Identifying the minimum-energy atomic configuration on a lattice: Lamarckian twist on Darwinian evolution

We examine how the two different mechanisms proposed historically for biological evolution compare for the determination of crystal structures from random initial lattice configurations. The Darwinian theory of evolution contends that the genetic makeup inherited at birth is the one passed on during mating to new offspring, in which case evolution is a product of environmental pressure and chance. In addition to this mechanism, Lamarck surmised that individuals can also pass on traits acquired during their lifetime. Here we show that the minimum-energy configurations of a binary A{sub 1-x}B{sub x} alloy in the full 0 {le} x {le} 1 concentration range can be found much faster if the conventional Darwinian genetic progression--mating configurations and letting the lowest-energy (fittest) offspring survive--is allowed to experience Lamarckian-style fitness improvements during its lifetime. Such improvements consist of A {leftrightarrow} B transmutations of some atomic sites (not just atomic relaxations) guided by 'virtual-atom' energy gradients. This hybrid evolution is shown to provide an efficient solution to a generalized Ising Hamiltonian, illustrated here by finding the ground states of face-centered-cubic Au{sub 1-x}Pd{sub x} using a cluster-expansion functional fitted to first-principles total energies. The statistical rate of success of the search strategies and their practical applicability are rigorously documented in terms of average number of evaluations required to find the solution out of 400 independent evolutionary runs with different random seeds. We show that all exact ground states of a 12-atom supercell (2{sup 12} configurations) can be found within 330 total-energy evaluations, whereas a 36-atom supercell (2{sup 36} configurations) requires on average 39,000 evaluations. Thus, this problem cannot be currently addressed with confidence using costly energy functionals [e.g., density-functional theory (DFT) based] unless it is limited to {le} 20 atoms. The computational cost can be reduced at the expense of accuracy: Searching for all approximate-minimum-energy configurations (within 3 meV) of a 12- or 36-atom supercell requires on average 30 or 580 total-energy evaluations, respectively. Thus it could be addressed even by costly energy functionals such as density-functional theory.