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.
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.
Abstract ................................................................................................. 359 4.1 Introduction .................................................................................. 359 4.2 Feynman’s Path Integral Quantum Formalism ............................ 364 4.2.1 Construction of Path Integral ........................................... 364 4.2.2 Propagator’s Equation ...................................................... 370 4.2.3 Recovering Wave Function’s Equation ............................ 373 4.3 Path Integrals for Basic Matter’s Structures ................................ 377 4.3.1 General Path Integral’s Properties ................................... 377 4.3.2 Path Integral for the Free Particle .................................... 384 4.3.3 Path Integral for Motion as the Harmonic Oscillator ...... 387 4.3.4 Path Integral Representation for the Bohr’s Atom ........... 394 4.3.5 Path Integral for Motion in the Quantum Well ................ 398 4.4 Density Matrix Approach Linking Path Integral Formalism ....... 402 4.4.1 On Mono-, Many-, and Reduced-Electronic Density Matrices .............................................................. 402 4.4.2 Canonical Density, Bloch Equation, and the Need of Path Integral ....................................................... 412 4.5 Roots of Self-Consistent Methods in Quantum Chemistry .......... 419 4.5.1 Molecular Orbital Approach ............................................ 420 4.5.2 Roothaan Approach .......................................................... 423 4.5.3 Introducing Semi-Empirical Approximations .................. 428 4.5.3.1 NDO Methods ................................................... 429 4.5.3.2 NDDO Methods ................................................ 432 4.5.4 Ab initio Methods: The Hartree-Fock Approach ............. 436 4.5.4.1 Hartree-Fock Orbital Energy ............................ 438 4.5.4.2 About Correlation Energy ................................. 442 4.5.4.3 Koopman’s Orbital Theorem with Hartree-Fock Picture ......................................... 444 4.5.4.4 Chemical Reactivity Indices in Orbital Energy Representation ...................................... 451 4.5.4.5 Testing Koopmans Theorem by Chemical Harness Reactivity Index .................. 458 4.6 Density Functional Theory: Observable Quantum Chemistry ..... 468 4.6.1 Hohenberg-Kohn theorems .............................................. 470 4.6.2 Optimized Energy-Electronegativity Connection ............ 475 4.6.3 Popular Energetic Density Functionals ............................ 480 4.6.3.1 Density Functionals of Kinetic Energy ............. 481 4.6.3.2 Density Functionals of Exchange Energy ......... 484 4.6.3.3 Density Functionals of Correlation Energy ...... 490 4.6.3.4 Density Functionals of Exchange-Correlation Energy ........................... 496 4.7 Observable Quantum Chemistry: Extending Heisenberg’s Uncertainty ............................................................. 502 4.7.1 Periodic Path Integrals ..................................................... 502 4.7.1.1 Effective Partition Function .............................. 502 4.7.1.2 Periodic Quantum Paths .................................... 505 4.7.2 Heisenberg Uncertainty Reloaded ................................... 508 4.7.3 Extending Heisenberg Uncertainty ...................................511 4.7.3.1 Averaging Quantum Fluctuations ......................511 4.7.3.2 Observed Quantum Evolution .......................... 514 4.7.3.3 Free Quantum Evolution ................................... 515 4.7.3.4 Free vs. Observed Quantum Evolution ............. 518 4.7.3.5 Spectral-IQ Method .......................................... 523 4.7.3.6 Spectral-IQ Results on Silica Sol-Gel-Based Mesosystems ..................................................... 527 4.8 Conclusion ................................................................................... 533 Keywords .............................................................................................. 535 References ............................................................................................. 536 Author’s Main References ........................................................... 536 Specific References ...................................................................... 537 Further Readings .......................................................................... 549 ABSTRACT Basing on the first principles of Quantum mechanics as exposed in the previous chapters and sections, special chapters of quantum theory are here unfolded in order to further extend and caching the quantum information from free to observed evolution within the matter systems with constraints (boundaries). As such, the Feynman path integral formalism is firstly exposed and then applied to atomic, quantum barrier and quantum harmonically vibration, followed by density matrix approach, opening the Hartree-Fock and Density Functional pictures of many-electronic systems, with a worthy perspective of electronic occupancies via Koopmans theorem, while ending with a further generalization of the Heisenberg observability and of its first application to mesosystems.
Abstract ................................................................................................. 164 3.1 Introduction .................................................................................. 165 3.2 The Wave-Function Continuity .................................................... 169 3.2.1 Quantum Tunneling and the Gamow Factor .................... 169 3.2.2 Nuclear Systems’ Lifetime by Alfa Disintegration .......... 176 3.3 Eigen-Energies and Eigen-Functions ........................................... 181 3.3.1 Atomic Hydrogenic States by Solving Laguerre Type Equations ................................................. 182 3.3.2 Molecular Vibrational States by Solving Hermite Type Equations .................................................. 200 3.3.3 Solid State Free Electronic States .................................... 209 3.4 Semiclassical Quantification of Energy ....................................... 216 3.4.1 Hydrogenic Atomic Systems ........................................... 219 3.4.2 Molecular Vibrational Systems ........................................ 222 3.4.3 Generalized |x|α Potential Systems ................................... 223 3.5 Variational Wave-Function and Energies ..................................... 228 3.5.1 Hydrogen’s Quantum Ground Level ................................ 228 3.5.2 Vibrational Ground Level ................................................ 231 3.5.3 Ground State Paradox of Free Electrons in Solids .......... 232 3.6 Causal Quantum Evolution .......................................................... 236 3.6.1 Schrödinger’s Picture ....................................................... 236 3.6.2 Unitary (“U”) Picture. Heisenberg Picture ...................... 246 3.6.3 Quantum Transitions: Interaction Picture ........................ 254 3.6.4 Green Function and Its Causal Properties ........................ 262 3.6.5 Free Particle’s Propagator ................................................ 271 3.7 Stationary Perturbations ............................................................... 274 3.7.1 General Perturbation Algorithm ....................................... 275 3.7.2 The Zeeman Effect ........................................................... 281 3.7.3 Nuclear Isotropic Corrections on Hydrogenic Atoms ...... 285 3.7.4 Harmonic Oscillator Perturbations .................................. 292 3.7.5 Quasi-Free Electronic Model of Solids ........................... 300 3.8 Quantum Measurements .............................................................. 304 3.8.1 Classical vs. Quantum Electric Resistance ...................... 306 3.8.2 Quantum Conservation Laws ............................................311 3.8.3 Quantum Elastic Scattering on Fixed Target ................... 318 3.8.4 More about Scattering: Born Approximation and Resonance Profile ..................................................... 332 3.8.5 From Yukawa Potential to Rutherford Scattering ............ 342 3.9 Conclusion ................................................................................... 353 Keywords .............................................................................................. 354 References ............................................................................................. 355 Author’s Main References ........................................................... 355 Specific References ...................................................................... 355 Further Readings .......................................................................... 355 ABSTRACT The phenomenological quantum mechanics is practically “re-storied” in a more formalized way, under the so-called extended quantum postulates, while opening and unfolding more complex analysis of the Nature’s phenomena, from nuclear, to atomic, to molecular, and to solid state and scattering observational quantum effects.
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 B̶ characterized by its mass (m(B̶)), velocity (v(B̶)), charge (e(B̶)), and life-time (t(B̶)). This is quantized either in ground or excited states of the chemical bond in terms of reduced Planck constant ħ, the bond energy E(bond) and length X(bond), respectively. 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 Ebond[kcal/mol]×Xbond[A0]=182019, providing this way the predictive framework in which the B̶ particle may be observed. Finally, its role in establishing the virtual states in Raman scattering was also established.
Contents 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.2 Delocalized orbital approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.2.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.3 Localized orbital approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.3.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.1 Introduction There are two computational approaches to the electronic structures of periodic and nonperiodic solids-the delocalized and localized orbital approaches [98]. The delocalized orbital approach or the so-called crystalline orbital (CO) theory [3-5,18,24,63,72,107,123] is strongly based on the periodic boundary conditions. It views an extended system of one-dimensional periodicity as a ring of K identical repeat units. A symmetry-adapted orbital such as a canonical Hartree-Fock (HF) orbital in this approach is delocalized over the entire ring and has the Bloch form [14]: ψpk(r) = K −1/2 ∑ ∑ m Cµpk exp (imka ) χµ (r − ma) , (6.1) where Cµpk is a CO coefficient, a is the lattice vector that outlines the unit cell, and χµ (r − ma) is the µth atomic orbital (AO) centered in the mth unit cell. Each orbital is characterized by k, which is the linear momentum (in atomic units) of an electron in this orbital, and can take one of K distinct values: ka π = 2m K , ∀m = 1, 2, . . . , K . (6.2) In the delocalized orbital approach, K is a crucial parameter with dual meaning. It is the number of k-vector sampling points in the first Brillouin zone (BZ) according to Equation (6.2). It is also the nominal size of the system, which should thus be commensurate with the number of the nearest neighbor unit cells included in the lattice sums of particle-particle interactions.
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).
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.