Feynman Path Integral Approach

The calculation of the integrals in Eq. (55) in the classical limit in the improved Condon approximation (for the nuclear subsystem) using the saddle point method leads to two coupled equations for the electron wave functions of the donor and the acceptor in the transitional configuration  

A direct variational method was used in Refs. 23 and 24 to go beyond the Condon approximation. Functions of the type 

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The calculation of the potential of mean force, AF(z), along the reaction coordinate z, requires statistical sampling by Monte Carlo or molecular dynamics simulations that incorporate nuclear quantum effects employing an adequate potential energy function. In our approach, we use combined QM/MM methods to describe the potential energy function and Feynman path integral approaches to model nuclear quantum effects. 

A dynamic description of the effect of relaxation on the probability of the adiabatic transition may be performed using various methods, e.g., a Feynman path integral approach similar to that presented in Section III (see also Refs. 81-84). Here we shall present the results for a simple model obtained by another method.85... 

A number of papers are devoted to the effect of dissipation on tunneling.81"83,103,104 Wolynes81 was one of the first to consider this problem using the Feynman path integral approach to calculate the correlation function of the reactive flux involved in the expression for the rate constant,... 

The first volume covers the concepts of nuclear, atomic, molecular and solids on the basis of quantum principles—from Planck, Bohr, Einstein, Schrodinger, Hartree-Fock, up to Feynman Path Integral approaches ... 

Gerry, C. C., Singh, V. A. (1979). Feynman path-integral approach to the Aharonov-Bohm effect. Phys. Rev. D 20,2550-2554. 

Polarization fluctuations of a certain type were considered in the configuration model presented above. In principle, fluctuations of a more complicated form may be considered in the same way. A more general approach was suggested in Refs. 23 and 24, where Eq. (16) for the transition probability has been written in a mixed representation using the Feynman path integrals for the nuclear subsystem and the functional integrals over the electron wave functions of the initial and final states t) and t) for the electron ... 

A similar procedure gives the corresponding result for the Wiener process the first term of which was obtained by Feynman and Hibbs4 using a path integral approach. 

Our simulations are based on well-established mixed quantum-classical methods in which the electron is described by a fully quantum-statistical mechanical approach whereas the solvent degrees of freedom are treated classically. Details of the method are described elsewhere [27,28], The extent of the electron localization in different supercritical environments can be conveniently probed by analyzing the behavior of the correlation length R(fih/2) of the electron, represented as polymer of pseudoparticles in the Feynman path integral representation of quantum mechanics. Using the simulation trajectories, R is computed from the mean squared displacement along the polymer path, R2(t - t ) = ( r(f) - r(t )l2), where r(t) represents the electron position at imaginary time t and 1/(3 is Boltzmann constant times the temperature. 

The techniques of discretized Feynman path integrals make the use of Eq. 41 practical for the more general case of quantized nuclear motion which is not restricted to harmonic behavior [36, 94, 99b[. Applications of this approach are discussed in Section 1.5 of this chapter. 

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. 

From the chemically point of view, the valence states are those situated in the chemical zone -and they are the main concern forthe chemical reactivity by employing the frontier or the outer electrons consequently, the semiclas-sical approximation that models the excited states was expressly presented either as an extension of the quantum Feynman path integral or as a specialization of the Feynman-Kleinert formalism for higher temperature treatment of quantum systems (see Section 2.5). However, due to the correspondences of Table 2.1 one may systematically characterize the semiclassical (or quantum chemical) approaches as one of the limiting situations (Putz, 2009) ... 

Basing on the first principles of Quantum mechanics as exposed in the previous chapters and sections, special chapters of quantum theory are here unfolded in order to further extend and caching the quantum information from free to observed evolution within the matter systems with constraints (boundaries). As such, the Feynman path integral formalism is firstly exposed and then applied to atomic, quantum barrier and quantum harmonically vibration, followed by density matrix approach, opening the Hartree-Fock and Density Functional pictures of many-electronic systems, with a worthy perspective of electronic occupancies via Koopmans theorem, while ending with a further generalization of the Heisenberg observability and of its first application to mesosystems. 

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