Electrons in a Central Potential

The subalgebra Cl+(E3) of Cl(E3) is the field of the Hamilton quaternions. We see that this field plays an important role in the theory of the Dirac electron in a central potential. 

We first consider a single electron in a central potential for which spherical polar coordinates (r, 6,(p) are appropriate. 

This section will close with a final eigenfucntion problem. The Schrodinger equation for the radial part (./ ) of the wavefunction of an electron in a central potential is ... 

It is known (Chap. A) that Koopmans theorem is not vahd for the wavefunctions and eigenvalues of strongly bound states in an atom or in the cores of a solid, i.e. for those states which are a solution of the Schrodinger (or Dirac) equation in a central potential. In them the ejection (or the emission) of one-electron in the electron system means a strong change in Coulomb and exchange interactions, with the consequent modification of the energy scheme as well as of the electronic wavefunction, in contradiction to Koopmans theorem. 

Methods for calculating collisions of an electron with an atom consist in expressing the many-electron amplitudes in terms of the states of a single electron in a fixed potential. In this chapter we summarise the solutions of the problem of an electron in different local, central potentials. We are interested in bound states and in unbound or scattering states. The one-electron scattering problem will serve as a model for formal scattering theory and for some of the methods used in many-body scattering problems. 

The first approximation to paramterize equation (1) is to assume that all electrons move in a central potential. If we then limit the analysis to a single configuration, we need discuss only the Coulomb and spin-orbit interaction between the equivalent f-electrons. With the aid of tensor operators the Coulomb interaction can be expressed as... 

A bound-state wavefunction for a single electron in a central field, corresponding either to an actual or hypothetical potential (for example the effective field used in an independent-particle model), is usually referred to as an atomic orbital. We denote the potential energy by V (r) (angle-independent), and the eigenvalue equation is thus... 

It is not found in other effectively one-electron systems such as the alkalis, where the orbital states 7 of a given principal quantum number n usually differ considerably in energy. In these atoms the valence electron moves in a central potential in which the Coulomb potential of the nucleus is modified by the screening effect of the inner... 

Wavefunctions in L-S coupling. We recall that in the central field approximation, discussed in section 3.9, each electron in a many-electron atom is considered to move in a central potential independently of the others, at least in the first approximation. The i electron may then be described by the function (equation (3.71)) in terms of... 

Important examples of chemical interest include particles that move in the central held on a circular orbit (V constant) particles in a hollow sphere V = 0) spherically oscillating particles (V = kr2), and an electron on a hydrogen atom (V = 1 /47re0r). The circular orbit is used to model molecular rotation, the hollow sphere to study electrons in an atomic valence state and the three-dimensional harmonic oscillator in the analysis of vibrational spectra. Constant potential in a non-central held dehnes the motion of a free particle in a rectangular potential box, used to simulate electronic motion in solids. 

When the nuclear charge changes due to radioactive decay and/or an inner-shell vacancy is produced, the bound electrons in the same atom or molecule experience the sudden change in the central potential and have a small but finite probability to be excited to an unoccupied bound state (shakeup) or ejected to the continuum (shakeoff). We calculated the shakeup-plu.s-shakeoff probabilities accompanying PI and EC using the method of Carlson and Nestor [45]. 

The problem of a single electron in an atom may be approximated by that of an electron in a local, central potential with the Coulomb form at large distances. The bound radial eigenstates u r) of an electron in such a potential may be expanded in a basis set fiif(r) of radial functions, each of which is square integrable and is normalised. 

In a scattering experiment a beam of electrons of momentum k hits a target. We consider the target to be represented by a potential V(r). Electrons are observed by a detector placed at polar and azimuthal angles 9,(f) measured from the direction of the incident beam, which is the z direction in a system of spherical polar coordinates (fig. 4.2). For a central potential the problem is axially symmetric. Relevant quantities do not depend on (f). The detector subtends a solid angle... 

In the previous part of the chapter we expressed the problem of an electron in a local, central potential in terms of radial equations and eigenstates of orbital angular momentum. In generalising to the case where the electron obeys the Dirac equation (3.154) we remember that spin and orbital angular momentum are coupled. 

We suppose that A is sufficiently small for considering its incidence as a perturbation of the Darwin solution corresponding to a state, in the central potential An, of energy E. The energy E of the electron will be then written... 

Thus, one might expect the one-electron atom orbitals (AOs, say, the familiar s, p, d etc. orbitals) to be the ones to use in expanding the orbitals in a singledeterminant model of the electronic structure of (at least) the rare-gas atoms which (presumably) have a central potential which is a combination of many interactions all described by Coulomb s law. The forms of these orbitals have been determined by the action of Coulomb s law and there are a huge number of them of all possible symmetry types, with which to msdce an expansion of apparently arbitrary accuracy. 

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