Negative continuum

In the numerical solution of the SCF orbital equations kinetic balance restrictions are not required, as this condition will be satisfied exactly. However, in the numerical solution of MCSCF equations for purely correlating orbitals, difficulties may arise if the orbital energy , becomes too negative (Bierori etal. 1994 Indelicate 1995,1996 Kim et al. 1998). Here it is suggested that we use projection operators to eliminate the functions that correspond to the negative continuum. 

Since this only affects the one-electron portion of the Hamiltonian, its implementation in DFT is straightforward for atomic calculations. However the eigenvalues of this relativistic Hamiltonian also correspond to a negative continuum [24]. A more sophisticated Hamiltonian is the non-virtual pair approximation or the projected Dirac-Coulomb-Breit Hamiltonian [24] ... 

As a matter of principle, the k-summation on the right-hand side of (107) runs over all KS-levels, including the negative continuum states. As soon as the... 

FIGURE 8.21 Baryon spectrum in a nucleus. Below the positive energy continuum exists the potential well of real nucleons. It has a depth of 50-60 MeV and shows the correct shell structure. The shell model of nuclei is realized here. However, from the negative continuum another potential well arises, in which about 40,000 bound particles are found, belonging to the vacuum. A part of the shell structure of the upper well and the lower (vacuum) well is depicted in the lower figures. 

Here, / is the boundary of the closed shells n>f indicating the unoccupied bound and the upper continuum electron states m electron vacuum polarization). All the vacuum polarization and the self-energy corrections to the sought-for values are omitted. Their numerical smallness compared... 

These terms refer to the positive part of the energy spectrum. For the negative continuum (the Dirac sea), the proportion of the components is reversed. 

Although Dirac s hole theory could account for an acceptable interpretation of the negative continuum states for the very first time, it is still plagued by a series of inconsistencies incompatible with a fundamental physical theory from today s point of view. First of all, it crucially demands the fermionic nature of the electrons since it heavily relies on the Pauli principle in order to... 

Similar to nonrelativistic Hartree-Fock theory, the Dirac-Roothaan Eqs. (10.61) are solved iteratively until self-consistency is reached. However, because of the properties of the one-electron Dirac Hamiltonian entering the Fock operator, molecular spinors representing unphysical negative-energy states (recall section 5.5) show up in this procedure. As many of these negative-continuum... 

P. Indelicato. Correlation and Negative Continuum Effects for the Relativistic Ml Transition in Two-Electron Ions using the Multiconfiguration Dirac-Fock Method. Phys. Rev. Lett, 77(16) (1996) 3323-3326. 

There is, however, a problem with this approach. The filling of the negative continuum implies that the vacuum state is infinitely charged, and as a consequence we would expect an infinite interaction between any state and the vacuum. We therefore need to take the reinterpretation one step further. What we actually measure is not the absolute properties of a state but the differences in properties between a state and the vacuum. The Dirac equation is therefore only the starting point for a theory that is an infinitely-many-body theory, even for a system with only one electron—or indeed for the vacuum itself ... 

Thus, the negative-energy electron solutions have now become positive-energy positron solutions, and because of this there is no more danger of radiative decay of bound electrons into the negative continuum. 

Here, C is the gauge constant, / is the boundary of the closed shells n > f indicating the vacant band and the upper continuum electron states matomic core and the states of a negative continuum (accounting for the electron vacuum polarization). The minimization of the functional ImSEninv leads to the Dirac-Kohn-Sham-like equations for the electron density that are numerically solved. Finally an optimal set of the IQP functions results. In concrete calculation it is sufficient to use the simplified procedure, which is reduced to the functional minimization using the variation of the correlation potential parameter b in Eq. 3.11 [20, 32]. The Dirac equations for the radial functions F and G (the large and small Dirac components respectively) are ... 

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