Selfenergy

Labzowsky, L., Goidenko, I., Tokman, M. and Pyykko, P. (1999) Calculated selfenergy contributions for an ns valence electron using the multiple-commutator method. Physical Review A, 59, 2707-2711. 

As mentioned in Section 2, the CPs of solids have to be calculated on the quasi-particle scheme. In order to calculate the quasi-particle states, non-local and energy-dependent self-energy in Equation (13) must be evaluated in a real system. In practice, the exact self-energy for real systems are impossible to compute, and we always resort to approximate forms. A more realistic but relatively simple approximation to the selfenergy is the GWA proposed by Hedin [7]. In the GW A, the self-energy operator in Equation (12) is... 

For many ionization energies and electron affinities, diagonal selfenergy approximations are inappropriate. Methods with nondiagonal self-energies allow Dyson orbitals to be written as linear combinations of reference-state orbitals. In most of these approximations, combinations of canonical, Hartree-Fock orbitals are used for this purpose, i.e. 

Fig. 3.19. Second order pertnrbation theory contribution with two one-loop selfenergy operators...

In this connection there is an important question concerning the infinite selfenergy of a point charge in classical as well as in quantum field theory. The latter uses a renormalization process to solve the problem, namely, by subtracting two infinities to end up with a finite result. Despite the success of such a procedure, a more physically satisfactory way is needed [80]. Possibly the present theory may provide such an alternative, by tackling the divergence problem in a more surveyable manner. The finite result of a difference between two infinities due to renormalization theory would then be replaced by a finite result obtained from the product of an infinity and a zero, as demonstrated by the present analysis. 

Most of the results presented in Table 9 do not take into account selfenergy corrections, which are necessary in order to describe, in a proper way, the one-particle excited states. In the fourth column of Table 9 we report the GW corrected band-gaps, for the smallest Ge-NWs in the [111],[110] directions, and for all the [100] Ge-NW. A complete discussion on this part can be found elsewhere [121], We can see (Table 9, fifth column) that the effect of the GW correction is an opening of the DFT-LDA gap, by an amount which... 

To remain consistent with this approximation, the bath selfenergy should also be treated to order t, more explicitly ... 

Contributions of the first and second orders in the uncutable selfenergy part X7 is given by the matrices ... 

Because second order is the first nonvanishing contribution, zero and first order electron propagator calculations correspond to KT results. In the diagonal, P3 selfenergy, terms with three virtual indices, such as... 

By using the expression of the exact Hamiltonian in Equation 3.20, the selfenergy of the ionic terms and the off-diagonal Hamiltonian matrix elements are readily obtained... 

So far, we have fairly extensively discussed the general aspects of static and dynamic relaxation of core holes. We have also discussed in detail methods for calculating the selfenergy (E). Knowing the self-energy, we know the spectral density of states function A (E) (Eq. (10)) which describes the X-ray photoelectron spectrum (XPS) in the sudden limit of very high photoelectron kinetic energy (Eq. (6)). We will now present numerical results for i(E) and Aj(E) and compare these with experimental XPS spectra and we will find many situations where atomic core holes behave in very unconventional ways. 

In this subsection, we describe a method where the electron propagator machinery developed for quantum chemical calculations of single molecules is applied [9-21]. We use a so-called noncrossing approximation where a selfenergy for interacting bridge electrons is divided into two independent parts [55-57] ... 

Other ions in the solution. The self-energy of a dipole embedded in a dielectric sphere is the key to Onsager s theory of the dielectric constant of dipolar fluids. Equally, in any theory for, say, the surface energy of water, or adsorption of molecule, the self-energy of a molecule as a function of its distance from an interface is involved. In adsorption proper, the same selfenergy for a molecule appears in the partition function of statistical mechaiucs from which the adsorption isotherm is derived. 

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