Gaussian function homogeneous

The assumption of a characteristic line pattern for lipids is justified by the fact that IMCL, EMCL, and other adipose tissue in subcutaneous fat tissue or in tibial bone marrow are known to consist mainly of a very similar composition of fatty acid triglycerides. The characteristic line pattern A is indirectly determined by recording a reference lipid spectrum (R) out of a lipid compartment with a relatively homogenous magnetic field described by a normal distribution, i.e., a Gaussian function G. That means R can be written as... 

The results just obtained for < y) are, however, rarely used in applications because (v ) and T are generally not known. The Gaussian dispersion parameters aj and al are, in a sense, generalizations of (Cj) and particle displacement variances o-y and a-] are not calculated by Eq. (8.8). Rather, they are treated as empirical dispersion coefficients the functional forms of which are determined by matching the Gaussian solution to data. In that way, the empirically determined a-y and deviations from stationary, homogeneous conditions which are inherent in the assumed Gaussian distribution. 

The free induction decay following 90° pulse has a line shape which generally follows the Weibull functions (Eq. (22)). In the homogeneous sample the FID is described by a single Weibull function, usually exponential (Lorentzian) (p = 1) or Gaussian (p = 2). The FID of heterogeneous systems, such as highly viscous and crosslinked polydimethylsiloxanes (PDMS) 84), hardened unsaturated polyesters 8S), and compatible crosslinked epoxy-rubber systems 52) are actually a sum of three... 

Figure 19 The VE prediction of the diffusive component of the vibrational coherence decay CpiD as a function of the solvent viscosity (ij = 1, 2, 4, 8, 16, 32 and oo cP) for typical parameters. At low viscosity, the decay is exponential, its rate is inversely proportional to the viscosity, and the corresponding Raman line is homogeneously broadened. At high viscosity, the decay becomes Gaussian, its decay time reaches a limiting value, and the Raman line is inhomogeneously broadened. (From Ref. 8.)...

The width (fwhm) of the electronic origin peak at 23,836 cm is 6.5 cm" at T = 1.2 K. Its lineshape can be well approximated by a Lorentzian Hneshape function, while a fit to a Gaussian lineshape is unsuccessful [74]. This indicates that the peak is homogeneously broadened [103,104]. Thus, the width of 6.5 cm" can be related to the lifetime of this Sj state to r(Si) = 8 x 10 s. (Compare Eq. (2).) Eurther, the high phosphorescence intensity that is observed from the Tj state when the Sj state is excited, indicates that the lifetime of this Sj state is largely determined by the intersystem crossing process from Sj to Tj. This would correspond to the rate of kjs = 1/r(Si) = 10 s. A value of the same order of magnitude has also been determined for Pd-phthalocyanine [144]. 

The unknown functions, Cij r),riij r) in (57) and (58) need to be parameterized in some way. In a first attempt we have chosen gaussians with variance and amplitude as new variational parameters [16]. This form was shown to be suitable for homogeneous electron gas [13]. Approximate analytical forms for ij r) and r]ij r), as well as for the two-body pseudopotential, have been obtained later in the framework of the Bohm-Pines collective coordinates approach [14]. This form is particularly suitable for the CEIMC because there are no parameters to be optimized. This trial function is faster than the pair product trial function with the LDA orbitals, has no problems when protons move around and its nodal structure has the same quality as the corresponding one for the LDA Slater determinant [14]. We have extensively used this form of the trial wave function for CEIMC calculations of metallic atomic hydrogen. 

In the case of electrochemical ET, the relevant overlap was between the gaussian density-of-states function of the reacting species in solution and the Fermi-Dirac distribution function of the charge carriers in the electrode (Fig. 4.21). Figure 4B. 1 shows the analogous density-of-states functions for a homogeneous ET reaction. The rate... 

In this section, we describe a theory for calculating observables resulting from incoherent excited state transport among chromophores randomly distributed in low concentration on isolated, flexible polymer chains. The pair correlation function used to describe the distribution of the chromophores is based on a Gaussian chain model. The method for calculating the excitation transfer dynamics is an extension to finite, inhomogeneous systems of a truncated cumulant expansion method developed by Huber for infinite, homogeneous systems (25.26). 

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