Two further approximations are of interest. When the fraction of reaction is small (as a practical matter, say f < 0.1, more conservatively f < 0.05) the right-hand-side of Equation 7.17 can be simplified using [Pg.210]

HyperChem s ab initio calculations solve the Roothaan equations (59) on page 225 without any further approximation apart from the use of a specific finite basis set. Therefore, ab initio calculations are generally more accurate than semi-empirical calculations. They certainly involve a more fundamental approach to solving the Schrodinger equation than do semi-empirical methods. [Pg.251]

Bearing in mind the discussion of the nature of the electronic non BO matrix elements (Q) given in Sec. I. C, the above rate expression can be further approximated by constraining Q and Q to the region Q =Q=Qo where the anion and neutral surfaces approach most closely [Pg.310]

In dilute solutions, the water is almost pure and its activity can be set equal to 1. With this approximation, we obtain the basicity constant, Kb. If we make the further approximation of replacing the activities of the solute species by the numerical values of their molar concentrations, we can write the basicity constant expression for ammonia as [Pg.528]

Mozumder (1971) calculated F(t) by the prescribed diffusion method. For the isolated ion-pair case, the solution appears in (7.28) for the multiple ion-pair case, further approximation was introduced in the nature of mean force acting on an electron, by which the problem was reduced to that of a collection of isolated [Pg.233]

If we believe that D = diag(u), that is, that errors are independent and proportional to the square root of the predicted value, then D" = diag(l/M )/o, where we may further approximate this result by estimating [Pg.80]

Again for illustration purpose, we supposedly have chosen Kc such that KcKvKpKm = 5, and xp is the mixing process time constant. Find, without trial-and-error and without further approximation, the maximum distance L that the photodetector can be placed downstream such that the system remains stable. (There are two ways to get the answer. The idea of using magnitude and phase [Pg.166]

In the Neglect of Diatomic Differential Overlap (NDDO) approximation there are no further approximations than those mentioned above. Using p and n to denote either an s-or p-type (pj, p or p ) orbital, the NDDO approximation is defined by the following equations. [Pg.82]

In different applications different approximations are introduced during evaluation of the matrix elements Hmn and Smn. Procedures in which the matrix elements are evaluated without further approximation, are called ab initio, which means from the beginning. The quantities e so obtained represent one-electron energies or orbital energies. [Pg.383]

The quantum mechanics methods in HyperChem differ in how they approximate the Schrodinger equation and how they compute potential energy. The ab initio method expands molecular orbitals into a linear combination of atomic orbitals (LCAO) and does not introduce any further approximation. [Pg.34]

This is a quadratic equation in [H + ] and may be solved in the usual manner. It can, however, be simplified by introducing the following further approximations. In a mixture of a weak acid and its salt, the dissociation of the acid is repressed by the common ion effect, and [H + ] may be taken as negligibly small by [Pg.46]

The parameter n increases exponentially with increasing

It is important to note that the only physical approximation here is the LDA. All other approximations are of a numerical nature and their convergence can be monitored and improved in a systematic way. Thus, this method allows a probe of the LDA limit for molecules and clusters without any further approximations such [Pg.53]

Equation (7.120) can be used to calculate II in a dilute solution where Raoult s law is a good approximation of the behavior of the solvent. This equation is often put in an alternate form by making further approximations that apply to the dilute solution. First, we write [Pg.371]

In Table 5 the insertion barrier at levels of theory higher than MP2 are also reported (runs 10-13). The MP3 and MP4 insertion barriers are both remarkably higher than the MP2 barrier. The CCSD insertion barrier also is quite larger than the MP2 barrier (5.2 kcal/mol above), but the perturbative inclusion of triple excitations in the couple cluster calculations reduces considerably the CCSD barrier, which is 8.7 kcal/mol (3.1 kcal/mol above the MP2 insertion barrier). The insertion barriers reported in Table 5 can be used to obtain a further approximation of the insertion barrier. In fact, the CCSD(T) barrier of 8.7 kcal/mol should be lowered by roughly 3 kcal/mol if [Pg.41]

The COSMO method is a solution of the Poisson equation designed primarily for the case of very high e [190], It takes advantage of an analytic solution for the case of a conductor (e = ). The difference between (l--)for the case of e = 80 and e = °°is only 1.3%, so this is a good approximation for water. Its use for the treatment of nonpolar solvents with e 2 depends on further approximations which have not yet been sufficiently tested to permit an evaluation of their efficacy. [Pg.28]

A study of the effect of the mesophase layer on the thermomechanical behaviour and the transfer mechanism of loads between phases of composites will be presented in this study. Suitable theoretical models shall be presented, where the mesophase is taken into consideration as an additional intermediate phase. To a first approximation the mesophase material is considered as a homogeneous isotropic one, while, in further approximations, more sophisticated models have been developed, in which the mesophase material is considered as an inhomogeneous material with progressively varying properties between inclusions and matrix. Thus, improvements of the basic Hashin-Rosen models have been incorporated, making the new models more flexible and suitable to describe the real behaviour of composites. [Pg.151]

All the treatments discussed above have been concerned with constant conditions, i.e., where in the accelerated tests the level of the degrading agents has been held constant throughout one exposure, and any extrapolation to service implicitly assumes that conditions there will also be constant. In real life, however, it is much more likely that service conditions will be variable or cyclic. Generally, therefore, further approximations have to be made. [Pg.127]

Here, the densities of the gaseous and solid fuels are denoted by pg and ps respectively and their specific heats by cpg and cps. D and A are the dispersion coefficient and the effective heat conductivity of the bed, respectively. The gas velocity in the pores is indicated by ug. The reaction source term is indicated with R, the enthalpy of reaction with AH, and the mass based stoichiometric coefficient with u. In Ref. [12] an asymptotic solution is found for high activation energies. Since this approximation is not always valid we solved the equations numerically without further approximations. Tables 8.1 and 8.2 give details of the model. [Pg.172]

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