Separability of the Coupled-Cluster Excited State Equations: The Case of Excitonic Couplings

Coupled-cluster theory provides an accurate description of electronic ground and excited states. While it has been rigorously established that the coupled-cluster ground state energy is size extensive and local excitations are size intensive in the thermodynamic limit, the separability of other properties in coupled-cluster theory is subject to known limitations. In particular, it was shown [Stanton, J. Chem. Phys. 1994, 101, 8928-8937] that multicenter two-particle density matrices are not asymptotically separable in general. In the present work, an analogous analysis is applied to the excitonic coupling of local excitations of two separate molecules, using both the equation-of-motion (EOM) and linear-response (LR) formalism. It is shown that for both formalisms the two-electron transition density associated with the excitonic coupling does not exactly separate into a product of local one-electron transition densities. Numerical examples are provided for Ne₂, (CO)₂, and (C₂H₄)₂, using coupled-cluster expansions up to single, double, triple, and quadruple excitations (CCSDTQ). Small but noticeable deviations on the order of 5-10% are reported for approximations like CCSD and its second-order approximate variant CC2. As soon as triple excitations are included, the separability error becomes significantly smaller. Exploratory computations for the coupling of larger chromophores such as perylene or cumarine dyes using the CC2 method indicate that the separability error can grow up to orders of 20%. While these deviations are typically below the error margin of this method, the separability error could be of relevance in the design and benchmarking of local coupled-cluster approaches for excited states.