Analytic calculations

Ebeling W and Grigoro M 1980 Analytical calculation of the equation of state and the critical point in a dense classical fluid of charged hard spheres Phys. (Leipzig) 37 21... 

The optimization of a transition structure will be much faster using methods for which the Hessian can be analytically calculated. For methods that incrementally compute the Hessian (i.e., the Berny algorithm), it is fastest to start with a Hessian from some simpler calculation, such as a semiempirical calculation. Occasionally, dilficulties are encountered due to these simpler methods giving a poor description of the Hessian. An option to compute the initial Hessian at the desired level of theory is often available to circumvent this problem at the expense of additional CPU time. 

HyperChem computes the Hessian using numerical second derivative of the total energy with respect to the nuclear positions based on the analytically calculated first derivatives in ab initio methods and any of the semi-empirical methods, except the Extended Hiickel. Vibration calculations in HyperChem using an ab initio method may take much longer than calculations using the semi-empirical methods. 

A balanced chemical reaction indicates the quantitative relationships between the moles of reactants and products. These stoichiometric relationships provide the basis for many analytical calculations. Consider, for example, the problem of determining the amount of oxalic acid, H2C2O4, in rhubarb. One method for this analysis uses the following reaction in which we oxidize oxalic acid to CO2. 

While the goal of the previous models is to carry out analytical calculations and gain insight into the physical picture, the multidimensional calculations are expected to give a quantitative description of concrete chemical systems. However at present we are just at the beginning of this process, and only a few examples of numerical multidimensional computations, mostly on rather idealized PES, have been performed so far. Nonetheless these pioneering studies have established a number of novel features of tunneling reactions, which do not show up in the effectively one-dimensional models. 

As long as the volume flow is kept near design point, both the deflection angle and pressure drop can be corrected. Temperature differential increase is limited by metallurgy, so it is neglected in analytical calculations. This evaluation is based on inlet pressure changes. The new volume at a different pressure is calculated by the ideal gas equation ... 

Interatomic potentials began with empirical formulations (empirical in the sense that analytical calculations based on them... no computers were being used yet... gave reasonable agreement with experiments). The most famous of these was the Lennard-Jones (1924) potential for noble gas atoms these were essentially van der Waals interactions. Another is the Weber potential for covalent interactions between silicon atoms (Stillinger and Weber 1985) to take into account the directed covalent bonds, interactions between three atoms have to be considered. This potential is well-tested and provides a good description of both the crystalline and... 

Zonal models are often used in analytical calculation of temperature, concentration, or humidity conditions in ventilated spaces. The space is divided in two or several zones, which typically have different target levels as described in Section 2.1 These typical zones can also be divided into additional subzones. 

The distribution function of the vectors normal to the surfaces,/(x), for the direction of the magnetic field B, in accord with the directions of the crystallographic axis (100) for the P, D, G surfaces, is presented in Fig. 6. The histograms for the P, D, G are practically the same, as they should be the differences between the histograms are of the order of a line width. The accuracy of the numerical results can be judged by comparing the histograms obtained in our calculation with the analytically calculated distribution function for the P, D, G surfaces [29]. The sohd line in Fig. 6(a) represents the result of analytical calculations [35]. 

This expression is simple enough to lead to analytical calculations. It has the advantage of taking into account the non pairwise character of the attractive part of the energy which is obvious from the presence of the square root in the second term of (13). Therefore, when the atom i has Zi... 

The numerical calculation of the potential-dependent microwave conductivity clearly describes this decay of the microwave signal toward higher potentials (Fig. 13). The simplified analytical calculation describes the phenomenon within 10% accuracy, at least for the case of silicon Schottky barriers, which serve as a good approximation for semiconduc-tor/electrolyte interfaces. The fact that the analytical expression derived for the potential-dependent microwave conductivity describes this phenomenon means that analysis of the mathematical formalism should... 

Monoisotopic molecular weight The molecular weight of an analyte, calculated by using the masses of the more/most abundant isotopes of each of the elements present. 

In a recent publication [22] we reported the implementation of dispersion coefficients for first hyperpolarizabiiities based on the coupled cluster quadratic response approach. In the present publication we extend the work of Refs. [22-24] to the analytic calculation of dispersion coefficients for cubic response properties, i.e. second hyperpolarizabiiities. We define the dispersion coefficients by a Taylor expansion of the cubic response function in its frequency arguments. Hence, this approach is... 

The present study demonstrates that the analytic calculation of hyperpolarizability dispersion coefficients provides an efficient alternative to the pointwise calculation of dispersion curves. The dispersion coefficients provide additional insight into non-linear optical properties and are transferable between the various optical processes, also to processes not investigated here as for example the ac-Kerr effect or coherent anti-Stokes Raman scattering (CARS), which depend on two independent laser frequencies and would be expensive to study with calculations ex-plictly frequency-dependent calculations. 

The final geometry optimizations with the DZ+D basis set and the analytic calculation of the force fields and MP-2 corrections were done with the program GRADSCfI on a CRAY-IA. There are 128 basis functions for the difluoro-benzenes, 132 basis functions for the dihydroxybenzenes and 136 basis functions for the diaminobenzenes. The calculations on the difluorobenzenes require about 14 x 10 non-zero 2e integrals whereas the calculations on the diaminobenzenes require about 40 x 10 non-zero 2e integrals. The integral sort step required for the... 

Figure 1.19 Semi-analytical calculation of the speed of mixing in an advanced interdigital micro mixer, named SuperFocus, and comparison with previously developed interdigital micro mixers [120],...

The foundation of our approach is the analytic calculations of the perturbed wave-functions for a hydrogenic atom in the presence of a constant and uniform electric field. The resolution into parabolic coordinates is derived from the early quantum calculation of the Stark effect (29). Let us recall that for an atom, in a given Stark eigenstate, we have ... 

Simple analytical calculations based on the previously discussed assumptions lead, among other results, to the determination of the film relative deposition rate as a function of the nitrogen content in the film, for several values of the interaction parameter q, as shown in Figure 18. 

Helgaker, T., Watson, M., Handy, N. C., 2000, Analytical Calculation of Nuclear Magnetic Resonance Indirect Spin-Spin Coupling Constants at the Generalized Gradient Aproximation and Hybrid Levels of Density Functional Theory , J. Chem. Phys., 113, 9402. 

Analytical calculations based upon equations of this complexity are not very fruitful, particularly as they depend upon detecting both radioactive... 

Models that are too complicated for the analytical methods of statistical mechanics can often be explored by computer simulations. In a certain sense these are a theoretician s experiment One can devise a model for a certain system, and then investigate with the aid of the computer its consequences. By varying the system parameters, or modifying features of the model, one can gain insight into the structure or dynamics of the system, which one could not obtain by other means. While computer simulations are not as intellectually satisfying as explicit calculations, they are often the only way to test a model. Sometimes they are also used to check the validity of approximations made in analytical calculations. 

In this example problem, we can analytically calculate the changes in C°pt in Equation (d) with respect to changes in the various cost parameters. Substitute >opt from Equation (/) into the total cost function... 

We have seen how analytical calculations in titrimetric analysis involve stoichiometry (Sections 4.5 and 4.6). We know that a balanced chemical equation is needed for basic stoichiometry. With redox reactions, balancing equations by inspection can be quite challenging, if not impossible. Thus, several special schemes have been derived for balancing redox equations. The ion-electron method for balancing redox equations takes into account the electrons that are transferred, since these must also be balanced. That is, the electrons given up must be equal to the electrons taken on. A review of the ion-electron method of balancing equations will therefore present a simple means of balancing redox equations. 

Following are some examples of analytical calculations involving redox titrations. 

The first problem of the theory of the intramolecular reactions is a calculation of dimensions of the intramolecularly cross-linked coils as a function of the degree of cross-linking. For the analytical calculation of such dependence one needs to know all possible topological structures for any number of cross-linkages and to have the calculation algorithm for each of... 

The early experiments of Bowman and Seery appeared to confirm this conclusion. Some of their results are shown in Fig. 8.3. In this figure the experimental points compared very well with the analytical calculations based on the Zeldovich mechanisms alone. The same computational program as that of Martenay [11] was used. Figure 8.3 also depicts another result frequently observed fuel-rich systems approach NO equilibrium much faster than do fuel-lean systems [12]. 

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