Path integrals approach techniques

The computation of quantum many-body effects requires additional effort compared to classical cases. This holds in particular if strong collective phenomena such as phase transitions are considered. The path integral approach to critical phenomena allows the computation of collective phenomena at constant temperature — a condition which is preferred experimentally. Due to the link of path integrals to the partition function in statistical physics, methods from the latter — such as Monte Carlo simulation techniques — can be used for efficient computation of quantum effects. 

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. 

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]. 

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... 

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. 

Fig. 6.1.4 Gradient paths for 3D reconstruction from projections. Only half a hemisphere is covered by the gradient paths, because signal for negative gradient values can be acquired by time inversion in echo techniques, (a) 3D space can be covered by a set of 2D projections, so that the 2D algorithm can be applied in two steps, (b) Optimization of the point density in 3D k space requires an integral approach to 3D reconstruction from projections. Adapted from [Lail] with permission from Institute of Physics.

Due to complexity of the real world, all QDT descriptions involve practically certain approximations or models. As theoretical construction is concerned, the infiuence functional path integral formulation of QDT may by far be the best [4]. The main obstacle of path integral formulation is however its formidable numerical implementation to multilevel systems. Alternative approach to QDT formulation is the reduced Liouville equation for p t). The formally exact reduced Liouville equation can in principle be constructed via Nakajima-Zwanzig-Mori projection operator techniques [5-14], resulting in general two prescriptions. One is the so-called chronological ordering prescription (COP), characterized by a time-ordered memory dissipation superoperator 7(t, r) and read as... 

To enhance convergence of free-particle sampling in centroid path integral simulations, a bisection sampling technique was used for a ring of beads by extending the original approach of Pollock and Ceperley for free particle... 

We have considered the stochastic dynamics of a particle interacting with its environment of two-level systems in the presence of an external potential field. The treatment is based on the canonical quantization procedure. This approach directly yields the dissipative term and the noise operators. It may be pertinent to mention that, although the calculation of dissipative effects is straightforward, the treatment of noise is not simple as far as the path integral techniques are concerned. 

The dynamics of PAH and PAH related systems have been studied in the past mostly in the ground state, usually to compute vibrational spectra, and without considering the effects of nuclei quantum delocalisation. We have however seen in this review that a variety of methods can now to be used to include these effects. In particular, the path integral molecular dynamics approach, which has received a growing interest in the recent years and whose first applications to PAH like systems have recently appeared in the literature, could become a standard method in the next years. Explicit dynamics in the excited states would also be interesting to go beyond the search for conical interstection techniques to analyse relaxation of an excited system. 

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