No-virtual-pair approximation

The DC or DCB Hamiltonians may lead to the admixture of negative-energy eigenstates of the Dirac Hamiltonian in an erroneous way [3,4]. The no-virtual-pair approximation [5,6] is invoked to correct this problem the negative-energy states are eliminated by the projection operator A+, leading to the projected Hamiltonians... 

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. 

The no virtual pair approximation. - According to Dirac s hole theory, those states lying below —me2 are taken to be filled according to the... 

Exclusion Principle. The energy associated with the filled vacuum is an unobservable constant which should be subtracted from a given physical model. Calculations which go beyond an independent particle model but are carried out using only the positive energy branch of the Dirac spectrum are said to be carried out within the no virtual pair approximation. Such calculations essentially follow the procedures adopted in non-relativistic studies. The relativistic and non-relativistic correlation energy calculations differ only in the model used to defined the reference independent particle model. 

Table 2 Relativistic and non-relativistic finite basis set many-body perturbation theory calculations for the argon ground state within the no virtual pair approximation...

Some work has been reported on relativistic coupled cluster methods most notably by Kaldor, Ishikawa and their collaborators.232 These calculations are carried out within the no virtual pair approximation and are therefore analogous to the non-relativistic formulation. Perturbative analysis of the relativistic electronic structure problem demonstrated the importance of the negative energy branch of the spectrum in the calculation of energies and other expectation values. 

Summation over the single-particle states is restricted to positive energy contributions, which has become known as the no virtual pair approximation. In practice, the interaction vu may be the complete energy-dependent, covariant interaction, vf2, the Coulomb interaction, gi2, or the transverse-gauge Breit operator, 312 -I- 612-... 

Only since the beginning of the 1970s have accurate calculations of QED effects in many electron atoms become feasible, which leads us beyond the no-virtual-pair approximation. Radiative corrections are regarded to be important only for inner-shell regions, for example, in accurate calculations of X-ray spectra. However, these corrections may become important in the atomic fine structure. For example, for a two-electron atom with Z = 60 radiative corrections contribute... 

The no-virtual-pair approximation (NVPA) is invoked so that negative-energy solutions of the SCF equations are discarded. 

Y. Watanabe, H. Nakano, and H. Tatewaki, Effect of removing the no-virtual pair approximation on the correlation energy of the He isoelectronic sequence. II. Point nuclear charge model, J. Chem. Phys. 132,124105 (7 pages) (2010). 

At the no-virtual-pair level of approximation, there is no numerical evidence that divergences in this theory occur even when four-component spinors are... 

The two relativistic four-component methods most widely used in calculations of superheavy elements are the no-(virtual)pair DF (Coulomb-Breit) coupled cluster technique (RCC) of Eliav, Kaldor, and Ishikawa for atoms (equation 3), and the Dirac-Slater discrete variational method (DS/DVM) by Fricke for atoms and molecules. " Fricke s DS/DVM code uses the Dirac equation (3) approximated by a Slater exchange potential (DFS), numerical relativistic atomic DS wavefunctions, and finite extension of the nuclei. DFS calculations for the superheavy elements from Z = 100 to Z = 173 have been tabulated by Fricke and Soff. A review on various local density functional methods applied in superheavy chemistry has been given by Pershina. ... 

Moreover, electron-positron pair creation and other proce.s.ses ( radiative corrections ) de.scribed by quantum electrodynamics which has quantized degrees of freedom for both the fermions and the electromagnetic field are usually not included in the theory, although the chaige-conjugated degrees of freedom are still there. Therefore the literature often refers to the no virtual pair or, in short, no pair approximatioa Very few calculations go beyond this approximation. " Nevertheless. the no-pair operator based on the DCB Hamiltonian provides an exellent approximation to the full theory, generally sufficient for the determination of relativistic effects in the electronic structure of neutral atoms and molecules. 

The no-virtual-pair Dirac-Coulomb-Breit Hamiltonian, correct to second order in the fine-structure constant a, provides the framework for four-component methods, the most accurate approximations in electronic structure calculations for heavy atomie and molecular systems, ineluding aetinides. Electron correlation is taken into aeeount by the powerful coupled eluster approaeh. The density of states in actinide systems necessitates simultaneous treatment of large manifolds, best achieved by Fock-space coupled eluster to avoid intruder states, which destroy the convergence of the CC iterations, while still treating a large number of states simultaneously, intermediate Hamiltonian sehemes are employed. 

Qvac is the total charge of the vacuum, which vanishes for free electrons, but is finite in the presence of an external field (the phenomenon of vacuum polarization). Note that whilst Q is conserved for all processes, the total number of particles need not be it is always possible to add virtual states incorporating electron-positron pairs without changing Q. The neglect of such terms in the total wavefunction of an n-electron system is called the no-pair approximation. 

This expression applies equally to the nonrelativistic case expressed in terms of spin-orbitals and the relativistic case expressed in terms of spinors. In nonrelativistic MP2, the sums over i, j and a, b span the range of occupied and virtual one-electron functions, respectively. In the relativistic formulation the sum over virtual functions can only include spinors with positive energies if we are to stay within the no-pair approximation. 

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