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).
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 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.
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).
Abstract ..................................................................................................... 2 1.1 Introduction ...................................................................................... 3 1.2 Valence Bond Theory ....................................................................... 5 1.2.1 Adiabatic Approximation ..................................................... 5 1.2.2 The Virial Theorem on the Molecular Systems .................... 9 1.2.3 The Chemical Bond Through the Valence Bond Method (VBM) ................................................................... 12 1.3 Molecular Orbital Theory of Bonding ........................................... 22 1.4 Sub-Quantum Chemistry: Chemical Bonding Field and Bondons ......................................................................... 34 1.4.1 Introducing Chemical Field ................................................ 35 1.4.2 Predicting the Bondonic’s Existence .................................. 38 1.4.3 Chemical Bonding by Bondons ......................................... 45 1.4.4 Chemical Bonding as a Bosonic Quantum Condensate ..... 55 1.4.4.1 ψ-Theory of Bose-Einstein Condensation ........... 55 1.4.4.2 DFT adapted to Bose-Einstein Condensation ...... 58 1.4.4.3 Local Density Approximation of Bose-Einstein Condensation ................................ 61 1.4.4.4 Heitler-London Bondonic Condensation ............. 64 1.4.4.5 Molecular Orbitals’ Bondonic Condensation ...... 70 1.5 Conclusion ..................................................................................... 82 Keywords ................................................................................................ 84 References ............................................................................................... 85 Author’s Main References ............................................................. 85 Specific References ........................................................................ 85 ABSTRACT The classical quantum methods in characterizing chemical bonding, i.e., the valence orbital theory (of Pauling) and the molecular orbital theory (of Heitler and London) are firstly exposed (and illustrated at least for H2 system) for didactical and benchmarking research purpose on fundamental nature of the chemical bonding. Then the “missing link” in characterizing chemical bonding field by its associated quantum particle, the bondon B , is revealed by employing the combined Bohmian quantum formalism within the Schrödinger quantum picture of electron motions and characterized by its mass (mB), velocity (vB), charge (eB), and life-time (t B); the mass-velocity-charge-time quaternion properties of bondons’ particles were used in discussing various paradigmatic types of chemical bond towards assessing their covalent, multiple bonding, metallic and ionic features. The bondonic picture was completed by discussing the relativistic charge and life-time (the actual zitterbewegung) problem, i.e., showing that the bondon equals the benchmark electronic charge through moving with almost light velocity. It carries negligible, although non-zero, mass in special bonding conditions and towards observable femtosecond life-time as the bonding length increases in the nanosystems and bonding energy decreases according with the bonding length-energy relationship of Heisenberg type E kcal mol X Abond bond[ / ] [ ]× = 0 182019, providing this way the predictive framework in which the B particle may be observed when either associate Ebond or length Xbond is known. However, being the bondon a boson, as the particle carrying the quantum information of the chemical wave/orbital/ field, further, possibility in assessing the density functional theory (DFT) for of the Kohn-Sham equation within Bose-Einstein condensation (BEC) and of its further reduction to the nonlinear Schrodinger equation under the GrossPitaevsky equation is responded in positive through employing specific DFTBEC exchange functional combined within the Hartree-Fock-Bogoliubov mean-field theory. This non-linear generalization of the Schrödinger equation is then employed in first providing the DFT-BEC connection at the level of the Thomas-Fermi approximation; such connections are further used towards generalizing the classical Heitler-London formalism and bonding-antibonding equations of homopolar chemical bonding with the aid of the mass quantification of the quantum particle of the chemical bonding field – the bondons. Actually, two branches of bosonic-bondonic condensate were identified as physical and chemical bonding BEC, both with bonding and antibonding features. These findings innovatively complete the already “classical” quantum treatment of chemical bonding through the valence orbital theory (dealing with resonances of molecular structures) and then to the orbital molecular theory (providing the hybridization and the bonding/anti-bonding paradox), here also exposed for their main features.
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).
By employing the combined Bohmian quantum formalism with the U(1) and SU(2) gauge transformations of the non-relativistic wave-function and the relativistic spinor, within the Schrödinger and Dirac quantum pictures of electron motions, the existence of the chemical field is revealed along the associate bondon particle B characterized by its mass ( Bm ), velocity ( Bv ), charge ( Be ), and life-time ( Bt ). This is quantized either in ground or excited states of the chemical bond in terms of reduced Planck constant ħ, the bond energy Ebond and length Xbond, respectively. The mass-velocity-charge-time properties of bondons’ particles were used in discussing various paradigmatic types of chemical bond towards assessing their covalent, multiple bonding, metallic and ionic features. Finally, its role in establishing the virtual states in Raman scattering was also established (Putz, 2010b).