Perturbation theory of relativistic corrections

Kutzelnigg, W. (1989) Perturbation theory of relativistic corrections 1. The non-relativistic limit of the Dirac equation and a direct perturbation expansion. Zeitschrifi fur Physik D, 11, 15-28. 

Kutzelnigg, W. (1990) Perturbation theory of relativistic corrections 2. Analysis and classification ofknown and other possible methods. Zeitschrift fur Physik D, 15, 27-50. 

We start this chapter with a discussion of the non-relativistic limit (nrl) of relativistic quantum theory (section 2). The Levy-Leblond equation will play a central role. We also discuss the nrl of electrodynamics and study how properties differ at their nrl from the respective results of standard non-relativistic quantum theory. We then present (section 3) the Foldy-Wouthuysen (FW) transformation, which still deserves some interest, although it is obsolete as a starting point for a perturbation theory of relativistic corrections. In this context we discuss the operator X, which relates the lower to the upper component of a Dirac bispinor, and give its perturbation expansion. The presentation of direct perturbation theory (DPT) is the central part of this chapter (section 4). We discuss the... 

Rutkowski. Regular perturbation theory of relativistic corrections Basic aspects. Phys. Rev. A, 53(1) (1996) 145-151. 

W. Kutzelnigg, Z. Phys. D, 11,15 (1989). Perturbation Theory of Relativistic Corrections. 1. The Non-Relativistic Limit of the Dirac Equation and a Direct Perturbation Expansion. 

The perturbation theory of relativistic QED, see for example [47,65], is the source of widely used methods of nonrelativistic many-body perturbation theory (MBPT) [66]. We demonstrate how it can also be used to formulate the theory of relativistic self-consistent fields as the first step in a more elaborate theory of MBPT incorporating radiative corrections. 

This chapter is devoted to the development of perturbation expansions in powers of 1 /c from the Dirac equation. In the previous chapter, the Pauli Hamiltonian was developed using the Foldy-Wouthuysen transformation. While this is an elegant method, it is probably simpler to make the derivation from the elimination of the small component with expansion of the denominator, and it is this approach that we use here. Another convenient approach is to make use of the modified Dirac equation in the limit of equality of the large and pseudo-large components. This approach enables us to draw on results from the modified Dirac approach in developing the two-electron terms of the Breit-Pauli Hamiltonian. We then demonstrate how the use of perturbation theory for relativistic corrections requires that multiple perturbation theory be employed for correlation effects and for properties. The last sections of this chapter are... 

Including also the next term of the expansion, Eq. (2.88), gives rise to additional operators including the mass-velocity, Darwin and one-electron spin-orbit operators, which can be used in perturbation theory calculations of relativistic corrections to the non-relativistic results of the Schrodinger equation and molecular properties. However, the expansion is based on the assumption that the scalar potential r) is small, which is not fulfilled for the inner electrons of heavy atoms, because close to the nucleus they are exposed to the strong Coulomb potential of the nucleus. For this situation the expansion is then no longer valid. Alternative expansions exist, which circumvent this... 

The expressions (4.22)-(4.23) found in chap. 4 for the isomer shift 5 in nonrelativ-istic form may be applied to lighter elements up to iron without causing too much of an error. In heavier elements, however, the wave function j/ is subject to considerable modification by relativistic effects, particularly near the nucleus (remember that the spin-orbit coupling coefficient increases with Z ). Therefore, the electron density at the nucleus l /(o)P will be modified as well and the aforementioned equations for the isomer shift require relativistic correction. This has been considered [1] in a somewhat restricted approach by using Dirac wave functions and first-order perturbation theory in this approximation the relativistic correction simply consists of a dimensionless factor S (Z), which is introduced in the above equations for S,... 

Table 8 Second-order many-body perturbation theory corrections to beryllium-like ions using non-relativistic (E ), Dirac-Coulomb (E ) and Dirac-Coulomb-Breit (E ) hamiltonians, obtained using the atomic precursor to BERTHA, known as SWIRLES. Basis sets are even-tempered S-spinors of dimension N= 17, with exponent sets, Xi generated by Xi = abi-i, with a = 0.413, and p = 1.376. Angular momenta in the range 0 < / < 6 have been included in the partial wave expansion of each second-order energy, and the total relativistic correction toE has been collected as Ef. All energies in hartree.

From this, we may deduce that the relativistic correction to the correlation energy is dominated by the contribution from the s electron pair, and that the total relativistic effect involving the exchange of a single transverse Breit photon is obtained to sufficient accuracy for our present purposes at second-order in many-body perturbation theory. 

Complete formal analysis of the recoil corrections in the framework of the relativistic two-particle equations, with derivation of all relevant kernels, perturbation theory contributions, and necessary subtraction terms may be performed along the same lines as was done for hyperfine splitting in [3]. However, these results may also be understood without a cumbersome formalism by starting with the simple scattering approximation. We will discuss recoil corrections below using this less rigorous but more physically transparent approach. 

Then the leading relativistic corrections of order a Za) may be easily obtained as a sum of the first and second order perturbation theory contributions corresponding to the diagrams in Fig. 7.7 [20]... 

In M0ller-Plesset theory, first-order perturbation theory does not improve on the HF energy because the zeroth-order Hamiltonian is not itself the HF Hamiltonian. However, first-order perturbation theory can be useful for estimating energetic effects associated with operators that extend the HF Hamiltonian. Typical examples of such terms include the mass-velocity and one-electron Darwin corrections that arise in relativistic quantum mechanics. It is fairly difficult to self-consistently optimize wavefunctions for systems where these tenns are explicitly included in the Hamiltonian, but an estimate of their energetic contributions may be had from simple first-order perturbation theory, since that energy is computed simply by taking the expectation values of the operators over the much more easily obtained HF wave functions. 

This expression excludes self-interaction. There have been a number of attempts to include into the Hartree-Fock equations the main terms of relativistic and correlation effects, however without great success, because the appropriate equations become much more complex. For a large variety of atoms and ions both these effects are fairly small. Therefore, they can be easily accounted for as corrections in the framework of first-order perturbation theory. Having in mind the constantly growing possibilities of computers, the Hartree-Fock self-consistent field method in various... 

The main ideas of the book are described in seven Parts divided into 33 Chapters, which are subdivided into Sections. In Part 1 we present the initial formulas to calculate the energy spectrum of a many-electron atom in non-relativistic and relativistic approximations, accounting for the relativistic effects as corrections and use perturbation theory in order to describe the energy spectra of an atom. Radiative and autoionizing... 

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