Vacuum contribution

Since the NJL model is not renormalizable, we need some regularization procedure to get a meaningful finite value for the vacuum contribution. Consider the sum of the negative energy over the Dirac sea,... 

Crowell [17] argues that the vacuum contribution to the virtual Bt3> field is a very small effect, about a millionth of the Lamb shift. 

The vacuum contribution caused by the externally applied electric field is represented by the term e0E, and the electrical polarization of the matter in the system is described by P, e.g. [2], This relation is independent of the nature of the polarization which could be pyroelectric polarization, by piezoelectric polarization or dielectric polarization (by an external electric field). 

Quantity Derivative expression3 Vacuum contribution Solvent contribution13... 

In addition the vacuum contributions in the functional dependence of E c on j are dropped , so that one is led to the RDFT analogue of the no-pair approximation applied in conventional relativistic many-body approaches (see e-g- [7]). An a posteriori perturbative evaluation of these corrections is possible and should be adequate, except in special circumstances as for instance the calculation of the structure of super-heavy atoms (with Z 137 [33]). The resulting RKS-equations are then still given by Eqs. (3.15-3.17), but f(x) and Tj are determined by the simpler expressions (3.7) and (3.11). 

In the vacuum contribution (168) we have some asymmetry between the indices m and j, since m runs from 1 to oo, whereas j runs from — oo to oo. This asymmetry can be eliminated if one differentiates both sides of equation (168) with respect to the independent variable x at a fixed value of u and performs the summation over the superscript n with the aid of the recurrence relations (87) and (88). It is easy to verify that all the summands with n > p are canceled, so the infinite series over n can be reduced to the finite sum from 1 to (p 1) ... 

If one aims at electronic stracture calculations for molecules or solids the most helpful, and at the same time most legitimate, simplification is the neglect of all vacuum contributions to the four current and the energy functional. The technical implementation of this neglect is immediately clear for all singleparticle quantities such as and 7i,... 

As common in relativistic electronic structure theory, one invokes the so-called no-sea approximation where one neglects all vacuum contributions of the filled negative energy continuum [41]. The only remaining effect of the sea is the restriction for electrons to occupy only states of positive energy. Then the density is constructed from DKS one-electron orbitals of a single-determinant A-electron wave function ... 

So far nobody has solved the indicated problem. The evaluation of the vacuum contributions (in and Tg) includes the full set of solutions of the Dirac equation and renormalization at each step of the self-consistency procedure. 

The double line stands for the free photon propagator D J, and the wiggly line for the lowest order fermion propagator of the RHEG,. The photon propagator can be split into a vacuum contribution and a direct contribution due to the occupied electron states... 

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