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AIP conference proceedings

Developing density functional theory for Bose-Einstein condensates. The case of chemical bonding

Mihai V. Putz

Published Jan 1, 2015• 0 citations
Publisher

Abstract

Since the nowadays growing interest in Bose-Einstein condensates due to the expanded experimental evidence on various atomic systems within optical lattices in weak and strong coupling regimes, the connection with Density Functional Theory is firstly advanced within the mean field framework at three levels of comprehension: the many-body normalization condition, Thomas-Fermi limit, and the chemical hardness closure with the inter-bosonic strength and universal Hohenberg-Kohn functional. As an application the traditional Heitler-London quantum mechanical description of the chemical bonding for homopolar atomic systems is reloaded within the non-linear Schrödinger (Gross-Pitaevsky) Hamiltonian; the results show that a two-fold energetic solution is registered either for bonding and antibonding states, with the bosonic contribution being driven by the square of the order parameter for the Bose-Einstein condensate density in free (gas) motion, while the associate wave functions remain as in classical molecular orbital model.

DOI
10.1063/1.4906737

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Systems ............................................................................................................ 229 Keywords ................................................................................................................. 231 9.1 INTRODUCTION Within the Bose-Einstein condensates (BEC) – density functional Theory (DFT) context, the question of whether the quantum chemical bonding homopolar picture may allow for such special modeling extension is here responded in positive through employing the Gross-Pitaevsky equation for the condensed wave function of bosons as an extension to the consecrated Heitler-London quantum description when bonding and anti-bonding electrons are considered as pairing to form bosons with 0 and 1 magnetic moments (spins) respectively. The results show that the same form of the quantum chemical bonding and anti-bonding wave functions might characterize the bosonic condensed wave functions, respectively, however experiencing different energetic states (with the contribution of a BECChemBond term experiencing the gap of the so called bosonic-fermionic degeneracy of the chemical bonding) relying on the correcting the fermionic energetic spectrum by some fraction of it with the coupling related with the chemical hardness (from fermionic side) reduced by the bosonic strength (from the bosonic side). The present Heitler-London-Bose-Einstein (superfluid) picture may eventually lead with a reformulation and the control of the frontier reactivity as well as the more subtle quantum concepts as the delocalization, aromaticity or other collective manifestations in complex macromolecules and nanosystems assimilated with diluted systems of bosons of fermions (Putz, 2011d, 2011e, 2011f).
Since the nowadays growing interest in Bose-Einstein condensates (BEC) due to the expanded experimental evidence on various atomic systems within optical lattices in weak and strong coupling regimes, its many-body quantum bases and equations are elementarily presented emphasizing on the order parameter, pseudo-potential and mean field approaches, while emphasizing the common places with the Density Functional Theory, like the integral normalization of density or the chemical potential, as the main vehicles of both theories of matter (Putz, 2011d, 2011e).
Potential ........................................................................................................... 143 6.7Exchange Density Functional of Homogeneous Systems-–Bosonic Case ..... 150 Keywords ................................................................................................................. 157 6.1 INTRODUCTION The basics of Density Functional Theory (DFT) and of Bose-Einstein Condensation (BEC) are merged such that the DFT enriches with the order parameter equation from the BEC modeling, while BEC acquires the new equation for the condensate density as provided by the DFT Hohenberg-KS analysis. Together, BEC-DFT may constitute as the next level of comprehension in modeling condensed matter in physics and chemical bond in chemistry, in particular, while providing the clue how the unified theory of physical-chemical phenomena should look like (Albus et al., 2003; Argaman and Band, 2011; Brand, 2004; Kim and Vetter, 1997; Nunes, 1999; Putz, 2011d; Zubarev, 2003).
The density functional theory originally developed by Hohenberg, Kohn, and Sham provides a rigorous conceptual framework for dealing with inhomogeneous interacting Fermi systems. We extend this approach to deal with inhomogeneous interacting Bose-condensed systems, limiting this presentation to setting up the formalism to deal with ground-state (T = 0) properties. The key new feature is that one must deal with energy functionals of both the local density n(r) and the local complex macroscopic wave function Φ(r) associated with the Bose broken-symmetry (the local condensate density is [Formula: see text]. Implementing the Kohn–Sham scheme, we reduce the problem to a gas of weakly interacting Bosons moving in self-consistent diagonal and off-diagonal one-body potentials. Our formalism should provide the basis for studies of the surface properties of liquid 4He as well as the properties of Bose-condensed atomic gases trapped in external potentials.
The density functional theory originally developed by Hohenberg, Kohn and Sham provides a rigorous conceptual framework for dealing with inhomogeneous interacting Fermi systems. We extend this approach to deal with inhomogeneous interacting Bose-condensed systems, limiting this presentation to setting up the formalism to deal with ground state (T = 0) properties. The key new feature is that one must deal with energy functionals of both the local density n(r) and the local complex macroscopic wavefunction Φ(r) associated with the Bose broken-symmetry (the local condensate density is nc(r) = |Φ(r) | 2). Implementing the Kohn-Sham scheme, we reduce the problem to a gas of weakly-interacting Bosons moving in self-consistent diagonal and off-diagonal one-body potentials. Our formalism should provide the basis for studies of the surface properties of liquid 4He as well as the properties of Bose-condensed atomic gases trapped in external potentials.