Moment method spectral moments

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. 

For He-Ar spectral moments have been computed from first principles, using advanced quantum chemical methods [278] details may be found in Chapters 4 and 5. We quote the results of the ab initio calculations of the moments in Table 3.1, columns 4 and 6. The agreement with measurement is satisfactory in view of the experimental uncertainties. We... 

Collision-induced dipoles manifest themselves mainly in collision-induced spectra, in the spectra and the properties of van der Waals molecules, and in certain virial dielectric properties. Dipole moments of a number of van der Waals complexes have been measured directly by molecular beam deflection and other techniques. Empirical models of induced dipole moments have been obtained from such measurements that are consistent with spectral moments, spectral line shapes, virial coefficients, etc. We will briefly review the methods and results obtained. 

Method of moments. In rare gas mixtures, the induced dipole consists of just one B component, with Ai AL = 0001, Eq. 4.14. Alternatively, one particular B(c) component may cause the overwhelming part of a measured spectrum, like the quadrupole-induced component in mixtures of small amounts of H2 in highly polarizable rare gases ((c) = Ai AL = 2023, Eq. 4.59) in a given spectral range, other components (like 0001, 2021,...) are often relatively insignificant. In such cases, one can write down more or less discriminating relationships between certain spectral moments of low order n that are obtainable from measurements of the collision-induced spectral profile, g Al(o>),... 

Whereas there is little doubt that the method of moments, as the procedure is called, is basically sound, it is obvious that for reliable results high-quality experimental data over a broad range of frequencies and temperatures are desirable. As importantly, reliable models of the interaction potential must be known. Since these requirements have rarely been met, ambiguous dipole models have sometimes been reported, especially if for the determination of the spectral moments substantial extrapolations to high or to low frequencies were involved. Furthermore, since for most works of the kind only two moments have been determined, refined dipole models that attempt to combine overlap and dispersion contributions cannot be obtained, because more than two parameters need to be determined in such case. As a consequence, empirical dipole models based on moments do not attempt to specify a dispersion component, or test theoretical values of the dispersion coefficient B(7) (Hunt 1985). 

Induced dipole autocorrelation functions of three-body systems have not yet been computed from first principles. Such work involves the solution of Schrodinger s equation of three interacting atoms. However, classical and semi-classical methods, especially molecular dynamics calculations, exist which offer some insight into three-body dynamics and interactions. Very useful expressions exist for the three-body spectral moments, with the lowest-order Wigner-Kirkwood quantum corrections which were discussed above. 

Irrespective of the viability of the bonding schemes outlined above, the mathematical problem of cataloguing the fullerene isomers that obey the starting condition of isolation of pentagon pairs is straightforwardly solved. One method is based on the spectral moments... 

The method of moments analysis has since been used extensively to derive the magnetic moments of the ground and/or excited states from MCD spectral data by determining the intensity of the MCD and UV-visible absorption bands. In instances where the bands overlap extensively, spectral band deconvolution techniques have also been used. 

By the moment method we mean the technique of directly using power moments to determine the Green s function and to reconstruct the spectral density. From a mathematical point of view, this problem goes back to the last century (see Chapter III), but the applications in several branches of physics have more recent origin. 

We would also like to mention the informational approach, which determines the most probable absorption lineshapes given a limited amount of information, that is, a finite number of spectral moments. The method is based on minimizing the informational entropy. ... 

Usually this method is used on an H-depleted molecular graph, truncated expansions being obtained considering only fragments up to a user-defined size. Some methods for -> log P estimations are based on cluster expansion. Moreover, a new method for the calculation of embedding frequencies for acyclic trees based on spectral moments of iterated line graph sequence was proposed recently [Estrada, 1999]. 

Dipole Moments. - The polarity and conformation in solution of 2-thiophosphoryl-l,3-dithianes (100) has been studied by the dipole moment method and using IR spectral data. In solution, they exist mainly as an equilibrium of two chair-like conformers with axial orientation of the phosphorus-containing substituent. The polarity and conformations of alkenylphosphine oxides (115) and related acetates (116) have also been determined. ... 

In favorable cases, the Ai, Bq, and Q parameters can be obtained from experimental spectra by the method of spectral moments. Spectral moments are defined by integrations given in Equation (7), where a(E) is either the absorption A(E) or the MCD... 

The current section of the chapter on numerical methods is devoted to an outline of the most frequently used numerical methods for solving the population balance equation either for the particle number distribution function or for a few moments of the number density function. The methods considered are the standard method of moments, the quadrature method of moments (QMOM), the direct quadrature method of moments (DQMOM), the sectional quadrature method of moments (SQMOM), the discrete fixed pivot method, the finite volume method, and the family of spectral weighted residual methods with emphasis on the least squares method. 

Tajimi H (1960) A standard method of determining the maximum response of a building structure during an earthquake. In Proceedings of 2nd world conference on Earthquake Engineering, vol 2, Tokyo Vanmarcke EH (1972) Properties of spectral moments with applications to random vibration. J Eng Mech ASCE 98 425 6... 

In the last four decades, Eq. 8 has been used by several authors to define the spectrum-compatible power-spectral density function. The methods proposed in literature mainly differentiate one from another for the hypothesis adopted to define the peak factor and for the approximations involved in the evaluation of the response spectral moments. 

However, calculation of m2 and m4 involves practical difficulties, as sample spacing and signal processing. For this work, the authors made trials to determine the second and fourth spectral moments, calculating the variation (dz/dx) and (d z/dx ) of the measured points by different methods, as described below. The profile file measured by the MAHR Concept Perthometer PGK equipment was used. The file contains 8064 points, 0.695 pm spaced. Because, in this equipment, the data contained in the file is supplied as measured, an additional filtering software has to be developed and it is described in Annex I. This user filtering process can be omitted if the equipment supplies the profile already filtered. 

The approach to the evaluation of vibrational spectra described above is based on classical simulations for which quantum corrections are possible. The incorporation of quantum effects directly in simulations of large molecular systems is one of the most challenging areas in theoretical chemistry today. The development of quantum simulation methods is particularly important in the area of molecular spectroscopy for which quantum effects can be important and where the goal is to use simulations to help understand the structural and dynamical origins of changes in spectral lineshapes with environmental variables such as the temperature. The direct evaluation of quantum time- correlation functions for anharmonic systems is extremely difficult. Our initial approach to the evaluation of finite temperature anharmonic effects on vibrational lineshapes is derived from the fact that the moments of the vibrational lineshape spectrum can be expressed as functions of expectation values of positional and momentum operators. These expectation values can be evaluated using extremely efficient quantum Monte-Carlo techniques. The main points are summarized below. 

A number of methods are employed to determine the conformations. These include the measurement of dipole moment or the study of spectral properties like I.R. studies or N.M.R. or electron diffraction or X-ray diffraction studies. Some of these methods are discussed below ... 

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