Alternatives to the electron density for describing Coulomb systems

Abstract

Stimulated by the difficulty of deriving effective kinetic energy functionals of the electron density, the authors consider using the local kinetic energy as the fundamental descriptor for molecular systems. In this ansatz, the electron density must be expressed as a functional of the local kinetic energy. There are similar results for other quantities, including the local temperature and the Kohn-Sham potential. One potential advantage of these approaches-and especially the approach based on the local temperature-is the chemical relevance of the fundamental descriptor. © 2007 American Institute of Physics. ͓DOI: 10.1063/1.2718950͔ I. MOTIVATIONAlthough the Kohn-Sham approach to density-functional theory ͑DFT͒ is now well established it bears remembering that almost 40 years separate the seminal papers of Thomas and Fermi from the breakthrough of Kohn and Sham. [1][2][3] Even with the rise of the Kohn-Sham approach, "orbitalfree" computational techniques have never entirely vanished from the scene because orbital-free DFT is much faster than conventional Kohn-Sham calculations. In orbital-free DFT one only varies a single function of the three spatial coordinates ͑the electron density͒ instead of using N threecoordinate functions ͑orbitals͒ or one six-coordinate function ͑the density matrix͒. ͑N is the number of electrons.͒ At present, however, orbital-free calculations are not really more "efficient" than Kohn-Sham calculations because while orbital-free calculations are computationally inexpensive, they are also very inaccurate. The problem is that the approximate kinetic energy functionals are ordinarily inadequate for describing bond breaking and other chemical processes in molecular systems. ͓Orbital-free methods have had more success in solid state materials, although seemingly only in cases where the electron density is low ͑most of the electrons are treated with a pseudopotential͒.4 ͔ Kohn-Sham theory circumvents the problem of evaluating the kinetic energy directly: instead the electron density is used to evaluate the Kohn-Sham potential, which is used to evaluate the Kohn-Sham orbitals, which are then used to evaluate the kinetic energy of the reference system of noninteracting electrons,Although T s ͓͔ is usually slightly smaller than the true kinetic energy, the Kohn-Sham kinetic energy functional is N representable by construction and, as such, avoids the "variational catastrophes" that afflict ordinary orbital-free DFT. 5-7The 80 years that has elapsed since the original paper of Thomas testifies to the fact that expressing the kinetic energy as a functional of the electron density is extraordinarily difficult. This raises the question: might it be easier to express the electron density as a functional of the local kinetic energy or another similar quantity-like the local temperature or the Kohn-Sham potential-that determines the kinetic energy? We do not yet know whether or not it is easier to express the electron density as a functional of these descriptors, but we can establish that it is possib...

The electrostatic potential: an overview

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Fisher and Shannon information in orbital‐free density functional theory

“…The shape function, or density per particle was the first quantity, that has been shown to be adequate for describing a Coulomb system. Reactivity indicators as Fukui function, local softness, softness kernel, electrostatic potential, local kinetic energy, and local temperature also provide a full description of a Coulomb system. Even broader families of descriptors for Coulomb systems have been presented.…”

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“…(11) According to a recent article [3], the quantities g(r) and q(r) are descriptors of a Coulomb system: any of them determines every property of the electron system. Earlier the shape function (that is, density per particle) [10], the Fukui function, the local softness, the softness kernel [11,12], the electrostatic potential, the local temperature, and the local kinetic energy [13] were proved to be capable of fully determining every property of a Coulomb system.…”