Stochastic Algorithms for Self-consistent Calculations of Electronic Structures

The convergence property of a stochastic algorithm for the self-consistent calculations (SCC) of electron structures is studied. The algorithm is formulated by rewriting the electron charges as a trace/diagonal of a matrix function, which is subsequently expressed as a statistical average. The function is further approximated by using a Krylov subspace approximation. As a result, each SCC iteration only samples one random vector without having to compute all the orbitals. We consider SCC iterations with damping and mixing, and we show with appropriate assumptions that the iterations converge in the mean-square sense, when the stochastic error has an almost sure bound. Otherwise, the convergence in probability is established.