Extended calculations

The theoretical results provided by the large basis sets II-V are much smaller than those from previous references [15-18] the present findings confirm that the second-hyperpolarizability is largely affected by the basis set characteristics. It is very difficult to assess the accuracy of a given CHF calculation of 2(ap iS, and it may well happen that smaller basis sets provide theoretical values of apparently better quality. Whereas the diagonal eomponents of the eleetrie dipole polarizability are quadratic properties for which the Hartree-Fock limit can be estimated with relative accuracy a posteriori, e.g., via extended calculations [38], it does not seem possible to establish a variational principle for, and/or upper and lower bounds to, either and atris-... 

So far, we have seen several ways of calculating the Gibbs free energy of a two-component mixture. To extend calculations to ternary and higher-order mixtures, we use empirical combinatory extensions of the binary properties. We summarize here only some of the most popular approaches. An extended comparative appraisal of the properties of ternary and higher-order mixtures can be found in Barron (1976), Grover (1977), Hillert (1980), Bertrand et al. (1983), Acree (1984), and Fei et al. (1986). 

Although the investigation of Segal and coworkers gives a first basis for the understanding of the ultraviolet absorption spectrum of 1, it leaves a number of open questions, which can only be answered by more extended calculations. Such calculations are possible and have been done for derivatives of 1 (see, e.g., the MR-CI calculations for fluoro- and methyl-cyclopropanone with more than 50,000 spin adapted configurations214) but not for 1 itself. Therefore, further calculations are needed to get additional information on the excited states of 1. 

Condensation calculations can be carried out to relatively low temperatures Grossman and Larimer (1974) extended calculations to below 400 K, and Lodders (2003) lists 50% condensation temperatures as low as 9 K , but the meaning of calculations below —1,000 K is unclear as gas-solid equilibrium is unlikely to be reached. 

The method of extended calculation of lattice energies described in the previous sections is restricted to salts the crystal structures of which are exactly known. For less simple structures than those of the alkali halides the extended calculation of lattice energies becomes increasingly laborious as the symmetry decreases. If however only the first two terms in the lattice energy are considered, that is, U is taken to be Uu — Vr, then adopting either an inverse power law or an exponential law for the repulsive term, U may be written as either... 

Kapustinskii s equation, of course, cannot be expected to produce values for the lattice energies as exact as those produced by extended calculation. Essentially the Kapustinskii formula ignores all contributions to the lattice energy save Us and Uu. If the ions are nonspherical there is considerable difficulty in choosing a repulsion envelope for the ion and assigning a radius. This difficulty also arises in the extended calculation of the lattice energy but only in calculation of Ur, which is an order of magnitude less than Um- In the Kapustinskii equation this difficulty arises in Um as well. 

The lattice energies of the alkali metal bifluorides have been recently computed by Waddington 127) by extended calculation. Values obtained are KHF2, 153.4 RbHF2, 147.3 CsHF2, 143.3 kcal/mole. They lead to a value for AH/ HF2 (g) of —150 5 kcal/mole. 

Abstract We describe in detail the determination of differential cross sections for the abstraction reaction of a halogen atom with molecular hydrogen. As an illustration we give some examples of differential cross sections for the F+H2—>HF+H and CI+H2—>HC1+H reactions, extending calculations described in earlier publications from our research groups. 

When these extended calculations are compared with experiment many compounds agree well (see Table 1). Significant deviations do occur, however for example, in compounds of metals in later groups where bonding is certainly less ionic (e.g. AgCl). 

More extended calculations are needed to derive the values of airway resistance and compliance of the lung. Processing of compliance of the ventilator and the hoses to the patient... 

Relativistic effects of the environment are included indirectly due to the solution of the KS-equations for the whole system. Special techniques have been developed for choosing the size and shape of the environment and of an initial electronic density on atomic centers. Application of this method to the calculations of interaction of Cu with Cu-clusters of different size shows good agreement with extended calculations of other workers (see [124] and references therein). The method is presently applied to the calculations of the interaction of the heaviest element 112 and its lighter homolog Hg with transition-metal clusters. First results are reported for those elements interacting with small gold clusters [125] (see Section 3.1). 

Figure 13. Kinetic falloff analysis of Y(CHsCF, F) results for F-for-F activated CH CF F (9) gas phase Y(CHsCF F) (O) liquid-phase Y(CHsCF F) left-side curve, Equation 94 and Table XV right-side curve, Equation 94 and Table XVI (----) extended calculated re-...

Commentarii degallica, see Gilbert, 2002, pp265-268 for extended calculations Schonfield, 1956, pp 1-7 Jeremiah 25 26 51 41... 

Extended calculations were carried out for conformations associated with the and G G states for purposes of application of... 

Just as several STOs are needed to give an accurate representation of Hartree-Fock AOs (Section 11.1), one needs more than one STO of a given n and / in the linear combination of STOs that is to accurately represent the Hartree-Fock MO. The primed and double-primed AOs in the extended-basis-set function are STOs with different orbital exponents. The 3dquantum number m = 0, that is, the 3do and 4/o AOs. The total energies found are -197.877 and -198.768 hartrees for the minimal and extended calculations, respectively. The experimental energy of F2 at Rg is -199.670 hartrees, so the error for the minimal calculation is twice that of the extended calculation. The extended-basis-set calculation is believed to give a wave function quite close to the true Hartree-Fock wave function. Therefore, the correlation energy in F2 is about -0.90 hartrees = -24.5 eV. 

Very extended calculations of various physical properties (including Im xisingle-impurity Anderson model in NCA. Figure 18 shows Im x (o), normalized to its maximum value and compared with a Lorentzian and experimental data for CePdj at 5 K from Galera et al. (1985a,b). For the... 

Fig. VI. 12 also shows the behavior expected at smaller. r, which results from more extended calculations. The plot is reminiscent of a phase transition at values of s slightly larger than a critical value Sg, there are two energy minima one at small X (coil) and one at large X (stretch). These minima are separated by a huge potential barrier, and the chain usually sticks to one and the same minimum. At large s s > Sg) the coil minimum ceases to exist, and the chain must go to the stretched state (if the gradient s is applied over a long enough time). Related mechanical models...

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