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Composite extrapolation procedures

A significant advantage of plane wave basis sets is that they are independent of the nuclear positions. This means that the problem of basis set superposition error (Section 5.10) does not occur, and the calculation of the gradient of the energy is easy, as it is given directly by the Hellman-Feynman force, i.e. there are no components associated with the change of basis function position ( Pulay forces ). [Pg.213]

Given a predefined target accuracy, the error from each of these four steps should be reduced below the desired tolerance. The error at a given level may be defined as the change that would occur if the calculation were taken to the infinite correlation, infinite basis limit. A typical target accuracy is 4kJ/mol ( 1 kcal/mol), the so-called [Pg.213]

The HF error depends only on the size of the basis set. The energy, however, behaves asymptotically as exp(-VZ), where L is the highest angular momentum in the basis set. For example, with a basis set of TZP quality (4s3p2dlf for first row elements) the results are already quite stable. Combined with the fact that an HF calculation is the least expensive ab initio method, this means that the HF error is rarely the limiting factor. [Pg.214]

The net effect of steps (3)-(5) is that a single calculation at the QCISD(T)/6-311-EG(3df,2p) is replaced by a series of calculations at lower levels, which in conabi-nation yields a comparable accuracy in significantly less computer time.  [Pg.215]

The main difference among the G2 models is the way the electron correlation beyond MP2 is estimated. The G2 method itself performs a series of MP4 and QCISD(T) calculations, G2(MP2) only does a single QCISD(T) calculation with the 6-311G(d,p) basis, while G2(MP2,SVP) (SVP stands for split valence polarization) reduces the basis set to only 6-31G(d). An even more pruned version, G2(MP2,SV), uses the unpolarized 6-31G basis for the QCISD(T) part, which increases the mean absolute deviation (MAD) to 9kJ/mol. That it is possible to achieve such good performance with this small a basis set for QCISD(T) partly reflects the importance of the large basis MP2 calculation and partly the absorption of errors in the empirical correction. [Pg.215]


Various attempts have been made to circumvent these problems and to eliminate junction potentials, including (1) extrapolation procedures designed to eliminate the difference between the compositions of the two solutions in the appropriate limit, (2) separation of the two solutions by means of a doublejunction salt bridge, (3) the use of double cells with dilute alkali metal amalgam connectors, and (4) the use of glass or other types of ion-specific electrodes as bridging reference electrodes. [Pg.177]

Voltages of the sample thermocouple corresponding to thermal arrests were converted to temperatures using N.B.S. Circular No. 561 (17). Table I hsts the temperatures of thermal arrests for the entire range of composition between 0 and 100 mole % UFg. The values listed in Table I are averages of several measurements, and the uncertainty values are standard deviations of the averages. The uncertainty values associated with the liquidus points obtained by the extrapolation procedure (i) have not been estimated. [Pg.314]

From 220° to 290 C, blends of different PS/PPE compositions were investigated. The free energy parameter X23 was calculated from Equation 6. The retention volumes of the probes in pure PS and PPE were obtained by the extrapolation procedure described above. [Pg.141]

The melting temperatures of syndiotactic polypropylene were studied as a function of syndiotactic pentad content, rrrr (see Section 2.3.4) values determined via NMR spectra see Figure 14.3 (5). The upper figure utilizes the Hoffman-Weeks extrapolation procedure see Section 6.8.6. The lower figure shows a second extrapolation of the data to 100% rrr content, arriving at a melting temperature of nearly 182°C. By comparison, an isotactic polypropylene with >99% mmmm composition has a melting temperature of 170°C (6). [Pg.760]

Another instance in which the constant-temperature method is used involves the direc t application of experimental KcO values obtained at the desired conditions of inlet temperatures, operating pressure, flow rates, and feed-stream compositions. The assumption here is that, regardless of any temperature profiles that may exist within the actu tower, the procedure of working the problem in reverse will yield a correct result. One should be cautious about extrapolating such data veiy far from the original basis and be carebil to use compatible equilibrium data. [Pg.1360]

C-H and N-H bond dissociation energies (BDEs) of various five- and six-membered ring aromatic compounds (including 1,2,5-oxadiazole) were calculated using composite ab initio CBS-Q, G3, and G3B3 methods. It was found that all these composite ab initio methods provided very similar BDEs, despite the fact that different geometries and different procedures in the extrapolation to complete incorporation of electron correlation and complete basis set limit were used. A good quantitive structure-activity relationship (QSAR) model for the C-H BDEs of aromatic compounds... [Pg.318]

In this book, the experts who have developed and tested many of the currently used electronic structure procedures present an authoritative overview of the theoretical tools for the computation of thermochemical properties of atoms and molecules. The first two chapters describe the highly accurate, computationally expensive approaches that combine high-level calculations with sophisticated extrapolation schemes. In chapters 3 and 4, the widely used G3 and CBS families of composite methods are discussed. The applications of the electron propagator theory to the estimation of energy changes that accompany electron detachment and attachment processes follow in chapter 5. The next two sections of the book focus on practical applications of the aforedescribed... [Pg.266]

In Chapters 16 and 17, we developed procedures for defining standard states for nonelectrolyte solutes and for determining the numeric values of the corresponding activities and activity coefficients from experimental measurements. The activity of the solute is defined by Equation (16.1) and by either Equation (16.3) or Equation (16.4) for the hypothetical unit mole fraction standard state (X2° = 1) or the hypothetical 1-molal standard state (m = 1), respectively. The activity of the solute is obtained from the activity of the solvent by use of the Gibbs-Duhem equation, as in Section 17.5. When the solute activity is plotted against the appropriate composition variable, the portion of the resulting curve in the dilute region in which the solute follows Henry s law is extrapolated to X2 = 1 or (m2/m°) = 1 to find the standard state. [Pg.439]

When ion-selective electrodes are used, it is important that the composition of the standard solution closely approximates the composition of the unknown. The medium in which the analyte exists is called the matrix. For complex or unknown matrixes, the standard addition method (Section 5-3) can be used. In this technique, the electrode is immersed in unknown and the potential is recorded. Then a small volume of standard solution is added, so as not to perturb the ionic strength of the unknown. The change in potential tells how the electrode responds to analyte and, therefore, how much analyte was in the unknown. It is best to add several successive aliquots and use a graphical procedure to extrapolate back to the concentration of unknown. Standard addition is best if the additions increase analyte to 1.5 to 3 times its original concentration. [Pg.317]

The literature authors prepared mixtures of enantiomerically pure 2-methyl-octanal (Md5-8.90°C) (isolated by preparative g.l.c. from the sample prepared by the above procedure from the aldimine of propanal and hexyl iodide) containing w/w amounts of 12.7, 25.7 and 44.6 per cent of hexyl iodide. The [a]D values of these mixtures were plotted against the weight per cent to give a linear relationship. The g.l.c. composition of the reaction product and its extrapolated specific rotation thus allowed an ee per cent value to be calculated. In a similar way the ee per cent of the reaction of the aldimine of octanal and methyl iodide was calculated from the mixtures of octanal with the optically pure 2-methyloctanal of 55.4,74.3 and 87.3 per cent and the plot weight per cent v. [a]D was again linear. [Pg.604]

These examples show that our knowledge of ion radical chemistry in homogenous soluction is far from complete and that extrapolation of this knowledge to ion radicals produced at electrodes is a risky procedure, especially if one contemplates the additional complexities involved. The composition of the medium in the vicinity of the electrode is not the same as in the bulk of the solution (Sect 5.2), the structure of the double-layer can at its best be the subject of educated guesses, and due allowance must be made for the possibility that reactions may take place between adsorbed intermediates. [Pg.48]


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