Alternative approximations

In analogy to the RHF reference wave function, these terms are calculated in the indirect way by modifying the intermediate 

Recently Adler et al. proposed another approximation to the CCSD(F12) model, called CCSD-F12b, where even more F12 terms are neglected [61, 62]. Besides the aspects that refer to the fixed-amplitudes approximation, several additional simplifications are 

Within the present work, both approximated schemes have been implemented. In the case of the CCSD-F12b model, the inclusion of the aibj xiyj terms would require programming the explicit contractions [Eqs. (227), (228)]. They were not necessary in the implementation of the CCSD(F12) model, because these terms were accounted for through the intermediates [Eqs. (230), (231)]. For these reasons they have not been included, and the model implemented in Turbomole is dubbed CCSD-F12b.  

Recently, Norinaga and Deutschmann modeled the chemical kinetics of the pyrolysis of the hydrocarbons ethylene, acetylene, and propylene under conditions relevant to the chemical vapor deposition (CVD) of pyrolytic carbon [63], For this modeling, they had developed a mechanism containing 227 species and 827 reactions. One of these was 

In the present work, we show that the production of ethane from the reaction of methane with methyl is grossly overestimated by the above rate constant. From transition-state theory (TST) and highly correlated ab initio calculations, we find rates that are between three (at 1500 K) and ten (at 300 K) orders of magnitudes slower than thc e used by Norinaga and Deutschmann [63]. This leads us to conclude that the role of this reaction in the mechanism must be reexamined. Furthermore, we show that the barrier height 

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The possibility of a nongradient component in the integrand introduces some difficulty and an alternative formulation has been proposed [273,283]. [At positions that are close to a ci, the alternative approximates well to the angle shown above in Eq. (73).]... 

An alternative approximation scheme, also proposed by Bom and Oppenheimer [5-7], employed the straightforward perturbation method. To tell the difference between these two different BO approximation, we call the latter the crude BOA (CBOA). A main purpose of this chapter is to study the original BO approximation, which is often referred to as the crude BO approximation and to develop this approximation into a practical method for computing potential energy suifaces of molecules. 

An alternative approximation to the adiabatic probability is to invoke an instantaneous equilibrium-like probability. In the context of the work theorem, Hatano and Sasa [72] analyzed a nonequilibrium probability distribution that had no memory, and others have also invoked a nonequilibrium probability distribution that is essentially a Boltzmann factor of the instantaneous value of the time-dependent potential [73, 74]. 

If the functional form of a molecular electron density is known, then various molecular properties affecting reactivity can be determined by quantum chemical computational techniques or alternative approximate methods. 

Thus, within the approximations on which scheme (1) is based, the yield Y/( 1 + Y) of ring product is solely determined by the value of the initial monomer concentration relative to the kintra/kdim ratio, which Stoll et al. termed the cyclisation constant C. C has units of mol 1 1 and represents the monomer concentration at which intra- and intermolecular processes occur at the same rate. That Stoll et al. disregarded the fact that two monomer units are used up in the intermolecular condensation, as well as that the monomer disappears by reaction with functional groups at the ends of polymer chains was pointed out by Morawetz and Goodman (1970), who proposed an alternative approximate solution to the problem, as based on... 

Any alternative approximation for the activity coefficients could be applied, the principle is the same. 

If the phase transition is a vaporization or sublimation, and if the vapor can be assumed to be an ideal gas, an alternative approximation applies ... 

A preferred sampling location is open to the free movement of the ambient air, at least 3 m above grass, 8 m fix>m shrubs, 40 m from large trees, and 120 m from heavy automotive traffic. The flat roof of a one-or two-story building is ideal. A probe that projects through a wall and ertends about 2 m beyond is an acceptable alternative approximating the roof. 

It will not always be possible to use the more accurate definition given in equation (A8.4) above, for example in cases where the required physical property data would be difficult to generate. In such cases, the following alternative approximate definitions of Omega have been proposed[3,4,51. However, the additional condition of applicability (h) in A8.2 needs to be considered before their use. Omega can be calculated by any of equations (A8.5) to (A8.10) below, which are essentially equivalent. In these equations, Omega is evaluated entirely at the conditions in the upstream reactor. ... 

A simple treatment, which circumvents this difficulty, uses an alternative approximate treatment of the image forces. Because the Born energy of ions in bulk water is given by... 

Alternatively, approximately 10 mg of a dried sample is accurately weighed and placed in a 1-ml micro-reaction vial fitted with a Teflon-lined screwcap. Acetic anhydride (0.1 ml) and 20(A of pyridine are added to the sample. For quantitative GC analysis, a measured amount of an appropriate internal standard is added to the mixture. The reaction vial is tightly sealed with the screwcap, and the resulting mixture is heated at 60°C for 2h with occasional shaking. The sample solution is now ready for GC and GC-MS analyses (Schultz et al. 1981). 

An alternative approximation to the derivative is the central-difference form... 

The SCF criteria were 10 relative change in the total potential in all cases. This implies 1-2 cV inaccuracy in the total energy, that is why we used an alternative approximation, the so called Slater s transition slate ( STS ) approach. 

Differences between different enzymes were supposed to be confined to differences in L they were not supposed to affect the form, or position of the dependence of the reaction rate on AG. However, for the dashed curve in Fig. 1.1, the part where the approximation by this equation would be reasonable, is at very small reaction rates. In the presumably functionally more relevant part (v/V between 0.1 and 0.9), this equation (i.e., a straight line through the origin) is a completely unsatisfactory approximation of the actual relationship between v and AG. Yet, around = 0.5 a different, though linear, approximation would seem possible. Below, we shall further substantiate this alternative approximation. 

Once a relaxation (retardation) spectrum is obtained from a relaxation (creep compliance) viscoelastic function, any other function can be obtained. Alternatively, approximate methods have been developed to calculate viscoelastic functions from one another (10). By taking into account... 

An alternative approximation to a complete EOMXCCSD method can be proposed if we replace the and H2 2) terms contributing to Hlf, Eq. (179), by their low-order counterparts obtained by using the formulas (cf. Figure 1),... 

An alternate approximation gives areas with a less than 1% error ... 

Let us assume that AH2 > AHi > AH3, as the energy of atomic disorder is normally larger than that needed for intrinsic electronic disordering in turn, the latter is higher with respect to the energetic effects related to defect charging (see Figures 3.3 and 3.4). Under this assumption, it is possible to consider two alternative approximations. 

AT, are state variables, and the solution requires that we know their values at time to - the same requirement as for all the other state variables in the simulation. The values, Tk, might be taken initially as the design values. Alternatively, approximate values could be chosen, and the model run to the steady state appropriate for the study under consideration. 

An alternative approximate solution to this problem can be obtained through the use of k -type coefficients. As mentioned in Example 2.3, the McCabe-Thiele method of analysis of distillation columns assumes equimolar counterdiffusion, which would justify the use of k-type coefficients regardless of the concentration levels. This would be exactly so only if the molar latent heats of vaporization of both components were equal. Although in this case there is a significant difference between the two heats of vaporization, we will show that the approximate solution is fairly close to the rigorous solution obtained above. It is easily shown that in this case, assuming steady-state equimolar counterdiffusion,... 

An alternative, approximate, approach is to use Slater s transition-state method.64 Then the energies of interest are calculated directly by performing calculations for configurations in which not one but only half an electron has been removed, added, or excited. 

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