Neglecting terms

The desired accuracy in prediction is seldom the maximal accuracy that can be obtained numerically. A good result is if a reactor volume can be estimated to within 10%. Often the calculated reactor volume is increased by 20% to 50% in order to compensate for uncertainties in rate constants and other parameters. In many cases, we can eliminate the terms in the equations that contribute only a few percent to the final result. If we can estimate whether omitting the terms will give an over- or underestimation of the final result, we can compensate for this uncertainty. 

First we need some tools for analyzing differential equations. The following method is very simple and usefiil, but gives only an order-of-magnitude estimation of the terms in a differential equation. It is always necessary to verify that the assumptions were justified after the equations have been solved and more accurate estimations of the derivatives are possible. 

In a general differential equation, it may be difficult to estimate the influence of different terms. One approach is to assume a solution and numerically approximate the derivatives. After solving the simplified equation by canceling unimportant terms, the initial assumptions can be confirmed. 

Find a criterion for neglecting the axial dispersion of a first-order reaction in a tubular reactor. A first-order reaction in a tubular reactor without dispersion is described by 

A coarse estimation of the derivatives is for the first-order derivative dC AC 0 - 

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The success of FMO theory is not because the neglected terms in the second-order perturbation expansion (eq. (15.1)) are especially small an actual calculation will reveal that they completely swamp the HOMO-LUMO contribution. The deeper reason is that the shapes of the HOMO and LUMO resemble features in the total electron density, which determines the reactivity. 

Neglecting terms of order greater than quadratic In B, we see that Eq. (A.2) simplifies to... 

Expanding u in Taylor series around the true 0 and neglecting terms of... 

This is a relationship between unknown field g and two measured quantities, namely, the distance 5 and time t, provided that we neglect terms proportional to the square of the coefficient a and those of higher order. Besides, this equation contains three unknown parameters, namely, the position of the mass. so at the moment when we start to measure time, the initial velocity, vo, at this moment and, finally, the rate of change of the gravitational field, a, along the vertical. Thus, in order to solve our problem and find the field we have to perform measurements of the distance. s at four instants. If so is known, the number of these measurements is reduced by one. In modern devices the coefficient of the last term on the right hand side of Equation (3.14) has a value around 100 pGal and it is defined by calculations as a correction factor s(vo, g, t, a). In the case when we can let so equal to zero, it is sufficient to make measurements at two instances only. 

Substituting the latter into the first and last equations of the set (3.74) and neglecting terms proportional to a>, we obtain after integration... 

For long tubes and cylindrical vessels this expression can be simplified by neglecting terms with the group (2L/nDo)2 in the denominator the equation then becomes ... 

This equation has been used by Sundstrom and coworkers [151] and adapted to the analysis of femtosecond spectral evolution as monitored by the bond-twisting events in barrierless isomerization in solution. The theoretical derivation of Aberg et al. establishes a link between the Smoluchowski equation with a sink and the Schrodinger equation of a solute coupled to a thermal bath. The reader is referred to this important work for further theoretical details and a thorough description of the experimental set up. It is sufficient to say here that the classical link is established via the Hamilton-Jacobi equation formalism. By using the standard ansatz Xn(X,t)= A(X,i)cxp(S(X,t)/i1l), where S(X,t) is the action of the dynamical system, and neglecting terms in once this... 

The neglected terms are of lower order by a factor of the order of [(B)-1 x (range of interaction)8]. In every M only the fully... 

The initial benchmark results obtained with the full CCSD-R12 method [34] testified that the various simplified CCSD-R12 methods reported earlier were highly accurate approximations to the full CCSD-R12 method unless the basis set was too small. The assumptions about the relative importance of diagrammatic terms made in these simplified methods were proven to be valid. However, these neglected terms do not increase the computational cost scaling of CCSD-R12 and there appears no need to eliminate them from full CCSD-R12, once they are implemented. In other words, it is important to distinguish whether a certain approximation is motivated by a compromise between accuracy and the computational cost or by that between accuracy and the development cost. The latter has become increasingly unjustifiable with the advent of computerized derivation and implementation. 

An alternative approach to our problem is Newton s method. The idea behind this method is illustrated in Fig. 3.5. If we define g(x) = f (x). then from a Taylor expansion g(x I It) = g(x) + hg (x). This expression neglects terms with higher orders of h, so it is accurate for small values of h. If we have evaluated our function at some position where g(x) A 0, our approximate equation suggests that a good estimate for a place where the function is zero is to define x = x I h = x g(x)/g x). Because the expression from the Taylor expansion is only approximate, x does not exactly define a place where g(.ri) = 0. Just as we did with the bisection method, we now have to repeat our calculation. For Newton s method, we repeat the calculation starting with the estimate from the previous step. 

Comparing Eq. (8) with Eq. (5), neglecting terms having powers higher than 2 in c gives... 

The activity coefficients for the solutes can be simplified by neglecting terms of the order v-JV. We will split the activity coefficient into the part for infinite dilution and the part depending on concentration. This results in... 

Considering only the first two previously neglected terms in (4.12) and (4.21), we have as the perturbation... 

We shall neglect terms higher than quadratic in (6.7) this is a good approximation if the vibrations are small. Equation (6.7) becomes... 

Actually, many other infrared transitions occur besides those allowed by the selection rule (6.74). The neglected terms in the expansion (6.66) will give transitions with a change of 2 or more in a given vibrational quantum number and transitions in which more than one vibrational quantum number changes moreover, anharmonicity corrections to the vibrational wave function will add to the probability of such transitions. Generally, the transitions (6.74) are the strongest. 

Let us consider as the basic equations those of the diffusion-controlled stochastic Lotka model which are derived in the superposition approximation, thus neglecting terms having a small parameter 5(t). 

Substituting the value of C in equation (17) with p = 2 and neglecting terms which tend to zero we have... 

To find the last line we expanded Rc(const,, dc(A)) about, , neglecting terms of order (/ e(A) — /3 )2. This is consistent with the linearization (10,31) of... 

The notion of canonical order needs some explanation. A priori in jtt-tli order of perturbation theory in u the leading neglected term is found to be proportional to... 

Then by neglecting terms of the second order in co, one finds... 

Let us return to the case where reaction takes place via a nonadiabatic transition. This situation typically occurs when the PES is constructed from a Hamiltonian in which one or more terms have been neglected. These terms then couple the initial and final states, thereby providing a mechanism for reaction to take place. The neglected terms may include, for example,... 

However, in most cases a simplified heat balance, which comprises the two first terms on the right-hand side of Equation 2.26, is sufficient for safety purposes. Let us consider a simplified heat balance, neglecting terms such as the heat input by the stirrer or heat losses. Then, the heat balance for a batch reactor can be written as... 

Note that using Eq. (7.167) is not possible as we have neglected terms proportional to k2 in solution (7.172).) This can in fact more cleverly be rewritten... 

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