The central problem in electronic structure theory is the computation of the eigenvalues of the electronic Hamiltonian—a semi-unbounded, self-adjoint operator acting on an \(L^2\)-type 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. The existing numerical analysis of coupled cluster methods 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.

电子结构理论的核心问题是电子哈密顿量的特征值的计算——电子哈密顿量是作用于反对称函数\(L^2\)型希尔伯特空间上的半无界自伴算子。耦合簇 (CC) 方法基于备受追捧的本征函数的非线性参数化并产生非线性方程组，是高精度量子化学模拟的首选方法。耦合聚类方法的现有数值分析依赖于CC 函数的局部强单调性特性，该特性仅在微扰状态下有效，即当所寻求的基态CC 解足够接近于零。在本文中，我们介绍了一种基于 CC 导数可逆性的单参考耦合聚类方法的新适定性分析。在最小假设下，即所寻求的特征函数是中间可归一化的，并且相关特征值是孤立的且非简并的，我们证明连续（无限维）CC 方程总是局部适定的。在相同的最小假设下，并且假设离散化足够好，我们证明离散 Full-CC 方程是局部适定的，并且我们推导出基于残差的误差估计，并保证为正常数。初步数值实验表明，我们估计中出现的常数比通过局部单调性方法获得的常数有了显着的改进。