The convergence property of a stochastic algorithm for the self-consistent field (SCF) calculations 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 SCF iteration only samples one random vector without having to compute all the orbitals. We consider the common practice of SCF iterations with damping and mixing. We prove with appropriate assumptions that the iterations converge in the mean-square sense, when the stochastic error has an almost sure bound. We also consider the scenario when such an assumption is weakened to a second moment condition, and prove the convergence in probability.

翻译：电子结构自相容场计算(SCF)的随机运算法的趋同特性正在研究中,算法是通过将电子电荷重写成矩阵函数的痕量/对角值来拟订的,该函数随后以统计平均数表示,该函数通过使用Krylov 子空间近似法进一步近似。因此,每个SCF循环法只对一个随机矢量进行抽样,而不必计算所有轨道。我们考虑了SCF循环的常见做法,即用阻断和混合法来计算。我们用适当的假设证明,在中方意义上,当随机误差几乎具有确定的约束性时,迭代相会汇合。我们还考虑了这种假设被削弱到第二时刻的情况,并证明了概率的趋同性。