We propose a comprehensive mathematical framework for Coupled-Cluster-type methods. These aim at accurately solving the many-body Schr\"odinger equation. The present work has two main aspects. First, we rigorously describe the discretization scheme involved in Coupled-Cluster methods using graph-based concepts. This allows us to discuss different methods in a unified and more transparent manner, including multireference methods. Second, we analyze the nonlinear equations of the single-reference Coupled-Cluster method using topological degree theory. We establish existence results and qualitative information about the solutions of these equations that also sheds light on some of the numerically observed behavior. For the truncated Coupled-Cluster method, we derive an energy error bound for approximate eigenstates of the Schr\"odinger equation.

翻译：我们为混合式方法提出了一个全面的数学框架。 这些数学框架旨在准确解决多体Schr\'odinger等式。 目前的工作有两个主要方面。 首先, 我们严格描述使用图形化概念的混合式方法所涉及的离散方案。 这使我们能够以统一和更加透明的方式讨论不同方法, 包括多种参考方法。 其次, 我们用地貌学理论分析单种参考混合- Cluster方法的非线性方程。 我们建立这些方程的存在结果和定性信息, 这些方程的解决方案也揭示了一些数字化行为。 对于纯种的混合式方法, 我们得出了一种能量差, 以Schr\'odiger等式的近似电子元为界。