Green function recursive

V. Mandelstam, H.S. Taylor, A simple recursion polynomial expansion of the Greens function with absorbing boundary conditions—Application to the reactive scattering, J. Chem. Phys. 103 (1995) 2903. 

There is considerable interest in the development and testing of spectral filters for use in Lanczos recursion procedures. Although progress has been made in the testing of a Green function filter for incorporation in Lanczos codes (74), it remains to be seen how well this will perform in difficult molecular problems having many coupled states (such as the benzene overtones). 

Recursion Polynomial Expansion of the Green s Function with Absorbing Boundary Conditions. Application to the Reactive Scattering. 

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... 

The memory function formalism leads to several advantages, both from a formal point of view and from a practical point of view. It makes transparent the relationship between the recursion method, the moment method, and the Lanczos metfiod on the one hand and the projective methods of nonequiUbrium statistical mechanics on the other. Also the ad hoc use of Padd iqiproximants of type [n/n +1], often adopted in the literature without true justification, now appears natural, since the approximants of the J-frac-tion (3.48) encountered in continued fraction expansions of autocorrelation functions are just of the type [n/n +1]. The mathematical apparatus of continued fractions can be profitably used to investigate properties of Green s functions and to embody in the formalism the physical information pertinent to specific models. Last but not least, the memory function formaUsm provides a new and simple PD algorithm to relate moments to continued fraction parameters. 

Summing up, we see that the traditional approach to impurity problems within the Green s-function formalism exploits the basic idea of splitting the problem into a perfect crystal described by the operator and a perturbation described by the operator U. The matrix elements of < are then calculated, usually by direct diagonalization of or by means of the recursion method. Following this traditional line of attack, one does not fully exploit the power of the memory function methods. They appear at most as an auxiliary (but not really essential) tool used to calculate the matrix elements of... 

For r e r, this relationship leads to a Fredholm integral equation of the second type for the field Ey within the region F. The Green s function for the background sequence, Gb, which is present in equation (9.21) can be calculated from a simple recursive formula. Details about this procedure can be found in a monograph by Berdichevsky and Zhdanov (1984). 

The derivation highlights two results First, the transition probability can be identified with the ensemble average of Green s function G " for a single realization. Second, when (p x, t, a) depends on a, Eq. (B6) becomes an integral equation in a which must be solved by recursive or other means, rather than an explicit solution for a. This issue does not arise when the source density is specified independently, for in.stance, by the locations and strengths of point sources of air pollutants. However, it is the norm rather than the exception in biogeochemical applications, as discu.ssed in Sections 2.4 and 3.5. 

To conclude the description of the power series Green s function (PSG), we wish to underscore how the recursive calculation proceeds. To compute the column of the ABC Green s function, denoted by Gj with elements given by (Gj)j/ = G(qj/,qj), one forms the dot product of the matrix in Eq. (3.20) with the column of the identity matrix. As such, the column of G(jE) is... 

The use of the Lanczos recursive method to define the local Hamiltonian and to calculate the density matrix is a common feature shared by several groups. Baroni and Giannozzi suggested a method based on a finite-difference representation of the Hamiltonian, and a recursive Green s function approach to calculate the electron density in real space." For the density at each point, the truncated finite length of the recursion defines the local Hamiltonian for that point. Aoki et al. constructed a bond-order potential method which determines each density matrix element with a recursion in the Green s function. ... 

EOMCC = equation-of-motion coupled-clu.ster FOD = fourth-order differencing MBGF = many-body Green s function RR = resonance Raman RRGM = recursive-residue-generation method SIL = short-time iterative Lanczos SOD = second-order differencing. 

G(R,Rq) is not known explicitly (or by quadrature) for any but the most simple (and uninteresting problems). But it is clearly related to the solution of a diffusion problem for a particle starting at Rq in a 3N dimension space and subject to absorption probability V(R) + Vq per unit time. We therefore expect to be able to sample points R from G(R,Rq) conditional on Rq. It turns out to be possible by means of a recursive random walk in which each step is drawn from a known Green s function for a simple subdomain of the full space for the wavefunction. References ( 4) and ( ) contain a thorough discussion of this essential technical point, and also of the methods which permit the accurate computation of the energy and other quantum expectations such as the structure function, momentum density, Bose-Einstein condensate fraction, and... 

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