Large amplitude vibrational motion in a one dimensional chain: Coherent state representation

Abstract

A study is made of the quantum mechanical motion of a one dimensional finite chain of anharmonic oscillators with free ends. It is shown that, for states which time evolve as coherent states (minimum uncertainty wave packets) of the normal mode vibrations, the motion is equivalent to a classical system with an effective potential interaction determined by convoluting the quantum wave packet and the potential energy. Some examples are discussed, with particular attention given to the Toda and Morse potentials, which are shown to be invariant in form under this convolution. The similarities between the classical Toda and Morse lattices are then utilized to infer the existence of compressional solitary waves in the Morse lattice from the well known soliton solutions of the Toda lattice. Further, for the Morse lattice an analytic expression is found for the first order perturbative correction to the Toda solitons for large amplitude vibrations. We also discuss the relation between the existence of such solitary waves and the rate of vibrational relaxation in molecular systems.

Thermal solitons and solectrons in nonlinear conducting chains

“…We assume the following soliton shape It seems pertinent to note that for all practical purposes a Gaussian appears as a valid approximation to (19) when using Morse or L-J potentials as shown by Rice and coworkers [22,23]. By introducing Eq.…”

“…Through a suitable Taylor expansion it provides the abovementioned anharmonic forces beyond Hooke's law. Rice and coworkers [22,23] explored soliton features in 1D lattices with Toda, Morse and L-J potentials including the standard (12-6) case and the (32-6) so-called standard-screw potential. More recently, Heeger and coworkers have used (topological) solitons to explain the electric conductivity of polymers [80] though in this case solitons come from the degeneracy of the ground state and not from an originally underlying lattice anharmonicity in trans-polyacetylene the case most studied by those authors.…”

“…It is now well established that if the underlying lattice dynamics involves an appropriate anharmonic interaction this results in the appearance of very stable supersonic acoustic (lattice) solitons [21][22][23][24][25][26][27]. Furthermore, it has also been shown that the lattice solitons in a Morse chain when coupled to free, conduction electrons bring a new form of dressed electrons or electro-soliton dynamic bound states [28][29][30][31][32][33][34][35][36].…”

Quantum soliton dynamics in vibrational chains: Comparison of fully correlated, mean field, and classical dynamics

“…The optimal coordinates for the mean field TDSCF are those which minimize the correlation, i.e., the normal coordinates of the chain a choice adopted by Dancz and Rice. 3 The classical-like local atomic coordinates are not optimal for describing the motion.…”

Electron pairing and Coulomb repulsion in one-dimensional anharmonic lattices

“…The period of oscillations in the Morse well results as 1/Ω Morse ≃ 0.1 − 0.5 ps. Noteworthy is that for the Toda potential [52], with about the same repulsive component, we know the exact analytical solutions that, for the Morse potential (46), can be used to quite a satisfactory level of approximation [24,53,54]. This potential energy of a lattice with Morse interaction can be expanded in a power series with respect to the displacements of the atoms from equilibrium positions.…”