Density quasi-classical

The quasi-classical description of the Q-branch becomes valid as soon as its rotational structure is washed out. There is no doubt that at this point its contour is close to a static one, and, consequently, asymmetric to a large extent. It is also established [136] that after narrowing of the contour its shape in the liquid is Lorentzian even in the far wings where the intensity is four orders less than in the centre (see Fig. 3.3). In this case it is more convenient to compare observed contours with calculated ones by their characteristic parameters. These are the half width at half height Aa)i/2 and the shift of the spectrum maximum ftW—< > = 5a>+A, which is usually assumed to be a sum of the rotational shift 5larger scale A determined by vibrational dephasing. 

Let us consider the quasi-classical formulation of impact theory. A rotational spectrum of ifth order at every value of co is a sum of spectral densities at a given frequency of all J-components of all branches... 

Figure 34. Quantum-mechanical (left) and quasi-classical (right) probability densities of the...

To study to what extent the mapping approach is able to reproduce the quantum results of Model 111, Eigs. 34 and 35 show the quasi-classical probability densities P (cp,f) for the two cases. The classical calculation for E = 0 is seen to accurately match the initial decay of the quantum-mechanical... 

As we discussed in Section II in relation to (2.41), a survival amplitude has a semiclassical behavior that is directly related to the periodic orbits by the Gutzwiller or the Berry-Tabor trace formulas, in contrast to the quasi-classical quantities (2.42) or (3.3). Therefore, we may expect the function (3.7) to present peaks on the intermediate time scale that are related to the classical periodic orbits. For such peaks to be located at the periodic orbits periods, we have to assume that die level density is well approximated as a sum over periodic orbits whose periods Tp = 3eSp and amplitudes vary slowly over the energy window [ - e, E + e]. A further assumption is that the energy window contains a sufficient number of energy levels. At short times, the semiclassical theory allows us to obtain... 

In conclusion, we have probed the local density of states in the vicinity of various N-S structures [19, 23] at very low temperature. The samples with a lateral geometry (and a reduced gap) showed a good agreement between the experiment and the quasi-classical theory. In contrast, N-S bilayers with a... 

The Perez model comes from an approach in which the source of mobility is the existence of quasi-punctual defects characterized by positive or negative fluctuations of packing density, whereas classical free volume theories take into account only the domains of low packing density, e.g., the holes. The model leads to the following equation for the complex modulus ... 

If eH 6 and atomic population at the surface of 1015 cm-2, it is clear that W"u corresponds to a very small coverage (<0 %) and densities of surface imperfections might be expected to exceed this value on most materials. It might therefore be imagined that no semiconductor - electrolyte interface would ever show quasi-classical behaviour in fact, at least in aqueous electrolytes, classical behaviour is often found and there are two reasons for this. 

Equations (1.105) and (1.106) are the Heitler-London counterpart of the corresponding quantities (Equations 3.4 and 3.5 on page 340 of Ruedenberg s paper (1962), which refers to a LCAO-MO wave function. Ruedenberg calls Equation (1.106) the modification of the quasi-classical density due to the interference effect , while we, more literally, speak of exchange[a(r)b(r)], [b(r)a(r)] and overlap[—Sa1(r)], [—Sfe2(r)] densities. Finally, it is worth noting that, while ... 

Expression (1) involves the Fourier components of the density operators with the momentum 2 pp.. To take account of the quasi-classical impurities one should multiply T (x) by exp[-fli(x, where tl(x) is a random... 

In principle, any electronic structure method that can produce a gradient and Hessian can be used in quasi-classical dynamics. However, density functional theory (DFT)-based methods are, in general (at the time of writing), limited to ground-state surfaces. CASSCF-based dynamics calculations can be used for excited-state computations, and we will focus our discussion on this method. In this case, the size of the molecule that can be studied is limited by the size of the active space (Section 2.2.1) at present, no more than 10 active elec-... 

In the case of coherent laser light, the pulses are characterized by well-defined phase relationships and slowly varying amplitudes (Haken, 1970). Such quasi-classical light pulses have spectral and temporal distributions that are also strictly related by a Fourier transformation, and are hence usually refered to as Fourier-transform-limited. They are required in the typical experiments of coherent optical spectroscopy, such as optical nutation, free induction decay, or photon echoes (Brewer, 1977). Here, the theoretical treatments generally adopt a semiclassical procedure, using a density matrix or Bloch formalism to describe the molecular system subject to a pulsed or continuous classical optical field, which generates a macroscopic sample polarization. In principle, a fully quantal description is possible if one represents the state of the field by the coherent or quasi-classical state vectors (Glauber, 1965 Freed and Villaeys, 1978). For our purpose, however. 

In essence, there are only two types of atomistic computational methodologies which are used for the prediction of materials properties, namely (1) empirical potential (or force field) approaches which describe the interactions between atoms in a quasi-classical form avoiding any details of the electronic structure and (2) quantum mechanical methods which take into account the motions and interactions of the electrons in a material. If the approaches are based solely on fundamental physical constants such as the mass and charge of an electron and no atom-specific parameters are introduced, then the methods are called ab initio or first principles . (In the chemical literature, the term ab initio is sometimes reserved for Hartree-Fock-based methods whereas in solid state physics it typically refers to density functional methods.) Since quantum mechanical methods are not biased towards any particular atom or bonding type, they provide powerful predictive capabilities. On the other hand, the computational effort involved in ab initio methods is several orders of magnitude larger then in the case of empirical potentials. Therefore, both approaches, empirical potential methods and quantum mechanical approaches, have their place. 

The extension of the CNT to homogeneous nucleation in atmospheric, essentially multicomponent, systems have faced significant problems due to difficulties in determining the activity coefficients, surface tension and density of binary and ternary solutions. The BHN and THN theories have been experiences a number of modifications and updates. At the present time, the updated quasi-steady state BHN model [16] and kinetic quasi-imary nucleation theory [24,66], and classical THN theory [25,33] and kinetic THN model constrained by the experimental data... 

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