Green’s function technique

To improve the latter a number of 0 N) methods have been recently proposed but practically all of them exploit Hamiltonian formalism. However, in Refs. 4,5 the locally self-consistent multiple scattering (LSMS) method based on the real space multiple scattering theory has been outlined, and in Ref. 6 its central idea in the form of the local interaction zone (LIZ) was incorporated into the Green s function technique, leading to the locally self-consistent Green s function method (LSGF). 

Figure 2. Total energies of ordered (LIq structure, squares), random (circles) and segregated (triangles) fee RhsoPdso alloys as a function of the number of neighboring shells included in the local interaction zone. Values obtained by the LSGF-CPA method are shown by filled symbols and full lines. The energies obtained by the reference calculations are shown by a dashed line (LMTO, ordered sample), a dotted line (LMTO-CPA, random sample), and a dot-dashed line (interface Green s function technique, segregated sample).

I.A. Abrikosov, A.M.N. Niklasson, S.I. Simak, B. Johansson, A.V. Ruban, and H.L. Skriver, Order-N Green s function technique for local environment effects in alloys, Phys. Rev. Lett. 76 4203 (1996). 

H. L. Skriver and N. M. Rosengaard, Self-consistent Green s function technique for surfaces and interfaces, Phys. Rev. B 43 9538 (1991). 

Inglesfield used a Green s function technique to write the one-electron wavefunctions as... 

Standard Green s function techniques are used in the following [46] to describe the dynamics of the protons and the ionic displacements. The equations of motions for the retarded Green s functions [[A (q) S (q))) are obtained from the Hamiltonian Eq. 1 where the operator A denotes 0p, 0k> u or... 

Equation (219) is a complicated equation and difficult to solve without a considerable simplification. Wilemski and Fixman [51] suggested the equation could be simplified considerably if the rate of the quenching reaction was slow compared with diffusion. The last term on the right-hand side perturbs the density, n, slightly under such circumstances and the equation can be solved by standard Green s function techniques (Appendix A) to give... 

Such purely mathematical problems as the existence and uniqueness of solutions of parabolic partial differential equations subject to free boundary conditions will not be discussed. These questions have been fully answered in recent years by the contributions of Evans (E2), Friedman (Fo, F6, F7), Kyner (K8, K9), Miranker (M8), Miranker and Keller (M9), Rubinstein (R7, R8, R9), Sestini (S5), and others, principally by application of fixed-point theorems and Green s function techniques. Readers concerned with these aspects should consult these authors for further references. 

Direct calculation of the ionization potential by LCAO-Xa, HAM/3 and Green s function techniques or via the Koopmans theorem by ab initio techniques. 

Fig. 3. The NEGF-DFT program flowchart. The basic steps of computational procedure are shown schematically in this figure. To calculate //ks[p] and p(r) self-consistently, we use the Green s function technique. In the block called Analysis , the transmission...

G. Csanak, H.S. Taylor, R. Yaris, in D.R. Bates, I. Esterman (Eds.), Green s Function Technique in Atomic and Molecular Physics in Advances in Atomic, Molecular Physics, and Optical Physics, Advances in Atomic and Molecular Physics, Vol. 7, Academic, New York, 1971, p. 287. 

The total Hamiltonian can be solved by Green s function techniques using the Hartree-Fock approximation for the Coulomb repulsion terms... 

The artificial requirement of the unit cell periodicity perpendicular to the interface may be avoided, if a Green s function technique is used. Such... 

On the other hand, Cederbaum et al. (24) have apphed a second quantization method, the Green s function technique, to the N2 problem, with excellent results. Unfortunately, second quantization methods are ahen to the one-electron orbital concept for example, it is clear that Cederbaum s successive approximations do not correspond to ever larger chunks of correlation energy being taken into account. 

There is long standing theoretical interest in the question of how this affects the electronic DOS [5.18,28-33], Due to the inapplicability of Bloch s theorem, calculations are extremely difficult. Using Green s function techniques, Ballentine [5.28] could show for liquid metals that distinct deviations from the free-electron-like behaviour may occur whenever v(K) is significantly large at a peak of S(K) (see Fig. 5.2a, c). The width of the so-called pseudo gap may then correspond to 2 n(/C) as in crystalline matter, and the depth to the intensity of S(K), the structural weight. Theoretical considerations by Nicholson and Schwartz [5.30] as well as recent work by Fresard [5.32], Beck et al. [5.33], and Hafner et al. [5.18] could also show the structural effects on the DOS. 

The first method for solving the MST problem in angular momentum representation was made by Korringa [43] and Kohn and Rostocker [44] separately. The method came to be called the KKR method for electronic structure calculations and used the Green s function technique from Chapter 3 to solve the electronic structure problem. The separation into potential- and structure dependent parts made the method conceptually clean and also speeded up calculations, since the structural dependent part could be calculated once and for all for each structure. Furthermore, the Green s function technique made the method very suitable for the treatment of disordered alloys, since the Coherent Potential Approximation [45] could easily be implemented. 

Development and application of Muffin-Tin Orbital based Green s function techniques to systems with magnetic and chemical disorder... 

In the previous sections, we have utilized Green s function techniques to eliminate some of the summations involved in the calculations of nonlinear susceptibilities. The general expression for R(t3,t2,t1) [Eq. (49) or (60)], involves four summations over molecular states a, b, c, d. In Eq. (80) we carried out two of these summations for harmonic molecules. It should be noted that for this particular model it is possible to carry out formally all the summations involved, resulting in a closed time-domain expression for R(t3,t2,t1). This expression, however, cannot be written in terms of simple products of functions of , r2 and t3. Therefore, calculating the frequency-domain response function / via Eq. (30) requires the performing of a triple Fourier transform (rather than three one-dimensional transforms). This formula is, therefore, useful for extremely short pulses when a time-domain expression is needed. Otherwise, it is more convenient to use the expressions of Section VI, whereby only two of the four summations were carried out, but the transformation to... 

Green s Function Technique in Atomic and Molecular Physics, Gy. Csanak, H. S. Taylor, and Robert Yaris... 

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