Path Integrals for Electronic Densities, Reactivity Indices, and Localization Functions in Quantum Systems

Abstract

The density matrix theory, the ancestor of density functional theory, provides the immediate framework for Path Integral (PI) development, allowing the canonical density be extended for the many-electronic systems through the density functional closure relationship. Yet, the use of path integral formalism for electronic density prescription presents several advantages: assures the inner quantum mechanical description of the system by parameterized paths; averages the quantum fluctuations; behaves as the propagator for time-space evolution of quantum information; resembles Schrödinger equation; allows quantum statistical description of the system through partition function computing. In this framework, four levels of path integral formalism were presented: the Feynman quantum mechanical, the semiclassical, the Feynman-Kleinert effective classical, and the Fokker-Planck non-equilibrium ones. In each case the density matrix or/and the canonical density were rigorously defined and presented. The practical specializations for quantum free and harmonic motions, for statistical high and low temperature limits, the smearing justification for the Bohr’s quantum stability postulate with the paradigmatic Hydrogen atomic excursion, along the quantum chemical calculation of semiclassical electronegativity and hardness, of chemical action and Mulliken electronegativity, as well as by the Markovian generalizations of Becke-Edgecombe electronic focalization functions – all advocate for the reliability of assuming PI formalism of quantum mechanics as a versatile one, suited for analytically and/or computationally modeling of a variety of fundamental physical and chemical reactivity concepts characterizing the (density driving) many-electronic systems.

Review of Feynman’s Path Integral in Quantum Statistics: from the Molecular Schrödinger Equation to Kleinert’s Variational Perturbation Theory

“…Based on the centroid density of path integrals [32,70,[84][85][86][87][88][89]92], Feynman and Hibbs suggested a completely different approach for constructing an effective semiclassical potential [32]. First, we recall that calculations of the quantum partition function requires a summation of all possible closed paths.…”

“…(2.3) is the integral equation of quantum mechanics, which are equivalent to the time-dependent Schrödinger differential equation [32,70], i.e.,…”

“…Kleinert's variational perturbation (KP) theory [34] for the centroid density [32,70,[84][85][86][87][88][89]92] of Feynman path integrals [2,[32][33][34][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75] provides a complete theoretical foundation for developing non-stochastic/non-sampling methods to systematically incorporate internuclear quantum-statistical effects in condensed phase systems. Similar to the complementary interplay between the rapidly growing quantum Monte Carlo simulations [146][147][148][149] and the well-established ab initio or density-functional theories (DFT) for electronic structure calculations [4,5,[25][26][27]29], non-sampling/non-stochastic pathintegral methods can complement the conventional Fourier or discretized path-integral Monte-Carlo (PIMC) [131,136,[139][140]…”

Bondonic characterization of extended nanosystems: Application to graphene’s nanoribbons

“…(15) the above critical temperature of pristine-defect coexistence in graphene as given by Eq. (12), along side the related bondonic action distance (8), bondonic N-normalized grand canonical internal energy (10) and the associate absolute caloric capacity (11) are respectively computed and represented in Figure 2. The results evidence the gaps that, at critical regime, generally affect all the physical quantities described by the bondonic model.…”

“…Considering a particle with mass M moving between the spacepoints x a and x b under the potential Vð xÞ with x ¼ ðx a þ x b Þ=2 the associate quantum statistical propagator may be expanded up to the second order in Dx = x b -x a [11]:…”