Two approximation

However, one can proceed beyond this zeroth approximation, and this was done independently by Guggenheim (1935) with his quasi-chemicaT approximation for simple mixtures and by Bethe (1935) for the order-disorder solid. These two approximations, which turned out to be identical, yield some enliancement to the probability of finding like or unlike pairs, depending on the sign of and on the coordmation number z of the lattice. (For the unphysical limit of z equal to infinity, they reduce to the mean-field results.)... 

In the derivation used here, it is clear that two approximations have been made—the configurations are incoherent, and the nuclear functions remain localized. Without these approximations, the wave function fonn Eq. (C.l) could be an exact solution of the Schrddinger equation, as it is in 2D MCTDH form (in fact is in what is termed a natural orbital form as only diagonal configurations are included [20]). 

The truncated octahedron and the rhombic dodecahedron provide periodic cells that are approximately spherical and so may be more appropriate for simulations of spherical molecules. The distance between adjacent cells in the truncated octahedron or the rhombic df)decahedron is larger than the conventional cube for a system with a given number of particles and so a simulation using one of the spherical cells will require fewer particles than a comparable simulation using a cubic cell. Of the two approximately spherical cells, the truncated octahedron is often preferred as it is somewhat easier to program. The hexagonal prism can be used to simulate molecules with a cylindrical shape such as DNA. 

In practice, it is best to purify a quantity, say one Winchester quart bottle, of technical 0 720 ether to cover the requirements of a group of students. The Winchester quart of ether is divided into two approximately equal volumes, and each is shaken vigorously in a large separatory funnel with 10-20 ml. of the above ferrous solution diluted with 100 ml. of water. The latter is removed, the ether transferred to the Winchester bottle, and 150-200 g. of anhydrous calcium chloride is added. The mixture is allowed to stand for at least 24 hours with occasional shaking. Both the water and the alcohol present are thus largely removed. The ether is then filtered through a large fluted filter paper into another clean dry Winchester bottle (CAUTION all flames in the vicinity must be... 

Researchers must be particularly cautious when using one estimated property as the input for another estimation technique. This is because possible error can increase significantly when two approximate techniques are combined. Unfortunately, there are some cases in which this is the only available method for computing a property. In this case, researchers are advised to work out the error propagation to determine an estimated error in the final answer. 

The mechanism of the cathode reaction for all three types of Mn02 can best be described by two approximately one-electron steps. 

SASA), a concept introduced by Lee and Richards [9], and the electrostatic free energy contribution on the basis of the Poisson-Boltzmann (PB) equation of macroscopic electrostatics, an idea that goes back to Born [10], Debye and Htickel [11], Kirkwood [12], and Onsager [13]. The combination of these two approximations forms the SASA/PB implicit solvent model. In the next section we analyze the microscopic significance of the nonpolar and electrostatic free energy contributions and describe the SASA/PB implicit solvent model. 

From (5.56) one can obtain an integro-differential equation for operator What we need is the mean particle position, <(Tz>, and in order to find it two approximations are made. First, in taking the bath averages we assume free bath dynamics. Second, we decouple the bath and pseudospin averages, guided by perturbation theory. The result is a Langevin-like equation for the expectation <(T2> [Dekker 1987a Meyer and Ernst 1987 Waxman 1985],... 

The dark-colored residue is divided into two approximately equal portions, and each is poured, while still hot, into a 4-I. beaker. After the liquid has cooled to about 70°, 1650 cc. of acetone is added to each portion. The mixture is stirred well, so that as much as possible of the dark-colored solid is brought into contact with the acetone. After standing in the ice box overnight, the jS-bromoethylamine hydrobromide is collected on a filter, washed with acetone until colorless (Note 3), and air-dried... 

In several works published after Flory (e.g. Tang and Yao7)) elimination of water is not taken into account either in the calculation of the conversion or in the general kinetic relation. However, these two approximations cancel each other and the final relation is correct. 

For large distortions, two approximations underlying the preceding calculations should be reconsidered. 

Isothermal a—time curves for the decomposition of U02(CH3C02)2 in air (513—573 K) [1018] showed two approximately linear regions, 0.0 < a < 0.2 and 0.2 < a < 0.9, for which the values of E were 107 and 165 kJ mole-1, respectively. In nitrogen, the earlier portion of the curve was not linear and E = 151 kJ mole-1 for the later interval. The zero-order kinetic behaviour was explained by growth of nuclei in thin, plate-like crystals, which were shown by microscopic and surface area measurements to fragment when a > 0.85. The proposed initial step in the decomposition was fission of bonds between the U02+ and the (OCO CH3) species [1018]. 

Phospholipids form bilayers in aqueous media. The molecules form two approximately parallel rows with tails on the inside and heads toward the outside, in contact with the solution. This arrangement, shown in Figure 12-18. is analogous to the vesicles in Figure 12-17. The bilayer forms a closed sac that contains the aqueous cytoplasm and all the cellular components. Thus, a cell can be viewed as a large and complex vesicle. 

Because ATeq is very iarge, the amount of the iimiting reactant present at equiiibrium, y, wiii be very smaii compared with the amounts of materiais present at compietion. That is the point of taking the reaction to compietion and then working back toward equiiibrium. This ieads to two approximations 10.0-y=10.0 and 5.0 + 0.5y=5.0... 

We expect y to have a small value, so we make two approximations ... 

The lowering of the energy in the ground state with respect to the FCI result is due to the tr Y term which is much too low in the two approximations. This error is compensated to a certain extent by errors in the opposite direction of the two other terms. 

Finally in the third state the two approximations give very similar energy values, both with higher energy than the FCI one. In each approximation, the error of the different terms compensate each other to a certain extent. 

This means that we have made two approximations. One of them is replacement of the total field by a normal field, and the second is a shift of the interval of integration. Since the secondary field is very small, as are the intervals BBi and AAi, we may expect that the errors caused by these approximations are relatively small. Besides, the normal field varies relatively slowly within the interval BiA. Correspondingly, U(4>,h) and U(4>,0) are values of the normal potential at points Ai and Bi. From Equations (2.290 and 2.291) we have... 

One of the few crystal structures solved for a metal complex of a heterocyclic thiosemicarbazone is bis(l-formylisoquinoline thiosemicarbazone) nickel(II) monohydrate, [Ni(21-H)2] (Fig. 5) [208]. The nickel(II) center was found to be NNS coordinated by two approximately planar thiosemicarbazone ligands. 

Chapters 7 and 8 discuss spin and identical particles, respectively, and each chapter introduces an additional postulate. The treatment in Chapter 7 is limited to spin one-half particles, since these are the particles of interest to chemists. Chapter 8 provides the link between quantum mechanics and statistical mechanics. To emphasize that link, the ffee-electron gas and Bose-Einstein condensation are discussed. Chapter 9 presents two approximation procedures, the variation method and perturbation theory, while Chapter 10 treats molecular structure and nuclear motion. 

Now in addition to all these Si and Ti minima which have counterparts in the ground state surface, minima (or funnels) in Si and Ti can also be expected at biradicaloid geometries. These are geometries at which the molecule, in the MO description, has two approximately... 

A riffler is a mechanical device, consisting of a metal box containing a number of equally spaced slots, which is used for dividing a sample into two approximately equal portions. The material to be subdivided is poured into the top of the box and emerges through the slots on opposite sides in two approximately equal portions. As with coning and quartering, the procedure is repeated until the desired sample size is obtained. 

We describe two approximate methods of determining the value of N in the TIS model from pulse-tracer experiments. One is based on the first moment or mean 0, and the other on the variance a as determined from the tracer data. 

While these two approximations are heuristic, the cases treated in Sections V-C and V-E correspond to well defined mathematical limiting procedures. In both models, one considers a dilute electrolyte (see Eq. (393)) with heavy ions (see Eq. (394)) the difference lies in the relation connecting these two limits. The small friction case, Eq. (395), corresponds to the plasma-dynamic model (Eq. (361)) ... 

The degree of dissociation is a = pic = 1 - qlc. We now consider the three special cases shown in Table 3. In order to ascertain which, if any, of the two approximations may be valid, we need to examine the magnitudes of K, Ky and c which are relevant to the systems under discussion. To determine a value of K which will be relevant to polymerizing systems, we need the dissociation constant of a carbonium salt with a large anion, in a solvent of e about 10, at 25 °C. The only relevant information is Longworth and Mason s value of K for triphenylmethyl perchlorate in ethylene dichloride [53], and values of K for two quaternary ammonium perchlorates [140] (see Table 4). 

The correlation methods in G3 theory are still computationally demanding and it is of interest to find modifications to reduce the computational requirements. Two approximate versions of G3 theory have been proposed to make the methods more widely applicable. The first is G3(MP3) [22] that eliminates the expensive MP4/2df,p calculation by evaluating the larger basis set effects at the MP3 level. It also eliminates the MP4/plus calculation,... 

---