Errors in calculations

Once there is an estimate for the error in calculating the adiabatic-to-diabatic tiansfomiation matrix it is possible to estimate the error in calculating the diabatic potentials. For this purpose, we apply Eq. (22). It is seen that the error is of the second order in , namely, of 0( ), just like for the adiabatic-to-diabatic transformation matrix. 

Table 7.1 presents us with something of a dilemma. We would obviously desire to explore i much of the phase space as possible but this may be compromised by the need for a sma time step. One possible approach is to use a multiple time step method. The underlyir rationale is that certain interactions evolve more rapidly with rime than other interaction The twin-range method (Section 6.7.1) is a crude type of multiple time step approach, i that interactions involving atoms between the lower and upper cutoff distance remai constant and change only when the neighbour list is updated. However, this approac can lead to an accumulation of numerical errors in calculated properties. A more soph sticated approach is to approximate the forces due to these atoms using a Taylor seri< expansion [Streett et al. 1978] ... 

Whether an adequate sampling of phase space is obtained Whether the system size is large enough to represent the bulk material Whether the errors in calculation have been estimated correctly... 

Another property which can be represented by generalized charts is fugacity, ( ). The fugacity of a substance can be regarded as a corrected vapor pressure. At low pressures (below atmospheric) the use of pressure in the place of fugacity leads to tittle error in calculations. The fugacity coefficient is defined by... 

Frequencies computed with methods other than Hartree-Fock are also scaled to similarly eliminate known systematic errors in calculated frequencies. The followng table lists the recommended scale factors for frequencies and for zero-point energies and for use in computing thermal energy corrections (the latter two items are discussed later in this chapter), for several important calculation types ... 

Thus, we can conclude such a method is generally likely to be fairly accurate when computing these thermochemical quantities, but when it is wrong, it can be off by quite a lot. A method with smaller maximum errors is obviously preferable (since the errors in calculations on new and unknown systems are by definition uncertain). 

The major source of error In calculating the free energies of Pu0(g) and Pu02(g) from Battles et al. probably results from the derived equations for the partial pressures of 0(g) and Pu(g) as a consequence of uncertainties In Ionization cross sections. The thermodynamic assessments of Ackermann et al. Involve extrapolations of oxygen potentials reported by Markin and Rand (4) In temperature of the order of 500 K. However, a second and third... 

Dicarbonyls. A third area of uncertainty is the treatment of dicarbonyls formed from aromatic or terpene hydrocarbon oxidation. (The simplest is glyoxal, CHOCHO, but a large number have been identified, 47. The yields and subsequent reactions of these compounds represent a major area of uncertainty in urban air photochemistry (186) and since they may be a significant source of HOjj through photolysis, inaccuracies in their portrayal may result in errors in calculated values of HO. and HO2.. 

Core electrons are highly relativistic and DFT methods may show systematic errors in calculating the charge density at the nucleus because of the inherent approximations. Fortunately, this does not hamper practical calculations of isomer shifts of unknown compounds, because only differences of li//(o)P are involved. In practice, the reliability of the results depends more on the number of compounds used for calibration and how wide the spread of their isomer shift values was. The isomer shift scale for several Mossbauer isotopes has been calibrated by this approach, among which are Au [1], Sn [4], and Fe [5-9]. For details on practical calculation of Mossbauer isomer shifts, see Chap. 5. 

Errors in Calculated Rate Constants Caused by Analytical Errors... 

Until recently, advances in calculating the free energy were not accompanied by comparable progress in rigorous error analysis and reduction. Although a variety of methods to estimate the error in calculated free energies were proposed [32, 106], they were usually somewhat heuristic or involved approximations that were not always sufficiently well supported. Only recently, considerable progress has been made on this front, in particular by Daniel Zuckerman and Thomas Woolf [107]. 

Figure 5 shows the variation of the hybrid theory with CMD for various Og. It is obvious that assuming an aerosol to be mono-disperse when it is in fact polydisperse leads to an underestimation of the attachment coefficient, leading in turn to large errors in calculation of theoretical unattached fraction. 

Uncertainty in the calculation, however, affects the reliability of values reported for saturation indices. Reaction log Ks for many minerals are determined by extrapolating the results of experiments conducted at high temperature to the conditions of interest. The error in this type of extrapolation shows up directly in the denominator of log Q/K. Error in calculating activity coefficients (see Chapter 8), on the other hand, directly affects the computed activity product Q. The effect is pronounced for reactions with large coefficients, such as those for clay minerals. 

Fig. 1. Effect of errors in calculating properties of a molecule on predictions of the course of a reaction reproduced from Ref. 2)...

Errors in the analytical laboratory are basically of two types determinate errors and indeterminate errors. Determinate errors, also called systematic errors, are errors that were known to have occurred, or at least were determined later to have occurred, in the course of the lab work. They may arise from avoidable sources, such as contamination, wrongly calibrated instruments, reagent impurities, instrumental malfunctions, poor sampling techniques, errors in calculations, etc. Results from laboratory work in which avoidable determinate errors are known to have occurred must be rejected or, if the error was a calculation error, recalculated. 

Table 4-1 Comparison of RMS errors in calculated bond lengths (A, top lines), and harmonic vibrational frequencies (cm-1, bottom lines).

Fig. 1. Error in calculated energies (in eV) for the d s —>d s excitations (the electronic states are given in Table 2). Dotted line CCSD solid line CCSD(T) dashed line CASPT2.

In the preceding example, the chords have been taken for equal intervals, because the curve changes slope only gradually and the data are given at integral temperatures at equal intervals. Under these circumstances, the method of numerical differentiation is actually preferable. In many cases, however, the intervals will not be equal nor will they occur at whole numbers. For the latter cases, the chord-area method of differentiation may be necessary, although considerable care is required to avoid numerical errors in calculations. 

Table I. Average errors in calculated properties (Continued) lolecule Angles (Degrees)...

The complexity and importance of combustion reactions have resulted in active research in computational chemistry. It is now possible to determine reaction rate coefficients from quantum mechanics and statistical mechanics using the ideas of reaction mechanisms as discussed in Chapter 4. These rate coefficient data are then used in large computer programs that calculate reactor performance in complex chain reaction systems. These computations can sometimes be done more economically than to carry out the relevant experiments. This is especially important for reactions that may be dangerous to carry out experimentally, because no one is hurt if a computer program blows up. On the other hand, errors in calculations can lead to inaccurate predictions, which can also be dangerous. 

Error propagation analysis provides a simple way of predicting error in calculated quantities. Ratioing experimental quantities is shown to be a source of large error. 

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