6 while it has no advantage over the conventional basis for k near the edge of the Brillouin zone (states near the top of the band). This in turn is consistent with the observation that properties of the superconducting state are ‘universal’, because they arise from long wavelength states near the bottom of the band that are insensitive to details of the electron-ion potential. In order for a sys- tem to be able to push the electronic occupation to the top of the band (Fig. 6 of ref. [6]) without undue energy cost requires the band to be almost full in the first place (hole conduction in the normal state), and is favored by a low curvature of the band near the top (large effective mass), a narrow band and a small ionic charge Z , which are all requirements that we discussed earlier in connec- tion with the theory of hole superconductivity but based on different arguments. We also found here that with the new basis the energy is lowered if parity is broken and a spin current exists, consistent with our earlier predictions[5, 9]. In a small cluster, a rigid electronic state where parity is broken should lead to an electric dipole moment as observed in the remarkable experiments of de Heer and coworkers[18]. The new basis proposed here may be of interest also for other unconventional theories of superconductivity be- sides the theory of hole superconductivity. As discussed here, the basis introduced in this paper is consistent with the proposal of ref. [6] that in the su- perconducting state the states near the top of the band / edge of the Brillouin zone become full and those near the bottom of the band / center of the zone, that resemble free-electron-like states, become empty or partially occu- pied (Fig. 6 of ref. [6]). This is because the new basis gives rise to energy lowering compared with the conven- tional basis for small k (near the bottom of the band), been tried in the past. For example, Froehlich[14] pro- posed a state where a shell of occupied states at the top of the normal Fermi distribution is moved outward, leaving behind an empty shell. This has a certain resemblance to the scenario proposed in Ref. [6]. Even in conventional BCS theory, the normal state Fermi surface is ‘blurred’ and Bloch states that are empty in the normal state (i.e. unfavorable for the electron-ion interaction) become par- tially occupied in the BCS state. However we are not aware of earlier attempts to describe the superconduct- ing state using single particle states that violate Bloch’s theorem as we propose in this paper, except for the early work of Kronig[15] and Heisenberg[16] that proposed the- ories of superconductivity based on an electronic ‘lattice’ that had no relation to the underlying ionic lattice. In addition, Ref.[17] used sine functions to take into account the effect of sufaces and impurities. [1] F. Bloch, Z. Physik 52, 555-600 (1929). [2] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 - A1138 (1965). [3] J. Bardeen, L. N. Cooper , and J. R. Schrieffer, Phys. Rev. 108, 1175(1957). [4] J.E. Hirsch, Phys.Rev. B 71 , 104522 (2005). [5] J.E. Hirsch, Europhys. Lett. 81 , 67003 (2008); J. Phys. Cond. Matt. 20 , 235233 (2008). [6] J.E. Hirsch, Int. J. Mod. Phys. 23 , 3035 (2009). [7] J.E. Hirsch, Phys.Lett.A 309, 457 (2003). [8] J.E. Hirsch, Phys.Lett.A 315, 474 (2003). [9] J.E. Hirsch, Phys.Rev. B 71 , 184521 (2005). [10] J.E. Hirsch and F. Marsiglio, Phys.Rev. B39, 11515 (1989). [18] S. Yin, S. Xu, A. Liang, J. Bowlan, R. Moro and W.A. de Heer, Jour. of Superconductivity and Novel Magnetism 21 , 265 (2008) and references therein. [17] K. Tanaka and F. Marsiglio, Phys.Rev. B 62 , 5345 (2000). [16] W. Heisenberg, Zeitschrift fur Naturforschung 2a , 185 (1947). [15] R. de L. Kronig, Z. Phys. 78 , 744 (1932). [14] H. Froelich, Phys. Rev. 79, 845 (1950). [13] P. Morel and P.W. Anderson, Phys. Rev. 125, 1263 (1962). [12] C.R. Hu, arXiv: 0711.3020 (2007). [11] J.E. Hirsch, Phys.Rev. B68, 184502 (2003).