Spectral moment definition

Units of spectral moments specified by other workers may differ from those implied here because frequency in wavenumbers, v, is often replaced by angular frequency, . Also, normalization by the product of densities, Q2, is sometimes suppressed, especially if many-body interactions are considered. Furthermore, factors of h/2kT or hc/2kT have been used to the right of Eq. 3.6 in the definition of yo-... 

Atomic spectral moments can be expressed in terms of the averages of the products of the relevant operators. Let Oi,C>2, --,Ok be the operators of interactions in definite shells or the operators of electronic transitions between definite shells in the second quantization form. The average of... 

We have already examined a few graph invariants which, according to the above definition, could be included among the topological indices of benzenoid molecules. These are the number of Kekule structures, the eigenvalues, spectral moments, (coefficients of) the characteristic and sextet polynomials, to mention just some. The total 7t-electron energy is surveyed elsewhere in this volume. 

Estrada, E. (1996). Spectral Moments of the Edge Adjacency Matrix of Molecular Graphs. 1. Definition and Applications to the Prediction of Physical Properties of Alkanes. J.Chem.Inf... 

Estrada, E. (1996) Spectral moments of the edge adjacency matrixofmolecular graphs. 1. Definition and applications to the prediction of physical properties of alkanes./. Chem. Inf. Comput. Sci., 36, 844-849. 

By definition the n-th spectral moment at any of those (very few) time instants considered, is given by another numerical integration ... 

In the analysis of stochastic processes, an important role is played by the so-called spectral moments (SM), introduced by Vanmarcke (1972), important for the definition of some characteristics of stochastic processes and in reliability analysis (Michaelov et al. (1999)). These quantities are defined as the moments of the one-sided PSD with respect to the frequency origin. Let Sy (co) be the PSD of the earthquake acceleration with 5y(ffl) = 5y(—co). Let Gy(co) be the one-sided PSD defined... 

The approximate solution given in Eq. 30 provides accurate results for low and intermediate frequencies. Therefore the author considered the solution proposed by Vanmarcke and Gasparini (1977) given by Eq. 10 with the adjusted spread factor as further improvement of Eq. 19. As the peak factor evaluation depends on the whole power-spectral density function, which is intrinsically related to the spectral characteristics of the unknown response process (see Eqs. 9, 13, 14, and 20), the author developed an algorithm which enables to update the spectral characteristics of the response process every frequency step of definition of the frequency domain. The iterative scheme mainly differs from the procedure proposed by Sundararajan (1980) as not only the zeroth-order spectral moment is updated at each step but also the peak factor. To this aim the... 

Ideally by Eqs. 54, 55, and 56 and the definition of the nonstationary spectral moments of the response, the evolutionary power-spectral density function pertinent to the codes prescriptions could be derivable. However this approach not yet attempted, to the best knowledge of the authors, it could be computationally burdensome, the convergence is not assured, and moreover it may lead to physically unacceptable results therefore alternative approaches are usually preferred. 

Here we focus on the effect of dipolar dispersion laws for high-frequency collective vibrations on the shift and width of their spectral line, with surface molecules inclined at an arbitrary angle 6 to the surface-normal direction. For definiteness, we consider the case of a triangular lattice and the ferroelectric ordering of dipole moments inherent in this lattice type.56,109 Lateral interactions of dynamic dipole moments p = pe (e = (sin os, sin6fcin , cos )) corresponding to collective vibrations on a simple two-dimensional lattice of adsorbed molecules cause these vibrations to collectivize in accordance with the dispersion law 121... 

The usefulness of spectral densities in nonequilibrium statistical mechanics, spectroscopy, and quantum mechanics is indicated in Section I. In Section II we discuss a number of known properties of spectral densities, which follow from only the form of their definitions, the equations of motion, and equilibrium properties of the system of interest. These properties, particularly the moments of spectral density, do not require an actual solution to the equations of motion, in order to be evaluated. Section III introduces methods which allow one to determine optimum error bounds for certain well-defined averages over spectral densities using only the equilibrium properties discussed in Section II. These averages have certain physical interpretations, such as the response to a damped harmonic perturbation, and the second-order perturbation energy. Finally, Section IV discusses extrapolation methods for estimating spectral densities themselves, from the equilibrium properties, combined with qualitative estimates of the way the spectral densities fall off at high frequencies. 

Spectroscopists have always known certain phenomena that are caused by collisions. A well-known example of such a process is the pressure broadening of allowed spectral lines. Pressure broadened lines are, however, not normally considered to be collision-induced, certainly not to that extent to which a specific line intensity may be understood in terms of an individual atomic or molecular dipole transition moment. The definition of collisional induction as we use it here implies a dipole component that arises from the interaction of two or more atoms or molecules, leading at high enough gas density to discernible spectral line intensities in excess of the sum of the absorption of the atoms/molecules of the complex. In other... 

Much effort has been put into the explanation of the spectral line shapes but it seems that the definite theory has yet to be established. In the meantime one can extract useful information from the first few moments of the spectral density, by applying the elegant theory developed by Van Kranendonk and Poll and Van Kranendonk This theory relates the first moment to the derivative of the dimer dipole moment with respect to the intermolecular distance. The zeroth moment yields information about the square of the dipole moment. As this review is not the place to go extensively into the Van Kranendonk theory, we only note that, once the intermolecular potential surface and the interaction dipole field are known, — for instance by ab initio calculations — it is relatively easy to compute the moments of the spectral density. Since these are directly observable, the experiment of pressure induced absorption may serve as a check on the correctness of ab initio calculations, not only of the interaction energy, but also of the interaction dipole. 

The spectral overlap integral J can be expressed in terms of either wavenumbers or wavelengths (Equation 2.36). The area covered by the emission spectrum of D is normalized by definition and the quantities / and lx are the normalized spectral radiant intensities of the donor D expressed in wavenumbers and wavelengths, respectively. Note that the spectral overlap integrals J defined here differ from those relevant for radiative energy transfer (Equation 2.33). Only the spectral distributions of the emission by D /,P and, are normalized, whereas the transition moment for excitation of A enters explicitly by way of the molar absorption coefficient sA. The integrals J" and Jx are equal, because the emission spectrum of D is normalized to unit area and the absorption coefficients sA are equal on both scales. 

As described above, the /3-carotene molecules are randomly oriented in the SC film, but the above discussion suggests that the short-range order similar to that in of the single crystal exists in the SC film. Therefore, we can now conclude definitely that the SC film is an amorphous film, and this conclusion supports the assumption by Babaev and Al perovich (1973). As for the LB film, the optical absorption spectrum measured on normal irradiation with light polarized parallel to the x-axis ofthe substrate showed exactly the same spectral pattern with that parallel to they-axis, although the intensifies of these spectra were apparently different. This shows that there is only one transition moment in the mixed LB film. Saito et al. (1991 a,b) reported the optical absorption spectra of various molecular aggregates in the... 

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