Path-integral techniques theory

Basically the perturbative techniques can be grouped into two classes time-local (TL) and time-nonlocal (TNL) techniques, based on the Nakajima-Zwanzig or the Hashitsume-Shibata-Takahashi identity, respectively. Within the TL methods the QME of the relevant system depends only on the actual state of the system, whereas within the TNL methods the QME also depends on the past evolution of the system. This chapter concentrates on the TL formalism but also shows comparisons between TL and TNL QMEs. An important way how to go beyond second-order in perturbation theory is the so-called hierarchical approach by Tanimura, Kubo, Shao, Yan and others [18-26], The hierarchical method originally developed by Tanimura and Kubo [18] (see also the review in Ref. [26]) is based on the path integral technique for treating a reduced system coupled to a thermal bath of harmonic oscillators. Most interestingly, Ishizaki and Tanimura [27] recently showed that for a quadratic potential the second-order TL approximation coincides with the exact result. Numerically a hint in this direction was already visible in simulations for individual and coupled damped harmonic oscillators [28]. 

Using bosonization procedures and real-time path integral techniques, the quantum conductance of a rigid (i.e., no lattice distortion) ID quantum wire has been computed. Analytical theories predict that, in the presence of even a single defect, the conductance of a strictly ID quantum wire would vanish at 0 K. Bosonization of this system yields an equivalent spin-boson model with an infinite number of tight-binding states, which turns out to be the same model as for CTCs discussed in... 

Memory effects play an important role for the description of dynamical effects in open quantum systems. As mentioned above, Meier and Tannor [32] developed a time-nonlocal scheme employing the numerical decomposition of the spectral density. The TL approach as discussed above as well as the approaches by Yan and coworkers [33-35] use similar techniques. Few systems exist for which exact solutions are available and can serve as test beds for the various theories. Among them is the damped harmonic oscillator for which a path-integral solution exists [1], In the simple model of an initially excited... 

We have obtained a broad range of detailed quantitative results relating to universality in the absence of the electron-phonon interaction by various theoretical techniques. The methods include perturbation theory, the coherent potential approximation (CPA), field theory, path integral methods, numerical calculations and the potential well analogy. The results include the density of states, the nature of the wave functions, the mean free path, the energy dependent... 

The present article presents an introduction to the path integral formulation of quantum dynamics and quantum statistical mechanics along with numerical procedures useful in these areas and in electronic structure theory. Section 2 describes the path integral formulation of the quantum mechanical propagator and its relation to the more conventional Schrddinger description. That section also derives the classical limit and discusses the connection with equilibrium properties in the canonical ensemble, Numerical techniques are described in Section 3. Selective chemical applications of the path integral approach are presented in Section 4 and Section 5 concludes. 

Integrals of Electron Repulsion Molecular Magnetic Properties Mpller-Plesset Perturbation Theory NMR Chemical Shift Computation Ab Initio Nonadiabatic Derivative Couplings Normal Modes Reaction Path Following Spectroscopy Computational Methods Time-dependent Multi-configurational Hartree Method Transition Structure Optimization Techniques. 

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