Random-walk statistics in moment-based O(N) tight binding and applications in carbon nanotubes.

A computational framework for a moment-based O(N) tight-binding atomistic method is presented, analyzed, and applied to the problem of electronic properties of deformed carbon nanotubes, where N is the number of atoms in the system. The moment-based approach is based on the maximum entropy and kernel polynomial methods for constructing the electronic density of states from local statistical information about the environment around individual atoms. Random-walk statistics are formally presented as the basis for several methods to collect the moments of the density of states in a computationally efficient manner. The computational complexity and accuracy of these methods are systematically analyzed. Using these methods for the problem of deformed carbon nanotubes, it is shown that the computational cost for some cases, per atom, scales as efficiently as O(M log M), where M is the desired number of moments in the expansion of the density of states. These methods are compared to other methods such as direct diagonalization and a Green's function approach.