Untangling Strongly and Weakly Interacting Configurations in Many-electron Wave Functions

Abstract: Accurate solution of the many-electron problem including correlations remains intractable except for few-electron systems. Describing interacting electrons as a superposition of independent electron configurations results in an apparent combinatorial scaling to achieve an accurate solution. Many approximate approaches for large systems have been introduced, yet controlling and quantifying the error remains a significant hurdle for comparison to experiment. Perturbation calculations are a popular means to estimate neglected energy contributions but give rise to two unanswered questions. How to select a reference state ensuring perturbation corrections will be well-behaved? What is the residual error relative to the exact solution? Conditions to specify these two elusive requisites are given by introducing a partitioning of the electronic configurations into reference and interaction spaces with sampling of neglected configurations using renormalization group ideas. This approach is applied to the molecular (non-relativistic Coulomb) Hamiltonian to identify continuous phase transitions in the energy as a function of the number of configurations. Identification of the minimum number of reference configurations such that the neglected space consists primarily of independent energy contributions is achieved. Balancing the magnitude of the neglected contributions between two calculation enables prediction of accurate physical properties for which calculations would otherwise not be achievable using an expansion consisting of a full set of configurations. Consistent with the use of the molecular Hamiltonian, the analysis is valid for any finite fermion system described by a Hamiltonian consisting of one- and two-body operators.