Semiclassical partition function

It is necessary to know the form of the partition functions to construct transition probabilities that properly sample the ensemble. For instructional purposes we record here the (semiclassical) partition functions that correspond to the osmotic and isomolar semigrand ensembles described above. In both ensembles one must average over moles of the Legendre-transformed species. Thus the osmotic semigrand ensemble partition function is... 

Usually sp (see Eqs (13) and (14)) is calculated numerically [66]. However, a variation of the numerical Strutinsky averaging method consists in using the semiclassical partition function and in expanding it in powers of T . With this method, for the case of an... 

In the semiclassical evaluation of the barrier partition function the integration goes along the whole imaginary axis in the c, plane (see fig. 21). 

In Section 2.5.3 we derived the semiclassic expression for the ceinonical partition function [see Eq. (2.110)] based on the assumption that at sufficiently high temperatures we may replace the Hamiltonian operator by its classic analog, the Hamiltonian function [see Elq. (2.100)]. In this section we will sketch a more refined treatment of the semiclassic theory developed in Section 2.5 originally due to Hill and presented in detail in his classical work on stati.stical mechanics [326]. Because of Hill s clear and detailed exposition and because we need the final result mainly as a justification to treat confined fluids by means of classic statistical thermodynamics, we will just briefly outhne the key ideas of Hill s treatment for reasons of completeness of the current work. 

In this case the classical partition function of reactants be used,instead of However, in a semiclassical approximation, we... 

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). 

SECOND ORDER EOR SEMICLASSICAL PROPAGATOR, PARTITION FUNCTION AND DENSITY... 

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 ... 

In the sum over states formula, excited states for the vibrational modes need to be included up to convergence. A more convenient integral expression is provided by classical or semiclassical theories. At high temperatures and low frequencies, the vibrational motion behaves increasingly classically and the semiclassical Wigner-Kirkwood expression is an excellent approximation to the quantum partition function [78] for low-frequency vibrations at pyrolysis temperatures. The semiclassical... 

Canonical partition function versus temperature for HOOH computed by three methods. The red curve is the semiclassical adiabatic method, the blue curve is the HO-RR method and the purple curve is the separable hindered rotor model. 

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