Partition Function Quantum Expansion

One may apply the Gauss theorem successively for integrals of gradient of a given long range defined quantity (. .), successively as (Putz Ori, 2014)  

Quantum Nanochemistry—Volume IV Quantum Solids and Orderability 

The second conceptual instrument used in the present analysis regards the physical-to-topological passage the present discussion follows (Putz Ori, 2012 2014)  

It is worth to remember here the basal properties of distance-based topological potentials making those mathematical object exceptionally suitable for determining delocalized bondonic properties (Putz Ori, 2012, 2014)  

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A first step toward quantum mechanical approximations for free energy calculations was made by Wigner and Kirkwood. A clear derivation of their method is given by Landau and Lifshitz [43]. They employ a plane-wave expansion to compute approximate canonical partition functions which then generate free energy models. The method produces an expansion of the free energy in powers of h. Here we just quote several of the results of their derivation. 

In order to obtain an expansion for 3(a,j8), one uses a formalism similar to the time-dependent perturbation method in quantum mechanics. The grand partition function (8) is rewritten as... 

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 ... 

It is well known that Zq , the kinetic part of the partition function, is irrelevant in the determination of the equation of state and in the case of equilibrium between two states. In this work we will be interested only in the configurational part of the distribution function Wn/zq ). Asymptotic expansions of the rotational part of for linear rotators and S3mimetric tops can be found elsewhere. We are now in a position to determine the general form of an expansion of Wn/Zq in powers of the quantum parameter A. 

The introduction of coupled cluster theory into quantum chemistry is attributed to Cfzek who, in 1966, published a seminal paper [61] in the Journal of Chemical Physics entitled On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell-Type Expansion Using Quantum-Field Theoretical Methods. The theory uses an exponential ansatz which was originally employed in statistical mechanics to compute the partition function of a non... 

Without going into the mathematical details we shall qualitatively overview and compare the possibilities for defining polyatomic fragments in quantum chemical calculations. Since the present-day techniques always use finite basis expansions for the development of the molecular wave function, the crucial question is how to partition this basis set in the most effective way. 

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