Many-body Dirac equation

A natural way to generalize the non-relativistic many-body Schrodinger equation is to combine the one-electron Dirac operators and Coulomb and Breit two-electron operators. However such an equation would have serious defects. One of them is the continuum dissolution first discussed by Brown and RavenhaU [36]. This means that the Schrodinger-type equation has no stable solutions due to the presence of the negative energy Dirac continuum. A constrained variational approach to the positive energy states becomes therefore necessary. 

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [9], The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit (DFB) functions. The spherical symmetry of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10], The radial four-spinor has the so-called large component the upper two places and the small component Q, in the lower two. The quantum number k (with k =j+ 1/2) comes from the spin-angular equation, and n is the principal quantum number, which counts the solutions of the radial equation with the same k. Defining... 

All that remains to be done for determining the fluctuation spectrum is to compute the conditional average, Eq. (31). However, this involves the full equations of motion of the many-body system and one can at best hope for a suitable approximate method. There are two such methods available. The first method is the Master Equation approach described above. Relying on the fact that the operator Q represents a macroscopic observable quantity, one assumes that on a coarse-grained level it constitutes a Markov process. The microscopic equations are then only required for computing the transition probabilities per unit time, W(q q ), for example by means of Dirac s time-dependent perturbation theory. Subsequently, one has to solve the Master Equation, as described in Section TV, to find both the spectral density of equilibrium fluctuations and the macroscopic phenomenological equation. 

Theoretical calculations of two-electron ion energy levels have been the topic of much research since the discovery of quantum mechanics. The contribution of relativistic effects via the Dirac equation and QED contributions has been intensely studied in the last three decades [1]. Two-electron systems provide a test-bed for quantum electrodynamics and relativistic effects calculations, and also for many body formalisms [2]. 

Since the Dirac equation is written for one electron, the real problem of ah initio methods for a many-electron system is an accurate treatment of the instantaneous electron-electron interaction, called electron correlation. The latter is of the order of magnitude of relativistic effects and may contribute to a very large extent to the binding energy and other properties. The DCB Hamiltonian (Equation 3) accounts for the correlation effects in the first order via the Vy term. Some higher order of magnitude correlation effects are taken into account by the configuration interaction (Cl), the many-body perturbation theory (MBPT) and by the presently most accurate coupled cluster (CC) technique. 

DFT is the ground-state limit of a more general OFT. The OEL equations of OFT do not necessarily contain local potential functions. Tests of locality fail for the effective exchange potential in the UHF exchange-only model. Dirac s derivation of TDHF theory can readily be extended to a TDOFT that includes electronic correlation. The exchange-only limit of this theory is consistent with TDHF and with the RPA in many-body theory. 

Later, in Sec. 4, we will give a detailed discussion of the need for the no-pair Hamiltonian in relativistic calculations, its limitations, and its relation to QED. To establish a foundation for our studies of few-electron systems, we start in Sec. 2 with a discussion of the one-electron central-field Dirac equation and radiative corrections to one-electron atoms. In Sec. 3 we describe many-body perturbation theory (MBPT) calculations of few-electron atoms, and finally, in Sec. 4 we turn to relativistic configuration-interaction (RCI) calculations. 

In order to establish a relativistic hyperfine Hamiltonian operator for a many-electron system one faces the problem of setting up a relativistic many-body Hamiltonian which cannot be written down in closed form. If one considers a one-electron system first one can obtain an exact expression for the hyperfine Hamiltonian starting from the one-electron Dirac equation in minimal coupling to the electromagnetic field ... 

The basic idea underlying the development of the various density functional theory (DFT) formulations is the hope of reducing complicated, many-body problems to effective one-body problems. The earlier, most popular approaches have indeed shown that a many-body system can be dealt with statistically as a one-body system by relating the local electron density p(r) to the total average potential, y(r), felt by the electron in the many-body situation. Such treatments, in fact, produced two well-known mean-field equations i.e. the Hartree-Fock-Slater (HFS) equation [14] and the Thomas-Fermi-Dirac (TFD) equation [15], It stemmed from such formulations that to base those equations on a density theory rather than on a wavefunction theory would avoid the full solution... 

A fully relativistic treatment of more than one particle would have to start from a full QED treatment of the system (Chapter 1), and perform a perturbation expansion in terms of the radiation frequency. There is no universally accepted way of doing this, and a full relativistic many-body equation has not yet been developed. For many-particle systems it is assumed that each electron can be described by a Dirac operator (ca n -I- P me and the many-electron operator is a sum of such terms, in analogy with the kinetic energy in non-relativistic theory. Furthermore, potential energy operators are added to form a total operator equivalent to the Hamiltonian operator in non-relativistic theory. Since this approach gives results that agree with experiments, the assumptions appear justified. 

Y. Ishikawa, H. M. Quiney. Relativistic many-body perturbation-theory calculations based on Dirac-Fock-Breit wave equations. Phys. Rev. A, 47 (1993) 1732-1739. 

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