Iterated Dirac Equation

For w = 1 or 2 they have the general form of a radial eigenvalue problem arising from some Hamiltonian. In fact, the radial parts of the nonrelativistic hydrogenic Hamiltonian, Klein-Gordon, and second-order iterated Dirac Hamiltonians with 1/r potential can all be expressed in this form for w = 1 and suitable choices of the parameters , rj, x. Similarly, the three-dimensional isotropic harmonic oscillator radial equation has this form for w = 2. 

The MS-Xa method was extended by Yang et al. to include relativistic effects starting from the Dirac equations [61] but this calculation could not be iterated to self-consistency for technical reasons. The results, in column 2 of Table 3 are rather poor, in that the HOMO-LUMO gap is far too small, and the highest filled levels are again derived from ng. 

No such singularities arise if one uses the direct perturbation theory (DPT) [12, 13, 15], which starts directly from the Dirac equation, with the Levy-Leblond equation [16] as zeroth-order (non-relativistic) approximation. Unfortunately, most practical applications of DPT were so far limited to the leading order, which is particularly easily implemented. This has sometimes led to the unjustified identification of DPT with its lowest order. Higher orders of DPT are straightforward, but have only occasionally been evaluated [17, 18, 19]. Even an infinite-order treatment of DPT is possible [12, 20], where one starts with a non-relativistic calculation and improves it iteratively towards the relativistic result. 

A straightforward elimination of the small components from the Dirac equation leads to the two-component Wood-Boring (WB) equation [81], which exactly yields the (electronic) eigenvalues of the Dirac Hamiltonian upon iterating the energy-dependent Hamiltonian... 

Eq. 20 can be solved iteratively and yields the same one-particle energies as the corresponding Dirac-equation. The radial functions P K r) correspond to the large components. In the many-electron case the correct nonlocal Hartree-Fock potential is used in Eq. 21, but a local approximation to it in Eqs. 22. Averaging over the relativistic quantum number k leads to a scalar-relativistic scheme. 

For a particle in an external field the situation is significantly more complicated, inasmuch as the terms in the Hamiltonian arising from such a field usually will not commute with the momentum-dependent terms of a transformation such as the one in Eq. [76]. As a result, there is no transformation in closed form that would exactly uncouple the Dirac equation. Instead, one must resort to iterative schemes involving a sequence of approximate transformations with successively smaller coupling between and This approach is very much in the spirit of a perturbation theory, and to some extent one can choose the parameter in which such a perturbation expansion is carried out. Several such methods have been discussed and compared by Kutzelnigg. - ... 

What this means for mean-field theory is that the lowest electron eigenvalue of the one-particle matrix that we are diagonalizing can never fall below the lowest eigenvalue of the positive-positive block of the matrix in any one iteration, and therefore there is no problem with variational collapse in a self-consistent field procedure, provided that the set of states in which we have formed the matrix represent the solutions of some one-particle Dirac equation. Failure to ensure a proper representation in a finite basis has been the occasion of problems that appear to exhibit variational collapse. Further discussion of this issue will be postponed to chapter 11, which covers finite basis methods. 

The answer lies in the interative way in which these ostensibly many body equations are solved. One really solves the one electron Dirac equation for each individual electron moving in a field produced by the electron distribution found in the previous iteration. The solutions to the one electron Dirac equation are, of course, well behaved, and only the positive energy solutions are kept in preparing the next iteration. When convergence is finally obtained, one has, in effect, solved the many body equation using projection operators constructed with the solutions of the many body equation. Not only does this intuitively seem to be a good way to define the A > but Mittleman has shown that using projection operators obtained from a Hartree-Fock... 

Dipole moment, 236, 270, 286 Dirac equation, 205, 207 Dirac-Fock, 213 Direct Cl methods, 109 Direct Inversion in the Iterative Subspace (DIIS), 73... 

Independently of the approximations used for the representation of the spinors (numerical or basis expansion), matrix equations are obtained for Equations (2.4) that must be solved iteratively, as the potential v(r) depends on the solution spinors. The quality of the resulting solutions can be assessed as in the nonrelativistic case by the use of the relativistic virial theorem (Kim 1967 Rutkowski et al. 1993), which has been generalized to allow for finite nuclear models (Matsuoka and Koga 2001). The extensive contributions by I. P. Grant to the development of the relativistic theory of many-electron systems has been paid tribute to recently (Karwowski 2001). The higher-order QED corrections, which need to be considered for heavy atoms in addition to the four-component Dirac description, have been reviewed in great detail (Mohr et al. 1998) and in Chapter 1 of this book. 

Equation (3.4d) has led mathematicians frequently to claim that the representation of the Wiener measure in (3.10) is undefined. Their complaint is reminiscent of the disrepute in which Dirac delta functions were held by mathematicians for a number of years. There are mathematically acceptable formulations, or notational transcriptions, of these functional integrals. These formulations may make for good mathematics, but they are physically unnecessary. When in doubt, we just remember that the functional integrals are defined in terms of the limit of an iterated integral. 

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