Dirac equation central potential

Ef ) so defined is different from the atomic eigenvalues of the Schrodinger (or Dirac) equation in a central potential, which we discussed in Part II, since the latter are defined for integral occupation of the orbitals.)... 

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. 

The Dirac-Coulomb equation for a central potential has the form... 

Experimental studies on the K0/Ka x-ray intensity ratio for 3d elements have shown [18-23] that this ratio changes under influence of the chemical environment of the 3d atom. Brunner et al. [22] explained their experimental results due to the change in screening of 3p electrons by 3d valence electrons as well as the polarization effect. Band et al. [24] used the scattered-wave (SW) Xa MO method [25] and calculated the chemical effect on the K0/Ka ratios for 3d elements. They performed the SW-Xa MO calculations for different chemical compounds of Cr and Mn. The spherically averaged self-consistent-field (SCF) potential and the total charge of valence electrons in the central atom, obtained by the MO calculations, were used to solve the Dirac equation for the central atom and the x-ray transition probabilities were calculated. 

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. 

The aim of this volume is twofold. First, it is an attempt to simplify and clarify the relativistic theory of the hydrogen-like atoms. For this purpose we have used the mathematical formalism, introduced in the Dirac theory of the electron by David Hestenes, based on the use of the real Clifford algebra Cl(M) associated with the Minkwoski space-time M, that is, the euclidean R4 space of signature (1,3). This algebra may be considered as the extension to this space of the theory of the Hamilton quaternions (which occupies an important place in the resolution of the Dirac equation for the central potential problem). 

In contrary, the real formalism we present further uses the same mathematical objects as the ones employed in this section. In particular, in the case of the solutions of the Dirac equation for the central potential problem, the wave function of the electron may be directly expressed by means of the same vectors as in (2.7), that is, those of the frame of the spherical coordinates in E3 = R3 °. 

Abstract. This chapter concerns a presentation of the Darwin solutions of the Dirac equation, in the Hestenes form of this equation, for the central potential problem. The passage from this presentation to that of complex spinor is entirely explicited. The nonrelativistic Pauli and Schrodinger theories are deduced as approximations of the Dirac theory. 

The derivation of the working equations of the CG method starts with the radial Dirac equation for one particle in a central-field potential V(r), e.g. the local potential approximation to the DHF equations... 

Although Dirac s equation does not directly admit of a completely self-consistent single-particle interpretation, such an interpretation is physically acceptable and of practical use, provided the potential varies little over distances of the order of the Compton wavelength (h/mc) of the particle in question. It allows, for instance, first-order relativistic corrections to the spectrum of the hydrogen atom and to the core-level densities of many-electron atoms. The latter aspect is of special chemical importance. The required calculations are invariably numerical in nature and this eliminates the need to investigate central-field solutions in the same detail as for Schrodinger s equation. A brief outline suffices. 

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