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Apple Academic Press eBooks

Quantum Mechanics for Quantum Chemistry

Mihai V. Putz

Published Mar 30, 2016• 1 citation
Publisher

Abstract

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.

DOI
10.1201/b21340-10

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Abstract ................................................................................................... 66 2.1 Introduction .................................................................................... 66 2.2 Wave Function Picture ................................................................... 67 2.2.1 Green and Dirac Functions ................................................ 67 2.2.2 Momentum and Energy Operators ..................................... 69 2.2.3 Klein-Gordon and Schrödinger Equations ......................... 72 2.2.4 Electronic and Photonic Spins ........................................... 74 2.2.5 Eigen-Functions and Eigen-Values .................................... 75 2.2.6 Hermitic Operators ............................................................ 76 2.2.7 Heisenberg Uncertainty Theorem ...................................... 81 2.2.8 Ehrenfest Theorem ............................................................. 85 2.2.9 Current Density Probability Conservation Theorem ......... 89 2.3 Classical to Quantum Mechanics’ Correspondence ....................... 92 2.3.1 Classical Euler-Lagrange, Hamilton, and Hamilton-Jacobi Equations ................................................ 92 2.3.2 Wave-Function Quantum Field .......................................... 97 2.3.3 Semi-classical Expansion and the WKB Approximation ................................................................. 102 2.3.4 From Field Internal Symmetry to Current Conservation .................................................................... 106 2.3.5 From Poisson Parentheses to Quantum Commutators and Heisenberg Picture ............................. 108 2.4 Bra-ket (Dirac) Formalism ............................................................115 2.4.1 Vectors in Hilbert Space....................................................115 2.4.2 Linear Operators in Hilbert Space ....................................118 2.4.3 Spectral Representations of Vectors and Operators ......... 126 2.4.4 Coordinate and Momentum Representations ................... 136 2.4.5 Energy Representation ..................................................... 148 2.4.6 Heisenberg Matrix Quantum Mechanics: The Harmonic Oscillator ................................................. 151 2.5 Conclusion ................................................................................... 159 Keywords .............................................................................................. 160 References ............................................................................................. 161 Author’s Main Reference ............................................................. 161 Further Readings .......................................................................... 161 ABSTRACT Although equivalent to certain points, various approaches of quantum phenomena may add new insights and perspective on studied systems either in isolate or interaction states. Here, the main formalisms are reviewed and illustrated, especially those making the ground-nous of the quantum mechanics itself, i.e., the undulatory quantum mechanics, while introducing the quantum propagator (Green function) concept, the semi-classical approach based on Euler-Lagrange and Hamiltonian, while the bra-ket Dirac picture opens the new perspectives of further developing quantum theory by abstract yet logical and formalized reasoning.

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Apple Academic Press eBooks•Mar 30, 2016•0 citations

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文伟, 白彦魁

量测量问题存在许多年并且启发一在物理和哲学的文学大，但是仍然在它上没有结论和一致。我们证明如果我们由考虑 Feynman 路径的相对论的效果扩大 Feynman 路径积分，它能被包含进量子论。根据这个扩大理论，我们推出不仅 Klein-Gordon 方程，而且 wave-function-collapse 方程。量测量的随机、即时的倒塌由于“仪器的潜在的噪音”或环境和“波浪的内部关联”分别地工作，这被显示出。因此，宏观的事明确地位由于自己，这不违背量力学。这个工作将为测量问题给新识别。

理论物理通讯：英文版•Jan 1, 2012•0 citations
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 ................................................................................................... 66 2.1 Introduction .................................................................................... 66 2.2 Wave Function Picture ................................................................... 67 2.2.1 Green and Dirac Functions ................................................ 67 2.2.2 Momentum and Energy Operators ..................................... 69 2.2.3 Klein-Gordon and Schrödinger Equations ......................... 72 2.2.4 Electronic and Photonic Spins ........................................... 74 2.2.5 Eigen-Functions and Eigen-Values .................................... 75 2.2.6 Hermitic Operators ............................................................ 76 2.2.7 Heisenberg Uncertainty Theorem ...................................... 81 2.2.8 Ehrenfest Theorem ............................................................. 85 2.2.9 Current Density Probability Conservation Theorem ......... 89 2.3 Classical to Quantum Mechanics’ Correspondence ....................... 92 2.3.1 Classical Euler-Lagrange, Hamilton, and Hamilton-Jacobi Equations ................................................ 92 2.3.2 Wave-Function Quantum Field .......................................... 97 2.3.3 Semi-classical Expansion and the WKB Approximation ................................................................. 102 2.3.4 From Field Internal Symmetry to Current Conservation .................................................................... 106 2.3.5 From Poisson Parentheses to Quantum Commutators and Heisenberg Picture ............................. 108 2.4 Bra-ket (Dirac) Formalism ............................................................115 2.4.1 Vectors in Hilbert Space....................................................115 2.4.2 Linear Operators in Hilbert Space ....................................118 2.4.3 Spectral Representations of Vectors and Operators ......... 126 2.4.4 Coordinate and Momentum Representations ................... 136 2.4.5 Energy Representation ..................................................... 148 2.4.6 Heisenberg Matrix Quantum Mechanics: The Harmonic Oscillator ................................................. 151 2.5 Conclusion ................................................................................... 159 Keywords .............................................................................................. 160 References ............................................................................................. 161 Author’s Main Reference ............................................................. 161 Further Readings .......................................................................... 161 ABSTRACT Although equivalent to certain points, various approaches of quantum phenomena may add new insights and perspective on studied systems either in isolate or interaction states. Here, the main formalisms are reviewed and illustrated, especially those making the ground-nous of the quantum mechanics itself, i.e., the undulatory quantum mechanics, while introducing the quantum propagator (Green function) concept, the semi-classical approach based on Euler-Lagrange and Hamiltonian, while the bra-ket Dirac picture opens the new perspectives of further developing quantum theory by abstract yet logical and formalized reasoning.
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.
Abstract ................................................................................................... 64 2.1 Introduction .................................................................................... 64 2.2 Atomic Valence Stability by Golden Ratio Imbalance................... 66 2.3 Reloading Quantum Path Integral Formalism for Chemistry ........ 68 2.4 Periodic Path Integrals ................................................................... 72 2.4.1 Survey on Matsubara Frequencies and the Quantum Periodic Paths ..................................................... 72 2.4.2 Matsubara Harmonic Partition Function ............................ 74 2.4.3 The Generalized Riemann’ Series ...................................... 78 2.4.4 Periodic Path Integral Measure .......................................... 81 2.5 Feynman-Kleinert Variational Formalism ..................................... 83 2.5.1 Feynman-Kleinert Partition Function ................................ 83 2.5.2 Feynman-Kleinert Optimum Potential ............................... 86 2.5.3 Quantum Smeared Effects and the Stability of Matter ...... 92 2.5.4 Ground State (β→∞, T→0K) Case .................................... 97 2.5.5 Excited State (β→0, T→∞) Case: Wigner Expansion ..... 101 2.6 Conclusion ................................................................................... 104 Keywords .............................................................................................. 105 References ............................................................................................. 105 Author’s Main References ........................................................... 105 Specific References ...................................................................... 106 ABSTRACT The alternative quantum mechanical description of total energy given by path integrals specialized to Feynman-Kleinert formalism, while recovering the Bohr quantification of Hydrogen atom in a great extend, yet within a more general framework in which the stability issue is solved by existing quantum fluctuation, here explicitly modeled by periodic paths and Matsubara frequencies; both ground and exited states are treated in quantum statistical perspective.
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.
量测量问题存在许多年并且启发一在物理和哲学的文学大，但是仍然在它上没有结论和一致。我们证明如果我们由考虑 Feynman 路径的相对论的效果扩大 Feynman 路径积分，它能被包含进量子论。根据这个扩大理论，我们推出不仅 Klein-Gordon 方程，而且 wave-function-collapse 方程。量测量的随机、即时的倒塌由于“仪器的潜在的噪音”或环境和“波浪的内部关联”分别地工作，这被显示出。因此，宏观的事明确地位由于自己，这不违背量力学。这个工作将为测量问题给新识别。