The Green Function Method

Rabitz and co-workers suggested (Hwang et al. 1978 Kramer et al. 1981, 1984 Hwang 1982 Rabitz et al. 1983) a numerical method based on the Green function for the calculation of the sensitivity matrix. The Green function method is defined as 

Here I is the mxm unit matrix. An element of matrix G(f, s) shows the effect of changing variable Yj at time s t on the value of variable T, at time t  

If i = j, then the effect of changing variable i on the value of the same variable is obtained. 

The Green function has a very clear physical meaning it shows the effect of changing the initial concentrations on the model solution therefore, its elements were also called initial concentration sensitivity coefficients. Using the notation of Eq. (5.11), gij(t,s) shows the effect of changing the concentration of species j at time s on the calculated concentration of species i at time t. This effect can be very small (this is typical when at time t the system is close to the equilibrium or the stationary point) or can be very large, such as when species j is an autocatalyst. Therefore, the Green function is not only an auxiliary variable for the calculation of the sensitivity matrix, but it can be used directly for the analysis of reaction mechanisms (Nikolaev et al. 2007). 

For the application of Eq. (5.8), the computational time is proportional to the number of variables (not to the number of parameters as in the brute force method), and therefore, this method is advantageous in situations where models have many parameters and a small number of variables (Edelson and Allara 1980). However, usually methods based on the solution of Eq. (5.6), to be discussed in the next subsection, are more effective than those that are based on the application of the Green function. 

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To explain the Green function method for the formulation of Dx, D and D, of the fuzzy cylinder [19], we first consider the transverse diffusion process of a test fuzzy cylinder in the solution. As in the case of rodlike polymers [107], we imagine two hypothetical planes which are perpendicular to the axis of the cylinder and touch the bases of the cylinder (see Fig. 15a). The two planes move and rotate as the cylinder moves longitudinally and rotationally. Thus, we can consider the motion of the cylinder to be restricted to transverse diffusion inside the laminar region between the two planes. When some other fuzzy cylinders enter this laminar region, they may hinder the transverse diffusion of the test cylinder. When the test fuzzy cylinder and the portions of such other cylinders are projected onto one of the hypothetical planes, the transverse diffusion process of the test cylinder appears as a two-dimensional translational diffusion of a circle (the projection of the test cylinder) hindered by ribbon-like obstacles (cf. Fig. 15a). 

This two-dimensional diffusion can be treated by the Green function method, which is expounded in detail in Appendix C, and shows that Dx can be... 

For an infinitely thin rodlike polymer for which d/L = de/Le = 0, we have fi = F 0 = Fx0 = D 0/D = 1, and Eq. (46) reduces to Teraoka and Hay-aka wa s original expression [107] of Dx for rodlike polymers. At high concentrations, the results from the Green function method approach the one from the cage model [107], Teraoka [110] calculated stochastic geometry and probability of the entanglement for infinitely thin rods by use of the cage model, and evaluated px to be... 

When the diffusion time is short enough, the translation on the spherical surface is approximately identical with that on the tangent plane to the spherical surface. The latter is the two-dimensional diffusion process treated by the Green function method in Appendix C, and we can use Eq. (C17) again. Since the rotational diffusion coefficient Dr is related to the translational diffusion coefficient D<2) in Eq. (Cl7) by Dr = D(2)/(Le/2)2, we have... 

The longitudinal diffusion coefficient D has been formulated by the hole theory in Sect. 6.3.2. If the similarity ratio X in this theory is chosen to be 0.025 for the rod with the axial ratio 50, Eq. (58) with Eq. (56) gives the solid curve in Fig. 16a. Though it fits closely the simulation data, the chosen X is not definitive because the change in D(l is small and the definition of the effective axial ratio is ambiguous. Though not shown here, Eq. (53) for D, by the Green function method describes the simulation data equally well if P and C, are chosen to be 1000 and 1, respectively. 

Equations (46)-(48) lead to an expression of Dx/Dx0, in which Le = L and dc = d for rodlike polymers. Since Le/de = L/d = 50, f can be equated to unity in a good approximation. Estimating Djo/Dj from Eq. (58) with X = 0.025 and Fjo/Fxo from Eqs. (B4) and (B5) in Appendix B with Le/de = L/d = 50, we have calculated Dx/Dx0 as a function of the reduced concentration L3c The results are compared with Bitsanis et al. s simulation data in Fig. 16a. It can be seen that the theoretical solid curve for D /D 0 deviates downward slightly from the simulation data points, implying that the Green function method for Dx overestimates the entanglement effect compared to Bitsanis et al. s simulation. 

The rotational diffusion coefficient can be calculated from Eqs. (50) (52) formulated by the Green function method. The Dr/Dr0 values obtained by Doi et al. for infinitely thin rods should be compared with the theory in which Le = L and D]]0/D, = F,0/F10 = fr = b The theoretical solid curve in Fig. 16b shows a favorable comparison with the simulation data. 

Recently, Doi 44) has shown that both conformational and dimensional problems of interrupted helices can be solved more simply by the use of a mathematical technique called the Green function method. His treatments may be viewed as an extension of the theory of de Gennes (45), who demonstrated the power of this method in a study of the branched structure of dAT copolymers. 

For the special case of non relativistic Hydrogen, the multiphoton transition rate can be obtained exactly using methods based on Green function techniques, which avoid summations over intermediate states. This approach was introduced in order to treat time independent problems, and later extended to time dependent ones [2]. In the Green function method, the evaluation of the infinite sums over intermediate states is reduced to the solution of a linear differential equation. For systems other than Hydrogen, this method can also be used, but the associated differential equation has to be integrated numerically. The two-photon transition rate can also be evaluated exactly by performing explicitly the summation over the intermediate states. 

We have calculated the second- and fourth-order dipole susceptibilities of an excited helium atom. Numerical results have been obtained for the ls2p Pq-and ls2p f2-states of helium. For the accurate calculations of these quantities we have used the model potential method. The interaction of the helium atoms with the external electric held F is treated as a perturbation to the second- and to the fourth orders. The simple analytical expressions have been derived which can be used to estimate of the second- and higher-order matrix elements. The present set of numerical data, which is based on the Green function method, has smaller estimated uncertainties in ones than previous works. This method is developed to high-order of the perturbation theory and it is shown specihcally that the continuum contribution is surprisingly large and corresponds about 23% for the scalar part of polarizability. 

Using the Green function method and some decoupling approximations corresponding to the RPA and the Zeeeman reduced splitting smaller than Debye phonon quantum, it is possible to get the dispersion equation... 

In this case the phonon-assisted mobility is most suitably treated with the Green function method. The mobility is represented by the graphs as shown in Fig. 2. Topologically different graphs contribute to a factor... 

The Green function method has been applied in various problems... 

After the transition to k-space by varying the crystal by Fourier transformation of creation and annihilation of electrons and crystal use the Green function method, the band stmctirre of single-walled CNTs with impurities adsorbed hydrogen atoms takes the form [8, 9] ... 

Several methods are used to calculate these coefficients, such as the finite difference approximation, the direct method and its modifications, the Green-function method, and the method of polynomial approximations. 

The Green-function method appeared to be very useful for displaying the chemical trends in defect energy levels [727,728]. However, the calculation of other defective-crystal properties (defect-formation energy, lattice relaxation, local-states localization) requires approaches based on molecular cluster or supercell models. Only recently have these models been used in the first-principles calculations to study point defects in SrTiOs. 

A useful application of the Green function method is the conversion of a homogeneous differential equation into an integral equation. For example, the equation... 

J.T. Hwang, E.P. Dougherty, S. Rabitz and H. Rabitz, "The Greenes Function Method of Sensitivity Analysis in Chemical Kinetics", J, Chem. Phys. 5180 (1978)... 

The DFT studies represent a natural step towards a more detailed, parameter-free understanding of the properties of the DMS. One possibility is to employ the supercell approach , in which big cells are needed to simulate experimentally observed low concentrations of magnetic atoms and other impurities. Alternatively, one can employ the Green function methods combined with the coherent potential approximation (CP.A) in the framework of the Korringa-Kohn-Rostoker (KKR) method or the tight-binding... 

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