  We develop computational tools for spectral analysis of stochastic networks
representing energy landscapes of atomic and molecular clusters. Physical
meaning and some properties of eigenvalues, left and right eigenvectors, and
eigencurrents are discussed. We propose an approach to compute a collection of
eigenpairs and corresponding eigencurrents describing the most important
relaxation processes taking place in the system on its way to the equilibrium.
It is suitable for large and complex stochastic networks where pairwise
transition rates, given by the Arrhenius law, vary by orders of magnitude. The
proposed methodology is applied to the network representing the
Lennard-Jones-38 cluster created by Wales's group. Its energy landscape has a
double funnel structure with a deep and narrow face-centered cubic funnel and a
shallower and wider icosahedral funnel. Contrary to the expectations, there is
no appreciable spectral gap separating the eigenvalue corresponding to the
escape from the icosahedral funnel. We provide a detailed description of the
escape process from the icosahedral funnel using the eigencurrent and
demonstrate a superexponential growth of the corresponding eigenvalue. The
proposed spectral approach is compared to the methodology of the Transition
Path Theory. Finally, we discuss whether the Lennard-Jones-38 cluster is
metastable from the points of view of a mathematician and a chemical physicist,
and make a connection with experimental works.

Spectral Theory