Approximations discrete spectrum

The concept of probability density may also be utilized to describe the discrete spectrum or, particularly, to solve the inverse problem of the approximate reconstruction of the spectrum from its envelope line using a certain number of its lower moments. The density function of the energy levels of the discrete spectrum may have the form... 

In practical applications, the continuum is often approximated by a discrete spectrum. To this end, one conveniently introduces a potential wall at long internuclear separations and solves for the artifically bound states.171,172 Alternatively, basis set expansion techniques can be employed.195,196 In either case, the density of states depends on external conditions, that is, the size of the box or the number of basis functions. This dependence on external conditions has to be accounted for by the energy normalization. Instead of employing a single continuum wave function with proper energy E in Eq. [240], one samples over the discrete levels with energy E -... 

The use of this approximation of the Dirac theory is interesting for the study of the transitions in the discrete spectrum, also in the photoeffect for states of the continuum whose energy is close to me2, but it is no longer acceptable for the states of high energy. 

The first approximation coincide with the Pauli approximation only for the discrete spectrum, but not for the continuum. One can expect that the first approximation has a weak incidence on the result, independently of the level of energy considered in the continuum, but the second one is directly related to the value of the number n in respect with Za and may lead to important differences for the weak values of n, that is, the high values of the energy. So in what follows, we mainly use the first approximation, the second one being devoted only to the verification of the results, by a passage to the well known nonrelativistic expressions (see [5], Sect. 71) of the matrix elements in the dipole approximation. 

The interest of a calculation with the dipole approximation is to show, by comparison, the incidence of the retardation. It may be considered as negliglide for the discrete spectrum and the values of the energy E in the continuum close to the freedom energy. But this incidence becomes important and even considerable for the high values of E. 

We can deduce from relations established in Sect. 9.2 that a direct passage of the vectors T- -(k) of the transitions sl/2 — pl/2 and sl/2 — pl/2 to a vector T- -(k) of a transition s —p is not possible. In other words, one of the effect of the retardation is to break the possibility to find an equivalence between the Pauli approximation and the Schrodinger theory, and the reason lies on the incidence of the retardation on the spherical parts of the Dirac wave functions, related to the presence of the spin. The incidence is already sensible, in the transitions of the discrete spectrum (see (9.38), (9.39), (9.40)) and this incidence may be amplified in the contribution of the continuum, independently of the incidence of the chosen values for the radial functions. 

The interest of the calculation of the contribution of the discrete spectrum is to refine the usual calculation in which, for the low energy contribution, an approximative formula is used. 

In the case of neutral atoms and of small molecules in fhe fixed-nuclei Born-Oppenheimer approximation, the electronic spectrum has the characteristics that are shown schematically in Figure 6.3. In the case of negative ions the discrete spectrum consists of only a few sfafes. However, uncovering all of fhem is experimentally and theoretically hard due to the high total spin that they normally have, e.g.. Ref. [108] and references fherein. 

The SI approach relies on the fact that a discretized spectrum deriving from an calculation and covering a large energy range, including the continuum, provides a reasonable approximation of, at least, the lowest order spectral moments S provided the extension of the basis set is sufficiently good ... 

As a result of these nonideaKties, the inference of a discrete spectrum using Eqs. 4.41a, b is an ill-posed problem. Although a set of parameters can be found that provides an approximate description of the rheological behavior that is suitable for some purposes, there is no unique discrete spectrum corresponding to a given set of data. 

In the molecular approximation used in (14) only the L = 3W — 6 (W is the number of atoms) discrete intramolecular vibrations of the molecular complex in vacuo are considered. In general these vibrations correspond to the L highest optical branches of the phonon spectrum. The intermolecular vibrations, which correspond to the three acoustical branches and to the three lowest optical branches are disregarded, i.e., the center of mass and - in case of small amplitudes - the inertial tensor of the complex are assumed to be fixed in space... 

The absorbance spectrum in Figure 54-1 is made from synthetic data, but mimics the behavior of real data in that both are represented by data points collected at discrete and (usually) uniform intervals. Therefore the calculation of a derivative from actual data is really the computation of finite differences, usually between adjacent data points. We will now remove the quotation marks from around the term, and simply call all the finite-difference approximations a derivative. As we shall see, however, often data points that are more widely spread are used. If the data points are sufficiently close together, then the approximation to the true derivative can be quite good. Nevertheless, a true derivative can never be measured when real data is involved. 

In the zeroth approximation, in the energy region of interest, the spectrum of states consists of one discrete state overlapping a dense but... 

The summation in (24.32) must be performed over all levels a J of the discrete and continuous spectrum. Moreover, transitions of all electrons must be accounted for. This sum rule is too general, because we are usually interested in the transitions of one outer electron. There are also certain sum rules (already approximate), for such transitions, e.g. (K0 stands for closed shells) ... 

Although the actual form of f(w, q) is different from formula (4.27), the latter leads to reasonable results when we use it to calculate the cross sections of inelastic collisions and the ionization losses.120 As one of the reasons for using approximation (4.27), one can consider the fact that the data concerning the Bethe surfaces for molecules are very scant, while there is extensive information about the optical oscillator strengths of molecules both in the discrete and in the continuous regions of the spectrum (see Refs. 119 and 121). 

Mathematically, the problem is easily solved. Remember that the minimization of Eq. (70) is the criterion for a white power spectrum. If C(to), or C[k] in the discrete case, is the aimed-at colored power spectrum, then the pseudostochastic binary sequence whose power spectrum is the best approximation to C[k] must fulfil k C[k] X[k] 2 = minimum. In the language of the the spin-1/2-system, the new energy function is... 

The major component of a-la is unglycosylated. However, there is a minor glycosylated form (approximately 10%) with a mass spectrum that contains at least 15 discrete peaks (Slangen and Visser 1999). These arise from the glycosylation of asparagine 45. 

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