Exact analytic multi-quanta states of the Davydov Dimer

TL;DR

This paper develops two methods to find exact multi-quanta eigenstates of the Davydov dimer, providing analytical solutions that advance understanding of energy transfer in proteins.

Contribution

It introduces two novel approaches—differential equations integration and algebraic methods—to derive exact eigenstates of the discrete Davydov dimer model.

Findings

Derived general eigenvalue expressions for any number of quanta.

Explicitly obtained eigenvectors for one to four quanta.

Provided examples for five and six quanta cases.

Abstract

The Davydov model describes amide I energy transfer in proteins without dispersion or dissipation. In spite of five decades of study, there are few exact analytical results, especially for the discrete version of this model. Here we develop two methods to determine the exact orthonormal, multi-quanta, eigenstates of the Davydov dimer. The first method involves the integration of a system of ordinary differential equations and the second method applies purely algebraic methods to this problem. We obtain the general expression of the eigenvalues for any number of quanta and also, as examples, apply the methods to the detailed derivation of the eigenvectors for one to four quanta, plus a brief example in the case of $n=5$ and $n=6$.