Comparisons of calculational methods

TABLE 6.3-1 Comparison of Calculation Methods for Four-Component Adiabatic Absorption System... 

Computer codes are used for the calculational procedures which provide highly detailed data, eg, the Ruby code (70). Rapid, short-form methods yielding very good first approximations, such as the Kamlet equations, are also available (71—74). Both modeling approaches show good agreement with experimental data obtained ia measures of performance. A comparison of calculated and experimental explosive detonation velocities is shown ia Table 5. 

A considerable variety of experimental methods has been applied to the problem of determining numerical values for barriers hindering internal rotation. One of the oldest and most successful has been the comparison of calculated and observed thermodynamic quantities such as heat capacity and entropy.27 Statistical mechanics provides the theoretical framework for the calculation of thermodynamic quantities of gaseous molecules when the mass, principal moments of inertia, and vibration frequencies are known, at least for molecules showing no internal rotation. The theory has been extended to many cases in which hindered internal rotation is... 

Table 6.6 Comparison of the methods to calculate In Kar at various temperatures.

Muller and Stock [227] used the vibronic coupling model Hamiltonian, Section III.D, to compare surface hopping and Ehrenfest dynamics with exact calculations for a number of model cases. The results again show that the semiclassical methods are able to provide a qualitative, if not quantitative, description of the dynamics. A large-scale comparison of mixed method and quantum dynamics has been made in a study of the pyrazine absorption spectrum, including all 24 degrees of freedom [228]. Here a method related to Ehrenfest dynamics was used with reasonable success, showing that these methods are indeed able to reproduce the main features of the dynamics of non-adiabatic molecular systems. 

The most significant assumptions made in these various electron energy calculations are indicated in the various acronyms listed in Table 6.2. These can be permutated in many combinations and a proper comparison of these methods is beyond the scope of the present article. Excellent review articles (Pettifor 1977, Turchi and Sluiter 1993, de Fontaine 1996) are available if further detail is required. Other references of particular interest are those which compare the results... 

Figure 3.16. Comparison of calculating the second derivative of a sample vector with a linear baseline (a), using a simple difference b), and the Gorry method with a window width of 11 (c).

Corrected DFT methods the standard DFT calculation is supplemented by an empirical atom-atom term that reproduces correlation effects. They perform well because the missing effect in pure DFT is exactly pinpointed and circumscribed. Examples are a comparison of calculated lattice energies with experimental thermochemical data [46], including periodic orbital treatment, or the accurate reproduction of organic crystal structures without distortion [47]. 

The computational method for detn of deton velocities, described in Ref 32, was used in detn of velocities, temps, densities and pressures of detonation front for 16 pure organic expls. Comparisons of calculated velocities with experimental values were possible for 12 expls. Agreement was satisfactory for 7 of these, while for remaining 5 expls the ealed values for velocities were lower than the exptl values... 

Explosive Performance, Comparison of Two Methods for Its Evaluation. Until, the middle of 1950 s, the choice of expls for blasting a particular rock was made on the basis of the following methods a) Trial and error gained from previous experience in similar rocks b) Calculation of meaningful performance parameters of expls from their chemical compns by means of laws of thermochemistry or thermodynamics or c) Laboratory determination of expl characteristics... 

The steady-state methods involve theoretical analysis of magnetic resonance spectra observed under steady-state conditions. This typically involves assumptions regarding the adequacy of magnetic resonance line shape theory, some model for molecular motions and distances of closest approach on collision, and a comparison of calculated spectra for various assumed diffusion constants, and observed spectra. In general, the agreement between diffusion constants calculated using the transient and steady-state methods has been excellent. 

Table 183 Comparison of Different Methods to Calculate the Molar Volume Vt of Organic Compounds. Values in cm3mol-1...

Available data processing is a special starting procedure in the further-comparison of three methods in line with established priorities by having recourse to the model problem (35) that we have set up on a square grid h,j = h2 = h in the square G with the unit sides (/2 = l2 = 1). Plain calculations show that... 

Figure 21-11 Comparison of calculated exact" ab initio energies of H2 as a function of internuclear distance, r, with the energies calculated for simple MO and simple VB methods. The dissociation energy calculated by the ab initio procedure is in close agreement with the experimental value of 102 kcal. The zero of the energy scale in this figure is the energy of widely separated hydrogen atoms.

There is a need for rovibrational averaging of theoretical shielding prior to comparison of calculated values with experiment. This becomes even more important as the theoretical methods begin to yield more accurate results. 

X-ray and neutron diffraction patterns can be detected when a wave is scattered by a periodic structure of atoms in an ordered array such as a crystal or a fiber. The diffraction patterns can be interpreted directly to give information about the size of the unit cell, information about the symmetry of the molecule, and, in the case of fibers, information about periodicity. The determination of the complete structure of a molecule requires the phase information as well as the intensity and frequency information. The phase can be determined using the method of multiple isomor-phous replacement where heavy metals or groups containing heavy element are incorporated into the diffracting crystals. The final coordinates of biomacromolecules are then deduced using knowledge about the primary structure and are refined by processes that include comparisons of calculated and observed diffraction patterns. Three-dimensional structures of proteins and their complexes (Blundell and Johnson, 1976), nucleic acids, and viruses have been determined by X-ray and neutron diffractions. 

Figure 5. Comparisons of calculated and experimental volumes per molecule in A-5 for the liquids in Table 1 and TIP4P water. Calculated values are from computer simulation using the OPLS potential method. Reprinted with permission from W. L. Jorgensen and J. Tirado-Rives, J Am. Chem. Soc. 110, 1657 (1988) [8], Copyright 1988 American Chemical Society.

Table 12-3. Comparison of calculated structural parameters for the kaolinite(O)—H2O system obtained from geometry optimizations of a cluster model (ONIOM(B3LYP/SVP PM3 method) and the periodic DFT(PW91) approach) (static relaxation and MD simulation) [73], Bond lengths and interatomic distances are in A, angles are in degrees. Superscript w stands for water, subscripts distinguish O and H atoms of three different surface OH groups...

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