Empirical Wave Parameter Calculation

1 Empirical Wave Parameter Calculation Planning, construction, and employment, as well as operation of maritime platforms (including ships and drilling rigs) require information about the sea state conditions. Schmager (1974, 1985a) developed empirical formulas for the calculation of the sea state parameters of the Baltic Sea. They are compiled in Table 7.4. 

A quick overview of the sea state parameters in the Baltic Sea is provided by the wave diagram (see Fig. 7.11). If the fetch (see Section 7.1.3.4) is known, the use of the diagram does not present much difficulty. For a fetch of 80 km and a wind speed of 45 kn the following information about the sea state conditions can be found in Table 7.5. 

In this atlas, a wave height is depicted that was established as a standard in Soviet specialized literature and that was defined as the wave height at the lower limit of the waves comprising the top 3%. These wave heights must be multiplied with the factor 0.76 to establish a relationship with the significant wave height. 

In accordance with Fig. 7.12, waves with a height of 55 dm are to be expected at the MARNET position north of Arkona. This corresponds to a significant wave height of 4.2 m. 

TABLE 7.5 Determining Sea State Parameters with the Help of the Wave Diagram 

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Here, is the distance between atoms i andj, C(/ is a dispersion coefficient for atoms i andj, which can be calculated directly from tabulated properties of the individual atoms, and /dampF y) is a damping function to avoid unphysical behavior of the dispersion term for small distances. The only empirical parameter in this expression is S, a scaling factor that is applied uniformly to all pairs of atoms. In applications of DFT-D, this scaling factor has been estimated separately for each functional of interest by optimizing its value with respect to collections of molecular complexes in which dispersion interactions are important. There are no fundamental barriers to applying the ideas of DFT-D within plane-wave DFT calculations. In the work by Neumann and Perrin mentioned above, they showed that adding dispersion corrections to forces... 

It should also be mentioned that there are a number of empirical expressions (Held, 1983, 1990 Lehmann, 1973) that one can use to calculate, with certain accuracy, the shock wave parameters such as shock wave pressure in different materials, shock wave velocity, pressure positive phase duration, and shock wave impulse. 

The broadening Fj is proportional to the probability of the excited state k) decaying into any of the other states, and it is related to the lifetime of the excited state as r. = l/Fj . For Fjt = 0, the lifetime is infinite and O Eq. 5.14 is recovered from O Eq. 5.20. Unfortunately, it is not possible to account for the finite lifetime of each individual excited state in approximate theories based on the response equations (O Eq. 5.4). We would be forced to use a sum-over-states expression, which is computationally intractable. Moreover, the lifetimes caimot be adequately determined within a semiclassical radiation theory as employed here and a fully quantized description of the electromagnetic field is required. In addition, aU decay mechanisms would have to be taken into account, for example, radiative decay, thermal excitations, and collision-induced transitions. Damped response theory for approximate electronic wave functions is therefore based on two simplifying assumptions (1) all broadening parameters are assumed to be identical, Fi = F2 = = r, and (2) the value of F is treated as an empirical parameter. With a single empirical broadening parameter, the response equations take the same form as in O Eq. 5.4 with the substitution to to+iTjl, and the damped linear response function can be calculated from first-order wave function parameters, which are now inherently complex. For absorption spectra, this leads to a Lorentzian line-shape function which is identical for all transitions. 

The main difficulty in the theoretical study of clusters of heavy atoms is that the number of electrons is large and grows rapidly with cluster size. Consequently, ab initio "brute force" calculations soon meet insuperable computational problems. To simplify the approach, conserving atomic concept as far as possible, it is useful to exploit the classical separation of the electrons into "core" and "valence" electrons and to treat explicitly only the wavefunction of the latter. A convenient way of doing so, without introducing empirical parameters, is provided by the use of generalyzed product function, in which the total electronic wave function is built up as antisymmetrized product of many group functions [2-6]. 

The application of quantum-mechanical methods to the prediction of electronic structure has yielded much detailed information about atomic and molecular properties.13 Particularly in the past few years, the availability of high-speed computers with large storage capacities has made it possible to examine both atomic and molecular systems using an ab initio variational approach wherein no empirical parameters are employed.14 Variational calculations for molecules employ a Hamiltonian based on the nonrelativistic electrostatic nuclei-electron interaction and a wave function formed by antisymmetrizing a suitable many-electron function of spatial and spin coordinates. For most applications it is also necessary that the wave function represent a particular spin eigenstate and that it have appropriate geometric symmetry. 

Usually, for both theoretical and semi-empirical determination of energy spectra, radial integrals that do not depend on term energy of the configuration are used. More exact values of the energy levels are obtained while utilizing the radial wave functions, which depend on term. Therefore, there have been attempts to account for this dependence in semi-empirical calculations. Usually the Slater parameters are multiplied by the energy dependent coefficient... 

The main advantage of the effective potential method consists in the relative simplicity of the calculations, conditioned by the comparatively small number of semi-empirical parameters, as well as the analytical form of the potential and wave functions such methods usually ensure fairly high accuracy of the calculated values of the energy levels and oscillator strengths. However, these methods, as a rule, can be successfully applied only for one- and two-valent atoms and ions. Therefore, the semi-empirical approach of least squares fitting is much more universal and powerful than model potential methods it combines naturally and easily the accounting for relativistic and correlation effects. 

Let us also notice that slow variations of K with Z imply that the gauge condition K may be treated as a semi-empirical parameter in practical calculations to reproduce, with a chosen K, the accurate oscillator strength values for the whole isoelectronic sequence. Thus, dependence of transition quantities on K may serve as the criterion of the accuracy of wave functions used instead of the comparison of two forms of 1-transition operators. In particular, the relative quantities of the coefficients of the equation fEi = aK2 + bK +c (the smaller the a value, the more exact the result), the position of the minimum of the parabola Kf = 0 (the larger the K value for which / = 0, the more exact is the approximation used, in the ideal case / = 0 for K = +oo) may also help to estimate the accuracy of the method utilized. 

The energies of the d-d-excitations in this model are obtained by diagonalizing the matrix of the Hamiltonian constructed in the basis of rid-electronic wave functions (nd is the number of d-electrons). Matrix elements of the Hamiltonian are expressed through the parameters describing the crystal field and those of the Coulomb repulsion of d-electrons, which are Slater-Condon parameters Fk, k = 0,2,4, or the Racah parameters A, B, and C. In the simplest version of the CFT these quantities are considered empirical parameters and determined by fitting the calculated excitation energies to the experimental ones. 

In principle one can use this wave function to calculate the magnetic and electric hyperfine parameters. In practice, it is interesting to follow the arguments of Gerry, Merer, Sassenberg and Steimle [61 ] as they attempt to find a semi-empirical description of the bonding in CuO which also gives a reasonable quantitative interpretation of the molecular constants. It is, evidently, not easy to find a satisfactory compromise between the physically visual semi-empirical model, and the frill blown ab initio calculations. 

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