By S S Schweber

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G. A. Rodríguez-Rosario, A. Aspuru-Guzik, Phys. Rev. Lett. 104, 043001 (2010) 85. K. Yabana, S. Sugiyama, Y. Shinohara, T. F. Bertsch, Phys. Rev. U. Gross, and Kieron Burke Abstract This chapter introduces thermal density functional theory, starting from the ground-state theory and assuming a background in quantum mechanics and statistical mechanics. We review the foundations of density functional theory (DFT) by illustrating some of its key reformulations. The basics of DFT for thermal ensembles are explained in this context, as are tools useful for analysis and development of approximations.

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However, any one of the degenerate ground-state densities still determines the potential uniquely. The first part of the HK theorem needs to be modified in light of this alteration of the mapping between wavefunctions and densities. To begin, note that two degenerate subspaces, sets of ground states of two different potentials, are disjoint. E1 E2 /«: (23) For this identity to be true, the eigenstate « must vanish in the region where the two potentials differ by more than an additive constant. This region has measure greater than zero.