Skyrme functional

In atomic nuclei, SRPA was derived [9,10,19] for the demanding Skyrme functional involving a variety of densities and currents (see [20] for the recent review on Skyrme forces). SRPA calculations for isoscalar and isovector giant resonances (nuclear counterparts of electronic plasmons) in doubly magic nuclei demonstrated high accuracy of the method [10]. 

The paper is organized as follows. In Section 2, derivation of the the SRPA formalism is done. Relations of SRPA with other alternative approaches are commented. In Sec. 3, the method to calculate SRPA strength function (counterpart of the linear response theory) is outlined. In Section 4, the particular SRPA versions for the electronic Kohn-Sham and nuclear Skyrme functionals are specified and the origin and role of time-odd currents in functionals are scrutinized. In Sec. 5, the practical SRPA realization is discussed. Some examples demonstrating accuracy of the method in atomic clusters and nuclei are presented. The summary is done in Sec. 6. In Appendix A, densities and currents for Skyrme functional are listed. In Appendix B, the optimal ways to calculate SRPA basic values are discussed. 

Although Skyrme forces are relatively simple, they are still much more demanding than the Coulomb interaction. In particular, they deal with a variety of diverse densities and currents. The Skyrme functional reads [12,36,37]... 

Comparison of the Kohn-Sham and Skyrme functionals leads to a natural question why these two functionals exploit, for the time-dependent problem, so different sets of basic densities and currents If the Kohn-Sham functional is content with one density, the Skyrme forces operate with a diverse set of densities and currents, both T-even and T-odd. Then, should we consider T-odd densities as genuine for the description of dynamics of finite many-body systems or they are a pequliarity of nuclear forces This question is very nontrivial and still poorly studied. We present below some comments which, at least partly, clarify this point. 

As compared with the Kohn-Sham functional for electronic systems, the nuclear Skyrme functional is less genuine. The main (Coulomb) interaction in the Kohn-Sham problem is well known and only exchange and corellations should be modeled. Instead, in the nuclear case, even the basic interaction is unknown and should be approximated, e.g. by the simple contact interaction in Skyrme forces. 

The crudeness of Skyrme forces has certain consequences. For example, the Skyrme functional has no any exchange-correlation term since the relevant effects are supposed to be already included into numerous Skyrme fitting parameters. Besides, the Skyrme functional may accept a diverse set of T-even and T-dd densities and currents. One may say that T-odd densities appear in the Skyrme functional partly because of its specific construction. Indeed, other effective nuclear forces (Gogny [40], Landau-Migdal [41]) do not exploit T-odd densities and currents for description of nuclear dynamics. 

The particular SRPA versions for electronic Kohn-Sham and nuclear Skyrme functional were considered and examples of the calculations for the dipole plasmon in atomic clusters and giant resonances in atomic nuclei were presented. SRPA was compared with alternative methods, in particular with EOM-CC. It would be interesting to combine advantages of SRPA and couled-cluster approach in one powerful method. 

At present, antioxidants are extensively studied as supplements for the treatment diabetic patients. Several clinical trials have been carried out with vitamin E. In 1991, Ceriello et al. [136] showed that supplementation of vitamin E to insulin-requiring diabetic patients reduced protein glycosylation without changing plasma glucose, probably due to the inhibition of the Maillard reaction. Then, Paolisso et al. [137] found that vitamin E decreased glucose level and improved insulin action in noninsulin-dependent diabetic patients. Recently, Jain et al. [138] showed that vitamin E supplementation increased glutathione level and diminished lipid peroxidation and HbAi level in erythrocytes of type 1 diabetic children. Similarly, Skyrme-Jones et al. [139] demonstrated that vitamin E supplementation improved endothelial vasodilator function in type 1 diabetic children supposedly due to the suppression of LDL oxidation. Devaraj et al. [140] used the urinary F2-isoprostane test for the estimate of LDL oxidation in type 2 diabetics. They also found that LDL oxidation decreased after vitamin E supplementation to patients. 

We consider here the example of the Skyrme energy functional where the potential energy is modeled as... 

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