Accuracy calculated equilibria

Nowadays a wide variety of quantum-chemical programs are disposable, which permit to calculate with high accuracy the equilibrium geometry of the molecules and their energy of formation. Theoretical methods have been developed for analytical calculation of the first and second derivatives of energy [8,9], so that the force-constant matrix FHT and the harmonic frequencies can be extracted from the quantum-mechanical calculations. Since as a rule the molecular orbitals (MO) obtained by the quantum-mechanical methods are spread around the entire molecule, the corresponding quantum-mechanical force fields incorporate the important effects of the off-diagonal interactions. 

A method is described for calculating equilibrium vapor compositions from boiling point vs. liquid composition data in ternary systems composed of two liquid components and a salt added to saturation. The procedure is tested on the ethanolr-water system containing each of a unde range of inorganic salts at saturation. The results suggest that good quality T-II-x data will yield y values of comparable accuracy. 

The accuracy of calculated equilibrium states depends critically on the data sources used. Accurate predictions of heat capacities are often... 

Interpretation of the validity of the near-equilibrium concept is dependent on the accuracy of intramitochondrial free NAD/NADH measurements and the difference between extra- and intramitochondrial phosphorylation potentials. In a series of studies, Wilson and associates [40,41,225,226] have presented evidence in support of their hypothesis. Utilizing rat liver mitochondria, Forman and Wilson [225] compared the mass action ratios to calculated equilibrium constants under conditions promoting either forward (net ATP synthesis) or reversed (net ATP hydrolysis) electron transport. Since the mass action ratios calculated under various conditions were similar to the calculated K, these findings were said to support a near-equi-... 

The method of successive approximations is often faster to apply than the quadratic formula. Keep in mind that the accuracy of a result is limited both by the accuracy of the input data (values of and initial concentrations) and by the fact that solutions are not ideal. It is pointless to calculate equilibrium concentrations to any degree of accuracy higher than 1% to 3%. 

The accuracy of the calculated equilibrium constants for gaseous reactions was good because of the high accuracy of experimentally determined thermodynamic data for gases. As a result, the values obtained for reactions such as NO - V2 N2 + 02, NO + CO - Vz N2 + C02, and CO+ H20 C02 + H2 agreed well with the values reported else-... 

The experimental equilibrium constants from the various studies and their source are listed in Table VIII-6. The calculated equilibrium constants at zero ionic strength based on these literature data are discussed below and are summarised in Table VIII-7. Table VIII-6 demonstrates that the agreement of the experimental values is excellent in the cases where the experimental conditions are the same, indicating both a high accuracy and the absence of systematic errors. The uncertainty in the calculated equilibrium constants at zero ionic strength is much larger than the experimental values but this is a result of the uncertainties in the interaction coefficients. 

As with Cl calculations, MP calculations with small basis sets are of little practical value, and MP calculations should use a 6-31G or larger basis set for useful results. For DZP basis sets, MP2 calculations on closed-shell molecules typically yield 85% to 95% of the basis-set correlation energy [R. J. Bartlett, Ann. Rev. Phys. Chem., 32, 359 (1981)] and substantially improve the accuracy of equilibrium-geometry and vibrational-frequency predictions. 

In calculations on molecules within the matrix Hartree-Fock approximation, it is found to be important to add polarization functions to double-zeta basis sets. Such basis functions do not improve the energies of the isolated component atomic species but contribute significantly to calculated bond energies and to the accuracy of calculated equilibrium bond angles. Double-zeta plus polarization basis sets (usually designated DZP or DZ + P) became widespread in quantum chemistry in the 1970s. In such a basis set the hydrogen atom is described by two s functions and one set of p functions the... 

These values are plotted in Fig. 3-3. An a of 1.313 was chosen to calculate equilibrium curve. Within the accuracy of the calculations, this curve was coincident with the curve obtained in Part 2. 

Only two factors are involved in the accuracy of calculated equilibria, i.e. the closeness of the calculated equilibrium temperature at a given pressure to the (accurately) measured temperature or temperature bracket. They are the accuracy of the thermochemical data employed, and the adequacy of the calculation procedure. (We will assume that both the thermochemical and experimental data refer to the same mineral compositions, state of ordering, and so on.)... 

O Figure 11-1, taken from Bak et al. (2000), illustrates the accuracy of equilibrium geometry dipole moments calculated for a series of small molecules. In Bak et al. (2000), the dipole moments of 11 molecules were studied at various levels of approximation using a sequence of basis sets, with the CCSD(T)/aug-cc-pVQZ results providing the benchmark values. The normal distribution of errors with respect to the reference values, shown in O Fig. 11-1, demonstrates the systematic improvements of the computed results with increasing basis set and improved treatment of electron correlation effects. 

The equilibrium constant for other reactions can be calculated with fairly good accuracy from thermodynamic data such as those presented by Kelley (1937). Calculated equilibrium constants for the reactions in equations 13-1 and 13-2 are shown in Table 13-12. 

In the face of these challenges, it is a remarkable accomplishment of computational physics and chemistry that it is now possible to calculate equilibrium structures for many systems with sufficient accuracy and confidence that computed structures are becoming an indispensable source of new information. This capability is now well established for isolated molecules, as is evident from entries in this encyclopedia (see Geometry Optimization 1 and Geometry Optimization 2). The status in the area of solids and surfaces is illustrated by the following examples. 

Figure 17 shows results for the acetonitrile-n-heptane-benzene system. Here, however, the two-phase region is somewhat smaller ternary equilibrium calculations using binary data alone considerably overestimate the two-phase region. Upon including a single ternary tie line, satisfactory ternary representation is obtained. Unfortunately, there is some loss of accuracy in the representation of the binary VLB (particularly for the acetonitrile-benzene system where the shift of the aceotrope is evident) but the loss is not severe. 

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