Partition function path integral expression

Eq. (4.8) is the approximate path integral expression for the partition function used in the DPI method. The exact path integral expressions arising from Eq. (4.8) are obtained in the limit that JVf i-+ oo. Noting that... 

Equilibrium properties can be determined from the partition function Zq and this quantity can, in turn, be computed using Feynman s path integral approach to quantum mechanics in imaginary time [86]. In this representation of quantum mechanics, quantum particles are mapped onto closed paths r(f) in imaginary time f, 0 f )8ft. The path integral expression for the canonical partition function of a quantum particle is given by the P 00 limit of the quantum path discretized into P segments. 

The statistical mechanics of distinguishable quantum particles is called Boltzmann statistics. For a collection of N such particles, the path integral expression for the partition function is given by equation (4), with r = ri, T2,. .. ryv - Consider as an example the case of N noninteracting free particles all having the same mass m. The Hamiltonian is H = which consists only of a nondiagonal part Hi = H. Applying equation (5) to this case yields... 

As seen from our discussion in Chapter 3, which dealt with onedimensional problems, in many relevant cases one actually does not need the knowledge of the behavior of the system in real time to find the rate constant. As a matter of fact, the rate constant is expressible solely in terms of the equilibrium partition function imaginary-time path integrals. This approximation is closely related to the key assumptions of TST, and it is not always valid, as mentioned in Section 2.3. The general real-time description of a particle coupled to a heat bath is the Feynman-Vernon... 

In the path integral approach, the transition amplitude between two states of the system can be calculated by summing amplitudes for all possible paths between them. By inserting a sequence of sums over sets of intermediate states into the expression for the partition function, Eq. (48) becomes... 

Path integrals are particularly useful for describing the quantum mechanics of an equilibrium system because the canonical distribution for a single quantum particle in the path integral picture becomes isomorphic with that for a classical ring polymer of quasiparticles [17-19, 26] (cf. Fig. 1). In the discretized path-integral representation, the partition function for a quantum particle is given by the expression... 

As it was derived in the previous Section 4.6.5 the Mulliken density functional electronegativity requires the knowledge of the electronic density under the external potential influence. Being exposed all the ingredients for the analytical expression for the partition function with only the external potential dependence, the electronic density computed through out of Feymnan-Kleinert path integral algorithm takes the form, see also Eq. (2.11) ... 

The strength of the path integral (PI) formalism [23-27] lies in the fact that it allows computing the partition function without finding the eigenstates of the Hamiltonian. Let us therefore consider a molecular system with D degrees of freedom with masses m Starting from the exact expression Q P) = Tr(e ), one obtains PI... 

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