ABSTRACT
Abstract ................................................................................................. 364 5.1 Introduction .................................................................................. 364 5.2 Quantum Algebra of Electronic Systems ..................................... 366 5.2.1 Introducing Kubo-Martin-Schwinger (KMS) States ........ 366 5.2.2 Basic Algebraic Structures and Transformations ............. 373 5.2.3 Chemical Potential and Reactive Equilibrium ................. 395 5.2.4 Non-Reactive Systems ..................................................... 399 5.2.5 Reactive Systems ............................................................. 402 5.3 Formalization of Thomas-Fermi Theory ...................................... 406 5.3.1 Thomas-Fermi Theorems ................................................. 406 5.3.2 Electronic Localization Problem ...................................... 415 5.4 Fokker-Planck Dynamics of Quantum Systems .......................... 422 5.4.1 Non-Equilibrium Dynamic Level .................................... 422 5.4.2 Properties of Fokker-Planck Equation ............................. 427 5.4.3 Fokker-Planck to Schrödinger
Equation Transformation .................................................. 434 5.4.4 Non-Equilibrium Lagrangian ........................................... 441 5.4.5 The Non-Equilibrium Harmonic Solution ........................ 447 5.4.6 The Non-Equilibrium Generalized
Anharmonic Solution ....................................................... 450
5.5 Electronic Localization Functions (ELF) by Markovian Path Integral .............................................................. 460
5.5.1 Thom’s Catastrophe Concepts for Chemical Bonding ..... 460 5.5.2 Fokker-Planck Modeling of the
Electronic Localization .................................................... 463 5.5.3 Markovian Shape for Electronic
Localization Functions ..................................................... 466 5.5.4 Markovian Specialization for Electronic
Localization Functions ..................................................... 470 5.5.5 Electronic Localization in Atom and in
Bonding Atoms ................................................................. 473 5.6 Conclusion ................................................................................... 484 Keywords .............................................................................................. 487 References ............................................................................................. 488 Author’s Main References ........................................................... 488 Specific References ...................................................................... 488
ABSTRACT
This chapter reviews and advances the two related quantum ways for quantum description of valence/interacting/exchanged electrons among atoms at their turn involved in binding or molecular systems:
• by abstract formalization within quantum algebra of open systems; and
• by analytical formulation within stochastic/dissipative systems. This way one models the electronic distribution, exchange and local-
ization in chemical (inherently open) system, dealing therefore with that chemistry is at its ultimate description: the science of moving electrons from one state to another (either by intra-or inter-atomic framework).
