Green s function matrix

We conclude this section by giving the equation for the alloy Green s function matrix, which is relevant for electronic structure calculations. Elaborating Eq. (23) one finds... 

These are iterative procedures that allow the calculation and the Green s function matrix elements without explicit diagonalization of the Hamiltonian [18]. In the present case the Hamiltonian is factorized into double chains, and the renormalization method can be conveniently and efficiently applied, since its implementation simply requires the handling and the inversion of small matrices of rank two. For more details and elaboration of the renormalization procedures, see, for example, Ref. [23]. 

Nonvanishing blocks Gg correspond to separate ligands A containing the unperturbed diagonal Green s function matrix elements (Go( ))ll corresponding to the LP L located on the ligand A ... 

Dougherty and Rabitz [8] point out that for many applications it is not necessary to compute the entire sensitivity matrix, but only those columns for species considered to be of interest, such as those susceptible to measurement. There are however, certain advantages to computing the entire Green s Function matrix, principally the ability of time scaling in cases where sensitivities are required at several points in time. [6] For the purpose of this paper, the entire matrix was computed. 

As opposed to the Lanczos method, in which the diagonali2 tion of (3.24) is performed, the recursion method focuses on the construction of the diagonal Green s-function matrix element... 

In spite of the apparently awkward aspect of whose determination seems to require the eigenvectors of H, it is possible to calculate it via this elegant trick. Consider the Green s-function matrix elements... 

Define a Green s function matrix G that expresses the solution of (4.A. 14) as... 

To apply the Green s function method, the model is first solved and the concentrations c,(0 are stored at frequent intervals. Then the matrix A is evaluated. Next, the Green s function matrix is obtained by solution of (4.A.16), which finally is used in (4.A.15) to generate kj t +tAt). For applications of the Green s function method see Dougherty et al. (1979), Kramer et al. (1982), and Cho et al. (1987). 

The transfer equation for the surface Green s function matrix... 

Note that for semi-infinite media in case of conservative scattering, a = 1, we find from Eqs. (33a), (33) and (30) that the energy flux integral for the surface Green s function matrix will be equal to zero. 

On taking into account Eq. (33), and also Eq. (28), we rewrite the transfer equation (20) for the surface Green s function matrix in the form... 

As a result, the original homogeneous transfer equation (20) for the surface Green s function matrix has been transformed into the transfer equation (37) with a modified phase matrix (36) and fictitious internal primary sources. When the transfer equation (37) is solved separately for the two internal primary source function vectors qi(r,M) and q2(r,w)givenby Eq. (39), the vector parameters g (0,//o) needed for compiling the complete solution can... 

In the following, it will be shown that the original surface Green s function matrix can be expressed immediately in terms of the surface Green s function matrix for the transformed transfer equation. 

Thus, Eq. (43) is the a-transformation formula for the surface Green s function matrix of a semi-infinite medium, which enables one to retrieve the original surface Green s function matrix, if the surface Green s function matrix of the transformed transfer equation (41) is known. With an appropriate choice of the free parameters in Eq. (36), it may be much easier to solve the transformed transfer equation (41) rather than to solve the original transport equation Eq. (20). 

Let us suppose now, that the effective single-scattering albedo of the modified phase matrix (m, v) has been reduced substantially by means of an appropriate choice of the free parameter vectors in Eq. (36). Then, with increasing optical depth, the surface Green s function matrix G (t, m 0, /Iq ) rapidly fades away, and Eq. (45) asymptotically yields... 

It is remarkable that the free function vectors Ci(m) and Cafw) determining the (p-transformation, do not appear explicitly in the a-transformation formulae for the albedo matrix or for the surface Green s function matrix, Eqs. (49) and (45), respectively. 

The surface Green s function matrix for a finite medium... 

If the reduced phase matrix W (m, v)has the syrmnetric form (53), the cp-transformation retains the minor symmetry of the Green s function matrix with respect to the midplane of the medium in the form... 

Then, obviously, the transformation formulae (59) and (61) can be formulated solely in terms of the surface Green s function matrix Gc (r, w 0, //q ) ... 

A special type of sensitivity coefficient probes the structural responses of a biomolecule to perturbations introduced to different parts of the biomolecule. In molecular mechanics, a Green s function matrix containing this information can be derived as follows. The x, y, or z component of a force F. acting on an atom of a molecule is given by... 

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