Green matrix method

A very effective method to describe scattering and transport is the Green function (GF) method. In the case of non-interacting systems and coherent transport single-particle GFs are used. In this section we consider the matrix Green function method for coherent transport through discrete-level systems. 

Other asymptotic forms consistent with unit Wronskian define different but equally valid Green functions, with different values of the asymptotic coefficient of u>i. In particular, if w k 2 exp i(kr — ln), this determines the outgoing-wave Green function, and the asymptotic coefficient of w is the single-channel F-matrix, F sin ij. This is the basis of the T-matrix method [342, 344], which has been used for electron-molecule scattering calculations [126], It is assumed that Avf is regular at the origin and that Ad vanishes more rapidly than r 2 for r — oo. 

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. 

The second reason to introduce the derivation (6 -9) is to note that all that is required to evaluate the absorption and emission probability F A (t, r) of (9) are matrix elements of the evolution operator exp(-i//r/h). (These matrix elements are the conventional probability amplitudes When considering a situation in which many different kinds of decay processes are involved, e.g. radiative and nonradiative decay, it is not always convenient to deal directly with the matrix elements of exp(-itfr/h), the af(t). Rather, it is simpler to introduce (imaginary) Laplace transforms 16) in the same manner that electrical engineers use them to solve ac circuit equations 33L Thus, if E is the transform variable conjugate to t, the transforms of af(t) are gf(E). The quantities gf (E) can also be labeled by the initial state k and are denoded by Gjk(E). It is customary in quantum mechanics to collect all these Gjk(E) into a matrix G(E). Since matrix methods in quantum mechanics imply some choice of basis set and all physical observables are independent of the chosen basis set, it is convenient to employ operator formulations. If G (E) is the operator whose matrix elements are Gjk(E), then it is well known that G(E) is the Green s function i6.3o.34) or resolvent operator... 

The Green s function is chosen to satisfy the boundary condition demanded by the R-matrix method. This is easily accomplished by choosing Rj (r) to be regular at the origin and requiring... 

However, to do this for every state is practically impossible in the case of really long chains (N> 1000, where N is the number of sites). Therefore, a technique is needed that gives a rough idea as to whether or not the states are localized in a certain eneigy region without explicitly calculating them. Such a method based on the calculation of Green matrix elements of the chain will be examined in the next section. 

Application of the above-described method to different poly-mers " has yielded reasonably good agreement with experiment only (see the next section) if one employs a good basis set and substitutes into the Green matrix elements (8.22) not the HF one-electron eneigies, but rather the quasi-particle energies, which contain also correlation contributions at least in the second order of many-body perturbation theory (see Section 5.3). 

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... 

The transition matrix relates the expansion coefficients of the incident and scattered fields. The existence of the transition matrix is postulated by the T-Matrix Ansatz and is a consequence of the series expansions of the incident and scattered fields and the linearity of the Maxwell equations. Historically, the transition matrix has been introduced within the null-field method formalism (see [253,256]), and for this reason, the null-field method has often been referred to as the T-matrix method. However, the null-field method is only one among many methods that can be used to compute the transition matrix. The transition matrix can also be derived in the framework of the method of moments [88], the separation of variables method [208], the discrete dipole approximation [151] and the point matching method [181]. Rother et al. [205] foimd a general relation between the surface Green function and the transition matrix for the exterior Maxwell problem, which in principle, allows to compute the transition matrix with the finite-difference technique. 

Flead and Silva used occupation numbers obtained from a periodic FIF density matrix for the substrate to define localized orbitals in the chemisorption region, which then defines a cluster subspace on which to carry out FIF calculations [181]. Contributions from the surroundings also only come from the bare slab, as in the Green s matrix approach. Increases in computational power and improvements in minimization teclmiques have made it easier to obtain the electronic properties of adsorbates by supercell slab teclmiques, leading to the Green s fiinction methods becommg less popular [182]. 

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