Pechukas dynamics

P. Pechukas, Dynamics of Molecular Collisions, Part B (W. H. Miller, ed.), Plenum, New York, 1976, chap. 6. 

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... 

The outstanding feature of the set (4.1.57) is that it is completely self-contained. Its structure does not depend on the specific form of the Hamiltonian (4.1.47). In fact, it describes the dynamics of energy levels for all Hamiltonian systems that can be split into two parts according to (4.1.47). The characteristics of a particular Hamiltonian enter only in the initial conditions. Thus we obtain the important result that if the system forgets the initial conditions after a while e, we can use the methods of equilibrium statistical mechanics to compute the statistical properties of the energy levels of the system. Pechukas (1983) used this approach to predict Wignerian statistics for the level fluctuations of chaotic systems. As discussed in the previous section, there is now ample evidence for the correctness of this prediction. 

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]. 

A number of studies have successfully applied the theory of nonlinear dynamics to studies of atomic and molecular motions some representative references follow. In perhaps the first such study, in 1955 DeVogelare and Boudart considered aspects of nonlinear dynamics in models of bimolecular exchange reactions. Poliak, Pechukas, and Child explored periodic orbits and their associated asymptotic limit sets in exchange reactions. - De Leon, Berne, and Rosenberg considered quasiperiodicity and chaos in unimolecular... 

Armed as we are now with the KAM theorem, the Center Manifold theorem, and the Stable Manifold theorem, we can begin to visualize the phase space of reaction dynamics. Returning to our original system (see Uncoupled Reaction Dynamics in Two Degrees of Freedom ), we now realize that the periodic orbit that sews together the half-tori to make up the separatrix is a hyperbolic periodic orbit, and it is not a fixed point of reflection. From our previous visualization of uncoupled phase-space dynamics, we know that the separatrix is completely nontwisted. In the terminology of Poliak and Pechukas, the hyperbolic periodic orbit is a repulsive PODS. ... 

The appearance and disappearance of various flux extrema as a function of energy is a manifestation of how the reaction mechanism changes as a function of energy e.g., at low energies other bottlenecks appear. These same phenomena have been seen by Pechukas and Poliak in a more dynamically based theory and by Garrett and... 

Pechukas P (1976) In Miller WH (ed) Dynamics of molecular collisions, part B, Plenum, New York, p 269... 

Pan YP, McAllister MA (1998) Characterization of low-barrier hydrogen bonds 4. Basis set arrd correlation effects an ab initio and DFT investigation. Theochem-J Mol Strac 427 221-227 Papadopoulos MG, Waite J (1991) The effect of basis set variation and correlation on the 2nd hyperpolatizability of H2O. Theochem-J Mol Strac 81 137-146 Pechukas P (1976) Statistical Approximations in colhsion theory. In Dynamics of Molectrlar Collisions. 

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. 

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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