Cellular automata in computational quantum chemistry

Abstract

Cellular Automata (CA) are discrete dynamical systems constructed from a large number of simple identical components with local interactions, which are together capable of complex self‐organizing behaviour. The complexity is generated by the cooperative effect of the many components. The use of CA in molecular and solid state dynamical calculations is relatively new and untested. Such methods, however, reduce the quantum molecular problem from one of solving differential equations to the computation of simple algebraic rules synchronously on a lattice of sites. We have demonstrated that the second quantized formalism of field operators in coordinate representation is readily implementable in terms of CA collision rules on a lattice. The Su‐Schrieffer‐Heeger (SSH) Hamiltonian for quasi‐one‐dimensional polymer chains has been examined in terms of CA rules. The CA consists of a 1‐D lattice of sites, each with a finite set of possible values, representing elements of electronic charge and phonon number, evolving in discrete time steps. At each time step, the value of each site is updated according to a stochastic rule, which specifies the new value of the site in terms of its own old value and those of its nearest neighbours. The resulting CA patterns show evidence of persistent propagating structures or solitonic behaviour. In another application, a quantum mechanical variational principle is implemented as a CA, to model the approach to a stationary atomic density in Thomas‐Fermi‐Dirac theory.