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Quantum Lattice Solitons
Alwyn Scott
ABSTRACT
A brief introduction to the theory of quantum solitons on a lattice is pre-
sented through three examples: (i) local modes of vibration on a small
molecule, (ii) a quantum discrete nonlinear Schr¨odinger soliton on a one-
dimensional lattice, and (iii) local modes on a molecular crystal.
1 Introduction
The aim of this chapter is to introduce the theory of quantum lattice soli-
tons by sketching applications to some basic problems arising in the theory
of molecular types of nonlinearity are considered: (i) in-
trinsic, which is caused by a readjustment of the electronic cloud as a
molecular bond is stretched, and (ii) extrinsic, arising from interactions of
a molecular vibration with the nearby structure of a first of
these is of interest in the context of local modes in small molecules, and
the second arises in relation to the formation of local modes on molecular
details are available in reference [12].
2 Local modes in the dihalomethanes
Methane is a biologically interesting molecule with the formula CH4, cor-
responding to a structure of four hydrogens attached at equal angles to a
tetravalent carbon the dihalomethanes—CF2H2, CCl2H2, CBr2H2,
and CI2H2—two of these hydrogens are replaced by the halides: fluorine,
chlorine, bromine, or molecules form little laboratories in
which interactions between the two CH oscillators can be investigated.
For dichloromethane, the picture is like this:
Cl
H←→ C↔H
Cl
where the two CH stretching oscillations are each anharmonic (nonlinear),
and they can interact with each other through both mechanical and electro-
. Christiansen, . Sørensen, and . Scott (Eds.): LNP 542, pp. 339−355, 2000.
Springer-Verlag Berlin Heidelberg 2000
340 Alwyn Scott
ic a sufficiently large value of the (intrinsic) anharmonic
parameter, it is