Quantum-Mechanical Derivation of the Davydov Equations for Multi-Quanta States

Abstract

Davydov and Kislukha¹ suggested in the 1970’s that nonlinear self-trapping could serve as a method of energy transport along quasi-one-dimensional chains of molecules. The problem was to explain how the energy released by hydrolysis of adenosine triphosphate and transferred to proteins in biological systems remains localized and moves along the protein chains at a reasonable rate to perform useful biological functions. The α-helix protein structure was considered, which consists of three chains of hydrogen-bonded peptide groups (HNCO) with associated side groups which contribute to the molecular mass but are assumed dynamically inert. The coupled fields which they suggested are relevant in this problem are a high frequency intramolecular vibration of the peptide groups (the Amide-I or C=O stretch mode, at about 1665 cm⁻¹), and the low frequency vibrations of the entire peptide groups (and associated side groups). These fields are coupled through the dependence of the Amide-I energy on the length of the hydrogen bond coupling neighboring peptide groups.² The Hamiltonian Davydov used to describe this situation is the same as that used for the polaron problem (the Froehlich Hamiltonian for electron-phonon interactions) with some changes in the meaning of the symbols. Davydov’s method of analysis^{1, 3} of this Hamiltonian led to connections with ideas of soliton propagation in other physical systems.⁴