We propose to improve the convergence properties of the single-reference coupled cluster (CC) method through an augmented Lagrangian formalism. The conventional CC method changes a linear high-dimensional eigenvalue problem with exponential size into a problem of determining the roots of a nonlinear system of equations that has a manageable size. However, current numerical procedures for solving this system of equations to get the lowest eigenvalue suffer from two practical issues: First, solving the CC equations may not converge, and second, when converging, they may converge to other -- potentially unphysical -- states, which are stationary points of the CC energy expression. We show that both issues can be dealt with when a suitably defined energy is minimized in addition to solving the original CC equations. We further propose an augmented Lagrangian method for coupled cluster (alm-CC) to solve the resulting constrained optimization problem. We numerically investigate the proposed augmented Lagrangian formulation showing that the convergence towards the ground state is significantly more stable and that the optimization procedure is less susceptible to local minima. Furthermore, the computational cost of alm-CC is comparable to the conventional CC method.

翻译：我们提出通过增广拉格朗日形式来改进单参考耦合簇（CC）方法的收敛性质。传统CC方法将指数规模的高维线性特征值问题转化为一个规模可处理的非线性方程组求根问题。然而，当前求解该方程组以获取最低特征值的数值过程面临两个实际问题：首先，CC方程可能不收敛；其次，即便收敛，也可能收敛至其他（可能非物理的）状态，即CC能量表达式的驻点。我们证明，在求解原始CC方程的同时，对适当定义的能量进行最小化处理可解决这两个问题。进一步，我们提出用于耦合簇的增广拉格朗日方法（alm-CC）来求解由此产生的约束优化问题。我们对所提出的增广拉格朗日形式进行数值研究，结果表明其向基态的收敛显著更稳定，且优化过程不易陷入局部极小值。此外，alm-CC的计算成本与常规CC方法相当。