Nuclei interaction with electrons

DFT is based on the notion that the electron density is uniquely defined by the external potential, which for a molecule or condensed solid is simply the interaction potential with the constituent nuclei. This was put forward as the Hohen-berg-Kohn theorem [9] which states that the ground-state electron density po(r) minimizes the energy functional, T[p(r)], which for a system of electrons interacting with nuclei is given by ... 

In this expression, the term hydrogen atoms A and B. The term interaction with the electrons interchanged. However, the term A2 represents both electrons 1 and 2 interacting with nucleus A. That means the structure described by the wave function is ionic, HA HB+. In an analogous way, the term B1 Bj2 represents both electrons interacting with nucleus B, which corresponds to the structure HA+ Hb . Therefore, what we have devised for a molecular wave function actually describes the hydrogen molecule as a "hybrid" (a valence bond term that is applied incorrectly) of... 

Figure 1.3 Energy levels for an electron interacting with a spin-1 /2 nucleus with A/hc — 0.1 cm-1. The arrows show the transitions induced by 0.315 cm-1 radiation.

Clearly, the eight hyperfine lines (7 = 7/2 for 51V) have different widths but careful examination also shows that the line spacing varies, increasing with increasing B. To understand the origin of this effect we must take a closer look at the solutions to Eqn. (3.1) for the case of an unpaired electron interacting with a single nucleus. This will lead us to a derivation of eqns (2.5) and (2.11) of Chapter 2. 

In the case of iron, magnetism is due to the unpaired electrons in the 3d-orbitals, which have all parallel spin. These electrons interact with all other electrons of the atom, also the s-electrons that have overlap with the nucleus. As the interaction between electrons with parallel spins is slightly less repulsive than between electrons with anti parallel spins, the s-electron cloud is polarized, which causes the large but also highly localized magnetic field at the nucleus. The field of any externally applied magnet adds vectorially to the internal magnetic field at the nucleus. 

Electron interacts with an atom by the Coulomb potential of the positive nucleus and electrons surrounding the nucleus. The relationship between the potential and the atomic charge is given by the Poisson equation ... 

The structure in which the electron forms a normal hydrogen atom with nucleus B, which then interacts with nucleus A, is just as stable a structure as the first ... 

ZA/r,A is the potential energy due to interaction of electron i with nucleus A at the varying distance r v(r,) is the external potential for the attraction of electron i to all the nuclei, and with it we can write the double summation more compactly. The density function p can be introduced into by using the fact [31] that... 

We start by considering the hydrogen atom, the simplest possible system, in which one electron interacts with a nucleus of unit positive charge. Only two terms are required from the master equation (3.161) in chapter 3, namely, those describing the kinetic energy of the electron and the electron-nuclear Coulomb potential energy. In the space-fixed axes system and SI units these terms are... 

Fig. 3. First derivative electron spin resonance spectra. (A) ESR spectrum of an unpaired electron. (B) ESR spectrum of an unpaired electron interacting with a nitroxide resulting in a nitrogen hyperfine coupling constant aN. (C) ESR spectrum of an unpaired electron interacting with a H nucleus and a l4N nucleus as is typical for PBN radical adducts. (D) ESR spectrum of an unpaired electron interacting with the l3C nucleus, the H nucleus and the 14N nucleus of the trichloromethyl radical adduct of PBN, where the carbon tetrachloride was labeled with 13C.

It is useful to discuss some preliminaries of quantum mechanics before the discussion of /-orbitals in lanthanides. Consider the simplest system namely, the hydrogen atom consisting of a single electron interacting with a proton. The Schrodinger equation for a particle of mass m, the electron in a central field produced by the nucleus is... 

In the problem of N identical electrons interacting with each other in the Coulomb potential of a nucleus, the Pauli exclusion principle plays a crucial role. If we consider the possible states of one electron interacting with the rest of the system the exclusion principle means that a state cannot be occupied by more than one electron. 

Kikuchi lines result from inelastic scattering of electrons in specimens. Generally, an electron scatters elastically when it interacts with an atomic nucleus. The mass of a nucleus is much larger than that of an electron. Thus, their interaction is similar to a ball hitting wall where the ball bounces without energy loss. However, when the electron interacts with an electron in an atomic shell, energy will transfer between the two electrons during collision, which is referred... 

For electrons in the multicharged ions or even for the valence electrons in heavy atoms the parameter aZ cannot be considered as small. In case of heavy atoms the reason is that the effective QED potentials of the electron interaction with the nucleus are rather short-range and the interaction occurs when the outer electrons penetrate deeply into the core. Therefore the methods described in Section III are not valid anymore and all-orders in aZ methods axe necessary. 

Fig. 4. Effect of an external magnetic field upon the energy levels of an electron interacting with a nucleus with I =, showing the two transitions. Energy at (a) = g Aj4 and at (b) E = —Sg A.

The previous considerations for the case of a single electron interacting with an atomic nucleus can be extended to the more general case of a polyatomic molecule. 

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