Bethe-Salpeter

Bethe-Guggenheim pair distribution approximation, 122 Bethe Salpeter s Hamiltonian operator, 240... 

On the other hand, functional derivatives of the Bethe-Salpeter equation allows to evaluate the nonlinear responses using the interaction kernels h only (which depend on the Hartree and exchange-correlation energies). The relations between the screened nonlinear responses and the bare ones are derived by using nonlinear PRF [32],... 

The account of two-particle correlations in nuclear matter can be performed considering the two-particle Green function in ladder approximation. The solution of the corresponding Bethe-Salpeter equation taking into account mean-field and Pauli blocking terms is equivalent to the solution of the wave equation... 

The calculation of n k) is an important and very nontrivial many-body problem and requires the approximate solution of a Bethe-Salpeter equation. Our present objective is to generate gradient corrections to the LDA so (A.7) can be simplified by assuming that Sn(r) is so slowly varying in space (in addition to being of small magnitude) that 8n k) is essentially zero except for very small k. 

Van der Horst JW, Bobbert PA, Michels MAJ, Bassler H (2001) Calculation of excitonic properties of conjugated polymers using the Bethe-Salpeter equation. J Chem Phys 114 6950... 

Bethe-Salpeter Equation and the Effective Dirac Equation 5... 

Quantum field theory provides an unambiguous way to find energy levels of any composite system. They are determined by the positions of the poles of the respective Green functions. This idea was first realized in the form of the Bethe-Salpeter (BS) equation for the two-particle Green function (see Fig. 1.2)... 

A weakly bound state is necessarily nonrelativistic, v Za (see discussion of the electron in the field of a Coulomb center above). Hence, there are two small parameters in a weakly bound state, namely, the fine structure constant a. and nonrelativistic velocity v Za. In the leading approximation weakly bound states are essentially quantum mechanical systems, and do not require quantum field theory for their description. But a nonrelativistic quantum mechanical description does not provide an unambiguous way for calculation of higher order corrections, when recoil and many particle effects become important. On the other hand the Bethe-Salpeter equation provides an explicit quantum field theory framework for discussion of bound states, both weakly and strongly bound. Just due to generality of the Bethe-Salpeter formalism separation of the basic nonrelativistic dynamics for weakly bound states becomes difficult, and systematic extraction of high order corrections over a and V Za becomes prohibitively complicated. 

Leading recoil corrections in Za (of order (Za) (m/M)") still may be taken into account with the help of the effective Dirac equation in the external field since these corrections are induced by the one-photon exchange. This is impossible for the higher order recoil terms which reflect the truly relativistic two-body nature of the bound state problem. Technically, respective contributions are induced by the Bethe-Salpeter kernels with at least two-photon exchanges and the whole machinery of relativistic QFT is necessary for their calculation. Calculation of the recoil corrections is simplified by the absence of ultraviolet divergences, connected with the purely radiative loops. 

The paper is organized as follows. A complete description of the theoretical methods used is given in Section 2 considering, in the different subsections, the Density Functional Theory (DFT) (Section 2.1), the A self-consistent (A-SCF) approach (Section 2.1.1), and the Many-Body perturbation theory (Section 2.2) through the GW (Section 2.2.1) and the Bethe-Salpeter (Section 2.2.2) methods. 

In particular we will show the dependence of the electronic gap on both wire size and orientation. Further, in some of the studied wires, self-energy corrections, by means of the GW method, and also electron-hole interaction, by solving the Bethe-Salpeter equation, will be included in order to have an appropriate description of the excited states. 

The most straightforward way refers to the Bethe-Salpeter equation, i.e. an equation for the two-body Green function. It may be solved for the Coulomb potential and a two-body perturbative theory can be developed starting from this solution. This method was rarely used in the bound state QED calculations, being very complicated. 

The electron and positron in positronium move with a typical velocity a and have a momentum mot and energy mo . If we are interested in a precision level where relativistic effects become important, the usual quantum mechanical treatment of the bound state has to be merged with the complete field theory description. Then, no matter how weak the coupling is, we deal with the problem of bound states in the field theory, which is rather complicated. One of the possible computational approaches (not always the most convenient) is the Bethe-Salpeter equation. 

In table 2 our result is compared with the UV spectroscopic result of Klein et al. [26], Also shown are the theoretical results of Zhang et al. [2], Plante et al. [27], and Chen et al. [28], The first of these uses perturbation theory, with matrix elements of effective operators derived from the Bethe-Salpeter equation, evaluated with high precision solutions of the non-relativistic Schrodinger equation. This yields a power series in a and In a. The calculations of Zhang et al. include terms up to O(o5 hi a) but omit terms of 0(ary) a.u. The calculations of Plante et al. use an all orders relativistic perturbation theory method, while those of Chen et al. use relativistic configuration interaction theory. These both obtain all structure terms, up to (Za)4 a.u., and use explicit QED corrections from Drake [29],... 

The standard approaches for two-fermion systems like Bethe-Salpeter or Dirac-Breit satisfy this condition. But usually this property is lost in the final results, because (nonrelativistic) approximations (e.g. Ei to ) are used for one or both particles to simplify the calculation. 

It now remains to expand the operators in (3.235) using the definitions given in (3.230) but before we do so we must draw attention to a difficulty with (3.235). The final term, containing the operator (00)2 is not obtained if a more sophisticated treatment starting from the Bethe Salpeter equation is used. The reader will recall our earlier comment that the interaction term in the Breit Hamiltonian is acceptable provided it is treated by first-order perturbation theory. Rather than launch into quantum electrodynamics at this stage, we shall proceed to develop (3.235) but will omit the (00)2 term without further comment. 

Gmn(r) is the one-exciton Green function. G(2) represents the time evolution of the two exciton states and satisfies the Bethe-Salpeter equation ... 

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