Configuration weights in coupled-cluster theory

We introduce a simple definition of the weight of any given Slater determinant in the coupled-cluster state, namely as the expectation value of the projection operator onto that determinant. The definition can be applied to any coupled-cluster formulation, including conventional coupled-cluster theory, perturbative coupled-cluster models, nonorthogonal orbital-optimized coupled-cluster theory, and extended coupled-cluster theory, allowing for wave-function analyses on par with configuration-interaction-based wave functions. Numerical experiments show that for single-reference systems the coupled-cluster weights are in excellent agreement with those obtained from the full configuration-interaction wave function. Moreover, the well-known insensitivity of the total energy obtained from truncated coupled-cluster models to the choice of orbital basis is clearly exposed by weights computed in the $\hat{T}_1$-transformed determinant basis. We demonstrate that the inseparability of the conventional linear parameterization of the bra (left state) for systems composed of noninteracting subsystems may lead to ill-behaved (negative or greater than unity) weights, an issue that can only be fully remedied by switching to extended coupled-cluster theory. The latter is corroborated by results obtained with quadratic coupled-cluster theory, which is shown numerically to yield a significant improvement.