Dirac equation exact solutions

Most problems in chemistry [all, according to Dirac (1929)] could be solved if we had a general method of obtaining exact solutions of the Schroedinger equation... 

Fig. 1. The ground state energies of a Z = 90 hydrogen-like atom obtainedfrom the Dirac (D) andfrom the Levy-Leblond (L) equations as functions of the nonlinear parameters. In the upper-rowfgures (Dl and LI) the a (abscissa) and (3 (ordinate) dependence ofE is displayed when L and S are set equal to the values corresponding to the exact solutions. In the lower-row figures (D2 and L2 ) the L (abscissa) and S (ordinate) dependence of E is displayed when a and 5 are set equal to the exact value. The arrows show directions of the gradient their length is proportional to the value of the gradient The solid line crossing the saddle corresponds to the functions andLl) or S = Sj (L) D2 and L2. For the definitions of thesefunctions see text.

Fig. 4. The same as in figures 1 and 3, but the values of L and S are not equal to the ones of the exact solutions. In the case of the Dirac equation L = S = i and in the case of the Levy-Leblond equation L = S = y have been taken. The saddlepoint coordinates (tXg, Pq are equal to (114,99) in the Dirac case and (66,84) in the Levy-Leblond case. The cross-sections of the energy surface by the planes a = p (broad solid line), and a = Uq (broken line) are plotted versus P, while the cross-sections by theplane P = Po (broken line with dots) and by the surface P = Pmax (thin solid line) areplotted versus a, in D2 (Dirac) and in L2 (Levy-Leblond). The scale o/P is shown in the horizontal axes. The scale of a has been chosen so that the curve for which P = Po und the onefor which a = Oq match at the saddle point The scale of a may be obtained by adding Oq - Po to the values o/P displayed in the axes of D2 and L2.

W. I. Fushchych and R. Z. Zhdanov, Symmetries and Exact Solutions of Nonlinear Dirac Equations, Mathematical Ukraina Publisher, Kiev, 1997. 

The Dirac-Coulomb-equation (i.e. the dimensionless equation with Coulomb potential) has exact solutions and can therefore be used as a good starting point for perturbation theory. 

The investigations of Asada et al. and Christensen - were carried out with linear-muffin-tin orbitals within the atomic sphere approximation (LMTO- AS A) Within the muffin-tin model suitable s, p and d basis functions (muffin-tin orbitals, MTO) are chosen. In contrast to the APW procedure the radial wave functions chosen in the linear MTO approach are not exact solutions of the radial Schrodinger (or Dirac) equation. Furthermore, in the atomic sphere approximation (ASA) the radii of the atomic spheres are of the Wigner-Seitz type (for metals the spheres have the volume of the Wigner-Seitz cell) and therefore the atomic spheres overlap. The ASA procedure is less accurate than the APW method. However, the advantage of the ASA-LMTO method is the drastic reduction of computer time compared to the APW procedure. 

An overview of the development of the finite difference Hartree-Fock method is presented. Some examples of it axe given construction of sequences of highly accurate basis sets, generation of exact solutions of diatomic states, Cl with numerical molecular orbitals, Dirac-Hartree-Fock method based on a second-order Dirac equation. 

The finite difference HF scheme can also be used to solve the Schrodinger equation of a one-electron diatomic system with an arbitrary potential. Thus the approach can be applied, for example, to the construction of exchange-correlation potentials employed by the density functional methods. The eigenvalues of several GaF39+ states have been reported and the Th 79+ system has been used to search for the influence of the finite charge distribution on the potential energy curve. It has been also indicated that the machinery of the finite difference HF method could be used to find exact solutions of the Dirac-Hartree-Fock equations based on a second-order Dirac equation. 

This so-called Hellmann-Feynman force (HFF) represents the electrostatic interaction between the negatively charged electrons and the nuclei as well as the interaction among the nuclei. Equation (4.58) would describe the forces correctly for an exact solution of the Dirac equation. However, in practical calculations we have to introduce approximations which have a rather large influence on the forces. There are two such artificial forces resulting from the following. 

In this section different methods to decouple the Dirac equation are discussed. A useful prerequisite for this reduction to two-component form is the analysis of the relationship between the large and small components of an exact eigen-solution of the Dirac equation. For every Dirac 4-spinor (f> given by equation... 

If and only if 0 is an exact solution of the Dirac equation (1), an apparently simple expression for X can inunediately be given in closed form by employing... 

If Eq. (93a) could be solved with Eq. (93b), the solution to the Dirac equation can be obtained exactly. However, Eq. (93a) has the total and potential energies in the denominator, and an appropriate approximation is needed. In our strategy, E — V in the denominator is replaced by the classical relativistic kinetic energy (relativistic substitutive correction)... 

The minimization of the energy functional with respect to the density for r > ro leads to the integral equation of the Thomas-Fermi-Dirac model restricted to this region. This was solved numerically with some constraints that must be imposed because of a wrong asymptotic behavior of the exact solution when r 00,... 

That is, the exact solution to the Dirac equation for the election in the external electric field produced by the proton. 

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