Influence functional, path integration

The influence functional theory, as it was formulated by Feyman and Vernon, relies on the additional assumption concerning factorization of the total (system and bath) density matrix in the past. Without this assumption the theory requires a triple path integral, with one thermal integration over the imaginary time axis [Grabert et al. 1988]. 

So far, one can be much more successful in calculating a rate constant when one knows in advance that it exists, than in answering the question of whether it exists. A considerable breakthrough in this area was the solution of the spin-boson problem, which, however, has only limited relevance to any problem in chemistry because it neglects the effects of intrawell dynamics (vibrational relaxation) and does not describe thermally activated transitions. A number of attempts have been made to go beyond the two-level system approximation, but the basic question of how vibrational relaxation affects the transition from coherent oscillations to exponential decay awaits a quantitative solution. Such a solution might be obtained by numerical computation of real-time path integrals for the density matrix using the influence functional technique. 

Other interesting treatments of the solid motion have been developed in which the motion of the solid s atoms is described by quantum mechanics [Billing and Cacciatore 1985, 1986]. This has been done for a harmonic solid in the context of treatment of the motion of the molecule by classical mechanics and use of a TDSCF formalism to couple the quantum and classical subsystems. The impetus for this approach is the fact that, if the entire solid is treated as a set of coupled harmonic oscillators, the quantum solution can be evaluated directly in an operator formalism. Then, the effect of solid atom motion can be incorporated as an added force on the gas molecule. Another advantage is the ability to treat the harmonic degrees of freedom of the solid and the harmonic electron -hole pair excitations on the same footing. The simplicity of such harmonic degrees of freedom can also be incorporated into the previously defined path-integral formalism in a simple manner to yield influence functionals (Feynman and Hibbs 1965). 

Variations on this surface hopping method that utilize Pechukas [106] formulation of mixed quantum-classical dynamics have been proposed [107,108]. Surface hopping algorithms [109-111] for non-adiabatic dynamics based on the quantum-classical Liouville equation [109,111-113] have been formulated. In these schemes the dynamics is fully prescribed by the quantum-classical Liouville operator and no additional assumptions about the nature of the classical evolution or the quantum transition probabilities are made. Quantum dynamics of condensed phase systems has also been carried out using techniques that are not based on surface hopping algorithms, in particular, centroid path integral dynamics [114] and influence functional methods [115]. 

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) ... 

Another approach (Newns 1985 Nourtier 1985) is based on the double path integral representation of the scattering probability. The surface dynamics is described in terms of an influence functional that can be found in explicit form in the linear coupling approximation. However, semiclassical evaluation of the path integrals leads to a complex classical trajectory problem that is non-local in time and too difficult for applications. 

A formalism presented in this part unifies both approaches. Path integrals over projectile variables in momentum representation are evaluated in the quasiclassical limit separately before and after the turning point in the spirit of generalized eikonal method. The Faddeev-Popov method (Popov 1983) is used to fix classical trajectories with respect to the symmetry of the problem. The influence functional is treated as a pre-exponential factor. 

What remains at this point is the path integral over an effective action that depends on the two electronic paths q r) and q x). This can be carried out by the MC method described in Section 3,1. But further improvement in the sign problem can actually be achieved using another level of blocking. The self-interactions between q(r) and q r) in the influence functional x can be simplified enormously by making a simple transformation of coordinates... 

The path integral approach was introduced by Feynman in a seminal paper published in 1948. It provides an alternative formulation of time-dependent quantum mechanics, equivalent to that of Schrodinger. Since its inception, the path integral has found innumerable applications in many areas of physics and chemistry. Its main attractions can be summarized as follows the path integral formulation offers an ideal way of obtaining the classical limit of quantum mechanics it provides a unified description of quantum dynamics and equilibrium quantum statistical mechanics it avoids the use of wavefunctions and thus is often the only viable approach to many-body problems and it leads to powerful influence functional methods for studying the dynamics of a low-dimensional system coupled to a harmonic bath. 

Here i, sf,.., and iQ, if,... denote discretizations of the forward and backward path employed in the path integral representation of the forward and reverse time evolution operators, respectively. The influence functional has the structure... 

In the absence of coupling to a bath the influence functional is equal to unity. In that limit the path integral variables are coupled only to their nearest neighbors in time, i.e is coupled only to j +i. This fact is a consequence of the Markovian nature of the dynamics for the quantum particle alone, as the wavefunction for the latter obeys the first-order SchrSdinger differential equation. Because of this structure, the path integral in the absence of influence functional interactions can be broken into a sequence of one-dimensional integrations which form the basis of iterative matrix multiplication schemes and their variants that are routinely used for wavefunction propagation of small molecules (see also Wave Packets). 

Where yP is defined as the angle between the epr and v. The direction of PR can be either parallel or perpendicular to the axis of the flight path. Only the terms are multiplied by a function of %p, which will influence the shape of the TOF spectra. Empirically, it was found that all of the data could be fitted by considering only the fio(22) parameter, which describes v j correlations. In order to separate the signal dependence on P(Etrans) versus its dependence on S, we first approximated both the beam and laboratory velocity distributions with single velocity vectors. This process allows us to remove the integral of Eq. 38. With this approximation, we can write an equation for / o(22) that is independent of P(Etrans). 

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