Green perturbation

If we know the Green function of the non-perturbed problem, we can found a solution by using the Lippman-Schwinger integral equation ... 

It should be observed that the subscript exact here refers to the lowest eigenvalue of the unrelativistic Hamiltonian the energy is here expressed in the unit Aci 00(l+m/Mz) 1 and Z is the atomic number. If the HE energies are taken from Green et al.,8 we get the correlation energies listed in the first column of Table I expressed in electron volts. The slow variation of this quantity is noticeable and may only partly be understood by means of perturbation theory. 

Gonzalez, D. S., Sawyer, A., and Ward, W. W. (1997). Spectral perturbations of mutants of recombinant Aequorea victoria green-fluorescent protein (GFP). Photochem. Photobiol. 65 21S. 

Ward WW, Prentice HJ, Roth AF, Cody CW, Reeves SC (1982) Spectral perturbations of the aequoria green-fluorescent protein. Photochem Photobiol 35 803-808... 

The interpretation of IETS is helpful in understanding molecular junctions. Several workers have developed techniques for doing so [97-102], some based on quite complex analyses of the full Green s function [99-101], others based on a much simpler analysis in which the fact that the response is so weak is used as the basis for perturbative expansion[98]. The results of these analyses fit the spectra well. 

Practically, any experimental study of an arbitrary system reduces to measuring the response of some physical characteristic A of the system to a probing external action which perturbs, in the general case, another characteristic B. The required response is then described by a conveniently calculable retarded Green s function (GF) that contains sufficiently complete information about the states of the system 144... 

An important and convenient characteristic of the system is its Green function (GF) that describes the response of the system to an instantaneous perturbation. We introduce the GF corresponding to Eq. (Al.ll) ... 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

In the standard choice BHF the self-consistency requirement (5) is restricted to hole states (k < kF, the Fermi momentum) only, while the free spectrum is kept for particle states k > kF- The resulting gap in the s.p. spectrum at k = kF is avoided in the continuous-choice BHF (ccBHF), where Eq. (5) is used for both hole and particle states. The continuous choice for the s.p. spectrum is closer in spirit to many-body Green s function perturbation theory (see below). Moreover, recent results indicate [6, 7] that the contribution of higher-order terms in the hole-line expansion is considerably smaller if the continuous choice is used. 

In colour vision there are three specific types of cone cell corresponding to red, green and blue receptors. The chromophore is the same for all three colours, being 11-cis-retinal bound to a protein which is structurally similar to opsin. Colour selectivity is achieved by positioning specific amino acid side chains along the chromophore so as to perturb the absorption spectrum of the chromophore. 

Following FerrelK, the second term in Equation 2 can be expressed as a Green-Kubo integral over a flux-flux correlation function. The transport is due to a velocity perturbation caused by two driving forces, the Brownian force and frictional force. The transport coefficient due to the segment-segment interaction can be calculated from the Kubo formula(9 ... 

Nakatsuji and Yasuda [56, 57] derived the 3- and 4-RDM expansions, in analogy with the Green function perturbation expansion. In their treatment the error played the role of the perturbation term. The algorithm that they obtained for the 3-RDM was analogous to the VCP one, but the matrix was decomposed into two terms one where two A elements are coupled and a higher-order one. Neither of these two terms can be evaluated exactly thus, in a sense, the difference with the VCP is just formal. However, the structure of the linked term suggested a procedure to approximate the A error, as will be seen later on. 

The wavefunctions in Eq. (2.34) are different from the wavefunctions of the free tip and free sample. The effect of the distortion potential (V = Us — Uso and V = Us - Uso), can be evaluated through time-independent perturbation. In the following, we present an approximate method based on the Green s function of the vacuum (see Appendix B). To first order, the distorted wavefunction i)i is related to the undistorted one, i]jo, by... 

Since in the FHCF(L) the effective crystal field is given in terms of the l-system Green s function, the natural way to go further with this technique is to apply the perturbation theory to obtain estimates of the /-system Green s function entering Fqs. (22) and/or (25). It was assumed and reasoned in [29] that the bare Green s function for the /-system has a block-diagonal form ... 

When some point obstacles or barriers appear in the path, additional perturbation terms must be incorporated in the diffusion equation and then the Green function is no longer given by Eq. (Cl). Once we know the perturbed Green function G, the effective diffusion coefficient D can be calculated from... 

The diffusion coefficient D0 in free space is obtained from Eqs. (C1)-(C3), so our task turns to finding the perturbed Green function. 

Suppose now a single reflecting point barrier is placed at position R in the path at time tR and removed at time tQ. Following Edwards and Evans [106], the Green function perturbed by this barrier can be expressed by... 

Next we consider the case where more than one barrier (or perturbation element) appears in the diffusion path, and the particle receives multiple perturbations. Teraoka and Hayakawa [107] assumed that the Green function for this case satifies the following Dyson equation ... 

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