### Abstract

A solid understanding of the Lieb functional FL is important because of its centrality in the foundations of electronic density functional theory. A basic question is whether directional derivatives of FL at an ensemble-V-representable density are given by (minus) the potential. A widely accepted purported proof that F_{L} is Gâteaux differentiate at EV-representable densities would say, "yes." But that proof is fallacious, as shown here. F _{L} is not Gâteaux differentiable in the normal sense, nor is it continuous. By means of a constructive approach, however, we are able to show that the derivative of FL at an EV-representable density ρ_{0} in the direction of ρ_{1} is given by the potential if ρ_{0} and ρ_{1} are everywhere strictly greater than zero, and they and the ground state wave function have square integrable derivatives through second order.

Original language | English (US) |
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Pages (from-to) | 1943-1953 |

Number of pages | 11 |

Journal | International Journal of Quantum Chemistry |

Volume | 107 |

Issue number | 10 |

DOIs | |

Publication status | Published - Aug 15 2007 |

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### All Science Journal Classification (ASJC) codes

- Atomic and Molecular Physics, and Optics
- Condensed Matter Physics
- Physical and Theoretical Chemistry