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The Wigner regime of a system of electrons in an external field is characterized by a low electron density and a high electron-interaction energy relative to the kinetic energy. The low-correlation regime is in turn described by a high electron density and an electron-interaction energy smaller than the kinetic energy. The Wigner regime of a nonuniform-electron-density system is investigated via quantal density functional theory (QDFT). Within QDFT, the contributions of electron correlations due to the Pauli exclusion principle, Coulomb repulsion, and correlation-kinetic effects are separately delineated and explicitly defined. The nonuniform-electron-density system studied is that of the Hooke's atom in the Wigner regime, for which the exact wave function is derived. As such, the results of the QDFT analysis are exact. It is observed that in comparison to the low-correlation case, not only is the electron-interaction energy greater than the kinetic energy as a fraction of the total energy, but so are its individual Hartree, Pauli, and Coulomb components. The ionization potential as a fraction of the total energy too is greater. But most significantly, in the Wigner regime, the correlation-kinetic energy as a fraction of both the electron-interaction and the total energy is substantially greater than in the low-correlation case. Hence, we propose that the Wigner regime now also be characterized by a high correlation-kinetic energy. The kinetic energy as a fraction of the total energy, however, is less than in the low-correlation case. This fact and the high correlation-kinetic energy value in the Wigner regime is explained by the concept of “quantal compression” of the kinetic energy density derived from QDFT. The corresponding results for the low-correlation case are in turn a consequence of a “quantal decompression” of the kinetic energy density. From the QDFT analysis, the exact values for the Kohn-Sham theory “exchange-correlation” and “correlation” energy functionals of the density and their respective functional derivatives are also obtained. These results ought to be of value in the construction and testing of approximate energy functionals valid for the Wigner regime.


This work was originally published in Physical Review A, available at



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