Title: Analysis of the Single Reference Coupled Cluster Method for Electronic Structure Calculations: The Full Coupled Cluster Equations Show Chinese title

Title: 单参考耦合簇方法用于电子结构计算的分析：全耦合簇方程

Abstract: 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. Show Chinese abstract

Abstract: 电子结构理论中的中心问题是计算电子哈密顿算子的本征值——一个作用于反对称函数希尔伯特空间上的无界自伴算子。 耦合簇（CC）方法基于所需本征函数的非线性参数化，并导致非线性方程组，它们是高精度量子化学模拟的首选方法，但其数值分析尚不完善。 现有的数值分析依赖于CC函数的局部强单调性性质，该性质仅在微扰区域中有效，即当所需的基态CC解足够接近零时。 本文中，我们基于CC导数的可逆性，引入了单参考耦合簇方法的新适定性分析。在假设所需的本征函数中间归一化且相关的本征值是孤立且非简并的最小假设下，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，并且在离散化足够精细的情况下，我们证明了离散全CC方程是局部适定的，并且推导出了具有保证正常数的残差型误差估计。 初步数值实验表明，我们估计中出现的常数相较于局部单调性方法得到的常数有显著改进。