuming for large systems. Despite these problems, the frozen Gaussian approximation is the basis of the spawning method that has been applied to... 

Van Gunsteren, W.F., Beutler, T.C., Praternali, F., King, P.M., Mark, A.E., Smith, P.E. Computation of free energy in practice Choice of approximations and accuracy limiting factors, in Computer Simulations of Biomolecular Systems, Vol 2, W.F. van Gunsteren, P.K. Weiner and A.J. Wilkinson, eds. Escom, Leiden (1993) 315-348. 

The many approaches to the challenging timestep problem in biomolecular dynamics have achieved success with similar final schemes. However, the individual routes taken to produce these methods — via implicit integration, harmonic approximation, other separating frameworks, and/or force splitting into frequency classes — have been quite different. Each path has encountered different problems along the way which only increased our understanding of the numerical, computational, and accuracy issues involved. This contribution reported on our experiences in this quest. LN has its roots in LIN, which... 

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