Self energy of an electron

The XC energy represents the correction to the Coulomb energy for the self-energy of an electron in a many-electron system. The latter is due to both the direct self-energy of the electron as well as the redistribution of electronic density around each electron because of the Pauli exclusion principle and the Coulomb interaction. As an example, we now discuss the case of Fermi hole and the exchange energy in Hartree-Fock (HF) theory [16]. For brevity, we restrict ourselves to closed-shell cases. 

Despite his reservations about the self-energy of an electron Madelung did offer some guidance on the hydrodynamic model for many-electron systems. Of the three possibilities that... 

U is the self-energy of an electron with a constant density within a sphere with radius R. If the system is a metal, the self-energy is equal to zero, since R 00. In the limit when the radius of the system tends to zero, the self-energy tends to infinity. 

From Figures 8.1(a) and 8.1(b) it would appear that the energy required to eject an electron from an orbital (atomic or molecular) is a direct measure of the orbital energy. This is approximately true and was originally proposed by Koopmans whose theorem can be stated as follows Tor a closed-shell molecule the ionization energy of an electron in a particular orbital is approximately equal to the negative of the orbital energy calculated by a self-consistent field (SCF see Section 7.1.1) method , or, for orbital i,... 

The second energy correction, known as the electron self-energy, corresponds to the self-interaction of an electron with the electromagnetic field. Here we encounter the first true divergence, which is removed by renormalization of the free-electron mass. The renormalized electron self energy for electron o, Es, is... 

The interelectronic energy of an electron in orbital i with two paired electrons in orbital / consists of two parts Jij for the different-spin interaction and Jy — Xy for the same-spin interaction, which together give 2 Jt] — Kij. Within the orbital i only Jn should appear but this term, due to relation (14), may be replaced by 2Ju —Ku. It is important to realize that this self-adjustment occurs only for occupied orbitals — thanks to the property (/< — K ) y)i = 0 — but not for virtual orbitals since for those the operator 2J —Kf is present, and an electron in a virtual orbital feels the full interaction of N electrons. For this reason it is often said that virtual-orbital solutions of (5) are appropriate for (N 4- l)-electron systems It would be natural to use an operator Hke (3) to obtain appropriate virtual orbitals for N-electron systems. This heis been done by Kelly in his extensive perturbation calculation of Be 29-65) by Hunt and Goddard in their calculation of the excited states of H2O >, and by Lefebvre-Brion et al. (frozen-core approximation) Goddard s method will serve to illustrate this general type of treatment. 

Hartree-Fock self-consistent field calculations indicate that the energy of an electron in the 4s orbital of vanadium lies above that of the 3d orbital in the ground state configuration, [Ar]3dMs. Explain why [Ar]3d 4s and [Ar]3d are less stable configurations than the ground state. 

Both types of procedure typically adopt a self-consistent field (SCF) procedure, in which an initial guess about the composition of the LCAO is successively refined until the solution remains unchanged in a cycle of calculation. For example, the potential energy of an electron at a point in the molecule depends on the locations of the nuclei and all the other electrons. Initially, we do not know the locations of those electrons (more specifically, we do not know the detailed form of the wavefunctions that describe their locations, the molecular orbitals they occupy). First, then, we guess the form of those wavefunctions—we guess... 

Figure 6.3 Feynman diagrams for GW and T-matrix self-energy of an excited electron, (a) GW-term, (b) T-matrix direct terms with multiple electron-electron scattering, (c) T-matrix direct terms with electron-hole scattering, and (d) T-matrix exchange terms.

The pseudopotential is derived from an all-electron SIC-LDA atomic potential. The relaxation correction takes into account the relaxation of the electronic system upon the excitation of an electron [44]- The authors speculate that ... the ability of the SIRC potential to produce considerably better band structures than DFT-LDA may reflect an extra nonlocality in the SIRC pseudopotential, related to the nonlocality or orbital dependence in the SIC all-electron potential. In addition, it may mimic some of the energy and the non-local space dependence of the self-energy operator occurring in the GW approximation of the electronic many body problem [45]. 

The values of the ESP at the nuclear positions, as obtained from the electron and Hartree-Fock structure amplitudes for the mentioned crystals (using a K-model and corrected on self-potential) are given in table 2. An analysis shows that the experimental values of the ESP are near to the ab initio calculated values. However, both set of values in crystals differ from their analogs for the free atoms [5]. It was shown earlier (Schwarz M.E. Chem. Phys. Lett. 1970, 6, 631) that this difference in the electrostatic potentials in the nuclear positions correlates well with the binding energy of Is-electrons. So an ED-data in principle contains an information on the bonding in crystals, which is usually obtaining by photoelectron spectroscopy. 

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