Hamiltonian relativistic many-body

B, 19, 2799 (1986). Diagonalisation of the Dirac Hamiltonian as a Basis for a Relativistic Many-Body Procedure. 

Since the relativistic many-body Hamiltonian cannot be expressed in closed potential form, which means it is unbound, projection one- and two-electron operators are used to solve this problem [39], The operator projects onto the space spanned by the positive-energy spectrum of the Dirac-Fock-Coulomb (DFC) operator. In this form, the no-pair Hamiltonian [40] is restricted then to contributions from the positive-energy spectrum and puts Coulomb and Breit interactions on the same footing in the SCF calculations. 

Second, the Hamiltonian operator for a relativistic many-body system does not have the simple, well-known form of that for the non-relativistic formulation, i.e. a sum of a sum of one-electron operators, describing the electronic kinetic energy and the electron-nucleus interactions, and a sum of two-electron terms associated with the Coulomb repulsion between the electrons. The relativistic many-electron Hamiltonian cannot be written in closed form it may be derived perturbatively from quantum electrodynamics.46... 

We may nevertheless ask whether it is possible to base RDFT on an approximate relativistic many-body approach, as, for example, the Dirac-Coulomb (DC) Hamiltonian,... 

An overview of the salient features of the relativistic many-body perturbation theory is given here concentrating on those features which differ from the familiar non-relativistic formulation and to its relation with quantum electrodynamics. Three aspects of the relativistic many-body perturbation theory are considered in more detail below the representation of the Dirac spectrum in the algebraic approximation is discussed the non-additivity of relativistic and electron correlation effects is considered and the use of the Dirac-Hartree-Fock-Coulomb-Breit reference Hamiltonian demonstrated effects which go beyond the no virtual pair approximation and the contribution made by the negative energy states are discussed. 

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 ... 

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, K. Koc. Relativistic many-body perturbation theory based on the no-pair Dirac-Coulomb-Breit Hamiltonian Relativistic correlation energies for the noble-gas sequence through Rn (Z=86), the group-llB atoms through Hg, and the ions of Ne isoelectronic sequence. Phys. Rev. A, 50(6) (1994) 4733-4742. 

The simplest relativistic many-body calculation one can do is probably a Hartree or Hartree-Fock type calculation using the many electron Hamiltonian obtained by extending eq. (7) as described above. Many such calculations have been done, but without including the projection operators A. Why is it that these calculations seem to give reasonable results where the Brown-Ravenhall analysis would seem to tell us they should not ... 

Heully JL, Lindgren I, Lindroth E, Lundqvist S, Martensson-Pendrill AM. Diagonahsation of the Dirac Hamiltonian as a basis for a relativistic many-body procedure. J Phys B At Mol Opt Phys. 

This potential is referred to in electromagnetism texts as the retarded potential. It gives a clue as to why a complete relativistic treatment of the many-body problem has never been given. A theory due to Darwin and Breit suggests that the Hamiltonian can indeed be written as a sum of nuclear-nuclear repulsions, electron-nuclear attractions and electron-electron repulsions. But these terms are only the leading terms in an infinite expansion. 

If the atom as a whole is also exposed to an external electric field Ez in the z-direction, then the total relativistic molecular many-body Hamiltonian can be written... 

In the many-body perturbation theory the non-relativistic Hamiltonian H is partitioned in the following way ... 

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.

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... 

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [10]. The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component... 

In order to improve the theoretical description of a many-body system one has to take into consideration the so-called correlation effects, i.e. to deal with the problem of accounting for the departures from the simple independent particle model, in which the electrons are assumed to move independently of each other in an average field due to the atomic nucleus and the other electrons. Making an additional assumption that this average potential is spherically symmetric we arrive at the central field concept (Hartree-Fock model), which forms the basis of the atomic shell structure and the chemical regularity of the elements. Of course, relativistic effects must also be accounted for as corrections, if they are small, or already at the very beginning starting with the relativistic Hamiltonian and relativistic wave functions. 

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. 

The total Hamiltonian describing the total interacting many-body problem—Dirac particles + radiation field + nucleus—may be obtained from the T 00-component of the energy-momentum tensor. The part of the Hamiltonian relevant for the relativistic description of the atomic many-body problem in the presence of the external electromagnetic held of the nucleus including radiative corrections and possible interactions... 

All these extensions of DFT to time-dependent, magnetic, relativistic and a multitude of other situations involve more complicated Hamiltonians than the basic ab initio many-electron Hamiltonian defined by Eqs. (2) to (6). Instead of attempting to achieve a more complete description of the many-body system under study by adding additional terms to the Hamiltonian, it is often advantageous to employ the opposite strategy, and reduce the complexity of the ab initio Hamiltonian by replacing it by simpler models, which focalize on specific aspects of the full many-body problem. Density-... 

Debashis Mukherjee is a Professor of Physical Chemistry and the Director of the Indian Association for the Cultivation of Science, Calcutta, India. He has been one of the earliest developers of a class of multi-reference coupled cluster theories and also of the coupled cluster based linear response theory. Other contributions by him are in the resolution of the size-extensivity problem for multi-reference theories using an incomplete model space and in the size-extensive intermediate Hamiltonian formalism. His research interests focus on the development and applications of non-relativistic and relativistic theories of many-body molecular electronic structure and theoretical spectroscopy, quantum many-body dynamics and statistical held theory of many-body systems. He is a member of the International Academy of the Quantum Molecular Science, a Fellow of the Third World Academy of Science, the Indian National Science Academy and the Indian Academy of Sciences. He is the recipient of the Shantiswarup Bhatnagar Prize of the Council of Scientihc and Industrial Research of the Government of India. 

Finally, there are contributions arising not only from the electron correlation but also from the terms originating in the spin-spin Hamiltonians derived from the relativistic quantum mechanics of many-body systems. 

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. 

The rules of perturbation theory associated with the relativistic no-pair Hamiltonian are identical to the well-known rules of nonrelativistic many-body perturbation theory, except for the restriction to positive-energy states. The nonrelativistic rules are explained in great detail, for example, in Lindgren and Morrison [30]. Let us start with a closed-shell system such as helium or beryllium in its ground state, and choose the background potential to be the Hartree-Fock potential. Expanding the energy in powers of V) as... 

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