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51: 1888 (1983). 34. 29 49. M. Levy, On long-range behavior and ionization potentials, 50. J. P. Perdew, What do the Kohn-Sham orbital energies mean? , chapter in this book. M. Levy, J. P. Perdew, and V. Sahni, Exact differential equation for the density of a many-particle system, unpublished manuscript (1983). This manuscript contains an extensive discussion and a convincing theorem which states that the Kohn-Sham effective potential tends asymptotically to zero. See also references 14, 22. 43.
2) has been known for a long time. An explicit construction is given in Ref. 26. ,!. h(x)h(x') * • Let A = -y(Xl,X'l) - y(X2,X'2) be an operator on the antisymmetric states. Its lowest eigenvalue is -1 -1/2 = - 3/2, which is doubly degenerate. If r = ~><~, then W must be a ground state since TrrA = - Try 2 = I1 -1/4 -1/4 = -3/2. But every ground state is of the form W = 2- det (f,p) where p = ag + bh, lal 2+lbl 2 = 1. h> D Hohenberg and Kohn assume that every P comes from some 1/1 that is a ground state for some v. 10) and we shall retain this definition for PEA N, where AN = {pip comes from a ground state}. 4)! 2, according to which there is a unique v (up to a constant) associated with p. We can also define VN = {vlHv has a ground state}. 12) It then follows easily that for vEVN E(v) = min { FHK(P) + J vP\PEA N }. 13) This is the HK variational principle, but it is important to note that it holds only for VEVN, which is unknown, and that the variation is restricted to the unknown set AN' We also do not know what FHKis, and that is a very serious problem.