Relation between moments

Examples. To determine the surface-volume mean diameter from a number distribution, put t = 0,r = 2,k= 1. 

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Several of the procedures for deriving structural parameters from moments of inertia make use of the method of least squares. Since the relation between moments of inertia and Cartesian coordinates or internal coordinates is nonlinear, an iterative least squares procedure must be used.18 In this procedure an initial estimate of the structural parameters is made and derivatives of the n moments of inertia with respect to each of the k coordinates are calculated based on this estimate. These derivatives make up a matrix D with n rows and k columns. We then define a vector X to be the changes in the k coordinates and a vector B to be the differences between the experimental moments and the calculated moments. We also define a weight matrix W to be the inverse of the ma-... 

These relations between moments can be taken into account (adding to the energy expression the zeros resulting from the Laplace equation (12.5)) and we may introduce what are known... 

The same idea was actually exploited by Neumann in several papers on dielectric properties [52, 69, 70]. Using a tin-foil reaction field the relation between the (frequency-dependent) relative dielectric constant e(tj) and the autocorrelation function of the total dipole moment M t] becomes particularly simple ... 

For more general laminated fiber-reinforced composite plates, the relations between forces, moments, middle-surface strains, and middle-surface curvatures. 

When an ionic solution contains neutral molecules, their presence may be inferred from the osmotic and thermodynamic properties of the solution. In addition there are two important effects that disclose the presence of neutral molecules (1) in many cases the absorption spectrum for visible or ultraviolet light is different for a neutral molecule in solution and for the ions into which it dissociates (2) historically, it has been mainly the electrical conductivity of solutions that has been studied to elucidate the relation between weak and strong electrolytes. For each ionic solution the conductivity problem may be stated as follows in this solution is it true that at any moment every ion responds to the applied field as a free ion, or must we say that a certain fraction of the solute fails to respond to the field as free ions, either because it consists of neutral undissociated molecules, or for some other reason ... 

MMl represents the mass and moment-of-inertia term that arises from the translational and rotational partition functions EXG, which may be approximated to unity at low temperatures, arises from excitation of vibrations, and finally ZPE is the vibrational zero-point-energy term. The relation between these terms and the isotopic enthalpy and entropy differences may be written... 

Coppens, P., Guru, T.N., Leung, P., Stevens, E.D., Becker, P. and Yang, Y. (1979) Atomic net charges and molecular dipole moments from spherical-atom X-ray refinement and the relation between atomic charge and shape, Acta Cryst. A, 35, 63-72. 

In order to describe second-order nonlinear optical effects, it is not sufficient to treat (> and x<2) as a scalar quantity. Instead the second-order polarizability and susceptibility must be treated as a third-rank tensors 3p and Xp with 27 components and the dipole moment, polarization, and electric field as vectors. As such, the relations between the dipole moment (polarization) vector and the electric field vector can be defined as ... 

It is interesting that there is a relation between the oscillator strength f (given by Eq. 2.6) and the square of the transition moment integral, which bridges the gap between the classical and quantum mechanical approaches ... 

Relation between emission anisotropy and angular distribution of the emission transition moments... 

The total fluorescence intensity at time t is obtained by summing over all molecules emitting at that time. Because there is no phase relation between the elementary emissions, the contributions of each molecule to the intensity components along Ox, Oy and Oz are proportional to the square of its transition moment components along each axis. Summation over all molecules leads to the following expressions for the fluorescence intensity components ... 

Finally, because y = cos dE, the relation between the emission anisotropy and the angular distribution of the emission transition moments can be written as... 

Having derived the symmetry relations between the expansion parameters in equation (55), we can proceed to fit the expansions through the ab initio dipole moment values. The expansion parameters in the expressions for and fiy are connected by symmetry relations since these two quantities have E symmetry in and so and fiy must be fitted together. The component ji, with A" symmetry, can be fitted separately. The variables p in equation (55) are chosen to reflect the properties of the potential surface, rather than those of the dipole moment surfaces. Therefore, the fittings of fi, fiy, fifi require more parameters than the fittings of the MB dipole moment representations. We fitted the 14,400 ab initio data points using 77 parameters for the component and 141 parameters for fi, fiy. The rms deviations attained were 0.00016 and 0.0003 D, respectively. 

For a molecular ion with charge number Q a transformation between isotopic variants becomes complicated in that the g factors are related directly to the electric dipolar moment and irreducible quantities for only one particular isotopic variant taken as standard for this species these factors become partitioned into contributions for atomic centres A and B separately. For another isotopic variant the same parameters independent of mass are still applicable, but an extra term must be taken into account to obtain the g factor and electric dipolar moment of that variant [19]. The effective atomic mass of each isotopic variant other than that taken as standard includes another term [19]. In this way the relations between rotational and vibrational g factors and and its derivative, equations (9) and (10), are maintained as for neutral molecules. Apart from the qualification mentioned below, each of these formulae applies individually to each particular isotopic variant, but, because the electric dipolar moment, referred to the centre of molecular mass of each variant, varies from one cationic variant to another because the dipolar moment depends upon the origin of coordinates, the coefficients in the radial function apply rigorously to only the standard isotopic species for any isotopic variant the extra term is required to yield the correct value of either g factor from the value for that standard species [19]. 

Although the relation between the vibrational g factor and the derivative of electric dipolar moment, equation (10), is formally equivalent to the relation between the rotational g factor and this dipolar moment, equation (9), there arises an important distinction. The derivative of the electrical dipolar moment involves the linear response of the ground-state wave function and thus a non-adiabatic expression for a sum over excited states similar to electronic contributions to the g factors. The vibrational g factor can hence not be partitioned in the same as was the rotational g factor into a contribution that depends only on the ground-state wave function and irreducible non-adiabatic contribution. Nevertheless g "(R) is treated as such. A detailed expression for ( ) in terms of quantum-mechanical operators and a sum over excited states, similar to equations (11) and (12), is not yet reported. 

Pope, S.B. 1994. On the relation between stochastic Lagrangian models of turbulence and second-moment closures. J. Physics Fluids 6(2) 973-85. 

Thus, the quasimoments are directly related to the moments /i of a distribution defined by nijk - - =, f(x)x xixk... dx. The relations between the two sets... 

A cumulant of rank s is a symmetric tensor with (s2 + 3s + 2)/2 unique elements for a three-dimensional distribution. Like the moments p and the quasimoments c, the cumulants are descriptors of the distribution. For a onedimensional distribution, the relations between the cumulants and the moments are defined by equating the two expansions ... 

Though the traceless moments can be derived from the unabridged moments, the converse is not the case because the information on the spherically averaged moments is no longer contained in the traceless moments. The general relations between the traceless moments and the unabridged moments follow from Eq. (7.3). For the quadrupole moments, we obtain with Eq. (7.2) ... 

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