Feynman-Kleinert Partition Function

Being equipped with the periodic path integral teehnique we can present one of the most effieient ways for approximate the elfective-elassieal partition function (2.21) it firstly unfolds for a general external potential like the exact integral (Putz, 2009)  

Quantum Nanochemistry— Volume II Quantum Atoms and Periodicity 

Since the exact solution for expression (2.57) is hard to formulate for an unspecified potential form, it may be eventually reformulated in a workable way by involving another partition function, the so-called Feynman-Kleinert (FK) partition function and its special average recipe, respectively as 

In Eqs. (2.58) and (2.59) the Feynman-Kleinert partition function takes the general form 

In fact, the Feynman-Kleinert action (2.61) is being to be involved in twofold optimization algorithm in providing the best approximation of the partition function (2.57). This will favor a close analogy with the double search for electronic density, in density functional theory (DFT), as will be latter discussed. 

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The optimization of the Feynman-Kleinert partition function (2.64) is performed employing the Jensen-Peierls inequality (Putz, 2009),... 

The original idea of approximating the quantum mechanical partition function by a classical one belongs to Feynman [Feynman and Vernon 1963 Feynman and Kleinert 1986]. Expanding an arbitrary /S-periodic orbit, entering into the partition-function path integral, in a Fourier series in Matsubara frequencies v . 

Remarkably, the partition function involvement in this density algorithm was widely and most extensive used by the Feynman-Kleinert approach which was proved to furnish meaningful approximations either for the ground state (as was the case for atomic Hydrogen and the Bohr s orbitalie proofed stability) as well for the higher temperature or excited or the valenee states (that resembles the semiclassical approximation). 

Feynman, R. P, Kleinert, H. (1986). Effective classical partition functions. Phys. Rev. A 34,5080-5084. 

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