Path Integral Semiclassical Expansion

Semiclassical derivation of the evolution amplitude employs some of the previously Feynman path integral ideas refined due to the works of Kleinert and collaborators (Feynman Hibbs, 1965 Kleinert, 2004 Dachen et al., 1974 Grosche, 1993 Manning Ezra, 1994). They can be 

Quantum Nanochemistry— Volume II Quantum Atoms and Periodicity 

In these conditions the quantum statistical path integral representation of quantum propagator, see Eq. (2.10) with Eq. (2.21), and Volume 1/ Chapter 4 of the present five-volume set, and takes the form 

There should be pointed out that the used re-parameterization is not modifying the value of the path integral but is intended to better visualizing of its properties, towards solving it. As such, from expression (3.43) now appears clearer than before that for the systems governed by smooth potentials, the series expansion may now be applied respecting the path fluctuation, here in the second order truncation  

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We are now fully convinced that Fe5mman-Kleinert path integral formulation works fine either at low and higher temperature limits, while recovering both the (Hydrogen) ground state and the semiclassical expansion with high fidelity, respectively. There nevertheless remains to stress on its further connection with the electronic density and consequently with the DFT towards the quantum chemical properties computation. These issues will be in addressed next. 

With these the coimected correlation function algorithm was provided and checked in details, being at disposition to be implemented in whatever order of semiclassical expansion of the path integral evolution amplitude (3.54) For exemplification, the next section will expose the analytic solution for the second order case. 

Understanding the electronic movement in physical atomic as being driven by the conneeted and correlated functions especially by the (temporally) causal Green-fimction/quantum propagators Describing the physical atom as a semiclassical description of quantum motion, i.e., merely quantum than classical yet with certain orders of Planck constant contributions in electronic orbits in atom Learning the difference between the second and the fourth order of path integral expansion of the quantum amplitude of electronic orbits as quantifies in the associated partition functions ... 

Yet, the learned lesson of WKB approximation, other - more powerful - forms of semiclassical eikonal expansion for the action will be considered in the forthcoming volume of this five-volume set with the occasion it will be also applied on the valence states (i.e., treated as semiclassical states) of atomic systems within the path integral formalism that will be soon in next exposed. 

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