Solving the Dirac Equation

The behavior of solutions of the time-dependent Dirac equation is rather strange. This can be shown explicitly for the Dirac equation in one space dimension. In this case two spinor components are sufficient, because in one space dimension the linearization E = cap + (3mc of the energy momentum relation 2 = -t- requires just one a matrix (there is only one component of 

While this equation might be useful to model a spectral gap in one-dimensional situations, we use it here just to illustrate some of the strange features of the Dirac equation. A numerical solution of this equation that starts from some reasonable initial conditions (Gaussian wave packets) reveals a quite unexpected 

---

Abstract. The relativistic periodically driven classical and quantum rotor problems are studied. Kinetical properties of the relativistic standard map is discussed. Quantum rotor is treated by solving the Dirac equation in the presence of the periodic -function potential. The relativistic quantum mapping which describes the evolution of the wave function is derived. The time-dependence of the energy are calculated. 

Until now we have described electrons states as obeying the wave equation without specifying which wave equation. In general it is necessary to solve the Dirac equation even for a local potential. If the local potential is spherically symmetric the Dirac equation reduces to ... 

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. 

Although relativistic effects can be included in the Schrodinger equation by addition of operators describing corrections to the non-relativistic wave function, it is perhaps more satisfying to include relativistic effects by solving the Dirac equation directly. The simplest approximate wave function is a single determinant constracted from four-... 

Relativistic effects remarkably influence the electronic structure and the chemical bonding of heavy atoms [15]. In order to calculate the relativistic effects a four-component relativistic formulation by solving the Dirac equation is essential [16]. 

For the same reasons as in the nonrelativistic case the availability of a numerical solver of the DHF equations for molecules would be very much desired. One possible way to proceed would be to deal with the DHF method cast in the form of the second-order equations instead of the system of first-order coupled equations and try to solve them by means of techniques used in the FD HF approach. The FD scheme was used by Laaksonen and Grant (50) and Sundholm (51) to solve the Dirac equation. Sundholm used the similar approach to perform Dirac-Hartree-Fock-Slater calculations for LiH, Li2, BH and CH+ systems (52,53). 

As a first step in solving the Dirac equation one may split off the time factor exp —iEt/%). That is, one inserts the expression... 

It is a simple matter to solve the Dirac equation in all four potentials, and the results for the 2 3/2 — 2si/2 transition in lithiumlike bismuth are collected in the first row of Table 1. We also give the 2si/2 and 2p i2 energies separately in Tables 2 and 3 respectively. We note that we will always drop any term contributing to the energy that affects only the core states. These cancel out of the transition we are studying, and also do not affect the valence removal energies. 

The third step is to generate radial wave functions and the corresponding potential parameters. To this end, the programme solves the Dirac equation without the spin-orbit interaction (Sect.9.6.1) using the trial potential. Hence, the programme includes the important relativistic mass-velocity and Darwin shifts. The potential parameters are calculated from (3.33-35) and then converted to standard parameters by the formulae in Sect. 4.6. The energy derivatives are calculated from the solutions of the Dirac equation at two energies, E + e and E - e, where e is some small fraction of the relevant bandwidth. 

C SOLVES THE DIRAC EQUATIONS WITHOUT SPIN-ORBIT COUPLING AND ... 

SOLVE THE DIRAC EQUATIONS FOR THREE ENERGIES IN ORDER TO EVALUATE ENERGY DERIVATIVES... 

We have arrived at the system of three rigorous equations which allow us to solve the Dirac equation in the two-component form to an arbitrary degree of accuracy. Starting from the two-component equation... 

In solving the Dirac equation, it would be desirable to use as much as possible of the well-established techniques known from nonrelativistic theory. However, as it has been discussed above we quickly encounter a problem with the variational... 

P. Falsaperla, G. Fonte, J. Z. Chen. Two methods for solving the Dirac equation without variational collapse. Phys. Reo. A, 56(2) (1997) 1240-1248. 

The developments of this section show that for energy solutions in the domain of interest to us, the Rayleigh quotient is bounded below, and there is therefore no danger of variational collapse when solving the Dirac equation in a kinetically balanced finite basis. For the Dirac-Hartree-Fock equations, the only addition is the electron-electron interaction, which is positive and therefore will not contribute to a variational collapse. 

---