Pechukas path integrals

Despite of its usefulness, however, incorporating quantum and classical mechanics in a consistent manner is a difficult problem. In the simplest case where nonadiabatic interactions are neghgible, the electronic state evolves adiabatically (i.e. keeps on a single PES) and the nuclear dynamics reduces to a Newtonian dynamics driven by the gradient of PES. This is what is called ab initio molecular dynamics (AIMD) approach (see Refe. [450, 451] for examples of chemical use of AIMD). 

On the other hand, there have been established no direct implementation of his formulation that is shown to be practical as well as of wide applicability (see also discussions below). Nevertheless it remains as an important theoretical reference or a start point for discussing further practical methods. 

Chemical Theory Beyond the Born-Oppenheimer Paradigm 

We first divide the dynamical variables into two groups those of heavy particle , which are treated in the coordinate representation R, and those of internal degrees of freedom, which are represented by state labels a, /3,. The former are to be treated in the classical approximation. In his original work, Pechukas considered atomic collision problems. Here we consider a general electron-nucleus coupled dynamics and set the former as the nuclear degrees of freedom and the latter as the electronic degrees of freedom. 

We then consider the transition amplitude of the system from a state R, a at time t to another R , / at time t . The kernel connecting the two end points is given by 

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Considering the semiclassical description of nonadiabatic dynamics, only the mapping approach [99, 100] and the equivalent formulation that is obtained by requantizing the classical electron analog model of Meyer and Miller [112] appear to be amenable to a numerical treatment via an initial-value representation [114, 116, 117, 121, 122]. Other semiclassical formulations such as Pechukas path-integral formulation [45] and the various connection... 

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]. 

In contrast to the Pechukas formulation, Eq. (4.41) and the nuclear dynamics driven by Eq. (4.43) can be solved in an explicit manner since it has no futme time dependence (although both have to be solved stepwise in a self-consistent maimer). Despite of computational easiness, however, Eq. (4.43) leads to imphysical dynamics when the electronic wavepacket bifurcates into multiple states each of which asymptotes to a distinct channel the nuclear motion is driven by an unphysical superposition of these distinct channels. In order to see what is wrong, we recall the path-integral formulation. First observe that the mean-field force Eq. (4.43) can be derived from the Pechukas force in Eq. (4.31) by a special assumption that the final state (T) is given by a unitary transformation... 

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