Propagator, particle-hole

In this section, we describe our model, and give a brief, self-contained account on the equations of the non-equilibrium Green function formalism. This is closely related to the electron and particle-hole propagators, which have been at the heart of Jens electronic structure research [7,8]. For more detailed and more general analysis, see some of the many excellent references [9-15]. We restrict ourselves to the study of stationary transport, and work in energy representation. We assume the existence of a well-defined self-energy. The aim is to solve the Dyson and the Keldysh equations for the electronic Green functions ... 

Both P and Q are sums of excitation operators (with weighting coefficients p and 9 )- Thus, P and Q applied to 0> create a polarization of 0> and we call P 6 a polarization propagator. In the special case where P and Q are both single particle-hole excitations, i.e. only one term in Eqs (5) and (6), we talk about the particle-hole propagator. It is important to note that only the residues of the polarization propagator and not of the particle-hole propagator determine transition moments (Oddershede, 1982). We must have the complete summations in Eqs (5) and (6) in order to represent the one-electron operator that induces the transition in question. 

In a recent publication [21], we have shown that already the simplest approximation to the self energy of the extended particle-hole propagator yields a well-behaved, hermitian approximation scheme that removes... 

The excitation propagator (particle-hole propagator) at the RPA level of approximation can be expressed as... 

As all four blocks of the matrix propagator contain the same information regarding excitation energies and transition moments (as is clear from the spectral representation), one can choose to concentrate on the so-called particle-hole propagator, which satisfies... 

It should be observed, however, that one cannot in general expect that an arbitrarily chosen basis B in the operator space should satisfy these three conditions, and this applies particularly to the particle-hole operator bases commonly used in the special propagator methods or the EOM approach. In such a case, the new operator basis B(1) defined by the linearly independent elements in the four matrices (2.31) may serve as a new starting basis. Due to the construction, the basis B(,) is automatically closed under adjunction (t). It is also clear that the basis B(l) is closed under multiplication due to the fact that one has the multiplication rule... 

In conclusion, a few words should be said about the equivalence between the ket-bra formalism frequently used in this article and the particle-hole formalism based on the ideas of second quantization T commonly used in the special propagator theories and the EOM method. Both formalisms are used to construct a basis for the operator space, and the essential difference is that the latter treats particles having specific symmetry properties—i.e., fermions or bosons—whereas the former is not yet adapted to any particular symmetry. In order to get a connection between the two schemes, it may be convenient in the ket-bra formalism to introduce a so-called Fock space for different numbers of particles... 

A full relativistic theory for coupling tensors within the polarization propagator approach at the RPA level was presented as a generalization of the nonrelativistic theory. Relativistic calculations using the PP formalism have three requirements, namely (i) all operators representing perturbations must be given in relativistic form (ii) the zeroth-order Hamiltonian must be the Dirac-Coulomb-Breit Hamiltonian, /foBC, or some approximation to it and (iii) the electronic states must be relativistic spin-orbitals within the particle-hole or normal ordered representation. Aucar and Oddershede used the particle-hole Dirac-Coulomb-Breit Hamiltonian in the no-pair approach as a starting point, Eq. (18),... 

The particle-hole part n s, s,(o)) and the hole-particle part II f,.,(w) of the well-known polarisation propagator [11] are defined by... 

To obtain a spectral representation of the propagator that contains a unit metric, one must transform the set of particle-hole and hole-particle operators to the representation where they give a diagonal metric with unit elements. This transformation is carried out using the excitation operators defined below ... 

Auger spectroscopy prepares a system in a core-hole state by ionizing radiation and measures the kinetic energy of secondary electrons produced when the highly excited core-hole state makes a radiationless transition to a continuum state with two valence-holes and a free electron. The initial photoelectron and the secondary (Auger) electron make this a two-electron detachment process leading to the two-particle two-hole propagator... 

This irreducible e-h interaction, via a Ward identity, also determines the particle-hole interaction in the Bethe-Salpeter equation (2.7) for the two-particle propagator. Ideally, we should... 

Finally, let us briefly turn to other relevant propagators not investigated above. The particle-hole or polarization... 

In order to calculate the excitations to the N-body system, one must consider the second-order Green s function. In particular the derivation of the polarization propagator of the particle-hole (PH) excitation is the term that needs to be outlined. This term describes the response of the system to a perturbation of the form... 

Here, we use short-hand notation 1 = (fi, q) etc. W is a screened potential, and the kernel lQri,cr2 is a two-particle propagator. In the case of multiple electron-hole scattering, the kernel (electron-hole propagator) is a product of electron and hole time-ordered Green functions... 

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