Dynamical self-energy

B Dyson equation and self energy C Static and dynamic self energy... 

The remaining w-dependent part of the self energy is called dynamic self energy MitcV... 

In the special case of systems without real two-body potentials V, where only interactions with external potentials or mean fields are considered, this is the exact solution for the primary block which determines the self energy. Since the dynamic self energy vanishes in this case, as will be shown shortly, the self energy is energy independent and given by... 

In the more general case of a correlated system, the dynamic self energy does not vanish and thus the eigenvalues of the primary block eire generally not the exact excitation energies. 

Concluding, we have shown that the present approximation to the matrix leads to the RPA. In terms of the self energy, the present approach consists of a first order static part of the self energy like in the FOSEP approximation and additionally to that part of the dynamic self energy comprising the AT-states which are in zeroth order degenerate to parts of the primary (static) block. 

The dynamic part DJ E), which can be found in the literature [8], possesses poles related to resonances of two-particle-hole and higher excitations and a branch cut above the ionisation threshold. It has an analytic structure similar to the dynamic self energy (42) from the formal point of view. This is not surprising since the projection method for deriving the Feshbach Hamiltonian is formally similar to the derivation of Dyson s equation of Sec. Ill or in Refs. [26,9]. There, a partitioning was performed yielding a projection to the primary space spanned by the orthonormal T-states while Feshbach, on the other hand, projected onto the space spanned by the non-orthonormal states aj 5 ). 

We elaborate on recent attempts to derive the local and energy-dependent density-functional potential v from the diagrammatic structure of many-body perturbation theory for the exact exchange-correlation energy, without explicit recourse to an extremal principle. The local v can be related to the nonlocal and dynamic self-energy E obtained from perturbation theory. 

The first gives the "unperturbed" one in terms of the self-consistent potential V + v, where v is the appropriate density -functional potential from eq. (2.13), l.e. v = 5e /6p. On the other hand, the "full" Green s function g is de ined as in eq.(2.1) through the non-local and dynamic self-energy operator E... 

Interestingly, the static part can be evaluated once the dynamic self-energy is known. ... 

The dynamic self-energy M(o)) can be written as the sum of two parts, M and M, analytical in the upper and lower half of the complex >-plane, respectively ... 

Here, e denotes the diagonal matrix of all one-particle (orbital) energies and 2(oo) stands for the static self-energy matrix. We recall that the latter can be computed exactly once the dynamic self-energy part is known. The one-particle GF is determined as the upper left block of the inverse of the matrix T(l — A. More precisely... 

Interestingly, an equivalent equation can also be obtained from the imaginary part of the dynamical self-energy as the electronic response to a weak, slow time-dependent perturbation. The lifetime associated with a given mode q is then defined as the inverse transition rates for the fundamental transition l- 0, Xg = l/r[ o, and can be related to the state energy broadening as =... 

---