Natural orbitals are often used to achieve a more compact representation of correlated wave-functions. Using natural orbitals computed as eigenstates of the virtual-virtual block of the state density matrix instead of the canonical Hartree-Fock orbitals results in smaller errors when the same fraction of virtual space is frozen. This strategy, termed frozen natural orbital (FNO) approach, is effective in reducing the cost of regular coupled-cluster (CC) calculations and some multistate methods, such as EOM-IP-CC (equation-of-motion CC for ionization potentials). This contribution extends the FNO approach to the EOM-SF-CC ansatz (EOM-CC with spin-flip). In contrast to EOM-IP-CCSD, EOM-SF-CCSD relies on high-spin open-shell references. Using FNOs computed for an open-shell reference leads to an erratic behavior of the EOM-SF-CC energies and properties due to an inconsistent truncation of the α and β orbital spaces. A general solution to problems arising in the EOM-CC calculations utilizing open-shell references, termed OSFNO (open-shell FNO), is proposed. By means of singular value decomposition (SVD) of the overlap matrix of the α and β orbitals, the OSFNO algorithm identifies the corresponding orbitals and determines virtual orbitals corresponding to the singly occupied space. This is followed by SVD of the singlet part of the state density matrix in the remaining virtual orbital subspace. The so-computed FNOs preserve the spin purity of the open-shell orbital subspace to the extent allowed by the original reference, thus facilitating a safe truncation of the virtual space. The performance of OSFNO is benchmarked for selected diradicals and triradicals.

中文翻译：

---

将冻结的自然轨道近似扩展为开壳参考：理论，实现和对单分子磁体的应用。

通常使用自然轨道来实现相关波函数的更紧凑表示。当冻结相同比例的虚拟空间时，使用计算为状态密度矩阵的虚拟-虚拟块的本征态的自然轨道而不是规范的Hartree-Fock轨道会产生较小的误差。这种称为冻结自然轨道（FNO）方法的策略可有效降低常规耦合簇（CC）计算和某些多状态方法（例如EOM-IP-CC（电离势的运动方程式CC））的成本。 。这一贡献将FNO方法扩展到了EOM-SF-CC ansatz（带有自旋翻转的EOM-CC）。与EOM-IP-CCSD相比，EOM-SF-CCSD依赖于高转速开壳参考。由于α和β轨道空间的截断不一致，使用为开壳参考计算的FNO导致EOM-SF-CC能量和特性的行为不稳定。提出了对使用称为OSFNO（开放式FNO）的开放式引用的EOM-CC计算中出现的问题的一般解决方案。通过对α和β轨道的重叠矩阵进行奇异值分解（SVD），OSFNO算法可以识别相应的轨道，并确定与单个占用空间相对应的虚拟轨道。接下来是其余虚拟轨道子空间中状态密度矩阵的单峰部分的SVD。如此计算的FNO将开壳轨道子空间的自旋纯度保持在原始参考所允许的范围内，从而有利于虚拟空间的安全截断。