Self energy, atomic calculation

Figure 2.35 shows the 3s radial function (radial part of the atomic orbital) of an arbitrary atom (Na in this case) according to a self-consistent atomic calculation. On each side, there are two radial nodes close to the origin characteristic for the 3s electron s high kinetic energy which itself reflects the large nuclear potential. 

The electrostatic energy is calculated using the distributed multipolar expansion introduced by Stone [39,40], with the expansion carried out through octopoles. The expansion centers are taken to be the atom centers and the bond midpoints. So, for water, there are five expansion points (three at the atom centers and two at the O-H bond midpoints), while in benzene there are 24 expansion points. The induction or polarization term is represented by the interaction of the induced dipole on one fragment with the static multipolar field on another fragment, expressed in terms of the distributed localized molecular orbital (LMO) dipole polarizabilities. That is, the number of polarizability points is equal to the number of bonds and lone pairs in the molecule. One can opt to include inner shells as well, but this is usually not useful. The induced dipoles are iterated to self-consistency, so some many body effects are included. 

We recently presented a correlation method based on the Wigner intracule, in which correlation energies are calculated directly from a Hartree-Fock waveftmction. We now describe a self-consistent form of this approach which we term the Hartree-Fock-Wigner method. The efficacy of the new scheme is demonstrated using a simple weight function to reproduce the correlation energies of the first- and second-row atoms with a mean absolute deviation of 2.5 m h. 

In most atomic programs (5) is actually solved self-consistently either in a local potential or by the relativistic Hartree-Fock method. There is, however, an important time-saving device that is often used in energy band calculations for actinides where the same radial Eq. (5) must be solved If (5.a) is substituted into (5.b) a single second order differential equation for the major component is obtained... 

Our method of calculation is based on an idea by Ivanov-Ivanova [11]. In an atomic system, the radiative shift and the relativistic part of the energy are, in principle, determined by one and the same physical field. It may be assumed that there exists some universal function that connects the self-energy correction and the relativistic energy. The self-energy correction for the states of a hydrogen-like ion was presented by Mohr [1] as ... 

We applied the generating functional approach to the periodic Anderson model. Calculation of the electron GFs gdd, 9ds, 9sd and gss reduces to calculation of only the d-electron GF. For this, an exact matrix equation was derived with the variational derivatives. Iterations with respect to the effective matrix element Aij(to) allow to construct a perturbation theory near the atomic limit. Along with the self-energy, the terminal part of the GF Q is very important. The first order correction for it describes the interaction of d-electrons with spin fluctuations. In the paramagnetic phase this term contains a logarithmic singularity near the Fermi-level and thus produces a Kondo-like resonance peak in the d-electron density of states. The spin susceptibility of d-electrons... 

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. 

Im. Z i(E) can be considered as a product of an ionic excitation density of states and an energy-dependent coupling constant. In model calculations one can independently vary the shape and the band with of the denstiy of states and the strength of the coupling constant. In the present case we can only vary these parameters indirectly by changing the atomic number Z. Since the self-energy involves the polarizability of the ionic system there must be an oscillator-strength sum rule such that... 

QED contributions to the Lamb shift consist of electron self-energy and vacuum polarization terms. In one-electron atoms the former is both the larger and the more difficult to calculate and has been the focus of much recent theoretical work. Up to Feynman diagrams including two-loops the self-energy contribution to a hydrogenic energy level can be written as [32]... 

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