Delta-Density method

Zimmerman, H. E. Alabugin, I. V. Energy Distribution and Redistribution and Chemical Reactivity. The Generalized Delta Overlap-Density Method for Ground State and Electron Transfer Reactions A New Quantitative Counterpart of Electron-Pushing. J. Am. Chem. Soc. 2001, 123, 2265-2270. 

Impulse (delta) response method The input signal is changed in the form of a delta function. This method is widely used in chemical engineering to investigate the residence time probability density distribution function. 

With much more powerful quantum mechanical computations available (i.e., Gaussian 98), the method was applied to a variety of photochemical reactions (note Scheme 1.12). The expression in Equation 1.12 for the delta-density matrix elements includes overlap integrals to take care of basis set definitions. Weinhold NHOs (i.e., hybrids) were used in order to permit easy analysis in terms of basis orbital pair bonds comprising orbital pairs. Note A refers to a reactant and B refers to the corresponding excited state in this study. 

How does one extract eigenpairs from Chebyshev vectors One possibility is to use the spectral method. The commonly used version of the spectral method is based on the time-energy conjugacy and extracts energy domain properties from those in the time domain.145,146 In particular, the energy wave function, obtained by applying the spectral density, or Dirac delta filter operator (8(E — H)), onto an arbitrary initial wave function ( (f)(0)))1 ... 

After P(Z) or P(Z) is obtained at each point in the flow by the methods that have now been described, it is usually possible to calculate any desired averages involving temperature and species concentrations at each point. As an illustration, let us assume that equations (3-82), (3-83) and (3-84) hold true. These relationships are shown schematically in Figure 10.3. A convenient approach is to use equation (9) to generate probability-density functions for Yq, and T from P(Z). If the shape of P(Z) is the second one shown for the jet in Figure 10.1, then the probability-density functions obtained are those illustrated in Figure 10.4. For the fuel, the contribution to P Z) for 0 < Z < Z collapses into a delta function at 7 = 0, as is readily... 

The probability density function, written as pif), describes the fraction of time that the fluctuating variable/ takes on a value between/ and/ + A/. The concept is illustrated in Fig. 5.7. The fluctuating values off are shown on the right side while p(f) is shown on the left side. The shape of the PDF depends on the nature of the turbulent fluctuations of/. Several different mathematical functions have been proposed to express the PDF. In presumed PDF methods, these different mathematical functions, such as clipped normal distribution, spiked distribution, double delta function and beta distribution, are assumed to represent the fluctuations in reactive mixing. The latter two are among the more popular distributions and are shown in Fig. 5.8. The double delta function is most readily computed, while the beta function is considered to be a better representation of experimentally observed PDF. The shape of these functions depends solely on the mean mixture fraction and its variance. The beta function is given as... 

A faithful representation of the spin-density projection P(x) would be obtained by this method of frequency encoding only if the spectrum F(x - x ) were a delta spike, or a narrow liquid-state NMR line. 

In the previous section, the Nerast-Planck equation was developed from the macroscopic flux density and mass conservation equations [(71) and (73), respectively]. The same equation can also be derived by statistical methods, which describe the probability of finding a particle within a volume region at a time t, given an initial distribution and a set of jump probabilities. For the simplest case, in one dimension, with equal probabilities of the particle making a jump to the right or to the left, the time evolution of an initial delta function in concentration at x = 0 is Gaussian ... 

---