The central problem in electronic structure theory is the computation of the eigenvalues of the electronic Hamiltonian -- an unbounded, self-adjoint operator acting on a Hilbert space of antisymmetric functions. Coupled cluster (CC) methods, which are based on a non-linear parameterisation of the sought-after eigenfunction and result in non-linear systems of equations, are the method of choice for high accuracy quantum chemical simulations but their numerical analysis is underdeveloped. The existing numerical analysis relies on a local, strong monotonicity property of the CC function that is valid only in a perturbative regime, i.e., when the sought-after ground state CC solution is sufficiently close to zero. In this article, we introduce a new well-posedness analysis for the single reference coupled cluster method based on the invertibility of the CC derivative. Under the minimal assumption that the sought-after eigenfunction is intermediately normalisable and the associated eigenvalue is isolated and non-degenerate, we prove that the continuous (infinite-dimensional) CC equations are always locally well-posed. Under the same minimal assumptions and provided that the discretisation is fine enough, we prove that the discrete Full-CC equations are locally well-posed, and we derive residual-based error estimates with guaranteed positive constants. Preliminary numerical experiments indicate that the constants that appear in our estimates are a significant improvement over those obtained from the local monotonicity approach.

翻译：电子结构理论的核心问题在于计算电子哈密顿量的本征值——这是一个作用于反称函数希尔伯特空间上的无界自伴算子。基于所求本征函数的非线性参数化并导致非线性方程组的耦合簇方法是高精度量子化学模拟的首选方法，但其数值分析尚不成熟。现有数值分析依赖于耦合簇函数的局部强单调性，该性质仅在微扰条件下（即所寻求的基态耦合簇解充分接近零时）成立。本文基于耦合簇导数的可逆性，提出了单参考耦合簇方法的一种新的适定性分析。在所求本征函数为中等可归一化且关联本征值孤立且非简并的最小假设下，我们证明了连续（无穷维）耦合簇方程总是局部适定的。基于相同的弱假设且离散化足够精细时，我们证明了离散完整耦合簇方程是局部适定的，并推导出具有保证正常数的基于残差的误差估计。初步数值实验表明，本文估计中的常数较之局部单调性方法所得结果有显著改进。